align:start position:0% foreign align:start position:0% align:start position:0% for every symmetry for every continuous align:start position:0% for every symmetry for every continuous align:start position:0% for every symmetry for every continuous symmetry there's a conserved current align:start position:0% symmetry there's a conserved current align:start position:0% symmetry there's a conserved current okay align:start position:0% okay align:start position:0% okay and then we also started talking about align:start position:0% and then we also started talking about align:start position:0% and then we also started talking about uh relativistic quantum mechanics align:start position:0% uh relativistic quantum mechanics align:start position:0% uh relativistic quantum mechanics uh um how we want to unify align:start position:0% uh um how we want to unify align:start position:0% uh um how we want to unify special relativity and quantum mechanics align:start position:0% special relativity and quantum mechanics align:start position:0% special relativity and quantum mechanics okay so the most immediate idea for that align:start position:0% okay so the most immediate idea for that align:start position:0% okay so the most immediate idea for that is uh what's called the right basic align:start position:0% is uh what's called the right basic align:start position:0% is uh what's called the right basic quantum mechanics and the most immediate align:start position:0% align:start position:0% generalization of the Schrodinger align:start position:0% generalization of the Schrodinger align:start position:0% generalization of the Schrodinger equation align:start position:0% equation align:start position:0% equation so if you have align:start position:0% align:start position:0% so at the end of last lecture we talked align:start position:0% so at the end of last lecture we talked align:start position:0% so at the end of last lecture we talked about say the most immediate realization align:start position:0% about say the most immediate realization align:start position:0% about say the most immediate realization of the Schrodinger equation align:start position:0% of the Schrodinger equation align:start position:0% of the Schrodinger equation which uh so the align:start position:0% which uh so the align:start position:0% which uh so the so if you have e square e equal to say P align:start position:0% so if you have e square e equal to say P align:start position:0% so if you have e square e equal to say P Square divided by 2m align:start position:0% Square divided by 2m align:start position:0% Square divided by 2m and then you go to non-registic align:start position:0% align:start position:0% quantum mechanics shielding the equation align:start position:0% quantum mechanics shielding the equation align:start position:0% quantum mechanics shielding the equation okay align:start position:0% align:start position:0% and now if you have e squared equal to p align:start position:0% and now if you have e squared equal to p align:start position:0% and now if you have e squared equal to p squared plus M Square align:start position:0% squared plus M Square align:start position:0% squared plus M Square for relativistic particle align:start position:0% for relativistic particle align:start position:0% for relativistic particle and then you get What's called the align:start position:0% and then you get What's called the align:start position:0% and then you get What's called the client golden equation align:start position:0% align:start position:0% and again this PSI has the imputation of align:start position:0% and again this PSI has the imputation of align:start position:0% and again this PSI has the imputation of the uh of the wave function so this align:start position:0% the uh of the wave function so this align:start position:0% the uh of the wave function so this describes and then this align:start position:0% describes and then this align:start position:0% describes and then this so if you as a generalization of this align:start position:0% so if you as a generalization of this align:start position:0% so if you as a generalization of this then this means to describe the align:start position:0% then this means to describe the align:start position:0% then this means to describe the um align:start position:0% um align:start position:0% um the quantum mechanics of a relativistic align:start position:0% the quantum mechanics of a relativistic align:start position:0% the quantum mechanics of a relativistic free particle say of mass m okay of mass align:start position:0% free particle say of mass m okay of mass align:start position:0% free particle say of mass m okay of mass m align:start position:0% m align:start position:0% m so here the PSI align:start position:0% so here the PSI align:start position:0% so here the PSI TX align:start position:0% TX align:start position:0% TX is the wave function align:start position:0% align:start position:0% of a relativistic particle align:start position:0% align:start position:0% relativistic particle align:start position:0% align:start position:0% of mass n align:start position:0% align:start position:0% okay align:start position:0% align:start position:0% and we also notice that this equation align:start position:0% and we also notice that this equation align:start position:0% and we also notice that this equation actually is the same align:start position:0% actually is the same align:start position:0% actually is the same as the simplest field series equation so align:start position:0% as the simplest field series equation so align:start position:0% as the simplest field series equation so we also talked about align:start position:0% we also talked about align:start position:0% we also talked about a simplest align:start position:0% align:start position:0% scalar field Theory align:start position:0% scalar field Theory align:start position:0% scalar field Theory classical so here is a a simple simple a align:start position:0% classical so here is a a simple simple a align:start position:0% classical so here is a a simple simple a classic simple scalar field Theory align:start position:0% align:start position:0% so this series you can write down an align:start position:0% so this series you can write down an align:start position:0% so this series you can write down an action align:start position:0% align:start position:0% of the form align:start position:0% align:start position:0% so this is the simplest align:start position:0% so this is the simplest align:start position:0% so this is the simplest see we can write down align:start position:0% see we can write down align:start position:0% see we can write down and then a relativistic Environ Theory align:start position:0% and then a relativistic Environ Theory align:start position:0% and then a relativistic Environ Theory and then equation of motion of this so align:start position:0% and then equation of motion of this so align:start position:0% and then equation of motion of this so this is a you view this as a classical align:start position:0% this is a you view this as a classical align:start position:0% this is a you view this as a classical field and again this has the equation align:start position:0% field and again this has the equation align:start position:0% field and again this has the equation motion you have to see exact the same align:start position:0% motion you have to see exact the same align:start position:0% motion you have to see exact the same form as this equation align:start position:0% form as this equation align:start position:0% form as this equation so but now here Phi align:start position:0% so but now here Phi align:start position:0% so but now here Phi again is the function of TX align:start position:0% again is the function of TX align:start position:0% again is the function of TX now has a completely different physical align:start position:0% now has a completely different physical align:start position:0% now has a completely different physical interpretation so here is the uh uh this align:start position:0% interpretation so here is the uh uh this align:start position:0% interpretation so here is the uh uh this is a classical field align:start position:0% align:start position:0% so this is a classical field align:start position:0% so this is a classical field align:start position:0% so this is a classical field okay so in this case align:start position:0% okay so in this case align:start position:0% okay so in this case the interpretation of the X in here align:start position:0% the interpretation of the X in here align:start position:0% the interpretation of the X in here and in here is very different okay so so align:start position:0% and in here is very different okay so so align:start position:0% and in here is very different okay so so not only Phi and beside the physical align:start position:0% not only Phi and beside the physical align:start position:0% not only Phi and beside the physical interpretation are different the align:start position:0% interpretation are different the align:start position:0% interpretation are different the physical interpretation of X also here align:start position:0% physical interpretation of X also here align:start position:0% physical interpretation of X also here are different here x is just a label align:start position:0% are different here x is just a label align:start position:0% are different here x is just a label is a label for the for the location in align:start position:0% is a label for the for the location in align:start position:0% is a label for the for the location in the space which which we Define this align:start position:0% the space which which we Define this align:start position:0% the space which which we Define this field align:start position:0% field align:start position:0% field but here the x is the eigenvalue align:start position:0% but here the x is the eigenvalue align:start position:0% but here the x is the eigenvalue of the position align:start position:0% of the position align:start position:0% of the position operator for this right basic particle align:start position:0% operator for this right basic particle align:start position:0% operator for this right basic particle okay and so they have very different align:start position:0% okay and so they have very different align:start position:0% okay and so they have very different physical imputation align:start position:0% align:start position:0% and so let me just label this equation align:start position:0% and so let me just label this equation align:start position:0% and so let me just label this equation by one align:start position:0% by one align:start position:0% by one enable this by two align:start position:0% enable this by two align:start position:0% enable this by two and this by 2 Prime align:start position:0% and this by 2 Prime align:start position:0% and this by 2 Prime okay align:start position:0% align:start position:0% so we also mentioned that this one has a align:start position:0% so we also mentioned that this one has a align:start position:0% so we also mentioned that this one has a the interpretation of this as the wave align:start position:0% the interpretation of this as the wave align:start position:0% the interpretation of this as the wave function for relativistic yeah foreign align:start position:0% align:start position:0% so the first he said align:start position:0% align:start position:0% as you will show in your PSAT 2 align:start position:0% as you will show in your PSAT 2 align:start position:0% as you will show in your PSAT 2 uh there's no sensitive align:start position:0% uh there's no sensitive align:start position:0% uh there's no sensitive no sensible way align:start position:0% align:start position:0% to Define align:start position:0% align:start position:0% a positive definite align:start position:0% align:start position:0% probability density okay align:start position:0% probability density okay align:start position:0% probability density okay so if you want to interpret align:start position:0% align:start position:0% this has a wave equation align:start position:0% this has a wave equation align:start position:0% this has a wave equation now you must have a way then you must align:start position:0% now you must have a way then you must align:start position:0% now you must have a way then you must have a probability density because in align:start position:0% have a probability density because in align:start position:0% have a probability density because in quantum mechanics probability should be align:start position:0% quantum mechanics probability should be align:start position:0% quantum mechanics probability should be conserved okay align:start position:0% conserved okay align:start position:0% conserved okay and the second difficulty align:start position:0% and the second difficulty align:start position:0% and the second difficulty is that the selective energy state align:start position:0% is that the selective energy state align:start position:0% is that the selective energy state because of the square because when you align:start position:0% because of the square because when you align:start position:0% because of the square because when you take the square roots then you get the align:start position:0% take the square roots then you get the align:start position:0% take the square roots then you get the minus sign and then this negative energy align:start position:0% minus sign and then this negative energy align:start position:0% minus sign and then this negative energy states align:start position:0% states align:start position:0% states which you cannot avoid in quantum align:start position:0% which you cannot avoid in quantum align:start position:0% which you cannot avoid in quantum mechanics align:start position:0% mechanics align:start position:0% mechanics even though classically you can just align:start position:0% even though classically you can just align:start position:0% even though classically you can just throw them away perhaps okay align:start position:0% throw them away perhaps okay align:start position:0% throw them away perhaps okay and the third thing we mentioned at the align:start position:0% and the third thing we mentioned at the align:start position:0% and the third thing we mentioned at the end align:start position:0% align:start position:0% he said for relativistic align:start position:0% he said for relativistic align:start position:0% he said for relativistic wave equation align:start position:0% wave equation align:start position:0% wave equation you can describe align:start position:0% align:start position:0% fixed number of particles align:start position:0% align:start position:0% so the particle number cannot change align:start position:0% so the particle number cannot change align:start position:0% so the particle number cannot change okay align:start position:0% okay align:start position:0% okay so so this way we so this equation if align:start position:0% so so this way we so this equation if align:start position:0% so so this way we so this equation if you describe a single particle align:start position:0% you describe a single particle align:start position:0% you describe a single particle and if you want to describe two align:start position:0% and if you want to describe two align:start position:0% and if you want to describe two particles align:start position:0% particles align:start position:0% particles then you leave the two write down a align:start position:0% then you leave the two write down a align:start position:0% then you leave the two write down a separate equation align:start position:0% separate equation align:start position:0% separate equation for different wave function align:start position:0% for different wave function align:start position:0% for different wave function so this is for the two particle wave align:start position:0% so this is for the two particle wave align:start position:0% so this is for the two particle wave function will be like this okay and Etc align:start position:0% function will be like this okay and Etc align:start position:0% function will be like this okay and Etc okay align:start position:0% okay align:start position:0% okay but this does not really make sense align:start position:0% but this does not really make sense align:start position:0% but this does not really make sense in a relativistic system align:start position:0% in a relativistic system align:start position:0% in a relativistic system because we know that in the relative is align:start position:0% because we know that in the relative is align:start position:0% because we know that in the relative is existing e equal to m c Square align:start position:0% existing e equal to m c Square align:start position:0% existing e equal to m c Square in any case you have enough energy then align:start position:0% in any case you have enough energy then align:start position:0% in any case you have enough energy then you should be able to create particles align:start position:0% you should be able to create particles align:start position:0% you should be able to create particles and then that means the lumbar particles align:start position:0% and then that means the lumbar particles align:start position:0% and then that means the lumbar particles in the given process is not conserved align:start position:0% in the given process is not conserved align:start position:0% in the given process is not conserved okay so if you want to use your quantum align:start position:0% okay so if you want to use your quantum align:start position:0% okay so if you want to use your quantum mechanics describe a process and then align:start position:0% mechanics describe a process and then align:start position:0% mechanics describe a process and then that's you cannot have a formalism which align:start position:0% that's you cannot have a formalism which align:start position:0% that's you cannot have a formalism which the number of particle is fixed which align:start position:0% the number of particle is fixed which align:start position:0% the number of particle is fixed which you cannot change align:start position:0% you cannot change align:start position:0% you cannot change and so so this is actually the most align:start position:0% and so so this is actually the most align:start position:0% and so so this is actually the most fundamental difficulty align:start position:0% fundamental difficulty align:start position:0% fundamental difficulty okay is that you cannot change the align:start position:0% okay is that you cannot change the align:start position:0% okay is that you cannot change the number of particles align:start position:0% number of particles align:start position:0% number of particles and related to this difficulty align:start position:0% and related to this difficulty align:start position:0% and related to this difficulty is this interpretation align:start position:0% is this interpretation align:start position:0% is this interpretation here we say now if you wanted to align:start position:0% here we say now if you wanted to align:start position:0% here we say now if you wanted to we say in here align:start position:0% we say in here align:start position:0% we say in here there's a fundamental asymmetry between align:start position:0% there's a fundamental asymmetry between align:start position:0% there's a fundamental asymmetry between the T and X okay also yeah maybe let me align:start position:0% the T and X okay also yeah maybe let me align:start position:0% the T and X okay also yeah maybe let me put it as four which is also fundamental align:start position:0% align:start position:0% no additional difficulty there's a align:start position:0% no additional difficulty there's a align:start position:0% no additional difficulty there's a fundamental asymmetry align:start position:0% align:start position:0% between align:start position:0% align:start position:0% T and X align:start position:0% align:start position:0% so so here in the wave equation T is align:start position:0% so so here in the wave equation T is align:start position:0% so so here in the wave equation T is just a parameter align:start position:0% just a parameter align:start position:0% just a parameter which we describe the evolution align:start position:0% which we describe the evolution align:start position:0% which we describe the evolution but the x is the eigenvalue align:start position:0% align:start position:0% over Quantum operators align:start position:0% over Quantum operators align:start position:0% over Quantum operators eigenvalues of quantum operators align:start position:0% eigenvalues of quantum operators align:start position:0% eigenvalues of quantum operators say a corresponding to say hats by align:start position:0% say a corresponding to say hats by align:start position:0% say a corresponding to say hats by putting ahead with the load the align:start position:0% putting ahead with the load the align:start position:0% putting ahead with the load the corresponding a Quantum operator yeah so align:start position:0% corresponding a Quantum operator yeah so align:start position:0% corresponding a Quantum operator yeah so so this is diagonal value of position align:start position:0% so this is diagonal value of position align:start position:0% so this is diagonal value of position operators align:start position:0% operators align:start position:0% operators and this AC major become even more align:start position:0% and this AC major become even more align:start position:0% and this AC major become even more prolonged so if you can see the two align:start position:0% prolonged so if you can see the two align:start position:0% prolonged so if you can see the two particles align:start position:0% particles align:start position:0% particles okay you have two x here but there's align:start position:0% okay you have two x here but there's align:start position:0% okay you have two x here but there's only one t okay align:start position:0% only one t okay align:start position:0% only one t okay but align:start position:0% but align:start position:0% but so uh so those because of those align:start position:0% so uh so those because of those align:start position:0% so uh so those because of those fundamental difficulties align:start position:0% fundamental difficulties align:start position:0% fundamental difficulties okay so if you align:start position:0% okay so if you align:start position:0% okay so if you connect to this one to four align:start position:0% align:start position:0% so we can we conclude align:start position:0% so we can we conclude align:start position:0% so we can we conclude that the align:start position:0% that the align:start position:0% that the um align:start position:0% um align:start position:0% um relativistic align:start position:0% align:start position:0% quantum mechanics defined in the sense align:start position:0% quantum mechanics defined in the sense align:start position:0% quantum mechanics defined in the sense that you write down a wave equation align:start position:0% that you write down a wave equation align:start position:0% that you write down a wave equation and for for wave function don't even it align:start position:0% and for for wave function don't even it align:start position:0% and for for wave function don't even it does not be a align:start position:0% does not be a align:start position:0% does not be a that's not align:start position:0% align:start position:0% cannot be align:start position:0% align:start position:0% a fundamental discussion okay align:start position:0% a fundamental discussion okay align:start position:0% a fundamental discussion okay but yeah but right this Quantum kind of align:start position:0% but yeah but right this Quantum kind of align:start position:0% but yeah but right this Quantum kind of just refers to this kind of wave align:start position:0% just refers to this kind of wave align:start position:0% just refers to this kind of wave equation okay align:start position:0% align:start position:0% so at most align:start position:0% so at most align:start position:0% so at most this can be approximate approximation align:start position:0% align:start position:0% at the most align:start position:0% align:start position:0% this is approximate description align:start position:0% align:start position:0% in situations align:start position:0% align:start position:0% say there's no align:start position:0% align:start position:0% there's no particle Creation with a align:start position:0% there's no particle Creation with a align:start position:0% there's no particle Creation with a violation align:start position:0% align:start position:0% so so in case which is your align:start position:0% so so in case which is your align:start position:0% so so in case which is your party Columbo is fixed align:start position:0% party Columbo is fixed align:start position:0% party Columbo is fixed and then the then you can use this as align:start position:0% and then the then you can use this as align:start position:0% and then the then you can use this as approximation okay but it cannot be a align:start position:0% approximation okay but it cannot be a align:start position:0% approximation okay but it cannot be a fundamental description align:start position:0% fundamental description align:start position:0% fundamental description for example later we will talk about the align:start position:0% for example later we will talk about the align:start position:0% for example later we will talk about the fumionic version of this wave equation align:start position:0% fumionic version of this wave equation align:start position:0% fumionic version of this wave equation so this will describe a particle without align:start position:0% so this will describe a particle without align:start position:0% so this will describe a particle without spin so later we will describe the the align:start position:0% spin so later we will describe the the align:start position:0% spin so later we will describe the the linear equation for electrons for spin align:start position:0% linear equation for electrons for spin align:start position:0% linear equation for electrons for spin half and then then that can indeed be align:start position:0% half and then then that can indeed be align:start position:0% half and then then that can indeed be used to describe electron in the align:start position:0% used to describe electron in the align:start position:0% used to describe electron in the hydrogen atom as how as far as you don't align:start position:0% hydrogen atom as how as far as you don't align:start position:0% hydrogen atom as how as far as you don't create new electrons Etc align:start position:0% create new electrons Etc align:start position:0% create new electrons Etc anyway so so so write this Quantum align:start position:0% anyway so so so write this Quantum align:start position:0% anyway so so so write this Quantum connect only be described as some kind align:start position:0% connect only be described as some kind align:start position:0% connect only be described as some kind of consider as approximate description align:start position:0% of consider as approximate description align:start position:0% of consider as approximate description okay but now if you want to you unify align:start position:0% align:start position:0% special relativity and quantum mechanics align:start position:0% special relativity and quantum mechanics align:start position:0% special relativity and quantum mechanics together align:start position:0% together align:start position:0% together it turns out that the right formulation align:start position:0% it turns out that the right formulation align:start position:0% it turns out that the right formulation is just Quantum field Theory okay align:start position:0% is just Quantum field Theory okay align:start position:0% is just Quantum field Theory okay so it turns out that the quantum field align:start position:0% so it turns out that the quantum field align:start position:0% so it turns out that the quantum field Theory align:start position:0% align:start position:0% a corner of your theory align:start position:0% a corner of your theory align:start position:0% a corner of your theory addresses these difficulties okay align:start position:0% align:start position:0% okay align:start position:0% okay align:start position:0% okay so it turns out align:start position:0% so it turns out align:start position:0% so it turns out it turns out the right way so if we want align:start position:0% it turns out the right way so if we want align:start position:0% it turns out the right way so if we want to describe quantum mechanics say of align:start position:0% to describe quantum mechanics say of align:start position:0% to describe quantum mechanics say of write a specific particles of mass m align:start position:0% write a specific particles of mass m align:start position:0% write a specific particles of mass m okay as we want to do here align:start position:0% okay as we want to do here align:start position:0% okay as we want to do here it turns out the proper thing to do align:start position:0% it turns out the proper thing to do align:start position:0% it turns out the proper thing to do which is a little bit unintuitive at align:start position:0% which is a little bit unintuitive at align:start position:0% which is a little bit unintuitive at first sight align:start position:0% first sight align:start position:0% first sight is to start with this field Siri okay align:start position:0% is to start with this field Siri okay align:start position:0% is to start with this field Siri okay which seemingly have nothing to do with align:start position:0% which seemingly have nothing to do with align:start position:0% which seemingly have nothing to do with write a basic particle but to start with align:start position:0% write a basic particle but to start with align:start position:0% write a basic particle but to start with this classical field Theory align:start position:0% this classical field Theory align:start position:0% this classical field Theory and then contact it okay it hands out align:start position:0% and then contact it okay it hands out align:start position:0% and then contact it okay it hands out turns out once you treat this Theory as align:start position:0% turns out once you treat this Theory as align:start position:0% turns out once you treat this Theory as a Quantum field Theory and this becomes align:start position:0% a Quantum field Theory and this becomes align:start position:0% a Quantum field Theory and this becomes a theory of arbitrary number of align:start position:0% a theory of arbitrary number of align:start position:0% a theory of arbitrary number of relativistic particles of mass m align:start position:0% relativistic particles of mass m align:start position:0% relativistic particles of mass m okay and so that's the non-intuitive align:start position:0% okay and so that's the non-intuitive align:start position:0% okay and so that's the non-intuitive part and and that's the uh one of the align:start position:0% part and and that's the uh one of the align:start position:0% part and and that's the uh one of the miracle say of the field theory is that align:start position:0% miracle say of the field theory is that align:start position:0% miracle say of the field theory is that automatically give you a formalism align:start position:0% automatically give you a formalism align:start position:0% automatically give you a formalism for for treating arbitrary lumbar align:start position:0% for for treating arbitrary lumbar align:start position:0% for for treating arbitrary lumbar particles okay align:start position:0% particles okay align:start position:0% particles okay and uh um yeah align:start position:0% align:start position:0% um and also in field Theory align:start position:0% um and also in field Theory align:start position:0% um and also in field Theory so both T and X are parameters okay even align:start position:0% so both T and X are parameters okay even align:start position:0% so both T and X are parameters okay even though X only labels your uh uh your align:start position:0% though X only labels your uh uh your align:start position:0% though X only labels your uh uh your location so both T and X are parameters align:start position:0% location so both T and X are parameters align:start position:0% location so both T and X are parameters and so you can easily to make them to be align:start position:0% and so you can easily to make them to be align:start position:0% and so you can easily to make them to be on equal ground to be compatible with align:start position:0% on equal ground to be compatible with align:start position:0% on equal ground to be compatible with special relativity align:start position:0% align:start position:0% um align:start position:0% um align:start position:0% um good align:start position:0% good align:start position:0% good so any questions on this align:start position:0% align:start position:0% okay so we will see that the uh the align:start position:0% okay so we will see that the uh the align:start position:0% okay so we will see that the uh the right framework is quantum field three align:start position:0% right framework is quantum field three align:start position:0% right framework is quantum field three okay align:start position:0% okay align:start position:0% okay so finally it's the last motivation for align:start position:0% so finally it's the last motivation for align:start position:0% so finally it's the last motivation for Quantum field Theory align:start position:0% Quantum field Theory align:start position:0% Quantum field Theory so we quickly uh uh describe the last uh align:start position:0% so we quickly uh uh describe the last uh align:start position:0% so we quickly uh uh describe the last uh so the fields here you can also arise align:start position:0% align:start position:0% as a limit align:start position:0% align:start position:0% of discrete systems align:start position:0% align:start position:0% okay align:start position:0% okay align:start position:0% okay and this is the uh uh the most relevant align:start position:0% and this is the uh uh the most relevant align:start position:0% and this is the uh uh the most relevant for this matter physics for example align:start position:0% align:start position:0% so so let's just consider align:start position:0% so so let's just consider align:start position:0% so so let's just consider say uh some yeah let's consider 803 align:start position:0% say uh some yeah let's consider 803 align:start position:0% say uh some yeah let's consider 803 example okay align:start position:0% example okay align:start position:0% example okay so let's imagine you have align:start position:0% so let's imagine you have align:start position:0% so let's imagine you have just number of particles a number of the align:start position:0% just number of particles a number of the align:start position:0% just number of particles a number of the atoms say on the on the Chain okay and align:start position:0% atoms say on the on the Chain okay and align:start position:0% atoms say on the on the Chain okay and then they're connected by some screens align:start position:0% then they're connected by some screens align:start position:0% then they're connected by some screens between them align:start position:0% align:start position:0% okay so so this is the uh uh the pro align:start position:0% okay so so this is the uh uh the pro align:start position:0% okay so so this is the uh uh the pro yeah consider this to be infinite okay align:start position:0% yeah consider this to be infinite okay align:start position:0% yeah consider this to be infinite okay and the spacing between them say is a align:start position:0% align:start position:0% item I fixed on some matches points align:start position:0% item I fixed on some matches points align:start position:0% item I fixed on some matches points and the lattice facing is a okay align:start position:0% and the lattice facing is a okay align:start position:0% and the lattice facing is a okay so align:start position:0% so align:start position:0% so yeah so we can label the other say align:start position:0% yeah so we can label the other say align:start position:0% yeah so we can label the other say each particle by their position for align:start position:0% each particle by their position for align:start position:0% each particle by their position for example this is x0 this is X1 this is X2 align:start position:0% example this is x0 this is X1 this is X2 align:start position:0% example this is x0 this is X1 this is X2 Etc okay and the typical particle is x n align:start position:0% align:start position:0% at the location of nth particle is x n align:start position:0% at the location of nth particle is x n align:start position:0% at the location of nth particle is x n and so we can also introduce the align:start position:0% and so we can also introduce the align:start position:0% and so we can also introduce the deviation align:start position:0% deviation align:start position:0% deviation between the equivalent position of each align:start position:0% between the equivalent position of each align:start position:0% between the equivalent position of each particle so let's call it n align:start position:0% particle so let's call it n align:start position:0% particle so let's call it n okay so now let's consider the Dynamics align:start position:0% okay so now let's consider the Dynamics align:start position:0% okay so now let's consider the Dynamics of ether M for this Theory align:start position:0% of ether M for this Theory align:start position:0% of ether M for this Theory and so this is the just deviation of the align:start position:0% and so this is the just deviation of the align:start position:0% and so this is the just deviation of the nth particle from its equivalent align:start position:0% nth particle from its equivalent align:start position:0% nth particle from its equivalent position okay so X and zero is its align:start position:0% position okay so X and zero is its align:start position:0% position okay so X and zero is its equivalent position align:start position:0% equivalent position align:start position:0% equivalent position so now so now if you write down the align:start position:0% so now so now if you write down the align:start position:0% so now so now if you write down the lagrangian for this system align:start position:0% lagrangian for this system align:start position:0% lagrangian for this system that we can easily do you just write T align:start position:0% that we can easily do you just write T align:start position:0% that we can easily do you just write T minus V the t is the kinetic energy and align:start position:0% minus V the t is the kinetic energy and align:start position:0% minus V the t is the kinetic energy and V is the uh um the potential energy so align:start position:0% V is the uh um the potential energy so align:start position:0% V is the uh um the potential energy so we can just write it as sum over n align:start position:0% we can just write it as sum over n align:start position:0% we can just write it as sum over n over o align:start position:0% over o align:start position:0% over o particles and then let's assume they align:start position:0% particles and then let's assume they align:start position:0% particles and then let's assume they have the same mass let's write mu align:start position:0% have the same mass let's write mu align:start position:0% have the same mass let's write mu ETA n dot Square okay so this is a align:start position:0% ETA n dot Square okay so this is a align:start position:0% ETA n dot Square okay so this is a kinetic term so so mu is the mass for align:start position:0% kinetic term so so mu is the mass for align:start position:0% kinetic term so so mu is the mass for each particle align:start position:0% each particle align:start position:0% each particle and then their their potential yeah align:start position:0% and then their their potential yeah align:start position:0% and then their their potential yeah let's assume at each point there is also align:start position:0% let's assume at each point there is also align:start position:0% let's assume at each point there is also a yeah let's just yeah then there's some align:start position:0% a yeah let's just yeah then there's some align:start position:0% a yeah let's just yeah then there's some uh uh uh interaction because each align:start position:0% uh uh uh interaction because each align:start position:0% uh uh uh interaction because each particle are connected by the spring and align:start position:0% particle are connected by the spring and align:start position:0% particle are connected by the spring and so they're a harmonic Force align:start position:0% so they're a harmonic Force align:start position:0% so they're a harmonic Force between align:start position:0% between align:start position:0% between neighboring particles okay align:start position:0% neighboring particles okay align:start position:0% neighboring particles okay and now let's imagine also there's a align:start position:0% and now let's imagine also there's a align:start position:0% and now let's imagine also there's a harmonic potential which trapped this align:start position:0% harmonic potential which trapped this align:start position:0% harmonic potential which trapped this particle itself align:start position:0% particle itself align:start position:0% particle itself at each location align:start position:0% at each location align:start position:0% at each location okay so this is a very simple uh spring align:start position:0% okay so this is a very simple uh spring align:start position:0% okay so this is a very simple uh spring and the particle problem which you align:start position:0% and the particle problem which you align:start position:0% and the particle problem which you encounter say in 803 align:start position:0% encounter say in 803 align:start position:0% encounter say in 803 okay is this problem clear align:start position:0% align:start position:0% okay I I assume most of you have seen align:start position:0% okay I I assume most of you have seen align:start position:0% okay I I assume most of you have seen this problem before align:start position:0% this problem before align:start position:0% this problem before and the and you're tasking 803 is align:start position:0% and the and you're tasking 803 is align:start position:0% and the and you're tasking 803 is actually to uh to find the lower modes align:start position:0% actually to uh to find the lower modes align:start position:0% actually to uh to find the lower modes say of this system okay align:start position:0% say of this system okay align:start position:0% say of this system okay and in 803 you also describe that we can align:start position:0% and in 803 you also describe that we can align:start position:0% and in 803 you also describe that we can in a go to zero limit align:start position:0% in a go to zero limit align:start position:0% in a go to zero limit so if the left is spacing is very small align:start position:0% so if the left is spacing is very small align:start position:0% so if the left is spacing is very small and if you're only interested in the align:start position:0% and if you're only interested in the align:start position:0% and if you're only interested in the behavior of the system at a very large align:start position:0% behavior of the system at a very large align:start position:0% behavior of the system at a very large distance say the distance much larger align:start position:0% distance say the distance much larger align:start position:0% distance say the distance much larger than a equal imagine lighter than a align:start position:0% than a equal imagine lighter than a align:start position:0% than a equal imagine lighter than a then you can essentially choose this align:start position:0% then you can essentially choose this align:start position:0% then you can essentially choose this season as a Continuum okay you don't align:start position:0% season as a Continuum okay you don't align:start position:0% season as a Continuum okay you don't have to resolve individual particles align:start position:0% have to resolve individual particles align:start position:0% have to resolve individual particles and so we can just enable the newer align:start position:0% and so we can just enable the newer align:start position:0% and so we can just enable the newer limits so you can choose the chain align:start position:0% align:start position:0% of particles align:start position:0% align:start position:0% okay align:start position:0% okay align:start position:0% okay and uh so you align:start position:0% and uh so you align:start position:0% and uh so you so each align:start position:0% so each align:start position:0% so each and T you replace it align:start position:0% and T you replace it align:start position:0% and T you replace it by ETA x t align:start position:0% by ETA x t align:start position:0% by ETA x t so X label is position align:start position:0% so X label is position align:start position:0% so X label is position okay the X label is position and T align:start position:0% okay the X label is position and T align:start position:0% okay the X label is position and T describes the Dynamics okay so so ETA is align:start position:0% describes the Dynamics okay so so ETA is align:start position:0% describes the Dynamics okay so so ETA is the deviation align:start position:0% the deviation align:start position:0% the deviation at the location X and it's depend on T align:start position:0% at the location X and it's depend on T align:start position:0% at the location X and it's depend on T okay so this is the oscillator align:start position:0% okay so this is the oscillator align:start position:0% okay so this is the oscillator and then sum over n align:start position:0% and then sum over n align:start position:0% and then sum over n in the lagrangian align:start position:0% in the lagrangian align:start position:0% in the lagrangian then we can replace it by integral align:start position:0% then we can replace it by integral align:start position:0% then we can replace it by integral over DX align:start position:0% over DX align:start position:0% over DX okay and now you just choose this as a align:start position:0% okay and now you just choose this as a align:start position:0% okay and now you just choose this as a one-dimensional continuum see some align:start position:0% one-dimensional continuum see some align:start position:0% one-dimensional continuum see some integration of the X but of course here align:start position:0% integration of the X but of course here align:start position:0% integration of the X but of course here there's a label a nothing spacing so so align:start position:0% there's a label a nothing spacing so so align:start position:0% there's a label a nothing spacing so so the uh so the implementational uh here align:start position:0% the uh so the implementational uh here align:start position:0% the uh so the implementational uh here the element is a so a times align:start position:0% the element is a so a times align:start position:0% the element is a so a times the sum Over N you can replace it by DX align:start position:0% the sum Over N you can replace it by DX align:start position:0% the sum Over N you can replace it by DX okay align:start position:0% okay align:start position:0% okay is the lattice spacing align:start position:0% is the lattice spacing align:start position:0% is the lattice spacing and now you can just write align:start position:0% and now you can just write align:start position:0% and now you can just write this Lagrange in terms of Continuum align:start position:0% this Lagrange in terms of Continuum align:start position:0% this Lagrange in terms of Continuum Theory align:start position:0% Theory align:start position:0% Theory okay now you can write this in the align:start position:0% okay now you can write this in the align:start position:0% okay now you can write this in the ground in a Continuum Theory and then align:start position:0% ground in a Continuum Theory and then align:start position:0% ground in a Continuum Theory and then let's just do it align:start position:0% let's just do it align:start position:0% let's just do it so um so we can write it yeah let me align:start position:0% so um so we can write it yeah let me align:start position:0% so um so we can write it yeah let me just write one more step align:start position:0% just write one more step align:start position:0% just write one more step so you can write it as sum over a we align:start position:0% so you can write it as sum over a we align:start position:0% so you can write it as sum over a we take the a factor out because of the a align:start position:0% take the a factor out because of the a align:start position:0% take the a factor out because of the a fact R have to be changed into align:start position:0% fact R have to be changed into align:start position:0% fact R have to be changed into integration align:start position:0% integration align:start position:0% integration and then you have one half mu divided by align:start position:0% and then you have one half mu divided by align:start position:0% and then you have one half mu divided by a align:start position:0% a align:start position:0% a ETA N squared align:start position:0% ETA N squared align:start position:0% ETA N squared minus one half Lambda a align:start position:0% align:start position:0% so so I just slightly rewrite align:start position:0% align:start position:0% it's lagranging align:start position:0% it's lagranging align:start position:0% it's lagranging so that it is easy to take the Continuum align:start position:0% so that it is easy to take the Continuum align:start position:0% so that it is easy to take the Continuum limit align:start position:0% limit align:start position:0% limit so we have taken the factor of a out align:start position:0% so we have taken the factor of a out align:start position:0% so we have taken the factor of a out but for this term align:start position:0% but for this term align:start position:0% but for this term because this contains the difference align:start position:0% because this contains the difference align:start position:0% because this contains the difference between the two and we also divided by a align:start position:0% between the two and we also divided by a align:start position:0% between the two and we also divided by a in the downstairs and then we need to align:start position:0% in the downstairs and then we need to align:start position:0% in the downstairs and then we need to multiply a upstairs and then there's a align:start position:0% multiply a upstairs and then there's a align:start position:0% multiply a upstairs and then there's a in the front okay align:start position:0% in the front okay align:start position:0% in the front okay another Continuum limit align:start position:0% another Continuum limit align:start position:0% another Continuum limit you can just replace this by integral align:start position:0% align:start position:0% and now I just align:start position:0% align:start position:0% you can just write it as one half align:start position:0% you can just write it as one half align:start position:0% you can just write it as one half you tilde align:start position:0% you tilde align:start position:0% you tilde is a start Square so now e to n just align:start position:0% is a start Square so now e to n just align:start position:0% is a start Square so now e to n just replace it by E to x t align:start position:0% align:start position:0% and here let me call it lamina tilde align:start position:0% and here let me call it lamina tilde align:start position:0% and here let me call it lamina tilde partial X is a square align:start position:0% partial X is a square align:start position:0% partial X is a square and this term we can just replace it by align:start position:0% and this term we can just replace it by align:start position:0% and this term we can just replace it by the derivative of ETA and then this is align:start position:0% the derivative of ETA and then this is align:start position:0% the derivative of ETA and then this is just become one half Sigma tilde is a align:start position:0% just become one half Sigma tilde is a align:start position:0% just become one half Sigma tilde is a square align:start position:0% square align:start position:0% square okay align:start position:0% okay align:start position:0% okay and the uh the MU tilde of course align:start position:0% and the uh the MU tilde of course align:start position:0% and the uh the MU tilde of course is Mu divided by a align:start position:0% is Mu divided by a align:start position:0% is Mu divided by a number is Lambda times a and sigma tilde align:start position:0% align:start position:0% this is the sigma divided by a align:start position:0% this is the sigma divided by a align:start position:0% this is the sigma divided by a okay so the Continuum limit is that align:start position:0% okay so the Continuum limit is that align:start position:0% okay so the Continuum limit is that those quantity has to be fixed okay the align:start position:0% those quantity has to be fixed okay the align:start position:0% those quantity has to be fixed okay the tier the quantity has to be fixed and align:start position:0% tier the quantity has to be fixed and align:start position:0% tier the quantity has to be fixed and then we have a continuum lagrangea align:start position:0% then we have a continuum lagrangea align:start position:0% then we have a continuum lagrangea okay and then we have a classical field align:start position:0% okay and then we have a classical field align:start position:0% okay and then we have a classical field Theory align:start position:0% Theory align:start position:0% Theory and this series is essentially the same align:start position:0% and this series is essentially the same align:start position:0% and this series is essentially the same as align:start position:0% align:start position:0% as that theory align:start position:0% as that theory align:start position:0% as that theory okay so if you take this Factor mu tilde align:start position:0% okay so if you take this Factor mu tilde align:start position:0% okay so if you take this Factor mu tilde out align:start position:0% out align:start position:0% out okay if we take this Factor mu tilde out align:start position:0% okay if we take this Factor mu tilde out align:start position:0% okay if we take this Factor mu tilde out okay so let me just take this part of align:start position:0% okay so let me just take this part of align:start position:0% okay so let me just take this part of mutilda out in the front just up to align:start position:0% mutilda out in the front just up to align:start position:0% mutilda out in the front just up to overall factor and here is Lambda tilde align:start position:0% overall factor and here is Lambda tilde align:start position:0% overall factor and here is Lambda tilde by a divided by mu tier that we could align:start position:0% by a divided by mu tier that we could align:start position:0% by a divided by mu tier that we could let's call it V Square align:start position:0% let's call it V Square align:start position:0% let's call it V Square and this becomes Sigma material divided align:start position:0% and this becomes Sigma material divided align:start position:0% and this becomes Sigma material divided by uh by mu till let's call it m Square align:start position:0% by uh by mu till let's call it m Square align:start position:0% by uh by mu till let's call it m Square so the V Square align:start position:0% so the V Square align:start position:0% so the V Square is equal to Mu tilde align:start position:0% is equal to Mu tilde align:start position:0% is equal to Mu tilde is equal to the Lambda tilde align:start position:0% is equal to the Lambda tilde align:start position:0% is equal to the Lambda tilde divided by mu tilde align:start position:0% divided by mu tilde align:start position:0% divided by mu tilde and the M Square align:start position:0% and the M Square align:start position:0% and the M Square is equal to Sigma material align:start position:0% is equal to Sigma material align:start position:0% is equal to Sigma material okay align:start position:0% okay align:start position:0% okay and then this is just essentially align:start position:0% and then this is just essentially align:start position:0% and then this is just essentially identical to that theory align:start position:0% align:start position:0% when V equal to one okay so when b equal align:start position:0% when V equal to one okay so when b equal align:start position:0% when V equal to one okay so when b equal to one align:start position:0% align:start position:0% become the same align:start position:0% become the same align:start position:0% become the same as just two okay equation two align:start position:0% as just two okay equation two align:start position:0% as just two okay equation two of course corresponding to relative K is align:start position:0% of course corresponding to relative K is align:start position:0% of course corresponding to relative K is a speed of light but in general uh this align:start position:0% a speed of light but in general uh this align:start position:0% a speed of light but in general uh this describer can yeah addition I can align:start position:0% describer can yeah addition I can align:start position:0% describer can yeah addition I can describe a Contin but in general this is align:start position:0% describe a Contin but in general this is align:start position:0% describe a Contin but in general this is a noun in general this can be just a align:start position:0% a noun in general this can be just a align:start position:0% a noun in general this can be just a long relativistic field series okay for align:start position:0% long relativistic field series okay for align:start position:0% long relativistic field series okay for other values of B okay align:start position:0% other values of B okay align:start position:0% other values of B okay foreign align:start position:0% foreign align:start position:0% foreign so this is align:start position:0% so this is align:start position:0% so this is so even though this example is very align:start position:0% so even though this example is very align:start position:0% so even though this example is very simple align:start position:0% simple align:start position:0% simple but this is actually a very general way align:start position:0% but this is actually a very general way align:start position:0% but this is actually a very general way that we can treat many condensed metal align:start position:0% that we can treat many condensed metal align:start position:0% that we can treat many condensed metal systems align:start position:0% systems align:start position:0% systems which often in Mobile lattice say align:start position:0% which often in Mobile lattice say align:start position:0% which often in Mobile lattice say because because solid you can imagine align:start position:0% because because solid you can imagine align:start position:0% because because solid you can imagine all the items on the lattice Etc and if align:start position:0% all the items on the lattice Etc and if align:start position:0% all the items on the lattice Etc and if you're only interested in the very align:start position:0% you're only interested in the very align:start position:0% you're only interested in the very microscopic Behavior then you can treat align:start position:0% microscopic Behavior then you can treat align:start position:0% microscopic Behavior then you can treat solid as a continuum align:start position:0% solid as a continuum align:start position:0% solid as a continuum and then and now you can uh now if align:start position:0% and then and now you can uh now if align:start position:0% and then and now you can uh now if you're interested in quantum mechanics align:start position:0% you're interested in quantum mechanics align:start position:0% you're interested in quantum mechanics of such a system then the quantum field align:start position:0% of such a system then the quantum field align:start position:0% of such a system then the quantum field theory that naturally Rises okay align:start position:0% align:start position:0% okay good any questions on this example align:start position:0% align:start position:0% yes align:start position:0% align:start position:0% sorry so the limit yeah yeah yeah align:start position:0% sorry so the limit yeah yeah yeah align:start position:0% sorry so the limit yeah yeah yeah like same windows view yeah yeah so what align:start position:0% like same windows view yeah yeah so what align:start position:0% like same windows view yeah yeah so what is that physically align:start position:0% is that physically align:start position:0% is that physically strength of yeah yeah it is align:start position:0% strength of yeah yeah it is align:start position:0% strength of yeah yeah it is corresponding to the the case that the align:start position:0% corresponding to the the case that the align:start position:0% corresponding to the the case that the um uh uh yeah it just tells you that align:start position:0% um uh uh yeah it just tells you that align:start position:0% um uh uh yeah it just tells you that relativistic limit is special happens at align:start position:0% relativistic limit is special happens at align:start position:0% relativistic limit is special happens at very special points but I guess why why align:start position:0% very special points but I guess why why align:start position:0% very special points but I guess why why is that the relative like to me lenders align:start position:0% is that the relative like to me lenders align:start position:0% is that the relative like to me lenders like the strength of your spring and align:start position:0% like the strength of your spring and align:start position:0% like the strength of your spring and then yeah use your mask right you've got align:start position:0% then yeah use your mask right you've got align:start position:0% then yeah use your mask right you've got to be comparable how does that yeah align:start position:0% to be comparable how does that yeah align:start position:0% to be comparable how does that yeah there's not much you can read from here align:start position:0% there's not much you can read from here align:start position:0% there's not much you can read from here yeah yeah uh it it just like when you align:start position:0% yeah yeah uh it it just like when you align:start position:0% yeah yeah uh it it just like when you choose some special parameters then you align:start position:0% choose some special parameters then you align:start position:0% choose some special parameters then you can uh have a relativistic limit align:start position:0% align:start position:0% other questions yes align:start position:0% align:start position:0% yeah so these are all um scalars right align:start position:0% yeah so these are all um scalars right align:start position:0% yeah so these are all um scalars right I could you can also have uh you can align:start position:0% I could you can also have uh you can align:start position:0% I could you can also have uh you can also you mean uh you can also have align:start position:0% also you mean uh you can also have align:start position:0% also you mean uh you can also have tensors or vectors yeah align:start position:0% tensors or vectors yeah align:start position:0% tensors or vectors yeah yeah align:start position:0% yeah align:start position:0% yeah so like what would you treat with this align:start position:0% so like what would you treat with this align:start position:0% so like what would you treat with this pronouns oh you can cheat yeah for align:start position:0% pronouns oh you can cheat yeah for align:start position:0% pronouns oh you can cheat yeah for example you can treat four rounds you align:start position:0% example you can treat four rounds you align:start position:0% example you can treat four rounds you can also choose spins align:start position:0% can also choose spins align:start position:0% can also choose spins and say for example if you have an icing align:start position:0% and say for example if you have an icing align:start position:0% and say for example if you have an icing model just consider lattice of spins align:start position:0% model just consider lattice of spins align:start position:0% model just consider lattice of spins and then the average Spin and then you align:start position:0% and then the average Spin and then you align:start position:0% and then the average Spin and then you can treat it as a scale of field and align:start position:0% can treat it as a scale of field and align:start position:0% can treat it as a scale of field and then again you can write down a field align:start position:0% then again you can write down a field align:start position:0% then again you can write down a field Theory yeah and actually the the align:start position:0% Theory yeah and actually the the align:start position:0% Theory yeah and actually the the Breakthrough align:start position:0% Breakthrough align:start position:0% Breakthrough of the phase transition align:start position:0% of the phase transition align:start position:0% of the phase transition metaphysics to to understand what phase align:start position:0% metaphysics to to understand what phase align:start position:0% metaphysics to to understand what phase transition is really about and describe align:start position:0% transition is really about and describe align:start position:0% transition is really about and describe the behavior near the phase transition align:start position:0% the behavior near the phase transition align:start position:0% the behavior near the phase transition and precisely coincided with the align:start position:0% and precisely coincided with the align:start position:0% and precisely coincided with the development of field Theory and uh yeah align:start position:0% development of field Theory and uh yeah align:start position:0% development of field Theory and uh yeah actually increased our understanding of align:start position:0% actually increased our understanding of align:start position:0% actually increased our understanding of quantum field Theory yeah align:start position:0% quantum field Theory yeah align:start position:0% quantum field Theory yeah foreign align:start position:0% foreign align:start position:0% foreign other questions align:start position:0% align:start position:0% okay good align:start position:0% okay good align:start position:0% okay good just to summarize align:start position:0% just to summarize align:start position:0% just to summarize what we have discussed so far align:start position:0% align:start position:0% or path align:start position:0% align:start position:0% to align:start position:0% to align:start position:0% to qft okay align:start position:0% qft okay align:start position:0% qft okay so we have described three parts three align:start position:0% so we have described three parts three align:start position:0% so we have described three parts three parts but they're pretty General first align:start position:0% parts but they're pretty General first align:start position:0% parts but they're pretty General first you say the quantum that line we offer align:start position:0% you say the quantum that line we offer align:start position:0% you say the quantum that line we offer increasing Quantum Dynamics align:start position:0% align:start position:0% of some classical fields align:start position:0% align:start position:0% say such as align:start position:0% say such as align:start position:0% say such as say Electric magnetic field align:start position:0% say Electric magnetic field align:start position:0% say Electric magnetic field or space-time metric if you are align:start position:0% or space-time metric if you are align:start position:0% or space-time metric if you are interested in gravity align:start position:0% align:start position:0% Etc okay so so so in this case we align:start position:0% Etc okay so so so in this case we align:start position:0% Etc okay so so so in this case we already have the classical Fields Theory align:start position:0% already have the classical Fields Theory align:start position:0% already have the classical Fields Theory but we know the word is quantum and we align:start position:0% but we know the word is quantum and we align:start position:0% but we know the word is quantum and we want to understand what's the quantum align:start position:0% want to understand what's the quantum align:start position:0% want to understand what's the quantum version of it align:start position:0% version of it align:start position:0% version of it and the second is that it unifies align:start position:0% align:start position:0% special relativity align:start position:0% special relativity align:start position:0% special relativity plus quantum mechanics okay so you read align:start position:0% plus quantum mechanics okay so you read align:start position:0% plus quantum mechanics okay so you read the field Theory to to unify them and align:start position:0% the field Theory to to unify them and align:start position:0% the field Theory to to unify them and the Third Way align:start position:0% the Third Way align:start position:0% the Third Way he said is the a large distance align:start position:0% he said is the a large distance align:start position:0% he said is the a large distance description align:start position:0% align:start position:0% of discrete systems align:start position:0% align:start position:0% okay align:start position:0% okay align:start position:0% okay so align:start position:0% so align:start position:0% so um align:start position:0% um align:start position:0% um yeah just combine all three elements align:start position:0% yeah just combine all three elements align:start position:0% yeah just combine all three elements together the cover many many areas of align:start position:0% together the cover many many areas of align:start position:0% together the cover many many areas of physics okay they cover many many areas align:start position:0% align:start position:0% good so so now we can just align:start position:0% good so so now we can just align:start position:0% good so so now we can just say a little bit about the plan for the align:start position:0% say a little bit about the plan for the align:start position:0% say a little bit about the plan for the whole semester align:start position:0% whole semester align:start position:0% whole semester okay so here is the plan align:start position:0% okay so here is the plan align:start position:0% okay so here is the plan so align:start position:0% so align:start position:0% so so this is like just rephrase of the align:start position:0% so this is like just rephrase of the align:start position:0% so this is like just rephrase of the outline align:start position:0% outline align:start position:0% outline which uh align:start position:0% align:start position:0% so so the first thing we do in chapter align:start position:0% so so the first thing we do in chapter align:start position:0% so so the first thing we do in chapter two so here is chapter one chapter two align:start position:0% align:start position:0% we discussed the simplest field Theory align:start position:0% align:start position:0% just Express this equation two okay the align:start position:0% just Express this equation two okay the align:start position:0% just Express this equation two okay the theory of Q okay 2 and the two prime align:start position:0% theory of Q okay 2 and the two prime align:start position:0% theory of Q okay 2 and the two prime yeah a a prime is is equation motion so align:start position:0% yeah a a prime is is equation motion so align:start position:0% yeah a a prime is is equation motion so so yeah we in physics we always start align:start position:0% so yeah we in physics we always start align:start position:0% so yeah we in physics we always start with a simplistic example okay we always align:start position:0% with a simplistic example okay we always align:start position:0% with a simplistic example okay we always start with simplistic example and uh and align:start position:0% start with simplistic example and uh and align:start position:0% start with simplistic example and uh and so that's the uh is the one we will align:start position:0% so that's the uh is the one we will align:start position:0% so that's the uh is the one we will start with align:start position:0% start with align:start position:0% start with so what we will see is that this align:start position:0% so what we will see is that this align:start position:0% so what we will see is that this describes align:start position:0% align:start position:0% that field we describe spin is align:start position:0% that field we describe spin is align:start position:0% that field we describe spin is there's no align:start position:0% there's no align:start position:0% there's no free align:start position:0% free align:start position:0% free massive particles okay align:start position:0% align:start position:0% okay so we will see when we quantize align:start position:0% okay so we will see when we quantize align:start position:0% okay so we will see when we quantize that theory two and then we get the align:start position:0% that theory two and then we get the align:start position:0% that theory two and then we get the theory of align:start position:0% theory of align:start position:0% theory of three Methodist or three spin is massive align:start position:0% three Methodist or three spin is massive align:start position:0% three Methodist or three spin is massive particles okay align:start position:0% particles okay align:start position:0% particles okay so you say oh that's a little bit boring align:start position:0% so you say oh that's a little bit boring align:start position:0% so you say oh that's a little bit boring because in this series three the align:start position:0% because in this series three the align:start position:0% because in this series three the particle by three means they don't align:start position:0% particle by three means they don't align:start position:0% particle by three means they don't intact okay the particle they just don't align:start position:0% intact okay the particle they just don't align:start position:0% intact okay the particle they just don't intact align:start position:0% intact align:start position:0% intact and then in chapter 3 we will add align:start position:0% and then in chapter 3 we will add align:start position:0% and then in chapter 3 we will add interactions we will describe align:start position:0% align:start position:0% how to align:start position:0% how to align:start position:0% how to treat align:start position:0% treat align:start position:0% treat interactions okay align:start position:0% interactions okay align:start position:0% interactions okay so we will introduce interactions align:start position:0% so we will introduce interactions align:start position:0% so we will introduce interactions and tell you how to treat the align:start position:0% and tell you how to treat the align:start position:0% and tell you how to treat the interactions between those particles align:start position:0% align:start position:0% then in chapter four align:start position:0% then in chapter four align:start position:0% then in chapter four we go to the real physics align:start position:0% we go to the real physics align:start position:0% we go to the real physics so this scale of fields align:start position:0% so this scale of fields align:start position:0% so this scale of fields is also real say for example can be used align:start position:0% is also real say for example can be used align:start position:0% is also real say for example can be used to describe the Hicks align:start position:0% to describe the Hicks align:start position:0% to describe the Hicks okay but the heat exposure maybe it's a align:start position:0% okay but the heat exposure maybe it's a align:start position:0% okay but the heat exposure maybe it's a little bit far from what we uh you align:start position:0% little bit far from what we uh you align:start position:0% little bit far from what we uh you normally think about so in chapter four align:start position:0% normally think about so in chapter four align:start position:0% normally think about so in chapter four we will go to something which is much align:start position:0% we will go to something which is much align:start position:0% we will go to something which is much closer we'll talk about the theory of align:start position:0% closer we'll talk about the theory of align:start position:0% closer we'll talk about the theory of electron so this is called the rock align:start position:0% electron so this is called the rock align:start position:0% electron so this is called the rock theory align:start position:0% align:start position:0% so this Theory describes three align:start position:0% so this Theory describes three align:start position:0% so this Theory describes three but spin half align:start position:0% but spin half align:start position:0% but spin half particles okay align:start position:0% align:start position:0% so this is a theory of electrons align:start position:0% so this is a theory of electrons align:start position:0% so this is a theory of electrons okay align:start position:0% okay align:start position:0% okay when we like these interactions okay so align:start position:0% when we like these interactions okay so align:start position:0% when we like these interactions okay so this is the three spring hard particles align:start position:0% align:start position:0% and then align:start position:0% align:start position:0% [Applause] align:start position:0% align:start position:0% we move on to the maximals here align:start position:0% we move on to the maximals here align:start position:0% we move on to the maximals here Maxwell's Theory align:start position:0% align:start position:0% so this is the theory of the quantum align:start position:0% so this is the theory of the quantum align:start position:0% so this is the theory of the quantum electric and magnetic field okay align:start position:0% align:start position:0% so we quantize the Maxwell Theory align:start position:0% so we quantize the Maxwell Theory align:start position:0% so we quantize the Maxwell Theory say without Source the vacuum box real align:start position:0% say without Source the vacuum box real align:start position:0% say without Source the vacuum box real Siri and you find you get free again align:start position:0% Siri and you find you get free again align:start position:0% Siri and you find you get free again there's no interaction align:start position:0% align:start position:0% is align:start position:0% align:start position:0% particle align:start position:0% particle align:start position:0% particle you get a series of align:start position:0% you get a series of align:start position:0% you get a series of massive spring one particle align:start position:0% align:start position:0% so this is what we call the photon align:start position:0% so this is what we call the photon align:start position:0% so this is what we call the photon okay so this is the Quantum for align:start position:0% okay so this is the Quantum for align:start position:0% okay so this is the Quantum for electromagnetic field align:start position:0% electromagnetic field align:start position:0% electromagnetic field okay align:start position:0% okay align:start position:0% okay and then first and then align:start position:0% align:start position:0% sorry did I so this should be chapter align:start position:0% sorry did I so this should be chapter align:start position:0% sorry did I so this should be chapter five now I think I lost my account align:start position:0% five now I think I lost my account align:start position:0% five now I think I lost my account so now go to chapter six align:start position:0% so now go to chapter six align:start position:0% so now go to chapter six we combine the four and five together align:start position:0% we combine the four and five together align:start position:0% we combine the four and five together okay align:start position:0% okay align:start position:0% okay combine align:start position:0% align:start position:0% electrons align:start position:0% electrons align:start position:0% electrons so Photon normally we if we don't align:start position:0% so Photon normally we if we don't align:start position:0% so Photon normally we if we don't developed by gamma align:start position:0% developed by gamma align:start position:0% developed by gamma combine the theory of electron and the align:start position:0% combine the theory of electron and the align:start position:0% combine the theory of electron and the photon together align:start position:0% photon together align:start position:0% photon together and then plus interactions align:start position:0% align:start position:0% between them align:start position:0% between them align:start position:0% between them and then we get the so-called align:start position:0% and then we get the so-called align:start position:0% and then we get the so-called quantum electrodynamics align:start position:0% align:start position:0% so this is called QED align:start position:0% align:start position:0% so QD is very general essentially covers align:start position:0% so QD is very general essentially covers align:start position:0% so QD is very general essentially covers all the quantum phenomena uh yeah a align:start position:0% all the quantum phenomena uh yeah a align:start position:0% all the quantum phenomena uh yeah a microscopic phenomena up to say big align:start position:0% microscopic phenomena up to say big align:start position:0% microscopic phenomena up to say big interactions and strong interactions if align:start position:0% interactions and strong interactions if align:start position:0% interactions and strong interactions if you could don't go inside the nucleus align:start position:0% you could don't go inside the nucleus align:start position:0% you could don't go inside the nucleus and uh uh or don't go to a very high align:start position:0% and uh uh or don't go to a very high align:start position:0% and uh uh or don't go to a very high energy and think that's covers align:start position:0% energy and think that's covers align:start position:0% energy and think that's covers essentially most of the physics align:start position:0% align:start position:0% yeah yeah and and then the align:start position:0% align:start position:0% end of this our our course align:start position:0% end of this our our course align:start position:0% end of this our our course so do you have any questions on this align:start position:0% align:start position:0% okay so this is a road map yes align:start position:0% align:start position:0% good other questions align:start position:0% good other questions align:start position:0% good other questions yes align:start position:0% align:start position:0% sorry align:start position:0% align:start position:0% yeah it's also Mercedes spin one but align:start position:0% yeah it's also Mercedes spin one but align:start position:0% yeah it's also Mercedes spin one but actually the interact with themselves align:start position:0% actually the interact with themselves align:start position:0% actually the interact with themselves and so good one is different so so glue align:start position:0% and so good one is different so so glue align:start position:0% and so good one is different so so glue on to describe gluons you have to wait align:start position:0% on to describe gluons you have to wait align:start position:0% on to describe gluons you have to wait for Quantum field City too align:start position:0% for Quantum field City too align:start position:0% for Quantum field City too and uh so so the thing about the photon align:start position:0% and uh so so the thing about the photon align:start position:0% and uh so so the thing about the photon is that the photons don't interact with align:start position:0% is that the photons don't interact with align:start position:0% is that the photons don't interact with itself but the gluons interact with align:start position:0% itself but the gluons interact with align:start position:0% itself but the gluons interact with itself okay yeah so so essentially we align:start position:0% itself okay yeah so so essentially we align:start position:0% itself okay yeah so so essentially we treat everything align:start position:0% treat everything align:start position:0% treat everything except gloves yeah align:start position:0% align:start position:0% other questions align:start position:0% other questions align:start position:0% other questions other questions align:start position:0% other questions align:start position:0% other questions okay good align:start position:0% okay good align:start position:0% okay good so now we can just move to chapter two align:start position:0% so now we can just move to chapter two align:start position:0% so now we can just move to chapter two now we are talking about this series align:start position:0% now we are talking about this series align:start position:0% now we are talking about this series okay so so actually I should not erase align:start position:0% okay so so actually I should not erase align:start position:0% okay so so actually I should not erase it align:start position:0% it align:start position:0% it so now we talk about this here align:start position:0% so now we talk about this here align:start position:0% so now we talk about this here so because because this series describes align:start position:0% so because because this series describes align:start position:0% so because because this series describes three particles align:start position:0% three particles align:start position:0% three particles so we call it free scalar field CV align:start position:0% align:start position:0% okay so so so this is the align:start position:0% okay so so so this is the align:start position:0% okay so so so this is the Theory we are interested in so so now we align:start position:0% Theory we are interested in so so now we align:start position:0% Theory we are interested in so so now we will describe how to quantize this align:start position:0% will describe how to quantize this align:start position:0% will describe how to quantize this series okay align:start position:0% align:start position:0% good align:start position:0% good align:start position:0% good so so first align:start position:0% so so first align:start position:0% so so first we will quickly align:start position:0% we will quickly align:start position:0% we will quickly go through align:start position:0% go through align:start position:0% go through the the quantitation of harmonic align:start position:0% the the quantitation of harmonic align:start position:0% the the quantitation of harmonic oscillator align:start position:0% oscillator align:start position:0% oscillator which you should already have done in align:start position:0% which you should already have done in align:start position:0% which you should already have done in your in your preset and so uh so we can align:start position:0% your in your preset and so uh so we can align:start position:0% your in your preset and so uh so we can do it relatively fast align:start position:0% do it relatively fast align:start position:0% do it relatively fast so align:start position:0% align:start position:0% the organization align:start position:0% align:start position:0% of pythonic oscillator align:start position:0% of pythonic oscillator align:start position:0% of pythonic oscillator in the Heisenberg picture align:start position:0% align:start position:0% so we will see that once we understand align:start position:0% so we will see that once we understand align:start position:0% so we will see that once we understand this example align:start position:0% this example align:start position:0% this example in the right way align:start position:0% in the right way align:start position:0% in the right way and then contacting this field Theory align:start position:0% and then contacting this field Theory align:start position:0% and then contacting this field Theory becomes trivial align:start position:0% becomes trivial align:start position:0% becomes trivial okay and the quantize in this field align:start position:0% okay and the quantize in this field align:start position:0% okay and the quantize in this field series become trigger align:start position:0% series become trigger align:start position:0% series become trigger okay so so let's start with a harmonic align:start position:0% okay so so let's start with a harmonic align:start position:0% okay so so let's start with a harmonic oscillator align:start position:0% align:start position:0% for Simplicity I take the mass to be one align:start position:0% for Simplicity I take the mass to be one align:start position:0% for Simplicity I take the mass to be one and take the frequency to be y okay align:start position:0% align:start position:0% yeah yeah let me put the frequency here align:start position:0% yeah yeah let me put the frequency here align:start position:0% yeah yeah let me put the frequency here it won't fit align:start position:0% it won't fit align:start position:0% it won't fit okay let's take the mass TDY y okay align:start position:0% okay let's take the mass TDY y okay align:start position:0% okay let's take the mass TDY y okay and so so for this series so so this is align:start position:0% and so so for this series so so this is align:start position:0% and so so for this series so so this is a simple harmonic oscillator which you align:start position:0% a simple harmonic oscillator which you align:start position:0% a simple harmonic oscillator which you have seen it align:start position:0% have seen it align:start position:0% have seen it uh uh uh maybe for the most of your align:start position:0% uh uh uh maybe for the most of your align:start position:0% uh uh uh maybe for the most of your intellectual life align:start position:0% intellectual life align:start position:0% intellectual life and the P will be x dot it's a momentum align:start position:0% and the P will be x dot it's a momentum align:start position:0% and the P will be x dot it's a momentum the conjugate momentum is x dot and so align:start position:0% the conjugate momentum is x dot and so align:start position:0% the conjugate momentum is x dot and so the hamiltonian is the P-Square divided align:start position:0% the hamiltonian is the P-Square divided align:start position:0% the hamiltonian is the P-Square divided by two align:start position:0% by two align:start position:0% by two of one half Omega Square x squared okay align:start position:0% of one half Omega Square x squared okay align:start position:0% of one half Omega Square x squared okay an equation motion align:start position:0% an equation motion align:start position:0% an equation motion is X dots double dot equal to x three x align:start position:0% align:start position:0% okay align:start position:0% align:start position:0% so align:start position:0% so align:start position:0% so so let's first look at this Theory uh align:start position:0% so let's first look at this Theory uh align:start position:0% so let's first look at this Theory uh look at harmonic considered as a align:start position:0% look at harmonic considered as a align:start position:0% look at harmonic considered as a classical Theory align:start position:0% align:start position:0% so for classical Theory align:start position:0% so for classical Theory align:start position:0% so for classical Theory we know how to solve this equation we align:start position:0% we know how to solve this equation we align:start position:0% we know how to solve this equation we just need to solve this equation align:start position:0% just need to solve this equation align:start position:0% just need to solve this equation so classical solution align:start position:0% align:start position:0% just given by x t align:start position:0% just given by x t align:start position:0% just given by x t equal to a align:start position:0% equal to a align:start position:0% equal to a cosine Omega t align:start position:0% cosine Omega t align:start position:0% cosine Omega t plus b sine of Omega t align:start position:0% plus b sine of Omega t align:start position:0% plus b sine of Omega t and a and a and b just some integration align:start position:0% and a and a and b just some integration align:start position:0% and a and a and b just some integration constant align:start position:0% align:start position:0% and for convenience I can also write it align:start position:0% and for convenience I can also write it align:start position:0% and for convenience I can also write it in the complex form align:start position:0% in the complex form align:start position:0% in the complex form bracket as following align:start position:0% bracket as following align:start position:0% bracket as following equal to a align:start position:0% equal to a align:start position:0% equal to a expansion minus sign Omega t align:start position:0% expansion minus sign Omega t align:start position:0% expansion minus sign Omega t plus a star align:start position:0% plus a star align:start position:0% plus a star expression I Omega T and A is some align:start position:0% expression I Omega T and A is some align:start position:0% expression I Omega T and A is some complex constant align:start position:0% complex constant align:start position:0% complex constant and again it's a integration constant I align:start position:0% and again it's a integration constant I align:start position:0% and again it's a integration constant I just rewrite the integration constant align:start position:0% just rewrite the integration constant align:start position:0% just rewrite the integration constant slightly differently align:start position:0% align:start position:0% okay align:start position:0% okay align:start position:0% okay now these are just integration constants align:start position:0% align:start position:0% so now align:start position:0% so now align:start position:0% so now yeah so this is a complete solution of align:start position:0% yeah so this is a complete solution of align:start position:0% yeah so this is a complete solution of the problem align:start position:0% the problem align:start position:0% the problem so now let's go to Quantum align:start position:0% align:start position:0% so when we go to Quantum align:start position:0% align:start position:0% and then then we replace align:start position:0% and then then we replace align:start position:0% and then then we replace then this classical dynamical variable align:start position:0% then this classical dynamical variable align:start position:0% then this classical dynamical variable then become align:start position:0% then become align:start position:0% then become the the Heisenberg operator align:start position:0% the the Heisenberg operator align:start position:0% the the Heisenberg operator becomes the quantum operator in align:start position:0% becomes the quantum operator in align:start position:0% becomes the quantum operator in particular in the in the Heisenberg align:start position:0% particular in the in the Heisenberg align:start position:0% particular in the in the Heisenberg picture and then then this operator will align:start position:0% picture and then then this operator will align:start position:0% picture and then then this operator will depend on time align:start position:0% depend on time align:start position:0% depend on time okay align:start position:0% okay align:start position:0% okay and now this equation align:start position:0% align:start position:0% become operating equation okay so now align:start position:0% become operating equation okay so now align:start position:0% become operating equation okay so now let's align:start position:0% let's align:start position:0% let's maybe I should label my equation align:start position:0% maybe I should label my equation align:start position:0% maybe I should label my equation so now this star align:start position:0% align:start position:0% become an operating equation align:start position:0% align:start position:0% now star is the operating equation align:start position:0% now star is the operating equation align:start position:0% now star is the operating equation for X hat align:start position:0% for X hat align:start position:0% for X hat so you have exact the same equation as a align:start position:0% so you have exact the same equation as a align:start position:0% so you have exact the same equation as a classical equation but not the align:start position:0% classical equation but not the align:start position:0% classical equation but not the interpretation is different align:start position:0% interpretation is different align:start position:0% interpretation is different and now now the X hat becomes the uh align:start position:0% and now now the X hat becomes the uh align:start position:0% and now now the X hat becomes the uh another X become the operating equation align:start position:0% another X become the operating equation align:start position:0% another X become the operating equation foreign align:start position:0% align:start position:0% so now the solution align:start position:0% align:start position:0% now let me call this star star align:start position:0% align:start position:0% so this still solves that equation align:start position:0% so this still solves that equation align:start position:0% so this still solves that equation okay so this still solves that equation align:start position:0% okay so this still solves that equation align:start position:0% okay so this still solves that equation except align:start position:0% except align:start position:0% except so these are just C numbers because align:start position:0% so these are just C numbers because align:start position:0% so these are just C numbers because because this is a function of T these align:start position:0% because this is a function of T these align:start position:0% because this is a function of T these are C numbers align:start position:0% are C numbers align:start position:0% are C numbers but now X becomes the uh so now Quantum align:start position:0% but now X becomes the uh so now Quantum align:start position:0% but now X becomes the uh so now Quantum mechanically align:start position:0% align:start position:0% so this becomes now become the quantum align:start position:0% so this becomes now become the quantum align:start position:0% so this becomes now become the quantum solution align:start position:0% align:start position:0% okay align:start position:0% okay align:start position:0% okay so now I've had so this still solves the align:start position:0% so now I've had so this still solves the align:start position:0% so now I've had so this still solves the equations so mechanically this becomes align:start position:0% equations so mechanically this becomes align:start position:0% equations so mechanically this becomes hat align:start position:0% align:start position:0% okay align:start position:0% okay align:start position:0% okay and so this still solves that equation align:start position:0% and so this still solves that equation align:start position:0% and so this still solves that equation but these are C numbers align:start position:0% but these are C numbers align:start position:0% but these are C numbers the left hand side is the operator align:start position:0% the left hand side is the operator align:start position:0% the left hand side is the operator and it can only be that a hat and B hat align:start position:0% and it can only be that a hat and B hat align:start position:0% and it can only be that a hat and B hat are operators and also a must be align:start position:0% are operators and also a must be align:start position:0% are operators and also a must be operators and the star will replace it align:start position:0% operators and the star will replace it align:start position:0% operators and the star will replace it by diagram align:start position:0% by diagram align:start position:0% by diagram okay so now align:start position:0% align:start position:0% say align:start position:0% say align:start position:0% say now a you just go to a hat align:start position:0% now a you just go to a hat align:start position:0% now a you just go to a hat and a star goes to a dagger it has align:start position:0% and a star goes to a dagger it has align:start position:0% and a star goes to a dagger it has dagger okay now these are align:start position:0% dagger okay now these are align:start position:0% dagger okay now these are these are integration constants for the align:start position:0% these are integration constants for the align:start position:0% these are integration constants for the operating equations align:start position:0% operating equations align:start position:0% operating equations so they are just count now they become align:start position:0% so they are just count now they become align:start position:0% so they are just count now they become constant operators align:start position:0% constant operators align:start position:0% constant operators okay so so so so so they're just align:start position:0% okay so so so so so they're just align:start position:0% okay so so so so so they're just constant align:start position:0% constant align:start position:0% constant Quantum operators align:start position:0% align:start position:0% okay they're just constant operators align:start position:0% okay they're just constant operators align:start position:0% okay they're just constant operators so they are integration constants for align:start position:0% so they are integration constants for align:start position:0% so they are integration constants for your for your Quantum operating align:start position:0% your for your Quantum operating align:start position:0% your for your Quantum operating equations align:start position:0% equations align:start position:0% equations so now the solution align:start position:0% so now the solution align:start position:0% so now the solution so the as another uh uh yeah so now this align:start position:0% so the as another uh uh yeah so now this align:start position:0% so the as another uh uh yeah so now this is your Quantum solution okay align:start position:0% align:start position:0% so this is the form we will often use align:start position:0% so this is the form we will often use align:start position:0% so this is the form we will often use okay align:start position:0% okay align:start position:0% okay you can also use that form but the align:start position:0% you can also use that form but the align:start position:0% you can also use that form but the equivalent but this is the form we often align:start position:0% equivalent but this is the form we often align:start position:0% equivalent but this is the form we often use okay align:start position:0% use okay align:start position:0% use okay you can also from here you take the align:start position:0% you can also from here you take the align:start position:0% you can also from here you take the derivative you can find the P so again align:start position:0% derivative you can find the P so again align:start position:0% derivative you can find the P so again this is become an operating equation you align:start position:0% this is become an operating equation you align:start position:0% this is become an operating equation you take the direction of X hat X and maybe align:start position:0% take the direction of X hat X and maybe align:start position:0% take the direction of X hat X and maybe you find P Etc align:start position:0% you find P Etc align:start position:0% you find P Etc okay align:start position:0% align:start position:0% yeah so you align:start position:0% yeah so you align:start position:0% yeah so you the P hats t align:start position:0% the P hats t align:start position:0% the P hats t just take a derivative align:start position:0% align:start position:0% okay and then you can just align:start position:0% okay and then you can just align:start position:0% okay and then you can just walk it out it's very easy align:start position:0% align:start position:0% so now align:start position:0% so now align:start position:0% so now so this equation just align:start position:0% so this equation just align:start position:0% so this equation just so we already solved the quantum Tobin align:start position:0% so we already solved the quantum Tobin align:start position:0% so we already solved the quantum Tobin okay so so because we find the full align:start position:0% okay so so because we find the full align:start position:0% okay so so because we find the full evolution full solution to the quantum align:start position:0% evolution full solution to the quantum align:start position:0% evolution full solution to the quantum operator equation align:start position:0% operator equation align:start position:0% operator equation except that we still leads to align:start position:0% except that we still leads to align:start position:0% except that we still leads to impose the canonical condition align:start position:0% align:start position:0% so this is just equal to I align:start position:0% so this is just equal to I align:start position:0% so this is just equal to I okay so so the standard so if you plug align:start position:0% okay so so the standard so if you plug align:start position:0% okay so so the standard so if you plug in the expression for x align:start position:0% in the expression for x align:start position:0% in the expression for x and the T and the P into here align:start position:0% and the T and the P into here align:start position:0% and the T and the P into here and then you find align:start position:0% and then you find align:start position:0% and then you find that a and a dagger align:start position:0% align:start position:0% accommodate is equal to one okay so this align:start position:0% accommodate is equal to one okay so this align:start position:0% accommodate is equal to one okay so this is your familiar align:start position:0% is your familiar align:start position:0% is your familiar a creation Annihilation operator for for align:start position:0% a creation Annihilation operator for for align:start position:0% a creation Annihilation operator for for harmonic oscillator align:start position:0% harmonic oscillator align:start position:0% harmonic oscillator okay for harmonic oscillator align:start position:0% okay for harmonic oscillator align:start position:0% okay for harmonic oscillator and uh align:start position:0% and uh align:start position:0% and uh and now we can also use the a to build align:start position:0% and now we can also use the a to build align:start position:0% and now we can also use the a to build the Hilbert space so because a are the align:start position:0% the Hilbert space so because a are the align:start position:0% the Hilbert space so because a are the yeah because all you operate now align:start position:0% yeah because all you operate now align:start position:0% yeah because all you operate now or because X and the T align:start position:0% or because X and the T align:start position:0% or because X and the T X and P are expressed in terms of a and align:start position:0% X and P are expressed in terms of a and align:start position:0% X and P are expressed in terms of a and a dagger so essentially any operator of align:start position:0% a dagger so essentially any operator of align:start position:0% a dagger so essentially any operator of this Theory can all be expressed in align:start position:0% this Theory can all be expressed in align:start position:0% this Theory can all be expressed in terms of being a diagram okay align:start position:0% align:start position:0% and then you can just use a and a gagger align:start position:0% and then you can just use a and a gagger align:start position:0% and then you can just use a and a gagger because A and negative essentially they align:start position:0% because A and negative essentially they align:start position:0% because A and negative essentially they they are fundamental building block of align:start position:0% they are fundamental building block of align:start position:0% they are fundamental building block of your full quantum theory and then we can align:start position:0% your full quantum theory and then we can align:start position:0% your full quantum theory and then we can also use that to build the helper space align:start position:0% also use that to build the helper space align:start position:0% also use that to build the helper space so the healable space align:start position:0% so the healable space align:start position:0% so the healable space is defined by the lowest state is align:start position:0% is defined by the lowest state is align:start position:0% is defined by the lowest state is evaluated by a align:start position:0% evaluated by a align:start position:0% evaluated by a and then and then the in the higher align:start position:0% and then and then the in the higher align:start position:0% and then and then the in the higher state align:start position:0% state align:start position:0% state uh align:start position:0% align:start position:0% obtained by acting a dagger align:start position:0% obtained by acting a dagger align:start position:0% obtained by acting a dagger on the on the on the ground state okay align:start position:0% on the on the on the ground state okay align:start position:0% on the on the on the ground state okay so so so this is your full series okay align:start position:0% so so so this is your full series okay align:start position:0% so so so this is your full series okay so this is your 4C align:start position:0% so this is your 4C align:start position:0% so this is your 4C and so now align:start position:0% align:start position:0% uh you can compute anything in this align:start position:0% uh you can compute anything in this align:start position:0% uh you can compute anything in this series just miss those knowledge okay align:start position:0% series just miss those knowledge okay align:start position:0% series just miss those knowledge okay just with lots of knowledge align:start position:0% just with lots of knowledge align:start position:0% just with lots of knowledge so any questions on this regarding the align:start position:0% so any questions on this regarding the align:start position:0% so any questions on this regarding the harmonic oscillator align:start position:0% align:start position:0% good okay align:start position:0% align:start position:0% so so so let me just summarize align:start position:0% so so so let me just summarize align:start position:0% so so so let me just summarize so this is maybe very familiar align:start position:0% so this is maybe very familiar align:start position:0% so this is maybe very familiar but let's summarize the rule we have align:start position:0% but let's summarize the rule we have align:start position:0% but let's summarize the rule we have been using Okay summarize the steps align:start position:0% align:start position:0% in context the harmonic oscillator align:start position:0% in context the harmonic oscillator align:start position:0% in context the harmonic oscillator and then the and then the same steps can align:start position:0% and then the and then the same steps can align:start position:0% and then the and then the same steps can be used to quantize the the field series align:start position:0% align:start position:0% steps align:start position:0% steps align:start position:0% steps of quantization align:start position:0% align:start position:0% so we make it a general align:start position:0% so we make it a general align:start position:0% so we make it a general so first align:start position:0% so first align:start position:0% so first so the zero step is that the classical align:start position:0% so the zero step is that the classical align:start position:0% so the zero step is that the classical equation motion the quantum quantum align:start position:0% align:start position:0% operator equation align:start position:0% operator equation align:start position:0% operator equation okay then the first step align:start position:0% okay then the first step align:start position:0% okay then the first step is to find the most General solution align:start position:0% align:start position:0% to align:start position:0% align:start position:0% find the most General solution to align:start position:0% find the most General solution to align:start position:0% find the most General solution to classical equation motion align:start position:0% classical equation motion align:start position:0% classical equation motion yeah just to equation motion align:start position:0% yeah just to equation motion align:start position:0% yeah just to equation motion okay align:start position:0% okay align:start position:0% okay and then align:start position:0% align:start position:0% when you've got the quantum align:start position:0% when you've got the quantum align:start position:0% when you've got the quantum and then you just promote align:start position:0% align:start position:0% the integration constants align:start position:0% align:start position:0% in your classical solution in one align:start position:0% in your classical solution in one align:start position:0% in your classical solution in one in the step one align:start position:0% align:start position:0% to align:start position:0% to align:start position:0% to constant operators align:start position:0% align:start position:0% constant Quantum operators okay align:start position:0% align:start position:0% so this gives the uh then then you have align:start position:0% so this gives the uh then then you have align:start position:0% so this gives the uh then then you have the full-time evolution align:start position:0% align:start position:0% at the quantum level okay okay align:start position:0% align:start position:0% now you know the how the quantum align:start position:0% now you know the how the quantum align:start position:0% now you know the how the quantum operator evolves align:start position:0% align:start position:0% and then align:start position:0% align:start position:0% you impose align:start position:0% align:start position:0% canonical condition align:start position:0% align:start position:0% s align:start position:0% align:start position:0% okay so that will tell you align:start position:0% okay so that will tell you align:start position:0% okay so that will tell you the commentators between those align:start position:0% the commentators between those align:start position:0% the commentators between those integration constant operators okay align:start position:0% integration constant operators okay align:start position:0% integration constant operators okay just as we do here align:start position:0% align:start position:0% and then align:start position:0% and then align:start position:0% and then um align:start position:0% align:start position:0% and then um align:start position:0% align:start position:0% constant operators align:start position:0% constant operators align:start position:0% constant operators in queue align:start position:0% in queue align:start position:0% in queue now now you know also load the align:start position:0% now now you know also load the align:start position:0% now now you know also load the commutation relation between them among align:start position:0% commutation relation between them among align:start position:0% commutation relation between them among them and then now can be used to align:start position:0% align:start position:0% generate the healable space align:start position:0% align:start position:0% okay align:start position:0% okay align:start position:0% okay so this step and one two four are very align:start position:0% so this step and one two four are very align:start position:0% so this step and one two four are very general align:start position:0% general align:start position:0% general and if you can do it align:start position:0% and if you can do it align:start position:0% and if you can do it and then you can then you can align:start position:0% and then you can then you can align:start position:0% and then you can then you can essentially uh do it applied to gen any align:start position:0% essentially uh do it applied to gen any align:start position:0% essentially uh do it applied to gen any system align:start position:0% system align:start position:0% system say one degree harmonic acid is one align:start position:0% say one degree harmonic acid is one align:start position:0% say one degree harmonic acid is one degrees three then you can apply it two align:start position:0% degrees three then you can apply it two align:start position:0% degrees three then you can apply it two degrees three degrees freedom and also align:start position:0% degrees three degrees freedom and also align:start position:0% degrees three degrees freedom and also to field series with infinite number align:start position:0% to field series with infinite number align:start position:0% to field series with infinite number equals freedom align:start position:0% equals freedom align:start position:0% equals freedom okay and now we will apply these two align:start position:0% okay and now we will apply these two align:start position:0% okay and now we will apply these two field series yes align:start position:0% align:start position:0% uh for this procedure you cannot align:start position:0% uh for this procedure you cannot align:start position:0% uh for this procedure you cannot but but uh but you can get the finite uh align:start position:0% but but uh but you can get the finite uh align:start position:0% but but uh but you can get the finite uh Dimension yeah align:start position:0% Dimension yeah align:start position:0% Dimension yeah because the finite here will space don't align:start position:0% because the finite here will space don't align:start position:0% because the finite here will space don't have the classical analog so here we align:start position:0% have the classical analog so here we align:start position:0% have the classical analog so here we start with the classical system and then align:start position:0% start with the classical system and then align:start position:0% start with the classical system and then we contact it and there's a the the the align:start position:0% we contact it and there's a the the the align:start position:0% we contact it and there's a the the the the quantum system with a finite align:start position:0% the quantum system with a finite align:start position:0% the quantum system with a finite Dimension healable space they align:start position:0% Dimension healable space they align:start position:0% Dimension healable space they essentially intrinsically quantum and uh align:start position:0% essentially intrinsically quantum and uh align:start position:0% essentially intrinsically quantum and uh yeah and like spin spin is an intrinsic align:start position:0% yeah and like spin spin is an intrinsic align:start position:0% yeah and like spin spin is an intrinsic Quantum scene yeah align:start position:0% Quantum scene yeah align:start position:0% Quantum scene yeah yes align:start position:0% align:start position:0% yeah yeah yeah it's just because they align:start position:0% yeah yeah yeah it's just because they align:start position:0% yeah yeah yeah it's just because they don't have yeah yeah the reason is they align:start position:0% don't have yeah yeah the reason is they align:start position:0% don't have yeah yeah the reason is they don't have classical counterpart yeah align:start position:0% align:start position:0% yes um is it always true align:start position:0% align:start position:0% yeah because because if you think about align:start position:0% yeah because because if you think about align:start position:0% yeah because because if you think about this way yeah that's a very good align:start position:0% this way yeah that's a very good align:start position:0% this way yeah that's a very good question because align:start position:0% question because align:start position:0% question because because let's just look at this harmonic align:start position:0% because let's just look at this harmonic align:start position:0% because let's just look at this harmonic oscillator and then you can try to think align:start position:0% oscillator and then you can try to think align:start position:0% oscillator and then you can try to think generalize it align:start position:0% generalize it align:start position:0% generalize it because align:start position:0% because align:start position:0% because they are integrating constant of the X align:start position:0% they are integrating constant of the X align:start position:0% they are integrating constant of the X and the p align:start position:0% and the p align:start position:0% and the p that any operator in your theory can all align:start position:0% that any operator in your theory can all align:start position:0% that any operator in your theory can all be expressed in terms of a and a dagger align:start position:0% be expressed in terms of a and a dagger align:start position:0% be expressed in terms of a and a dagger and then and then you'll hear the space align:start position:0% and then and then you'll hear the space align:start position:0% and then and then you'll hear the space must begin you must be able to generate align:start position:0% must begin you must be able to generate align:start position:0% must begin you must be able to generate the herbal space using them yeah yeah align:start position:0% the herbal space using them yeah yeah align:start position:0% the herbal space using them yeah yeah because they are the building product of align:start position:0% because they are the building product of align:start position:0% because they are the building product of your of your whole operators yeah align:start position:0% align:start position:0% yeah yeah the working State here is align:start position:0% yeah yeah the working State here is align:start position:0% yeah yeah the working State here is based on uh it's come from the uh uh uh align:start position:0% based on uh it's come from the uh uh uh align:start position:0% based on uh it's come from the uh uh uh uh from the energy right so so once we align:start position:0% uh from the energy right so so once we align:start position:0% uh from the energy right so so once we solve x and P align:start position:0% solve x and P align:start position:0% solve x and P and then you can write to the align:start position:0% and then you can write to the align:start position:0% and then you can write to the hamiltonian in terms of X and P and then align:start position:0% hamiltonian in terms of X and P and then align:start position:0% hamiltonian in terms of X and P and then you just look for the lowest energy align:start position:0% you just look for the lowest energy align:start position:0% you just look for the lowest energy State and then you find the lowest align:start position:0% State and then you find the lowest align:start position:0% State and then you find the lowest energy State just satisfy this equation align:start position:0% energy State just satisfy this equation align:start position:0% energy State just satisfy this equation yeah and then from there you can find align:start position:0% yeah and then from there you can find align:start position:0% yeah and then from there you can find other states align:start position:0% other states align:start position:0% other states yeah the same thing we are going to yeah align:start position:0% yeah the same thing we are going to yeah align:start position:0% yeah the same thing we are going to yeah the same strategy we are going to use align:start position:0% the same strategy we are going to use align:start position:0% the same strategy we are going to use for the uh of a Quantum velocity align:start position:0% for the uh of a Quantum velocity align:start position:0% for the uh of a Quantum velocity okay good align:start position:0% okay good align:start position:0% okay good okay good so so now become a mechanical align:start position:0% okay good so so now become a mechanical align:start position:0% okay good so so now become a mechanical we can just uh apply this to to this align:start position:0% we can just uh apply this to to this align:start position:0% we can just uh apply this to to this Theory align:start position:0% Theory align:start position:0% Theory okay we can just apply these two three align:start position:0% okay we can just apply these two three align:start position:0% okay we can just apply these two three Theory and now let me add here so here align:start position:0% Theory and now let me add here so here align:start position:0% Theory and now let me add here so here the canonical momentum density conjugate align:start position:0% the canonical momentum density conjugate align:start position:0% the canonical momentum density conjugate to Phi is called the pi before it's just align:start position:0% to Phi is called the pi before it's just align:start position:0% to Phi is called the pi before it's just the atom derivative of Phi align:start position:0% the atom derivative of Phi align:start position:0% the atom derivative of Phi and the hamiltonian density you can find align:start position:0% and the hamiltonian density you can find align:start position:0% and the hamiltonian density you can find it explicitly align:start position:0% it explicitly align:start position:0% it explicitly is pi square plus align:start position:0% is pi square plus align:start position:0% is pi square plus one half align:start position:0% align:start position:0% okay align:start position:0% okay align:start position:0% okay and then this is a classical equation align:start position:0% and then this is a classical equation align:start position:0% and then this is a classical equation motion okay align:start position:0% align:start position:0% now let's just solve this classical align:start position:0% now let's just solve this classical align:start position:0% now let's just solve this classical equation motion align:start position:0% align:start position:0% so this equation can be align:start position:0% so this equation can be align:start position:0% so this equation can be so this equation is easy to solve align:start position:0% so this equation is easy to solve align:start position:0% so this equation is easy to solve because the because the translation align:start position:0% because the because the translation align:start position:0% because the because the translation symmetry align:start position:0% symmetry align:start position:0% symmetry okay you can just do a free transform align:start position:0% okay you can just do a free transform align:start position:0% okay you can just do a free transform okay so we can fully transform align:start position:0% align:start position:0% so so now let's do the um align:start position:0% so so now let's do the um align:start position:0% so so now let's do the um you can freely transform align:start position:0% you can freely transform align:start position:0% you can freely transform so 2 Prime align:start position:0% so 2 Prime align:start position:0% so 2 Prime to be solved align:start position:0% align:start position:0% okay align:start position:0% okay align:start position:0% okay so we can just write 5X align:start position:0% align:start position:0% equal to expenditure align:start position:0% equal to expenditure align:start position:0% equal to expenditure minus IET align:start position:0% minus IET align:start position:0% minus IET plus I align:start position:0% plus I align:start position:0% plus I K dot X align:start position:0% K dot X align:start position:0% K dot X okay align:start position:0% align:start position:0% okay so and then you you can see that align:start position:0% okay so and then you you can see that align:start position:0% okay so and then you you can see that this align:start position:0% this align:start position:0% this just solves the align:start position:0% just solves the align:start position:0% just solves the so this is provide align:start position:0% align:start position:0% a basis align:start position:0% align:start position:0% of solutions align:start position:0% align:start position:0% 22 prime okay align:start position:0% 22 prime okay align:start position:0% 22 prime okay it's just a plain wave okay just plain align:start position:0% it's just a plain wave okay just plain align:start position:0% it's just a plain wave okay just plain weight align:start position:0% weight align:start position:0% weight oh align:start position:0% oh align:start position:0% oh so now we plug this into there align:start position:0% so now we plug this into there align:start position:0% so now we plug this into there then you just get the dispersion align:start position:0% then you just get the dispersion align:start position:0% then you just get the dispersion relation align:start position:0% relation align:start position:0% relation e Square align:start position:0% e Square align:start position:0% e Square should be M squared plus K Square align:start position:0% align:start position:0% okay so so we'll delote this as align:start position:0% okay so so we'll delote this as align:start position:0% okay so so we'll delote this as Omega K Square align:start position:0% Omega K Square align:start position:0% Omega K Square so Omega k align:start position:0% so Omega k align:start position:0% so Omega k defines to be align:start position:0% defines to be align:start position:0% defines to be just the K Square align:start position:0% just the K Square align:start position:0% just the K Square plus M squared align:start position:0% plus M squared align:start position:0% plus M squared okay align:start position:0% align:start position:0% so ee when you take the square root of e align:start position:0% so ee when you take the square root of e align:start position:0% so ee when you take the square root of e so you can take a plus minus Omega k align:start position:0% so you can take a plus minus Omega k align:start position:0% so you can take a plus minus Omega k okay can be plus minus Omega K so we align:start position:0% okay can be plus minus Omega K so we align:start position:0% okay can be plus minus Omega K so we normally call the solution align:start position:0% normally call the solution align:start position:0% normally call the solution we normally separate align:start position:0% align:start position:0% so for historical reasons okay align:start position:0% so for historical reasons okay align:start position:0% so for historical reasons okay we normally quote align:start position:0% we normally quote align:start position:0% we normally quote Define UK align:start position:0% Define UK align:start position:0% Define UK X to be align:start position:0% X to be align:start position:0% X to be experiential matters I Omega k align:start position:0% experiential matters I Omega k align:start position:0% experiential matters I Omega k t align:start position:0% align:start position:0% X so okay now we have inserted the align:start position:0% X so okay now we have inserted the align:start position:0% X so okay now we have inserted the positive root of E align:start position:0% positive root of E align:start position:0% positive root of E so this is normally called the positive align:start position:0% so this is normally called the positive align:start position:0% so this is normally called the positive Energy Solution align:start position:0% align:start position:0% even though this align:start position:0% align:start position:0% even though this name is Elizabeth align:start position:0% even though this name is Elizabeth align:start position:0% even though this name is Elizabeth misleading okay so so uh actually this align:start position:0% misleading okay so so uh actually this align:start position:0% misleading okay so so uh actually this we don't Define the energy actually uh align:start position:0% we don't Define the energy actually uh align:start position:0% we don't Define the energy actually uh yeah later we will see this is not align:start position:0% yeah later we will see this is not align:start position:0% yeah later we will see this is not really the energy of a particle and so align:start position:0% really the energy of a particle and so align:start position:0% really the energy of a particle and so so this is just a traditional name okay align:start position:0% so this is just a traditional name okay align:start position:0% so this is just a traditional name okay this is just a traditional name align:start position:0% this is just a traditional name align:start position:0% this is just a traditional name conventional name align:start position:0% conventional name align:start position:0% conventional name and then you can Define the compressed align:start position:0% and then you can Define the compressed align:start position:0% and then you can Define the compressed conjugate of K align:start position:0% conjugate of K align:start position:0% conjugate of K now you have align:start position:0% now you have align:start position:0% now you have then corresponding to you have align:start position:0% then corresponding to you have align:start position:0% then corresponding to you have minus align:start position:0% align:start position:0% Omega K in there align:start position:0% align:start position:0% so we align:start position:0% so we align:start position:0% so we yeah we take this conjugate align:start position:0% align:start position:0% and so this is called the reactive align:start position:0% and so this is called the reactive align:start position:0% and so this is called the reactive Energy Solution align:start position:0% align:start position:0% okay align:start position:0% okay align:start position:0% okay so all together align:start position:0% align:start position:0% therefore the former company side of align:start position:0% therefore the former company side of align:start position:0% therefore the former company side of solutions okay so complete side of align:start position:0% solutions okay so complete side of align:start position:0% solutions okay so complete side of solutions align:start position:0% align:start position:0% so align:start position:0% so align:start position:0% so complete the basis the complete side of align:start position:0% complete the basis the complete side of align:start position:0% complete the basis the complete side of yeah complete align:start position:0% align:start position:0% phases align:start position:0% align:start position:0% is formed by align:start position:0% align:start position:0% UK align:start position:0% UK align:start position:0% UK and the UK UK star align:start position:0% and the UK UK star align:start position:0% and the UK UK star for okay align:start position:0% align:start position:0% okay align:start position:0% align:start position:0% so these are the uh when you these are align:start position:0% so these are the uh when you these are align:start position:0% so these are the uh when you these are the complete set of solutions to that align:start position:0% the complete set of solutions to that align:start position:0% the complete set of solutions to that wave equation okay so that's just the uh align:start position:0% wave equation okay so that's just the uh align:start position:0% wave equation okay so that's just the uh um yeah the uh align:start position:0% um yeah the uh align:start position:0% um yeah the uh any questions on this align:start position:0% align:start position:0% so this is just like a classically this align:start position:0% so this is just like a classically this align:start position:0% so this is just like a classically this is like a wave okay just like a plane align:start position:0% is like a wave okay just like a plane align:start position:0% is like a wave okay just like a plane wave which you should also have seen in align:start position:0% wave which you should also have seen in align:start position:0% wave which you should also have seen in 803 you know align:start position:0% 803 you know align:start position:0% 803 you know or align:start position:0% or align:start position:0% or good align:start position:0% good align:start position:0% good so now we can find that so now we can align:start position:0% so now we can find that so now we can align:start position:0% so now we can find that so now we can write down the most General align:start position:0% write down the most General align:start position:0% write down the most General so this is a basis align:start position:0% align:start position:0% so these are the counterparts of the align:start position:0% so these are the counterparts of the align:start position:0% so these are the counterparts of the exponential plus minus the Omega T here align:start position:0% exponential plus minus the Omega T here align:start position:0% exponential plus minus the Omega T here okay so now we can write down the most align:start position:0% okay so now we can write down the most align:start position:0% okay so now we can write down the most General solutions by just putting the align:start position:0% General solutions by just putting the align:start position:0% General solutions by just putting the integration constant align:start position:0% align:start position:0% to the most General align:start position:0% align:start position:0% classical Solutions align:start position:0% align:start position:0% so you can just write Phi X align:start position:0% so you can just write Phi X align:start position:0% so you can just write Phi X equal to align:start position:0% equal to align:start position:0% equal to integrate of all possible value align:start position:0% align:start position:0% of K because this is for all K so we align:start position:0% of K because this is for all K so we align:start position:0% of K because this is for all K so we just sum up all of them align:start position:0% align:start position:0% and uh so we so align:start position:0% and uh so we so align:start position:0% and uh so we so so this factor is for for just for align:start position:0% so this factor is for for just for align:start position:0% so this factor is for for just for convenience okay it's just a convention align:start position:0% convenience okay it's just a convention align:start position:0% convenience okay it's just a convention you don't have to put it here it's just align:start position:0% you don't have to put it here it's just align:start position:0% you don't have to put it here it's just a convention align:start position:0% a convention align:start position:0% a convention and then and then we have align:start position:0% and then and then we have align:start position:0% and then and then we have a k align:start position:0% a k align:start position:0% a k u k align:start position:0% u k align:start position:0% u k plus a k star align:start position:0% plus a k star align:start position:0% plus a k star UK stop align:start position:0% UK stop align:start position:0% UK stop okay so this is just the most General align:start position:0% okay so this is just the most General align:start position:0% okay so this is just the most General set of solutions align:start position:0% set of solutions align:start position:0% set of solutions with AK and the AK Star as integration align:start position:0% with AK and the AK Star as integration align:start position:0% with AK and the AK Star as integration constant align:start position:0% align:start position:0% so this is a full set of the integration align:start position:0% so this is a full set of the integration align:start position:0% so this is a full set of the integration constant okay align:start position:0% align:start position:0% good align:start position:0% align:start position:0% so now when you go to Quantum level align:start position:0% align:start position:0% so now we can just follow the rule align:start position:0% so now we can just follow the rule align:start position:0% so now we can just follow the rule okay we find the the most General align:start position:0% okay we find the the most General align:start position:0% okay we find the the most General classical solution align:start position:0% classical solution align:start position:0% classical solution and in the column level we just promote align:start position:0% and in the column level we just promote align:start position:0% and in the column level we just promote the distribute operator align:start position:0% the distribute operator align:start position:0% the distribute operator you just put a hat there align:start position:0% align:start position:0% and change this to dagger align:start position:0% align:start position:0% okay so now this becomes align:start position:0% align:start position:0% your basis of quantum operators align:start position:0% align:start position:0% okay align:start position:0% okay align:start position:0% okay so these are the cons the full set of align:start position:0% so these are the cons the full set of align:start position:0% so these are the cons the full set of constant Quantum operators align:start position:0% align:start position:0% and this solves your Theory on align:start position:0% and this solves your Theory on align:start position:0% and this solves your Theory on okay so uh so this solves the the align:start position:0% okay so uh so this solves the the align:start position:0% okay so uh so this solves the the operator equation align:start position:0% operator equation align:start position:0% operator equation and dissolves the operating equation align:start position:0% align:start position:0% so now the Legacy align:start position:0% align:start position:0% is to impose the canonical align:start position:0% align:start position:0% commutation relation align:start position:0% align:start position:0% so firstly we have to uh now we have to align:start position:0% so firstly we have to uh now we have to align:start position:0% so firstly we have to uh now we have to do a little bit thinking okay align:start position:0% do a little bit thinking okay align:start position:0% do a little bit thinking okay so so far you just uh straightforward align:start position:0% so so far you just uh straightforward align:start position:0% so so far you just uh straightforward but now we have to do a little bit align:start position:0% but now we have to do a little bit align:start position:0% but now we have to do a little bit thinking align:start position:0% thinking align:start position:0% thinking so so for align:start position:0% align:start position:0% for finite align:start position:0% for finite align:start position:0% for finite for harmonic oscillator or for align:start position:0% for harmonic oscillator or for align:start position:0% for harmonic oscillator or for Quantum system of a single variable align:start position:0% align:start position:0% you just have X you you just have this align:start position:0% align:start position:0% you just have these and now we need to align:start position:0% you just have these and now we need to align:start position:0% you just have these and now we need to come up align:start position:0% come up align:start position:0% come up with the generalization of these two align:start position:0% with the generalization of these two align:start position:0% with the generalization of these two field Theory align:start position:0% field Theory align:start position:0% field Theory okay align:start position:0% okay align:start position:0% okay so so we need to come up with the align:start position:0% so so we need to come up with the align:start position:0% so so we need to come up with the generation of that to field City with align:start position:0% generation of that to field City with align:start position:0% generation of that to field City with corresponding to align:start position:0% align:start position:0% by X align:start position:0% by X align:start position:0% by X so now let me align:start position:0% so now let me align:start position:0% so now let me make it the time and the special align:start position:0% make it the time and the special align:start position:0% make it the time and the special coordinate separate align:start position:0% coordinate separate align:start position:0% coordinate separate and these conjugate the momentum is Phi align:start position:0% and these conjugate the momentum is Phi align:start position:0% and these conjugate the momentum is Phi it is pi okay align:start position:0% it is pi okay align:start position:0% it is pi okay conjugate the momentum density align:start position:0% align:start position:0% so we should do them at the same time align:start position:0% align:start position:0% remember align:start position:0% remember align:start position:0% remember T is the is the same Evolution operator align:start position:0% T is the is the same Evolution operator align:start position:0% T is the is the same Evolution operator so they have to be valued at the same align:start position:0% so they have to be valued at the same align:start position:0% so they have to be valued at the same time as it's a equal time a canonical of align:start position:0% time as it's a equal time a canonical of align:start position:0% time as it's a equal time a canonical of quantitative condition is always at the align:start position:0% quantitative condition is always at the align:start position:0% quantitative condition is always at the equal time align:start position:0% align:start position:0% but the x is a label of operators so so align:start position:0% but the x is a label of operators so so align:start position:0% but the x is a label of operators so so actually they don't have to be the same align:start position:0% actually they don't have to be the same align:start position:0% actually they don't have to be the same okay so here can be X here can be X align:start position:0% okay so here can be X here can be X align:start position:0% okay so here can be X here can be X Prime okay so now we have to come up align:start position:0% Prime okay so now we have to come up align:start position:0% Prime okay so now we have to come up with the generation of what is this align:start position:0% with the generation of what is this align:start position:0% with the generation of what is this quantity for filter okay align:start position:0% align:start position:0% so so now we just need to do a little align:start position:0% so so now we just need to do a little align:start position:0% so so now we just need to do a little bit guesswork okay you can easily guess align:start position:0% bit guesswork okay you can easily guess align:start position:0% bit guesswork okay you can easily guess it align:start position:0% align:start position:0% so before we do that you have any align:start position:0% so before we do that you have any align:start position:0% so before we do that you have any questions on this yes align:start position:0% align:start position:0% yeah yeah X is always so X is always align:start position:0% yeah yeah X is always so X is always align:start position:0% yeah yeah X is always so X is always here it's always just the label of the align:start position:0% here it's always just the label of the align:start position:0% here it's always just the label of the spatial location right yes yeah it's a align:start position:0% spatial location right yes yeah it's a align:start position:0% spatial location right yes yeah it's a label for the yeah yeah it's your field align:start position:0% label for the yeah yeah it's your field align:start position:0% label for the yeah yeah it's your field Theory enable yes so align:start position:0% align:start position:0% operators has been constant in time is align:start position:0% operators has been constant in time is align:start position:0% operators has been constant in time is there any way that you can get them align:start position:0% there any way that you can get them align:start position:0% there any way that you can get them where it's like the evolution is more align:start position:0% where it's like the evolution is more align:start position:0% where it's like the evolution is more complex rather than just a constant align:start position:0% complex rather than just a constant align:start position:0% complex rather than just a constant operator and it align:start position:0% operator and it align:start position:0% operator and it yeah so so normally if you have second align:start position:0% yeah so so normally if you have second align:start position:0% yeah so so normally if you have second order differential equation align:start position:0% order differential equation align:start position:0% order differential equation you always have some integration align:start position:0% you always have some integration align:start position:0% you always have some integration constant yeah that's it yeah align:start position:0% align:start position:0% yeah align:start position:0% yeah align:start position:0% yeah yeah align:start position:0% align:start position:0% other questions align:start position:0% align:start position:0% yes align:start position:0% align:start position:0% directly into the operator align:start position:0% directly into the operator align:start position:0% directly into the operator right yeah yeah so that's a very good align:start position:0% right yeah yeah so that's a very good align:start position:0% right yeah yeah so that's a very good the uh that's a very good uh align:start position:0% the uh that's a very good uh align:start position:0% the uh that's a very good uh [Music] align:start position:0% [Music] align:start position:0% [Music] questions align:start position:0% questions align:start position:0% questions so that's just extension align:start position:0% so that's just extension align:start position:0% so that's just extension of our usual procedure for the quantum align:start position:0% of our usual procedure for the quantum align:start position:0% of our usual procedure for the quantum mechanics align:start position:0% mechanics align:start position:0% mechanics so so the usual procedure when you even align:start position:0% so so the usual procedure when you even align:start position:0% so so the usual procedure when you even just for harmonic also the for single align:start position:0% just for harmonic also the for single align:start position:0% just for harmonic also the for single variable system you have this align:start position:0% variable system you have this align:start position:0% variable system you have this correspondence between the classical align:start position:0% correspondence between the classical align:start position:0% correspondence between the classical system the quantum system when you align:start position:0% system the quantum system when you align:start position:0% system the quantum system when you quantize the classical system then the align:start position:0% quantize the classical system then the align:start position:0% quantize the classical system then the classical equation working becomes a align:start position:0% classical equation working becomes a align:start position:0% classical equation working becomes a Quantum operating equation uh here we align:start position:0% Quantum operating equation uh here we align:start position:0% Quantum operating equation uh here we just use the same rule because Quantum align:start position:0% just use the same rule because Quantum align:start position:0% just use the same rule because Quantum field is just the theory of infinite align:start position:0% field is just the theory of infinite align:start position:0% field is just the theory of infinite number of degrees freedoms we are not align:start position:0% number of degrees freedoms we are not align:start position:0% number of degrees freedoms we are not changing the rule of quantum mechanics align:start position:0% changing the rule of quantum mechanics align:start position:0% changing the rule of quantum mechanics and so that's why we just again just align:start position:0% and so that's why we just again just align:start position:0% and so that's why we just again just promote the classical equation into the align:start position:0% promote the classical equation into the align:start position:0% promote the classical equation into the operating equation align:start position:0% align:start position:0% other questions align:start position:0% other questions align:start position:0% other questions yes align:start position:0% yes align:start position:0% yes one way of understanding like the align:start position:0% one way of understanding like the align:start position:0% one way of understanding like the Heisenberg equations for the align:start position:0% Heisenberg equations for the align:start position:0% Heisenberg equations for the Quantum mechanic system brackets for align:start position:0% Quantum mechanic system brackets for align:start position:0% Quantum mechanic system brackets for like the classroom is there something align:start position:0% like the classroom is there something align:start position:0% like the classroom is there something like that for field Theory as well align:start position:0% like that for field Theory as well align:start position:0% like that for field Theory as well um yeah yeah yeah there is yeah so so align:start position:0% um yeah yeah yeah there is yeah so so align:start position:0% um yeah yeah yeah there is yeah so so classically you can Define the possum align:start position:0% classically you can Define the possum align:start position:0% classically you can Define the possum bracket align:start position:0% bracket align:start position:0% bracket between the uh between the classical align:start position:0% between the uh between the classical align:start position:0% between the uh between the classical field variables and then and then align:start position:0% field variables and then and then align:start position:0% field variables and then and then Quantum mechanically just become commit align:start position:0% Quantum mechanically just become commit align:start position:0% Quantum mechanically just become commit uh uh uh quantum commutators align:start position:0% align:start position:0% yeah yeah you can also do that that's align:start position:0% yeah yeah you can also do that that's align:start position:0% yeah yeah you can also do that that's right yeah so so one way to come to this align:start position:0% right yeah so so one way to come to this align:start position:0% right yeah so so one way to come to this is you first describe first you need to align:start position:0% is you first describe first you need to align:start position:0% is you first describe first you need to generalize align:start position:0% generalize align:start position:0% generalize your standards possible brackets align:start position:0% your standards possible brackets align:start position:0% your standards possible brackets for final number because freedom to align:start position:0% for final number because freedom to align:start position:0% for final number because freedom to classical field Theory and then you can align:start position:0% classical field Theory and then you can align:start position:0% classical field Theory and then you can just generalize that to the uh to align:start position:0% just generalize that to the uh to align:start position:0% just generalize that to the uh to Quantum yeah indeed uh that's one route align:start position:0% Quantum yeah indeed uh that's one route align:start position:0% Quantum yeah indeed uh that's one route of doing it yeah align:start position:0% align:start position:0% okay other questions align:start position:0% align:start position:0% good so we'll be discussed the answer align:start position:0% good so we'll be discussed the answer align:start position:0% good so we'll be discussed the answer okay the answer is very easy to guess align:start position:0% align:start position:0% so so remember align:start position:0% align:start position:0% so um align:start position:0% so um align:start position:0% so um if you have a single X and P align:start position:0% if you have a single X and P align:start position:0% if you have a single X and P that's what you have okay align:start position:0% align:start position:0% but if you have more than one particles align:start position:0% align:start position:0% if you have more than one particles align:start position:0% align:start position:0% say just hint align:start position:0% align:start position:0% say you have multiple particles align:start position:0% align:start position:0% system in quantum mechanics and then you align:start position:0% system in quantum mechanics and then you align:start position:0% system in quantum mechanics and then you have x a align:start position:0% have x a align:start position:0% have x a and the PA as your dynamical variable align:start position:0% and the PA as your dynamical variable align:start position:0% and the PA as your dynamical variable so a equal to 1 to say ends it says the align:start position:0% so a equal to 1 to say ends it says the align:start position:0% so a equal to 1 to say ends it says the lumbar particles okay align:start position:0% lumbar particles okay align:start position:0% lumbar particles okay and then your canonical quantization align:start position:0% and then your canonical quantization align:start position:0% and then your canonical quantization condition just become x a align:start position:0% condition just become x a align:start position:0% condition just become x a t align:start position:0% t align:start position:0% t p b t align:start position:0% p b t align:start position:0% p b t equal to I Delta ad align:start position:0% equal to I Delta ad align:start position:0% equal to I Delta ad and the different XA is commute align:start position:0% align:start position:0% and different P commute align:start position:0% align:start position:0% okay given the P commutes okay align:start position:0% align:start position:0% so align:start position:0% so align:start position:0% so so now this A and B align:start position:0% so now this A and B align:start position:0% so now this A and B are essentially just replaced by x and x align:start position:0% are essentially just replaced by x and x align:start position:0% are essentially just replaced by x and x Prime align:start position:0% Prime align:start position:0% Prime so x and x Prime are just continual of align:start position:0% so x and x Prime are just continual of align:start position:0% so x and x Prime are just continual of those A and B okay remember we kind of align:start position:0% those A and B okay remember we kind of align:start position:0% those A and B okay remember we kind of emphasize the X and the X Prime are the align:start position:0% emphasize the X and the X Prime are the align:start position:0% emphasize the X and the X Prime are the labels of your degrees freedom align:start position:0% align:start position:0% so now you can just guess align:start position:0% align:start position:0% okay so we must have align:start position:0% okay so we must have align:start position:0% okay so we must have the following scene align:start position:0% align:start position:0% so so from here we must have align:start position:0% so so from here we must have align:start position:0% so so from here we must have Phi TX align:start position:0% Phi TX align:start position:0% Phi TX Phi TX Prime align:start position:0% Phi TX Prime align:start position:0% Phi TX Prime must be zero align:start position:0% must be zero align:start position:0% must be zero and the pi TX so Pi is the analog of P align:start position:0% and the pi TX so Pi is the analog of P align:start position:0% and the pi TX so Pi is the analog of P here align:start position:0% here align:start position:0% here yeah so those are operators align:start position:0% yeah so those are operators align:start position:0% yeah so those are operators TX Prime align:start position:0% TX Prime align:start position:0% TX Prime must be zero align:start position:0% must be zero align:start position:0% must be zero and then Phi align:start position:0% and then Phi align:start position:0% and then Phi TX align:start position:0% TX align:start position:0% TX which pi align:start position:0% which pi align:start position:0% which pi TX Prime align:start position:0% TX Prime align:start position:0% TX Prime should be something can only be zero but align:start position:0% should be something can only be zero but align:start position:0% should be something can only be zero but when X is not equal to X Prime align:start position:0% when X is not equal to X Prime align:start position:0% when X is not equal to X Prime can only be long zero when x equal to X align:start position:0% can only be long zero when x equal to X align:start position:0% can only be long zero when x equal to X Prime okay align:start position:0% align:start position:0% okay as a generation of this align:start position:0% okay as a generation of this align:start position:0% okay as a generation of this okay align:start position:0% okay align:start position:0% okay and uh so so you can now you can guess align:start position:0% and uh so so you can now you can guess align:start position:0% and uh so so you can now you can guess so what should this be align:start position:0% align:start position:0% so what align:start position:0% so what align:start position:0% so what yeah align:start position:0% yeah align:start position:0% yeah yeah I just used to be just directed huh align:start position:0% yeah I just used to be just directed huh align:start position:0% yeah I just used to be just directed huh okay align:start position:0% okay align:start position:0% okay but now you answer the question why has align:start position:0% but now you answer the question why has align:start position:0% but now you answer the question why has to be direct Delta or maybe y should not align:start position:0% to be direct Delta or maybe y should not align:start position:0% to be direct Delta or maybe y should not be align:start position:0% be align:start position:0% be say the derivatives of direct data okay align:start position:0% say the derivatives of direct data okay align:start position:0% say the derivatives of direct data okay say y should not be say 100th derivative align:start position:0% say y should not be say 100th derivative align:start position:0% say y should not be say 100th derivative of direct Delta align:start position:0% of direct Delta align:start position:0% of direct Delta and that question can be addressed just align:start position:0% and that question can be addressed just align:start position:0% and that question can be addressed just by a form dimensional analysis align:start position:0% by a form dimensional analysis align:start position:0% by a form dimensional analysis so so here we know it's somehow this align:start position:0% so so here we know it's somehow this align:start position:0% so so here we know it's somehow this must be really direct Delta and now align:start position:0% must be really direct Delta and now align:start position:0% must be really direct Delta and now let's decide align:start position:0% let's decide align:start position:0% let's decide so now you can do the dimension do a align:start position:0% so now you can do the dimension do a align:start position:0% so now you can do the dimension do a little bit dimensional analysis align:start position:0% little bit dimensional analysis align:start position:0% little bit dimensional analysis so so if you just write down the action align:start position:0% so so if you just write down the action align:start position:0% so so if you just write down the action yeah the action I have just erased sorry align:start position:0% yeah the action I have just erased sorry align:start position:0% yeah the action I have just erased sorry so so if you look back on the action let align:start position:0% so so if you look back on the action let align:start position:0% so so if you look back on the action let me just alternate the idea because I'm align:start position:0% me just alternate the idea because I'm align:start position:0% me just alternate the idea because I'm sure you can do dimensional analysis align:start position:0% sure you can do dimensional analysis align:start position:0% sure you can do dimensional analysis yourself align:start position:0% yourself align:start position:0% yourself so if you look at the action align:start position:0% so if you look at the action align:start position:0% so if you look at the action so the action is dimension is in the align:start position:0% so the action is dimension is in the align:start position:0% so the action is dimension is in the lateral unit we are using align:start position:0% lateral unit we are using align:start position:0% lateral unit we are using so so if uh so from that you can deduce align:start position:0% so so if uh so from that you can deduce align:start position:0% so so if uh so from that you can deduce the dimension of Phi align:start position:0% the dimension of Phi align:start position:0% the dimension of Phi should be 1 over l so one over the lens align:start position:0% should be 1 over l so one over the lens align:start position:0% should be 1 over l so one over the lens okay align:start position:0% okay align:start position:0% okay and from the fact that the pi align:start position:0% align:start position:0% where is pi or maybe how also erase is align:start position:0% where is pi or maybe how also erase is align:start position:0% where is pi or maybe how also erase is equal to Five Dot align:start position:0% equal to Five Dot align:start position:0% equal to Five Dot means Pi should be Dimension 1 over L align:start position:0% means Pi should be Dimension 1 over L align:start position:0% means Pi should be Dimension 1 over L Square align:start position:0% Square align:start position:0% Square okay because you take the derivative one align:start position:0% okay because you take the derivative one align:start position:0% okay because you take the derivative one time and then there's a lot of factor of align:start position:0% time and then there's a lot of factor of align:start position:0% time and then there's a lot of factor of L align:start position:0% L align:start position:0% L then that means on the right hand side align:start position:0% then that means on the right hand side align:start position:0% then that means on the right hand side here align:start position:0% here align:start position:0% here must be something 1 over L to a cube align:start position:0% must be something 1 over L to a cube align:start position:0% must be something 1 over L to a cube okay because there's no other parameters align:start position:0% okay because there's no other parameters align:start position:0% okay because there's no other parameters here okay yeah because uh here there align:start position:0% here okay yeah because uh here there align:start position:0% here okay yeah because uh here there should be a I okay and if it is a align:start position:0% should be a I okay and if it is a align:start position:0% should be a I okay and if it is a dimension a one over Cube then can only align:start position:0% dimension a one over Cube then can only align:start position:0% dimension a one over Cube then can only be the data function not 100 derivative align:start position:0% be the data function not 100 derivative align:start position:0% be the data function not 100 derivative data function so so so so this thing align:start position:0% data function so so so so this thing align:start position:0% data function so so so so this thing should be just a data function okay align:start position:0% should be just a data function okay align:start position:0% should be just a data function okay okay so so um align:start position:0% align:start position:0% so this you know the convention that align:start position:0% so this you know the convention that align:start position:0% so this you know the convention that there should be I align:start position:0% there should be I align:start position:0% there should be I and then it should be just the three align:start position:0% and then it should be just the three align:start position:0% and then it should be just the three data function align:start position:0% data function align:start position:0% data function and this indeed have the dimension one align:start position:0% and this indeed have the dimension one align:start position:0% and this indeed have the dimension one of uh L Cube okay align:start position:0% align:start position:0% good align:start position:0% good align:start position:0% good so now you can just plug align:start position:0% so now you can just plug align:start position:0% so now you can just plug so you have the expression for X for Phi align:start position:0% so you have the expression for X for Phi align:start position:0% so you have the expression for X for Phi you take the time derivative of this align:start position:0% you take the time derivative of this align:start position:0% you take the time derivative of this you get the expression for for pi align:start position:0% you get the expression for for pi align:start position:0% you get the expression for for pi and now you can just plug them into here align:start position:0% and now you can just plug them into here align:start position:0% and now you can just plug them into here you can just plug them into here okay align:start position:0% you can just plug them into here okay align:start position:0% you can just plug them into here okay and then you can find the commutation align:start position:0% and then you can find the commutation align:start position:0% and then you can find the commutation relation between those a case align:start position:0% relation between those a case align:start position:0% relation between those a case okay and so this is a slightly TD align:start position:0% okay and so this is a slightly TD align:start position:0% okay and so this is a slightly TD calculation align:start position:0% calculation align:start position:0% calculation which is align:start position:0% which is align:start position:0% which is however a little bit fun align:start position:0% however a little bit fun align:start position:0% however a little bit fun which of course I will leave you to do align:start position:0% which of course I will leave you to do align:start position:0% which of course I will leave you to do so so if you just plug them in and then align:start position:0% so so if you just plug them in and then align:start position:0% so so if you just plug them in and then you can deduce align:start position:0% you can deduce align:start position:0% you can deduce at the following commutation relation align:start position:0% at the following commutation relation align:start position:0% at the following commutation relation between A's align:start position:0% between A's align:start position:0% between A's so so this is the I I think this is in P align:start position:0% so so this is the I I think this is in P align:start position:0% so so this is the I I think this is in P said two uh but I can still change my align:start position:0% said two uh but I can still change my align:start position:0% said two uh but I can still change my mind yeah I wanted to put in pizza two align:start position:0% mind yeah I wanted to put in pizza two align:start position:0% mind yeah I wanted to put in pizza two so you find the the commentator between align:start position:0% so you find the the commentator between align:start position:0% so you find the the commentator between a align:start position:0% a align:start position:0% a and the commentator between a dagger align:start position:0% align:start position:0% yeah so so now I will suppress the Hat align:start position:0% yeah so so now I will suppress the Hat align:start position:0% yeah so so now I will suppress the Hat okay because at the right height I think align:start position:0% okay because at the right height I think align:start position:0% okay because at the right height I think over and over I will be too tired align:start position:0% align:start position:0% so these are zero align:start position:0% so these are zero align:start position:0% so these are zero okay so the commutation relation between align:start position:0% okay so the commutation relation between align:start position:0% okay so the commutation relation between a and zero and a align:start position:0% align:start position:0% you think align:start position:0% align:start position:0% between a align:start position:0% between a align:start position:0% between a and a dagger align:start position:0% align:start position:0% so this gives you align:start position:0% so this gives you align:start position:0% so this gives you two pair Cube align:start position:0% two pair Cube align:start position:0% two pair Cube that the function k align:start position:0% align:start position:0% okay so this is a three data function align:start position:0% okay so this is a three data function align:start position:0% okay so this is a three data function okay okay align:start position:0% align:start position:0% so again this is a straightforward align:start position:0% so again this is a straightforward align:start position:0% so again this is a straightforward generation align:start position:0% generation align:start position:0% generation so if you have multiple harmonic align:start position:0% so if you have multiple harmonic align:start position:0% so if you have multiple harmonic oscillators align:start position:0% oscillators align:start position:0% oscillators so if you have considered the multiple align:start position:0% so if you have considered the multiple align:start position:0% so if you have considered the multiple harmonic oscillators before align:start position:0% harmonic oscillators before align:start position:0% harmonic oscillators before and then the a between the different align:start position:0% and then the a between the different align:start position:0% and then the a between the different harmonicles because K Prime are just align:start position:0% harmonicles because K Prime are just align:start position:0% harmonicles because K Prime are just here just corresponding to align:start position:0% here just corresponding to align:start position:0% here just corresponding to essentially you have yeah here it just align:start position:0% essentially you have yeah here it just align:start position:0% essentially you have yeah here it just is essentially you have infinite number align:start position:0% is essentially you have infinite number align:start position:0% is essentially you have infinite number of harmonic oscillators and each one of align:start position:0% of harmonic oscillators and each one of align:start position:0% of harmonic oscillators and each one of the enabled by a k okay so this is just align:start position:0% the enabled by a k okay so this is just align:start position:0% the enabled by a k okay so this is just like essentially we find yeah let me align:start position:0% like essentially we find yeah let me align:start position:0% like essentially we find yeah let me just write it here align:start position:0% align:start position:0% [Applause] align:start position:0% [Applause] align:start position:0% [Applause] so from those commutation relations align:start position:0% align:start position:0% we conclude align:start position:0% align:start position:0% conclude align:start position:0% conclude align:start position:0% conclude this series will be quantize after we align:start position:0% this series will be quantize after we align:start position:0% this series will be quantize after we quantize it align:start position:0% align:start position:0% become an infinite number align:start position:0% align:start position:0% independent harmonic oscillators align:start position:0% independent harmonic oscillators align:start position:0% independent harmonic oscillators decoupled harmonical status align:start position:0% align:start position:0% harmonic oscillators labeled by align:start position:0% align:start position:0% continuous parameter k align:start position:0% align:start position:0% the K is yeah align:start position:0% the K is yeah align:start position:0% the K is yeah okay is the wave Lumber align:start position:0% align:start position:0% okay align:start position:0% okay align:start position:0% okay so for each k align:start position:0% so for each k align:start position:0% so for each k there is a align:start position:0% there is a align:start position:0% there is a an A and so between between a a align:start position:0% an A and so between between a a align:start position:0% an A and so between between a a themselves it's zero between a dag it's align:start position:0% themselves it's zero between a dag it's align:start position:0% themselves it's zero between a dag it's zero but a a dagger align:start position:0% zero but a a dagger align:start position:0% zero but a a dagger they not equal to zero align:start position:0% they not equal to zero align:start position:0% they not equal to zero and uh so this is again the continuing align:start position:0% and uh so this is again the continuing align:start position:0% and uh so this is again the continuing generation of one okay this is a align:start position:0% generation of one okay this is a align:start position:0% generation of one okay this is a Continuum generation of one uh because align:start position:0% Continuum generation of one uh because align:start position:0% Continuum generation of one uh because you have a continuous variables yes align:start position:0% align:start position:0% yeah then you cannot say for sure align:start position:0% align:start position:0% yeah yeah yeah align:start position:0% yeah yeah yeah align:start position:0% yeah yeah yeah no but but you see the conservation align:start position:0% no but but you see the conservation align:start position:0% no but but you see the conservation condition align:start position:0% condition align:start position:0% condition is in quantum mechanics quantum align:start position:0% is in quantum mechanics quantum align:start position:0% is in quantum mechanics quantum mechanics T and X are not on the same align:start position:0% mechanics T and X are not on the same align:start position:0% mechanics T and X are not on the same protein you can require your action to align:start position:0% protein you can require your action to align:start position:0% protein you can require your action to be x and t to be on the same 14 once you align:start position:0% be x and t to be on the same 14 once you align:start position:0% be x and t to be on the same 14 once you start the quantize your theory and then align:start position:0% start the quantize your theory and then align:start position:0% start the quantize your theory and then T will have a pronounced row align:start position:0% align:start position:0% because I wanted to so I couldn't write align:start position:0% because I wanted to so I couldn't write align:start position:0% because I wanted to so I couldn't write that like the commutation relation as align:start position:0% that like the commutation relation as align:start position:0% that like the commutation relation as functions in the four Vector x no Delta align:start position:0% functions in the four Vector x no Delta align:start position:0% functions in the four Vector x no Delta x no no no no no no no the canonical align:start position:0% x no no no no no no no the canonical align:start position:0% x no no no no no no no the canonical computation variation have to be imposed align:start position:0% computation variation have to be imposed align:start position:0% computation variation have to be imposed at equal time align:start position:0% align:start position:0% other questions align:start position:0% other questions align:start position:0% other questions good align:start position:0% good align:start position:0% good so um align:start position:0% align:start position:0% yeah so so so essentially we just get align:start position:0% yeah so so so essentially we just get align:start position:0% yeah so so so essentially we just get and now it's just trivial okay so so you align:start position:0% and now it's just trivial okay so so you align:start position:0% and now it's just trivial okay so so you can just build up your hillbow space align:start position:0% can just build up your hillbow space align:start position:0% can just build up your hillbow space essentially you just have infinite align:start position:0% essentially you just have infinite align:start position:0% essentially you just have infinite number harmonical signatures align:start position:0% number harmonical signatures align:start position:0% number harmonical signatures okay just have infinite number of align:start position:0% okay just have infinite number of align:start position:0% okay just have infinite number of harmonic concentrators align:start position:0% harmonic concentrators align:start position:0% harmonic concentrators and there's no surprise you get the align:start position:0% and there's no surprise you get the align:start position:0% and there's no surprise you get the infinite number of harmonic oscillators align:start position:0% infinite number of harmonic oscillators align:start position:0% infinite number of harmonic oscillators because we mentioned that this field align:start position:0% because we mentioned that this field align:start position:0% because we mentioned that this field Theory align:start position:0% Theory align:start position:0% Theory can be actually written as a Continuum align:start position:0% can be actually written as a Continuum align:start position:0% can be actually written as a Continuum limit align:start position:0% limit align:start position:0% limit of these particles on the chain align:start position:0% of these particles on the chain align:start position:0% of these particles on the chain which in these eight or three examples align:start position:0% which in these eight or three examples align:start position:0% which in these eight or three examples you know that is a harmonic oscillator align:start position:0% you know that is a harmonic oscillator align:start position:0% you know that is a harmonic oscillator once you uh find the lower mode they're align:start position:0% once you uh find the lower mode they're align:start position:0% once you uh find the lower mode they're all just from a bunch of harmonicles align:start position:0% all just from a bunch of harmonicles align:start position:0% all just from a bunch of harmonicles features and this is just a align:start position:0% features and this is just a align:start position:0% features and this is just a three-dimensional version of that okay align:start position:0% three-dimensional version of that okay align:start position:0% three-dimensional version of that okay and now we will align:start position:0% and now we will align:start position:0% and now we will yeah today we are running all the uh align:start position:0% align:start position:0% all the time so next time we will see align:start position:0% all the time so next time we will see align:start position:0% all the time so next time we will see that each excitations of the harmonic align:start position:0% that each excitations of the harmonic align:start position:0% that each excitations of the harmonic oscillator align:start position:0% oscillator align:start position:0% oscillator can be interpreted as a space-time align:start position:0% can be interpreted as a space-time align:start position:0% can be interpreted as a space-time particle okay so that's the crossing of align:start position:0% particle okay so that's the crossing of align:start position:0% particle okay so that's the crossing of it and now you have this infinite number align:start position:0% it and now you have this infinite number align:start position:0% it and now you have this infinite number of harmonic oscillator and now you can align:start position:0% of harmonic oscillator and now you can align:start position:0% of harmonic oscillator and now you can act and now you can Define the vacuum align:start position:0% act and now you can Define the vacuum align:start position:0% act and now you can Define the vacuum and then actually this creation align:start position:0% and then actually this creation align:start position:0% and then actually this creation operators on the vacuum and now you find align:start position:0% operators on the vacuum and now you find align:start position:0% operators on the vacuum and now you find each excitation actually equals one into align:start position:0% each excitation actually equals one into align:start position:0% each excitation actually equals one into a particle align:start position:0% a particle align:start position:0% a particle and and has the uh a corresponding to align:start position:0% and and has the uh a corresponding to align:start position:0% and and has the uh a corresponding to relativistic particle and that's how you align:start position:0% relativistic particle and that's how you align:start position:0% relativistic particle and that's how you can have actually arbitrary lumbar align:start position:0% can have actually arbitrary lumbar align:start position:0% can have actually arbitrary lumbar particles align:start position:0% particles align:start position:0% particles in this series and uh yeah because you align:start position:0% in this series and uh yeah because you align:start position:0% in this series and uh yeah because you can excite as many times as you want align:start position:0% can excite as many times as you want align:start position:0% can excite as many times as you want okay each excitation is a particle align:start position:0% okay each excitation is a particle align:start position:0% okay each excitation is a particle good good okay so so I think it's a good align:start position:0% good good okay so so I think it's a good align:start position:0% good good okay so so I think it's a good time yeah we are two minutes I think align:start position:0% time yeah we are two minutes I think align:start position:0% time yeah we are two minutes I think early but I think it's a very good place align:start position:0% early but I think it's a very good place align:start position:0% early but I think it's a very good place to to break