When we substitute from equations () to () and then from (), the exponential factors cancel out, so that the interface conditions reduce to the simultaneous equations


which are easily solved for and yielding


At "normal incidence" indicated by an additional subscript 0, these results become


At "grazing incidence" , we have hence and .


For the "p" polarization, the incident, reflected, and transmitted fields are parallel to the red arrows and may therefore be described by their components in the directions of those arrows. Let those components be (redefining the symbols for the new context). Let the reflection and transmission coefficients be and . Then, if the incident field is taken to have unit amplitude, we have


If the fields are in the directions of the red arrows, then, in order for to form a right-handed orthogonal triad, the respective fields must be in the  direction ("into the page") and may therefore be described by their components in that direction. This is consistent with the adopted sign convention, namely that a positive reflection or transmission coefficient is one that preserves the direction of the transverse field the field in the case of the "p" polarization. The agreement of the "other" field with the red arrows reveals an alternative definition of the sign convention: that a positive reflection or transmission coefficient is one for which the field vector in the plane of incidence points towards the same medium before and after reflection or transmission.


So, for the incident, reflected, and transmitted fields, let the respective components in the  direction be .  Then, since


At the interface, the tangential components of the and fields must be continuous; that is,


When we substitute from equations () and () and then from (), the exponential factors again cancel out, so that the interface conditions reduce to


At "grazing incidence" , we again have hence and .


Comparing () and () with () and (), we see that at "normal" incidence, under the adopted sign convention, the transmission coefficients for the two polarizations are equal, whereas the reflection coefficients have equal magnitudes but opposite signs. While this clash of signs is a disadvantage of the convention, the attendant advantage is that the signs agree at "grazing" incidence.


From equations () and (), taking squared magnitudes, we find that the "reflectivity" (ratio of reflected power to incident power) is


for the p polarization. Note that when comparing the powers of two such waves in the same medium and with the same cos"θ", the impedance and geometric factors mentioned above are identical and cancel out. But in computing the power "transmission" (below), these factors must be taken into account.


The simplest way to obtain the power transmission coefficient ("transmissivity", the ratio of transmitted power to incident power "in the direction normal to the interface", i.e. the direction) is to use (conservation of energy). In this way we find


In the case of an interface between two lossless media (for which ϵ and μ are "real" and positive), one can obtain these results directly using the squared magnitudes of the amplitude transmission coefficients that we found earlier in equations () and (). But, for given amplitude (as noted above), the component of the Poynting vector in the direction is proportional to the geometric factor and inversely proportional to the wave impedance . Applying these corrections to each wave, we obtain two ratios multiplying the square of the amplitude transmission coefficient:


for the p polarization. The last two equations apply only to lossless dielectrics, and only at incidence angles smaller than the critical angle (beyond which, of course, ).


From equations () and (), we see that two dissimilar media will have the same refractive index, but different admittances, if the ratio of their permeabilities is the inverse of the ratio of their permittivities. In that unusual situation we have (that is, the transmitted ray is undeviated), so that the cosines in equations (), (), (), (), and () to () cancel out, and all the reflection and transmission ratios become independent of the angle of incidence; in other words, the ratios for normal incidence become applicable to all angles of incidence. When extended to spherical reflection or scattering, this results in the Kerker effect for Mie scattering.


Since the Fresnel equations were developed for optics, they are usually given for non-magnetic materials. Dividing () by ()) yields


For non-magnetic media we can substitute the vacuum permeability "μ"0 for "μ", so that


that is, the admittances are simply proportional to the corresponding refractive indices. When we make these substitutions in equations () to () and equations () to (), the factor "cμ"0 cancels out. For the amplitude coefficients we obtain:


For the case of normal incidence these reduce to:


The power transmissions can then be found from .


For equal permeabilities (e.g., non-magnetic media), if and are "complementary", we can substitute for and for so that the numerator in equation () becomes which is zero (by Snell's law). Hence and only the s-polarized component is reflected. This is what happens at the Brewster angle. Substituting for in Snell's law, we readily obtain


This switch of polarizations has an analog in the old mechanical theory of light waves (see "§History", above). One could predict reflection coefficients that agreed with observation by supposing (like Fresnel) that different refractive indices were due to different "densities" and that the vibrations were "normal" to what was then called the plane of polarization, or by supposing (like MacCullagh and Neumann) that different refractive indices were due to different "elasticities" and that the vibrations were "parallel" to that plane. Thus the condition of equal permittivities and unequal permeabilities, although not realistic, is of some historical interest.


The second law of thermodynamics establishes the concept of entropy as a physical property of a thermodynamic system. Entropy predicts the direction of spontaneous processes, and determines whether they are irreversible or impossible, despite obeying the requirement of conservation of energy, which is established in the first law of thermodynamics. The second law may be formulated by the observation that the entropy of isolated systems left to spontaneous evolution cannot decrease, as they always arrive at a state of thermodynamic equilibrium, where the entropy is highest. If all processes in the system are reversible, the entropy is constant.


An increase in entropy accounts for the irreversibility of natural processes, often referred to in the concept of the arrow of time.


Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanics provides a microscopic explanation of the law in terms of probability distributions of the states of large assemblies of atoms or molecules.


The second law has been expressed in many ways. Its first formulation, which preceded the proper definition of entropy and was based on caloric theory, is Carnot's theorem, credited to the French scientist Sadi Carnot, who in 1824 showed that the efficiency of conversion of heat to work in a heat engine has an upper limit. The first rigorous definition of the second law based on the concept of entropy came from German scientist Rudolph Clausius in the 1850s including his statement that heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.


The second law of thermodynamics can also be used to define the concept of thermodynamic temperature, but this is usually delegated to the zeroth law of thermodynamics.


The first law of thermodynamics provides the definition of the internal energy of a thermodynamic system, and expresses the law of conservation of energy. The second law is concerned with the direction of natural processes. It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such phenomena are accounted for in terms of entropy. If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, , increases.


In a fictive reversible process, an infinitesimal increment in the entropy () of a system is defined to result from an infinitesimal transfer of heat () to a closed system (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature () of the system in equilibrium and the surroundings which supply the heat:


Different notations are used for infinitesimal amounts of heat () and infinitesimal amounts of entropy () because entropy is a function of state, while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality


This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature () and the temperature of the surroundings ("surr"). Note that the equality still applies for pure heat flow,


which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry. Introducing a set of internal variables formula_4 to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure "P" and temperature "T") from the chemical equilibrium state, one can record the equality


The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of "the second principle of thermodynamics" – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.


The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body.


The second law of thermodynamics may be expressed in many specific ways, the most prominent classical statements being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.


In modern terms, Carnot's principle may be stated more precisely:


The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work. His formulation of the second law, which was published in German in 1854, is known as the "Clausius statement":


Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.


The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.


Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.


Lord Kelvin expressed the second law in several wordings.


Equivalence of the Clausius and the Kelvin statements.


Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.


Relation between Kelvin's statement and Planck's proposition.


It is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law, as for example in the text by ter Haar and Wergeland.


The Kelvin–Planck statement (or the "heat engine statement") of the second law of thermodynamics states that


Planck stated the second law as follows.


Rather like Planck's statement is that of Uhlenbeck and Ford for "irreversible phenomena".


Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:


In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.


With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, formula_9.


Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.


In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle


This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible."


Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.


A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:


Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.


Statement for a system that has a known expression of its internal energy as a function of its extensive state variables.


The second law has been shown to be equivalent to the internal energy "U" being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...).


Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.


Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:


In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realize the Carnot cycle's reversibility and was condemned to be less efficient.


Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.


The Clausius theorem (1854) states that in a cyclic process


The equality holds in the reversible case and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.


For an arbitrary heat engine, the efficiency is:


where "W"n is for the net work done per cycle. Thus the efficiency depends only on "q""C"/"q""H".


Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, any reversible heat engine operating between temperatures "T"1 and "T"2 must have the same efficiency, that is to say, the efficiency is the function of temperatures only:


In addition, a reversible heat engine operating between temperatures "T"1 and "T"3 must have the same efficiency as one consisting of two cycles, one between "T"1 and another (intermediate) temperature "T"2, and the second between "T"2 and"T"3. This can only be the case if


Now consider the case where formula_12 is a fixed reference temperature: the temperature of the triple point of water. Then for any "T"2 and "T"3,


Therefore, if thermodynamic temperature is defined by


then the function "f", viewed as a function of thermodynamic temperature, is simply


and the reference temperature "T"1 will have the value 273.16. (Any reference temperature and any positive numerical value could be usedthe choice here corresponds to the Kelvin scale.)


According to the Clausius equality, for a reversible process


That means the line integral formula_17 is path independent for reversible processes.


So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies


With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the third law of thermodynamics, which states that "S" = 0 at absolute zero for perfect crystals.


For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.


Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop,


where the equality holds if the transformation is reversible.


Notice that if the process is an adiabatic process, then formula_21, so formula_22.


An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an "unlimited" heat reservoir at temperature "TR" and pressure "PR" so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain "TR"; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain "PR".


Whatever changes to "dS" and "dSR" occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy "S"tot of the isolated total system must not decrease:


According to the first law of thermodynamics, the change "dU" in the internal energy of the sub-system is the sum of the heat "δq" added to the sub-system, "less" any work "δw" done "by" the sub-system, "plus" any net chemical energy entering the sub-system "d" ∑"μiRNi", so that:


where "μ""iR" are the chemical potentials of chemical species in the external surroundings.


Now the heat leaving the reservoir and entering the sub-system is


where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.


It therefore follows that any net work "δw" done by the sub-system must obey


It is useful to separate the work "δw" done by the subsystem into the "useful" work "δwu" that can be done "by" the sub-system, over and beyond the work "pR dV" done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:


It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the "availability" or "exergy" "E" of the subsystem,


The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,


i.e. the change in the subsystem's exergy plus the useful work done "by" the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done "on" the system) must be less than or equal to zero.


In sum, if a proper "infinite-reservoir-like" reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in "E" for an irreversible process and no change for a reversible process.


This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. ("Also, see process engineer"). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found ("See second law efficiency".)


This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.


For a spontaneous chemical process in a closed system at constant temperature and pressure without non-"PV" work, the Clausius inequality Δ"S" > "Q/T"surr transforms into a condition for the change in Gibbs free energy


or d"G" < 0. For a similar process at constant temperature and volume, the change in Helmholtz free energy must be negative, formula_33. Thus, a negative value of the change in free energy ("G" or "A") is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products. The chemical equilibrium condition at constant "T" and "p" without electrical work is d"G" = 0.


The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.


Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow "spontaneously" from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).


Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.


The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.


There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a macroscopic system. This doctrine is obsolescent.


In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:


where "Q" is heat, "T" is temperature and "N" is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:


The entropy of the universe tends to a maximum.


This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.


In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:


The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:


with formula_38 the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature formula_39 it gives the so-called dissipated energy formula_40.


The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:


The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.


For open systems (also allowing exchange of matter):


Here formula_47 is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.


The first mechanical argument of the Kinetic theory of gases that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to James Clerk Maxwell in 1860; Ludwig Boltzmann with his H-theorem of 1872 also argued that due to collisions gases should over time tend toward the Maxwell–Boltzmann distribution.


Due to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though other scenarios have also been suggested.


Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of formula_48 is:


where formula_50 is the number of quantum states in a small interval between formula_48 and formula_52. Here formula_53 is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of formula_53. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on formula_53.


Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then formula_56 will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that formula_56 is maximized as that is the most probable situation in equilibrium.


If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that formula_56 is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).


Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number formula_56 of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of formula_60. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the quantity increases monotonically as a function of time during the intermediate out of equilibrium state.


Derivation of the entropy change for reversible processes.


The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:


See here for the justification for this definition. Suppose that the system has some external parameter, "x", that can be changed. In general, the energy eigenstates of the system will depend on "x". According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.


The generalized force, "X", corresponding to the external variable "x" is defined such that formula_63 is the work performed by the system if "x" is increased by an amount "dx". For example, if "x" is the volume, then "X" is the pressure. The generalized force for a system known to be in energy eigenstate formula_64 is given by:


Since the system can be in any energy eigenstate within an interval of formula_53, we define the generalized force for the system as the expectation value of the above expression:


To evaluate the average, we partition the formula_50 energy eigenstates by counting how many of them have a value for formula_69 within a range between formula_70 and formula_71. Calling this number formula_72, we have:


The average defining the generalized force can now be written:


We can relate this to the derivative of the entropy with respect to "x" at constant energy "E" as follows. Suppose we change "x" to "x" + "dx". Then formula_50 will change because the energy eigenstates depend on "x", causing energy eigenstates to move into or out of the range between formula_48 and formula_77. Let's focus again on the energy eigenstates for which formula_78 lies within the range between formula_70 and formula_71. Since these energy eigenstates increase in energy by "Y dx", all such energy eigenstates that are in the interval ranging from "E" – "Y" "dx" to "E" move from below "E" to above "E". There are


such energy eigenstates. If formula_82, all these energy eigenstates will move into the range between formula_48 and formula_77 and contribute to an increase in formula_56. The number of energy eigenstates that move from below formula_77 to above formula_77 is given by formula_88. The difference


is thus the net contribution to the increase in formula_56. Note that if "Y dx" is larger than formula_53 there will be the energy eigenstates that move from below "E" to above formula_77. They are counted in both formula_93 and formula_88, therefore the above expression is also valid in that case.


Expressing the above expression as a derivative with respect to "E" and summing over "Y" yields the expression:


The logarithmic derivative of formula_56 with respect to "x" is thus given by:


The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanish in the thermodynamic limit. We have thus found that:


Derivation for systems described by the canonical ensemble.


If a system is in thermal contact with a heat bath at some temperature "T" then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:


Here "Z" is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:


Inserting the formula for formula_104 for the canonical ensemble in here gives:


As elaborated above, it is thought that the second law of thermodynamics is a result of the very low-entropy initial conditions at the Big Bang. From a statistical point of view, these were very special conditions. On the other hand, they were quite simple, as the universe - or at least the part thereof from which the observable universe developed - seem to have been extremely uniform.


This may seem somewhat paradoxical, since in many physical systems uniform conditions (e.g. mixed rather than separated gases) have high entropy. The paradox is solved once realizing that gravitational systems have negative heat capacity, so that when gravity is important, uniform conditions (e.g. gas of uniform density) in fact have lower entropy compared to non-uniform ones (e.g. black holes in empty space). Yet another approach is that the universe had high (or even maximal) entropy given its size, but as the universe grew it rapidly came out of thermodynamic equilibrium, its entropy only slightly increased compared to the increase in maximal possible entropy, and thus it has arrived at a very low entropy when compared to the much larger possible maximum given its later size.


As for the reason why initial conditions were such, one suggestion is that cosmological inflation was enough to wipe off non-smoothness, while another is that the universe was created spontaneously where the mechanism of creation implies low-entropy initial conditions.


There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. The thermodynamics of living organisms has been considered by many authors, such as Erwin Schrödinger, Léon Brillouin and Isaac Asimov.


Furthermore, the ability of living organisms to grow and increase in complexity, as well as to form correlations with their environment in the form of adaption and memory, is not opposed to the second law – rather, it is akin to general results following from it: Under some definitions, an increase in entropy also results in an increase in complexity, and for a finite system interacting with finite reservoirs, an increase in entropy is equivalent to an increase in correlations between the system and the reservoirs.


Living organisms may be considered as open systems, because matter passes into and out from them. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to unsettled current debate or research.


Commonly, systems for which gravity is not important have a positive heat capacity, meaning that their temperature rises with their internal energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature decreases while the sink temperature is increased; hence temperature differences tend to diminish over time.


This is not always the case for systems in which the gravitational force is important: systems that are bound by their own gravity, such as stars, can have negative heat capacities. As they contract, both their total energy and their entropy decrease but their internal temperature may increase. This can be significant for protostars and even gas giant planets such as Jupiter.


As gravity is the most important force operating on cosmological scales, it may be difficult or impossible to apply the second law to the universe as a whole.


The theory of classical or equilibrium thermodynamics is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.


In all cases, the assumption of thermodynamic equilibrium, once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.


It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.


There are intermediate cases, in which the assumption of local thermodynamic equilibrium is a very good approximation, but strictly speaking it is still an approximation, not theoretically ideal.


For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.


The physics of macroscopically observable fluctuations is beyond the scope of this article.


The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with symmetries observed in the fundamental laws of physics (particularly CPT symmetry) since the second law applies statistically on time-asymmetric boundary conditions. The second law has been related to the difference between moving forwards and backwards in time, or to the principle that cause precedes effect (the causal arrow of time, or causality).


Irreversibility in thermodynamic processes is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.


Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.


James Clerk Maxwell imagined one container divided into two parts, "A" and "B". Both parts are filled with the same gas at equal temperatures and placed next to each other, separated by a wall. Observing the molecules on both sides, an imaginary demon guards a microscopic trapdoor in the wall. When a faster-than-average molecule from "A" flies towards the trapdoor, the demon opens it, and the molecule will fly from "A" to "B". The average speed of the molecules in "B" will have increased while in "A" they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in "A" and increases in "B", contrary to the second law of thermodynamics.


One response to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.


Maxwell's 'demon' repeatedly alters the permeability of the wall between "A" and "B". It is therefore performing thermodynamic operations on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.


In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-½ massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way.


The equation also implied the existence of a new form of matter, "antimatter", previously unsuspected and unobserved and which was experimentally confirmed several years later. It also provided a "theoretical" justification for the introduction of several component wave functions in Pauli's phenomenological theory of spin. The wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation.


Although Dirac did not at first fully appreciate the importance of his results, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represents one of the great triumphs of theoretical physics. This accomplishment has been described as fully on a par with the works of Newton, Maxwell, and Einstein before him. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-½ particles.


The Dirac equation appears on the floor of Westminster Abbey on the plaque commemorating Paul Dirac's life, which was unveiled on 13 November 1995.


The Dirac equation in the form originally proposed by Dirac is:


where is the wave function for the electron of rest mass with spacetime coordinates . The are the components of the momentum, understood to be the momentum operator in the Schrödinger equation. Also, is the speed of light, and is the reduced Planck constant. These fundamental physical constants reflect special relativity and quantum mechanics, respectively.


Dirac's purpose in casting this equation was to explain the behavior of the relativistically moving electron, and so to allow the atom to be treated in a manner consistent with relativity. His rather modest hope was that the corrections introduced this way might have a bearing on the problem of atomic spectra.


Up until that time, attempts to make the old quantum theory of the atom compatible with the theory of relativity, attempts based on discretizing the angular momentum stored in the electron's possibly non-circular orbit of the atomic nucleus, had failed – and the new quantum mechanics of Heisenberg, Pauli, Jordan, Schrödinger, and Dirac himself had not developed sufficiently to treat this problem. Although Dirac's original intentions were satisfied, his equation had far deeper implications for the structure of matter and introduced new mathematical classes of objects that are now essential elements of fundamental physics.


The new elements in this equation are the four matrices , , and , and the four-component wave function . There are four components in because the evaluation of it at any given point in configuration space is a bispinor. It is interpreted as a superposition of a spin-up electron, a spin-down electron, a spin-up positron, and a spin-down positron (see below for further discussion).


The matrices and are all Hermitian and are involutory:


These matrices and the form of the wave function have a deep mathematical significance. The algebraic structure represented by the gamma matrices had been created some 50 years earlier by the English mathematician W. K. Clifford. In turn, Clifford's ideas had emerged from the mid-19th-century work of the German mathematician Hermann Grassmann in his "Lineale Ausdehnungslehre" ("Theory of Linear Extensions"). The latter had been regarded as well-nigh incomprehensible by most of his contemporaries. The appearance of something so seemingly abstract, at such a late date, and in such a direct physical manner, is one of the most remarkable chapters in the history of physics.


The single symbolic equation thus unravels into four coupled linear first-order partial differential equations for the four quantities that make up the wave function. The equation can be written more explicitly in Planck units as:


which makes it clearer that it is a set of four partial differential equations with four unknown functions.


The Dirac equation is superficially similar to the Schrödinger equation for a massive free particle:


The left side represents the square of the momentum operator divided by twice the mass, which is the non-relativistic kinetic energy. Because relativity treats space and time as a whole, a relativistic generalization of this equation requires that space and time derivatives must enter symmetrically as they do in the Maxwell equations that govern the behavior of light — the equations must be differentially of the "same order" in space and time. In relativity, the momentum and the energies are the space and time parts of a spacetime vector, the four-momentum, and they are related by the relativistically invariant relation


which says that the length of this four-vector is proportional to the rest mass . Substituting the operator equivalents of the energy and momentum from the Schrödinger theory, we get the Klein–Gordon equation describing the propagation of waves, constructed from relativistically invariant objects,


with the wave function being a relativistic scalar: a complex number which has the same numerical value in all frames of reference. Space and time derivatives both enter to second order. This has a telling consequence for the interpretation of the equation. Because the equation is second order in the time derivative, one must specify initial values both of the wave function itself and of its first time-derivative in order to solve definite problems. Since both may be specified more or less arbitrarily, the wave function cannot maintain its former role of determining the probability density of finding the electron in a given state of motion. In the Schrödinger theory, the probability density is given by the positive definite expression


and this density is convected according to the probability current vector


with the conservation of probability current and density following from the continuity equation:


The fact that the density is positive definite and convected according to this continuity equation implies that we may integrate the density over a certain domain and set the total to 1, and this condition will be maintained by the conservation law. A proper relativistic theory with a probability density current must also share this feature. Now, if we wish to maintain the notion of a convected density, then we must generalize the Schrödinger expression of the density and current so that space and time derivatives again enter symmetrically in relation to the scalar wave function. We are allowed to keep the Schrödinger expression for the current, but must replace the probability density by the symmetrically formed expression


which now becomes the 4th component of a spacetime vector, and the entire probability 4-current density has the relativistically covariant expression


The continuity equation is as before. Everything is compatible with relativity now, but we see immediately that the expression for the density is no longer positive definite – the initial values of both and may be freely chosen, and the density may thus become negative, something that is impossible for a legitimate probability density. Thus, we cannot get a simple generalization of the Schrödinger equation under the naive assumption that the wave function is a relativistic scalar, and the equation it satisfies, second order in time.


Although it is not a successful relativistic generalization of the Schrödinger equation, this equation is resurrected in the context of quantum field theory, where it is known as the Klein–Gordon equation, and describes a spinless particle field (e.g. pi meson or Higgs boson). Historically, Schrödinger himself arrived at this equation before the one that bears his name but soon discarded it. In the context of quantum field theory, the indefinite density is understood to correspond to the "charge" density, which can be positive or negative, and not the probability density.


Dirac thus thought to try an equation that was "first order" in both space and time. One could, for example, formally (i.e. by abuse of notation) take the relativistic expression for the energy


replace by its operator equivalent, expand the square root in an infinite series of derivative operators, set up an eigenvalue problem, then solve the equation formally by iterations. Most physicists had little faith in such a process, even if it were technically possible.


As the story goes, Dirac was staring into the fireplace at Cambridge, pondering this problem, when he hit upon the idea of taking the square root of the wave operator thus:


On multiplying out the right side we see that, in order to get all the cross-terms such as to vanish, we must assume


Dirac, who had just then been intensely involved with working out the foundations of Heisenberg's matrix mechanics, immediately understood that these conditions could be met if , , and are "matrices", with the implication that the wave function has "multiple components". This immediately explained the appearance of two-component wave functions in Pauli's phenomenological theory of spin, something that up until then had been regarded as mysterious, even to Pauli himself. However, one needs at least matrices to set up a system with the properties required — so the wave function had "four" components, not two, as in the Pauli theory, or one, as in the bare Schrödinger theory. The four-component wave function represents a new class of mathematical object in physical theories that makes its first appearance here.


Given the factorization in terms of these matrices, one can now write down immediately an equation


with formula_19 to be determined. Applying again the matrix operator on both sides yields


On taking formula_21 we find that all the components of the wave function "individually" satisfy the relativistic energy–momentum relation. Thus the sought-for equation that is first-order in both space and time is


we get the Dirac equation as written above.


To demonstrate the relativistic invariance of the equation, it is advantageous to cast it into a form in which the space and time derivatives appear on an equal footing. New matrices are introduced as follows:


and the equation takes the form (remembering the definition of the covariant components of the 4-gradient and especially that = )


where there is an implied summation over the values of the twice-repeated index , and is the 4-gradient. In practice one often writes the gamma matrices in terms of 2 × 2 sub-matrices taken from the Pauli matrices and the 2 × 2 identity matrix. Explicitly the standard representation is


The complete system is summarized using the Minkowski metric on spacetime in the form


denotes the anticommutator. These are the defining relations of a Clifford algebra over a pseudo-orthogonal 4-dimensional space with metric signature . The specific Clifford algebra employed in the Dirac equation is known today as the Dirac algebra. Although not recognized as such by Dirac at the time the equation was formulated, in hindsight the introduction of this "geometric algebra" represents an enormous stride forward in the development of quantum theory.


The Dirac equation may now be interpreted as an eigenvalue equation, where the rest mass is proportional to an eigenvalue of the 4-momentum operator, the proportionality constant being the speed of light:


Using formula_31 (formula_32 is pronounced "d-slash"), according to Feynman slash notation, the Dirac equation becomes:


In practice, physicists often use units of measure such that , known as natural units. The equation then takes the simple form


A fundamental theorem states that if two distinct sets of matrices are given that both satisfy the Clifford relations, then they are connected to each other by a similarity transformation:


If in addition the matrices are all unitary, as are the Dirac set, then itself is unitary;


The transformation is unique up to a multiplicative factor of absolute value 1. Let us now imagine a Lorentz transformation to have been performed on the space and time coordinates, and on the derivative operators, which form a covariant vector. For the operator to remain invariant, the gammas must transform among themselves as a contravariant vector with respect to their spacetime index. These new gammas will themselves satisfy the Clifford relations, because of the orthogonality of the Lorentz transformation. By the fundamental theorem, we may replace the new set by the old set subject to a unitary transformation. In the new frame, remembering that the rest mass is a relativistic scalar, the Dirac equation will then take the form


If we now define the transformed spinor


then we have the transformed Dirac equation in a way that demonstrates manifest relativistic invariance:


Thus, once we settle on any unitary representation of the gammas, it is final provided we transform the spinor according to the unitary transformation that corresponds to the given Lorentz transformation.


The various representations of the Dirac matrices employed will bring into focus particular aspects of the physical content in the Dirac wave function (see below). The representation shown here is known as the "standard" representation – in it, the wave function's upper two components go over into Pauli's 2 spinor wave function in the limit of low energies and small velocities in comparison to light.


The considerations above reveal the origin of the gammas in "geometry", hearkening back to Grassmann's original motivation – they represent a fixed basis of unit vectors in spacetime. Similarly, products of the gammas such as represent "oriented surface elements", and so on. With this in mind, we can find the form of the unit volume element on spacetime in terms of the gammas as follows. By definition, it is


For this to be an invariant, the epsilon symbol must be a tensor, and so must contain a factor of , where is the determinant of the metric tensor. Since this is negative, that factor is "imaginary". Thus


This matrix is given the special symbol , owing to its importance when one is considering improper transformations of space-time, that is, those that change the orientation of the basis vectors. In the standard representation, it is


This matrix will also be found to anticommute with the other four Dirac matrices:


It takes a leading role when questions of "parity" arise because the volume element as a directed magnitude changes sign under a space-time reflection. Taking the positive square root above thus amounts to choosing a handedness convention on spacetime.


where is the conjugate transpose of , and noticing that


we obtain, by taking the Hermitian conjugate of the Dirac equation and multiplying from the right by , the adjoint equation:


where is understood to act to the left. Multiplying the Dirac equation by from the left, and the adjoint equation by from the right, and adding, produces the law of conservation of the Dirac current:


Now we see the great advantage of the first-order equation over the one Schrödinger had tried – this is the conserved current density required by relativistic invariance, only now its 4th component is "positive definite" and thus suitable for the role of a probability density:


Because the probability density now appears as the fourth component of a relativistic vector and not a simple scalar as in the Schrödinger equation, it will be subject to the usual effects of the Lorentz transformations such as time dilation. Thus, for example, atomic processes that are observed as rates, will necessarily be adjusted in a way consistent with relativity, while those involving the measurement of energy and momentum, which themselves form a relativistic vector, will undergo parallel adjustment which preserves the relativistic covariance of the observed values. The Dirac current itself is then the spacetime-covariant four-vector:


See Dirac spinor for details of solutions to the Dirac equation. Note that since the Dirac operator acts on 4-tuples of square-integrable functions, its solutions should be members of the same Hilbert space. The fact that the energies of the solutions do not have a lower bound is unexpected – see the hole theory section below for more details.


Here and formula_52 represent the components of the electromagnetic four-potential in their standard SI units, and the three sigmas are the Pauli matrices. On squaring out the first term, a residual interaction with the magnetic field is found, along with the usual classical Hamiltonian of a charged particle interacting with an applied field in SI units:


This Hamiltonian is now a matrix, so the Schrödinger equation based on it must use a two-component wave function. On introducing the external electromagnetic 4-vector potential into the Dirac equation in a similar way, known as minimal coupling, it takes the form:


A second application of the Dirac operator will now reproduce the Pauli term exactly as before, because the spatial Dirac matrices multiplied by , have the same squaring and commutation properties as the Pauli matrices. What is more, the value of the gyromagnetic ratio of the electron, standing in front of Pauli's new term, is explained from first principles. This was a major achievement of the Dirac equation and gave physicists great faith in its overall correctness. There is more however. The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner. First the equation is written in the form of coupled equations for 2-spinors with the SI units restored:


Assuming the field is weak and the motion of the electron non-relativistic, we have the total energy of the electron approximately equal to its rest energy, and the momentum going over to the classical value,


and so the second equation may be written


which is of order – thus at typical energies and velocities, the bottom components of the Dirac spinor in the standard representation are much suppressed in comparison to the top components. Substituting this expression into the first equation gives after some rearrangement


It should be strongly emphasized that this separation of the Dirac spinor into large and small components depends explicitly on a low-energy approximation. The entire Dirac spinor represents an "irreducible" whole, and the components we have just neglected to arrive at the Pauli theory will bring in new phenomena in the relativistic regime – antimatter and the idea of creation and annihilation of particles.


In the limit , the Dirac equation reduces to the Weyl equation, which describes relativistic massless spin- particles.


Both the Dirac equation and the Adjoint Dirac equation can be obtained from (varying) the action with a specific Lagrangian density that is given by:


If one varies this with respect to one gets the Adjoint Dirac equation. Meanwhile, if one varies this with respect to one gets the Dirac equation.


The critical physical question in a quantum theory is—what are the physically observable quantities defined by the theory? According to the postulates of quantum mechanics, such quantities are defined by Hermitian operators that act on the Hilbert space of possible states of a system. The eigenvalues of these operators are then the possible results of measuring the corresponding physical quantity. In the Schrödinger theory, the simplest such object is the overall Hamiltonian, which represents the total energy of the system. If we wish to maintain this interpretation on passing to the Dirac theory, we must take the Hamiltonian to be


where, as always, there is an implied summation over the twice-repeated index . This looks promising, because we see by inspection the rest energy of the particle and, in the case of , the energy of a charge placed in an electric potential . What about the term involving the vector potential? In classical electrodynamics, the energy of a charge moving in an applied potential is


Thus, the Dirac Hamiltonian is fundamentally distinguished from its classical counterpart, and we must take great care to correctly identify what is observable in this theory. Much of the apparently paradoxical behavior implied by the Dirac equation amounts to a misidentification of these observables.


The negative solutions to the equation are problematic, for it was assumed that the particle has a positive energy. Mathematically speaking, however, there seems to be no reason for us to reject the negative-energy solutions. Since they exist, we cannot simply ignore them, for once we include the interaction between the electron and the electromagnetic field, any electron placed in a positive-energy eigenstate would decay into negative-energy eigenstates of successively lower energy. Real electrons obviously do not behave in this way, or they would disappear by emitting energy in the form of photons.


To cope with this problem, Dirac introduced the hypothesis, known as hole theory, that the vacuum is the many-body quantum state in which all the negative-energy electron eigenstates are occupied. This description of the vacuum as a "sea" of electrons is called the Dirac sea. Since the Pauli exclusion principle forbids electrons from occupying the same state, any additional electron would be forced to occupy a positive-energy eigenstate, and positive-energy electrons would be forbidden from decaying into negative-energy eigenstates.


Dirac further reasoned that if the negative-energy eigenstates are incompletely filled, each unoccupied eigenstate – called a hole – would behave like a positively charged particle. The hole possesses a "positive" energy since energy is required to create a particle–hole pair from the vacuum. As noted above, Dirac initially thought that the hole might be the proton, but Hermann Weyl pointed out that the hole should behave as if it had the same mass as an electron, whereas the proton is over 1800 times heavier. The hole was eventually identified as the positron, experimentally discovered by Carl Anderson in 1932.


It is not entirely satisfactory to describe the "vacuum" using an infinite sea of negative-energy electrons. The infinitely negative contributions from the sea of negative-energy electrons have to be canceled by an infinite positive "bare" energy and the contribution to the charge density and current coming from the sea of negative-energy electrons is exactly canceled by an infinite positive "jellium" background so that the net electric charge density of the vacuum is zero. In quantum field theory, a Bogoliubov transformation on the creation and annihilation operators (turning an occupied negative-energy electron state into an unoccupied positive energy positron state and an unoccupied negative-energy electron state into an occupied positive energy positron state) allows us to bypass the Dirac sea formalism even though, formally, it is equivalent to it.


In certain applications of condensed matter physics, however, the underlying concepts of "hole theory" are valid. The sea of conduction electrons in an electrical conductor, called a Fermi sea, contains electrons with energies up to the chemical potential of the system. An unfilled state in the Fermi sea behaves like a positively charged electron, though it is referred to as a "hole" rather than a "positron". The negative charge of the Fermi sea is balanced by the positively charged ionic lattice of the material.


In quantum field theories such as quantum electrodynamics, the Dirac field is subject to a process of second quantization, which resolves some of the paradoxical features of the equation.


The Dirac equation is Lorentz covariant. Articulating this helps illuminate not only the Dirac equation, but also the Majorana spinor and Elko spinor, which although closely related, have subtle and important differences.


Understanding Lorentz covariance is simplified by keeping in mind the geometric character of the process. Let formula_65 be a single, fixed point in the spacetime manifold. Its location can be expressed in multiple coordinate systems. In the physics literature, these are written as formula_66 and formula_67, with the understanding that both formula_66 and formula_67 describe "the same" point formula_65, but in different local frames of reference (a frame of reference over a small extended patch of spacetime).


One can imagine formula_65 as having a fiber of different coordinate frames above it. In geometric terms, one says that spacetime can be characterized as a fiber bundle, and specifically, the frame bundle. The difference between two points formula_66 and formula_67 in the same fiber is a combination of rotations and Lorentz boosts. A choice of coordinate frame is a (local) section through that bundle.


The presentation here follows that of Itzykson and Zuber. It is very nearly identical to that of Bjorken and Drell. A similar derivation in a general relativistic setting can be found in Weinberg. Under a Lorentz transformation formula_74 the Dirac spinor to transform as


It can be shown that an explicit expression for formula_76 is given by


where formula_78 parameterizes the Lorentz transformation, and formula_79 is the 4x4 matrix


This matrix can be interpreted as the intrinsic angular momentum of the Dirac field. That it deserves this interpretation arises by contrasting it to the generator formula_81 of Lorentz transformations, having the form


This can be interpreted as the total angular momentum. It acts on the spinor field as


Note the formula_66 above does "not" have a prime on it: the above is obtained by transforming formula_85 obtaining the change to formula_86 and then returning to the original coordinate system formula_87.


The geometrical interpretation of the above is that the frame field is affine, having no preferred origin. The generator formula_81 generates the symmetries of this space: it provides a relabelling of a fixed point formula_89 The generator formula_79 generates a movement from one point in the fiber to another: a movement from formula_85 with both formula_66 and formula_67 still corresponding to the same spacetime point formula_94 These perhaps obtuse remarks can be elucidated with explicit algebra.


Let formula_95 be a Lorentz transformation. The Dirac equation is


If the Dirac equation is to be covariant, then it should have exactly the same form in all Lorentz frames:


The two spinors formula_98 and formula_99 should both describe the same physical field, and so should be related by a transformation that does not change any physical observables (charge, current, mass, "etc.") The transformation should encode only the change of coordinate frame. It can be shown that such a transformation is a 4x4 unitary matrix. Thus, one may presume that the relation between the two frames can be written as


Inserting this into the transformed equation, the result is


The original Dirac equation is then regained if


An explicit expression for formula_104 (equal to the expression given above) can be obtained by considering an infinitessimal Lorentz transformation


where formula_106 is the metric tensor and formula_107 is antisymmetric. After plugging and chugging, one obtains


which is the (infinitessimal) form for formula_76 above. To obtain the affine relabelling, write


After properly antisymmetrizing, one obtains the generator of symmetries formula_81 given earlier. Thus, both formula_81 and formula_79 can be said to be the "generators of Lorentz transformations", but with a subtle distinction: the first corresponds to a relabelling of points on the affine frame bundle, which forces a translation along the fiber of the spinor on the spin bundle, while the second corresponds to translations along the fiber of the spin bundle (taken as a movement formula_85 along the frame bundle, as well as a movement formula_115 along the fiber of the spin bundle.) Weinberg provides additional arguments for the physical interpretation of these as total and intrinsic angular momentum.


The Dirac equation can be formulated in a number of other ways.


This article has developed the Dirac equation in flat spacetime according to special relativity. It is possible to formulate the Dirac equation in curved spacetime.


This article developed the Dirac equation using four vectors and Schrödinger operators. The Dirac equation in the algebra of physical space uses a Clifford algebra over the real numbers, a type of geometric algebra.


In quantum field theory, the nonlinear Dirac equation is a model of self-interacting Dirac fermions.


This model is widely considered in quantum physics as a toy model of self-interacting electrons.


The nonlinear Dirac equation appears in the Einstein-Cartan-Sciama-Kibble theory of gravity, which extends general relativity to matter with intrinsic angular momentum (spin). This theory removes a constraint of the symmetry of the affine connection and treats its antisymmetric part, the torsion tensor, as a variable in varying the action. In the resulting field equations, the torsion tensor is a homogeneous, linear function of the spin tensor. The minimal coupling between torsion and Dirac spinors thus generates an axial-axial, spin–spin interaction in fermionic matter, which becomes significant only at extremely high densities. Consequently, the Dirac equation becomes nonlinear (cubic) in the spinor field, which causes fermions to be spatially extended and may remove the ultraviolet divergence in quantum field theory.


Two common examples are the massive Thirring model and the Soler model.


The Thirring model was originally formulated as a model in (1 + 1) space-time dimensions and is characterized by the Lagrangian density


where is the spinor field, is the Dirac adjoint spinor,


(Feynman slash notation is used), is the coupling constant, is the mass, and are the "two"-dimensional gamma matrices, finally is an index.


The Soler model was originally formulated in (3 + 1) space-time dimensions. It is characterized by the Lagrangian density


is now the four-gradient operator contracted with the "four"-dimensional Dirac gamma matrices , so therein .


In Einstein-Cartan theory the Lagrangian density for a Dirac spinor field is given by (formula_5)


is the Fock-Ivanenko covariant derivative of a spinor with respect to the affine connection, formula_8 is the spin connection, formula_9 is the determinant of the metric tensor formula_10, and the Dirac matrices satisfy


The Einstein-Cartan field equations for the spin connection yield an algebraic constraint between the spin connection and the spinor field rather than a partial differential equation, which allows the spin connection to be explicitly eliminated from the theory. The final result is a nonlinear Dirac equation containing an effective "spin-spin" self-interaction,


where formula_13 is the general-relativistic covariant derivative of a spinor, and formula_14 is the Einstein gravitational constant, formula_15. The cubic term in this equation becomes significant at densities on the order of formula_16.


In mathematical physics, the Dirac equation in curved spacetime generalizes the original Dirac equation to curved space.


It can be written by using vierbein fields and the gravitational spin connection. The vierbein defines a local rest frame, allowing the constant Gamma matrices to act at each spacetime point. In this way, Dirac's equation takes the following form in curved spacetime:


Here is the vierbein and is the covariant derivative for fermionic fields, defined as follows


where is the commutator of Gamma matrices:


Note that here Latin indices denote the "Lorentzian" vierbein labels while Greek indices denote manifold coordinate indices.


The optical response of a semiconductor follows if one can determine its macroscopic polarization formula_1 as a function of the electric field formula_2 that excites it. The connection between formula_1 and the microscopic polarization formula_4 is given by


where the sum involves crystal-momenta formula_6 of all relevant electronic states. In semiconductor optics, one typically excites transitions between a valence and a conduction band. In this connection, formula_7 is the dipole matrix element between the conduction and valence band and formula_8 defines the corresponding transition amplitude.


The derivation of the SBEs starts from a system Hamiltonian that fully includes the free-particles, Coulomb interaction, dipole interaction between classical light and electronic states, as well as the phonon contributions. Like almost always in many-body physics, it is most convenient to apply the second-quantization formalism after the appropriate system Hamiltonian formula_9 is identified. One can then derive the quantum dynamics of relevant observables formula_10 by using the Heisenberg equation of motion


Due to the many-body interactions within formula_9, the dynamics of the observable formula_10 couples to new observables and the equation structure cannot be closed. This is the well-known BBGKY hierarchy problem that can be systematically truncated with different methods such as the cluster-expansion approach.


At operator level, the microscopic polarization is defined by an expectation value for a single electronic transition between a valence and a conduction band. In second quantization, conduction-band electrons are defined by fermionic creation and annihilation operators formula_14 and formula_15, respectively. An analogous identification, i.e., formula_16 and formula_17, is made for the valence band electrons. The corresponding electronic interband transition then becomes


that describe transition amplitudes for moving an electron from conduction to valence band (formula_19 term) or vice versa (formula_8 term). At the same time, an electron distribution follows from


It is also convenient to follow the distribution of electronic vacancies, i.e., the holes,


that are left to the valence band due to optical excitation processes.


The quantum dynamics of optical excitations yields an integro-differential equations that constitute the SBEs


+ \mathrm{i} \hbar \left. \frac{\partial}{\partial t} P_{\mathbf{k}} \right|_{\mathrm{scatter}}\,


as well as the renormalized carrier energy


where formula_27 corresponds to the energy of free electron–hole pairs and formula_28 is the Coulomb matrix element, given here in terms of the carrier wave vector formula_29.


The symbolically denoted formula_30 contributions stem from the hierarchical coupling due to many-body interactions. Conceptually, formula_31, formula_32, and formula_33 are single-particle expectation values while the hierarchical coupling originates from two-particle correlations such as polarization-density correlations or polarization-phonon correlations. Physically, these two-particle correlations introduce several nontrivial effects such as screening of Coulomb interaction, Boltzmann-type scattering of formula_34 and formula_35 toward Fermi–Dirac distribution, excitation-induced dephasing, and further renormalization of energies due to correlations.


All these correlation effects can be systematically included by solving also the dynamics of two-particle correlations. At this level of sophistication, one can use the SBEs to predict optical response of semiconductors without phenomenological parameters, which gives the SBEs a very high degree of predictability. Indeed, one can use the SBEs in order to predict suitable laser designs through the accurate knowledge they produce about the semiconductor's gain spectrum. One can even use the SBEs to deduce existence of correlations, such as bound excitons, from quantitative measurements.


The presented SBEs are formulated in the momentum space since carrier's crystal momentum follows from formula_36. An equivalent set of equations can also be formulated in position space. However, especially, the correlation computations are much simpler to be performed in the momentum space.


The formula_31 dynamic shows a structure where an individual formula_31 is coupled to "all" other microscopic polarizations due to the Coulomb interaction formula_28. Therefore, the transition amplitude formula_31 is collectively modified by the presence of other transition amplitudes. Only if one sets formula_28 to zero, one finds isolated transitions within each formula_42 state that follow exactly the same dynamics as the optical Bloch equations predict. Therefore, already the Coulomb interaction among formula_31 produces a new solid-state effect compared with optical transitions in simple atoms.


Conceptually, formula_31 is just a transition amplitude for exciting an electron from valence to conduction band. At the same time, the homogeneous part of formula_31 dynamics yields an eigenvalue problem that can be expressed through the generalized Wannier equation. The eigenstates of the Wannier equation is analogous to bound solutions of the hydrogen problem of quantum mechanics. These are often referred to as exciton solutions and they formally describe Coulombic binding by oppositely charged electrons and holes.


However, a real exciton is a true two-particle correlation because one must then have a correlation between one electron to another hole. Therefore, the appearance of exciton resonances in the polarization does not signify the presence of excitons because formula_31 is a single-particle transition amplitude. The excitonic resonances are a direct consequence of Coulomb coupling among all transitions possible in the system. In other words, the single-particle transitions themselves are influenced by Coulomb interaction making it possible to detect exciton resonance in optical response even when true excitons are not present.


Therefore, it is often customary to specify optical resonances as exciton"ic" instead of exciton resonances. The actual role of excitons on optical response can only be deduced by quantitative changes to induce to the linewidth and energy shift of excitonic resonances.


The solutions of the Wannier equation can produces valuable insight to the basic properties of a semiconductor's optical response. In particular, one can solve the steady-state solutions of the SBEs to predict optical absorption spectrum analytically with the so-called Elliott formula. In this form, one can verify that an unexcited semiconductor shows several excitonic absorption resonances well below the fundamental bandgap energy. Obviously, this situation cannot be probing excitons because the initial many-body system does not contain electrons and holes to begin with. Furthermore, the probing can, in principle, be performed so gently that one essentially does not excite electron–hole pairs. This gedanken experiment illustrates nicely why one can detect excitonic resonances without having excitons in the system, all due to virtue of Coulomb coupling among transition amplitudes.


The SBEs are particularly useful when solving the light propagation through a semiconductor structure. In this case, one needs to solve the SBEs together with the Maxwell's equations driven by the optical polarization. This self-consistent set is called the Maxwell–SBEs and is frequently applied to analyze present-day experiments and to simulate device designs.


At this level, the SBEs provide an extremely versatile method that describes linear as well as nonlinear phenomena such as excitonic effects, propagation effects, semiconductor microcavity effects, four-wave-mixing, polaritons in semiconductor microcavities, gain spectroscopy, and so on. One can also generalize the SBEs by including excitation with terahertz (THz) fields that are typically resonant with intraband transitions. One can also quantize the light field and investigate quantum-optical effects that result. In this situation, the SBEs become coupled to the semiconductor luminescence equations.


The Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum.


In convective (or Lagrangian) form the Cauchy momentum equation is written as:


Note that only we use column vectors (in the Cartesian coordinate system) above for clarity, but the equation is written using physical components (which are neither covariants ("column") nor contravariants ("row") ). However, if we chose a non-orthogonal curvilinear coordinate system, then we should calculate and write equations in covariant ("row vectors") or contravariant ("column vectors") form.


After an appropriate change of variables, it can also be written in conservation form:


where is the momentum density at a given space-time point, is the flux associated to the momentum density, and contains all of the body forces per unit volume.


Let us start with the generalized momentum conservation principle which can be written as follows: "The change in system momentum is proportional to the resulting force acting on this system". It is expressed by the formula:


where formula_13 is momentum in time t, formula_14 is force averaged over formula_15. After dividing by formula_15 and passing to the limit formula_17 we get (derivative):


Let us analyse each side of the equation above.


We split the forces into body forces formula_19 and surface forces formula_20


Surface forces act on walls of the cubic fluid element. For each wall, the X component of these forces was marked in the figure with a cubic element (in the form of a product of stress and surface area e.g. formula_22).


Adding forces (their X components) acting on each of the cube walls, we get:


After ordering formula_24 and performing similar reasoning for components formula_25 (they have not been shown in the figure, but these would be vectors parallel to the Y and Z axes, respectively) we get:


We can then write it in the symbolic operational form:


There are mass forces acting on the inside of the control volume. We can write them using the acceleration field formula_30 (e.g. gravitational acceleration):


Let us calculate momentum of the cube:


Because we assume that tested mass (cube) formula_33 is constant in time, so


Divide both sides by formula_38, and because formula_39 we get:


Applying Newton's second law (th component) to a control volume in the continuum being modeled gives:


Then, based on the Reynolds transport theorem and using material derivative notation, one can write


where represents the control volume. Since this equation must hold for any control volume, it must be true that the integrand is zero, from this the Cauchy momentum equation follows. The main step (not done above) in deriving this equation is establishing that the derivative of the stress tensor is one of the forces that constitutes .


The Cauchy momentum equation can also be put in the following form:


where is the momentum density at the point considered in the continuum (for which the continuity equation holds), is the flux associated to the momentum density, and contains all of the body forces per unit volume. is the dyad of the velocity.


Here and have same number of dimensions as the flow speed and the body acceleration, while , being a tensor, has .


In the Eulerian forms it is apparent that the assumption of no deviatoric stress brings Cauchy equations to the Euler equations.


A significant feature of the Navier–Stokes equations is the presence of convective acceleration: the effect of time-independent acceleration of a flow with respect to space. While individual continuum particles indeed experience time dependent acceleration, the convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle.


Regardless of what kind of continuum is being dealt with, convective acceleration is a nonlinear effect. Convective acceleration is present in most flows (exceptions include one-dimensional incompressible flow), but its dynamic effect is disregarded in creeping flow (also called Stokes flow). Convective acceleration is represented by the nonlinear quantity , which may be interpreted either as or as , with the tensor derivative of the velocity vector . Both interpretations give the same result.


The convection term formula_44 can be written as , where is the advection operator. This representation can be contrasted to the one in terms of the tensor derivative.


The tensor derivative is the component-by-component derivative of the velocity vector, defined by , so that


where the Feynman subscript notation is used, which means the subscripted gradient operates only on the factor .


Lamb in his famous classical book Hydrodynamics (1895), used this identity to change the convective term of the flow velocity in rotational form, i.e. without a tensor derivative:


where the vector formula_48 is called the Lamb vector. The Cauchy momentum equation becomes:


In fact, in case of an external conservative field, by defining its potential :


And by projecting the momentum equation on the flow direction, i.e. along a "streamline", the cross product disappears due to a vector calculus identity of the triple scalar product:


If the stress tensor is isotropic, then only the pressure enters: formula_55 (where is the identity tensor), and the Euler momentum equation in the steady incompressible case becomes:


that is, "the mass conservation for a steady incompressible flow states that the density along a streamline is constant". This leads to a considerable simplification of the Euler momentum equation:


in fact, the above equation can be simply written as:


That is, "the momentum balance for a steady inviscid and incompressible flow in an external conservative field states that the total head along a streamline is constant".


The Lamb form is also useful in irrotational flow, where the curl of the velocity (called vorticity) is equal to zero. In that case, the convection term in formula_44 reduces to


The effect of stress in the continuum flow is represented by the and terms; these are gradients of surface forces, analogous to stresses in a solid. Here is the pressure gradient and arises from the isotropic part of the Cauchy stress tensor. This part is given by the normal stresses that occur in almost all situations. The anisotropic part of the stress tensor gives rise to , which usually describes viscous forces; for incompressible flow, this is only a shear effect. Thus, is the deviatoric stress tensor, and the stress tensor is equal to:


where is the identity matrix in the space considered and the shear tensor.


All non-relativistic momentum conservation equations, such as the Navier–Stokes equation, can be derived by beginning with the Cauchy momentum equation and specifying the stress tensor through a constitutive relation. By expressing the shear tensor in terms of viscosity and fluid velocity, and assuming constant density and viscosity, the Cauchy momentum equation will lead to the Navier–Stokes equations. By assuming inviscid flow, the Navier–Stokes equations can further simplify to the Euler equations.


The divergence of the stress tensor can be written as


The effect of the pressure gradient on the flow is to accelerate the flow in the direction from high pressure to low pressure.


As written in the Cauchy momentum equation, the stress terms and are yet unknown, so this equation alone cannot be used to solve problems. Besides the equations of motion—Newton's second law—a force model is needed relating the stresses to the flow motion. For this reason, assumptions based on natural observations are often applied to specify the stresses in terms of the other flow variables, such as velocity and density.


The vector field represents body forces per unit mass. Typically, these consist of only gravity acceleration, but may include others, such as electromagnetic forces. In non-inertial coordinate frames, other "inertial accelerations" associated with rotating coordinates may arise.


Often, these forces may be represented as the gradient of some scalar quantity , with in which case they are called conservative forces. Gravity in the direction, for example, is the gradient of . Because pressure from such gravitation arises only as a gradient, we may include it in the pressure term as a body force . The pressure and force terms on the right-hand side of the Navier–Stokes equation become


It is also possible to include external influences into the stress term formula_66 rather than the body force term. This may even include antisymmetric stresses (inputs of angular momentum), in contrast to the usually symmetrical internal contributions to the stress tensor.


In order to make the equations dimensionless, a characteristic length and a characteristic velocity need to be defined. These should be chosen such that the dimensionless variables are all of order one. The following dimensionless variables are thus obtained:


Substitution of these inverted relations in the Euler momentum equations yields:


and by dividing for the first coefficient:


and the coefficient of skin-friction or the one usually referred as 'drag' co-efficient in the field of aerodynamics:


by passing respectively to the conservative variables, i.e. the momentum density and the force density:


the equations are finally expressed (now omitting the indexes):


2 \nabla \cdot \boldsymbol \tau + \frac 1 {\mathrm{Fr}} \mathbf g


Cauchy equations in the Froude limit (corresponding to negligible external field) are named free Cauchy equations:


and can be eventually conservation equations. The limit of high Froude numbers (low external field) is thus notable for such equations and is studied with perturbation theory.


Finally in convective form the equations are:


For asymmetric stress tensors, equations in general take the following forms:


Below, we write the main equation in pressure-tau form assuming that the stress tensor is symmetrical (formula_75):


Plastic limit theorems in continuum mechanics provide two bounds that can be used to determine whether material failure is possible by means of plastic deformation for a given external loading scenario. According to the theorems, to find the range within which the true solution must lie, it is necessary to find both a stress field that balances the external forces and a velocity field or flow pattern that corresponds to those stresses. If the upper and lower bounds provided by the velocity field and stress field coincide, the exact value of the collapse load is determined.


The two plastic limit theorems apply to any elastic-perfectly plastic body or assemblage of bodies.


If an equilibrium distribution of stress can be found which balances the applied load and nowhere violates the yield criterion, the body (or bodies) will not fail, or will be just at the point of failure.


The body (or bodies) will collapse if there is any compatible pattern of plastic deformation for which the rate of work done by the external loads exceeds the internal plastic dissipation.


In particle physics, the baryon number is a strictly conserved additive quantum number of a system. It is defined as


where "n"q is the number of quarks, and "n" is the number of antiquarks. Baryons (three quarks) have a baryon number of +1, mesons (one quark, one antiquark) have a baryon number of 0, and antibaryons (three antiquarks) have a baryon number of −1. Exotic hadrons like pentaquarks (four quarks, one antiquark) and tetraquarks (two quarks, two antiquarks) are also classified as baryons and mesons depending on their baryon number.


Quarks carry not only electric charge, but also charges such as color charge and weak isospin. Because of a phenomenon known as "color confinement", a hadron cannot have a net color charge; that is, the total color charge of a particle has to be zero ("white"). A quark can have one of three "colors", dubbed "red", "green", and "blue"; while an antiquark may be either "anti-red", "anti-green" or "anti-blue".


For normal hadrons, a white color can thus be achieved in one of three ways:


The baryon number was defined long before the quark model was established, so rather than changing the definitions, particle physicists simply gave quarks one third the baryon number. Nowadays it might be more accurate to speak of the conservation of quark number.


In theory, exotic hadrons can be formed by adding pairs of quarks and antiquarks, provided that each pair has a matching color/anticolor. For example, a pentaquark (four quarks, one antiquark) could have the individual quark colors: red, green, blue, blue, and antiblue. In 2015, the LHCb collaboration at CERN reported results consistent with pentaquark states in the decay of bottom Lambda baryons ().


Particles without any quarks have a baryon number of zero. Such particles are


The baryon number is conserved in all the interactions of the Standard Model, with one possible exception. 'Conserved' means that the sum of the baryon number of all incoming particles is the same as the sum of the baryon numbers of all particles resulting from the reaction. The one exception is the hypothesized Adler–Bell–Jackiw anomaly in electroweak interactions; however, sphalerons are not all that common and could occur at high energy and temperature levels and can explain electroweak baryogenesis and leptogenesis. Electroweak sphalerons can only change the baryon and/or lepton number by 3 or multiples of 3 (collision of three baryons into three leptons/antileptons and vice versa). No experimental evidence of sphalerons has yet been observed.


The hypothetical concepts of grand unified theory (GUT) models and supersymmetry allows for the changing of a baryon into leptons and antiquarks (see "B" − "L"), thus violating the conservation of both baryon and lepton numbers. Proton decay would be an example of such a process taking place, but has never been observed.


The conservation of baryon number is not consistent with the physics of black hole evaporation via Hawking radiation. It is expected in general that quantum gravitational effects violate the conservation of all charges associated to global symmetries. The violation of conservation of baryon number led John Archibald Wheeler to speculate on a principle of mutability for all physical properties.