The projective representations of any group are isomorphic to the ordinary representations of a central extension of the group. For example, projective representations of the 3-dimensional rotation group, which is the special orthogonal group SO(3), are ordinary representations of the special unitary group SU(2) (see Representation theory of SU(2)). Projective representations of the rotation group that are not representations are called spinors and so quantum states may transform not only as tensors but also as spinors.


If one adds to this a classification by parity, these can be extended, for example, into notions of


which also have negative determinant and form a valid parity transformation. Then, combining them with rotations (or successively performing "x"-, "y"-, and "z"-reflections) one can recover the particular parity transformation defined earlier. The first parity transformation given does not work in an even number of dimensions, though, because it results in a positive determinant. In even dimensions only the latter example of a parity transformation (or any reflection of an odd number of coordinates) can be used.


Parity forms the abelian group formula_3 due to the relation formula_4. All Abelian groups have only one-dimensional irreducible representations. For formula_3, there are two irreducible representations: one is even under parity, formula_6, the other is odd, formula_7. These are useful in quantum mechanics. However, as is elaborated below, in quantum mechanics states need not transform under actual representations of parity but only under projective representations and so in principle a parity transformation may rotate a state by any phase.


Newton's equation of motion formula_8 (if the mass is constant) equates two vectors, and hence is invariant under parity. The law of gravity also involves only vectors and is also, therefore, invariant under parity.


However, angular momentum formula_9 is an axial vector,


In classical electrodynamics, the charge density formula_11 is a scalar, the electric field, formula_12, and current formula_13 are vectors, but the magnetic field, formula_14 is an axial vector. However, Maxwell's equations are invariant under parity because the curl of an axial vector is a vector.


Effect of spatial inversion on some variables of classical physics.


Classical variables, predominantly scalar quantities, which do not change upon spatial inversion include:


Classical variables, predominantly vector quantities, which have their sign flipped by spatial inversion include:


In quantum mechanics, spacetime transformations act on quantum states. The parity transformation, formula_40, is a unitary operator, in general acting on a state formula_41 as follows: formula_42.


For electronic wavefunctions, even states are usually indicated by a subscript g for "gerade" (German: even) and odd states by a subscript u for "ungerade" (German: odd). For example, the lowest energy level of the hydrogen molecule ion (H2+) is labelled formula_56 and the next-closest (higher) energy level is labelled formula_57.


The wave functions of a particle moving into an external potential, which is centrosymmetric (potential energy invariant with respect to a space inversion, symmetric to the origin), either remain invariable or change signs: these two possible states are called the even state or odd state of the wave functions.


The law of conservation of parity of particle (not true for the beta decay of nuclei) states that, if an isolated ensemble of particles has a definite parity, then the parity remains invariable in the process of ensemble evolution.


The parity of the states of a particle moving in a spherically symmetric external field is determined by the angular momentum, and the particle state is defined by three quantum numbers: total energy, angular momentum and the projection of angular momentum.


When parity generates the Abelian group ℤ2, one can always take linear combinations of quantum states such that they are either even or odd under parity (see the figure). Thus the parity of such states is ±1. The parity of a multiparticle state is the product of the parities of each state; in other words parity is a multiplicative quantum number.


In quantum mechanics, Hamiltonians are invariant (symmetric) under a parity transformation if formula_58 commutes with the Hamiltonian. In non-relativistic quantum mechanics, this happens for any scalar potential, i.e., formula_59, hence the potential is spherically symmetric. The following facts can be easily proven:


Some of the non-degenerate eigenfunctions of formula_70 are unaffected (invariant) by parity formula_58 and the others are merely reversed in sign when the Hamiltonian operator and the parity operator commute:


where formula_77 is a constant, the eigenvalue of formula_58,


The overall parity of a many-particle system is the product of the parities of the one-particle states. It is −1 if an odd number of particles are in odd-parity states, and +1 otherwise. Different notations are in use to denote the parity of nuclei, atoms, and molecules.


Atomic orbitals have parity (−1)ℓ, where the exponent ℓ is the azimuthal quantum number. The parity is odd for orbitals p, f, … with ℓ = 1, 3, …, and an atomic state has odd parity if an odd number of electrons occupy these orbitals. For example, the ground state of the nitrogen atom has the electron configuration 1s22s22p3, and is identified by the term symbol 4So, where the superscript o denotes odd parity. However the third excited term at about 83,300 cm−1 above the ground state has electron configuration 1s22s22p23s has even parity since there are only two 2p electrons, and its term symbol is 4P (without an o superscript).


The complete (rotational-vibrational-electronic-nuclear spin) electromagnetic Hamiltonian of any molecule commutes with (or is invariant to) the parity operation P (or E*, in the notation introduced by Longuet-Higgins) and its eigenvalues can be given the parity symmetry label + or - as they are even or odd, respectively. The parity operation involves the inversion of electronic and nuclear spatial coordinates at the molecular center of mass.


does not commute with the point group inversion operation i because of the effect of the nuclear hyperfine Hamiltonian. The nuclear hyperfine Hamiltonian can mix the rotational levels of g and u vibronic states (called ortho-para mixing) and give rise to ortho-para transitions


If we can show that the vacuum state is invariant under parity, formula_80, the Hamiltonian is parity invariant formula_81 and the quantization conditions remain unchanged under parity, then it follows that every state has good parity, and this parity is conserved in any reaction.


To show that quantum electrodynamics is invariant under parity, we have to prove that the action is invariant and the quantization is also invariant. For simplicity we will assume that canonical quantization is used; the vacuum state is then invariant under parity by construction. The invariance of the action follows from the classical invariance of Maxwell's equations. The invariance of the canonical quantization procedure can be worked out, and turns out to depend on the transformation of the annihilation operator:


where p denotes the momentum of a photon and ± refers to its polarization state. This is equivalent to the statement that the photon has odd intrinsic parity. Similarly all vector bosons can be shown to have odd intrinsic parity, and all axial-vectors to have even intrinsic parity.


A straightforward extension of these arguments to scalar field theories shows that scalars have even parity, since


This is true even for a complex scalar field. ("Details of spinors are dealt with in the article on the "Dirac equation", where it is shown that fermions and antifermions have opposite intrinsic parity.")


With fermions, there is a slight complication because there is more than one spin group.


In the Standard Model of fundamental interactions there are precisely three global internal U(1) symmetry groups available, with charges equal to the baryon number "B", the lepton number "L" and the electric charge "Q". The product of the parity operator with any combination of these rotations is another parity operator. It is conventional to choose one specific combination of these rotations to define a standard parity operator, and other parity operators are related to the standard one by internal rotations. One way to fix a standard parity operator is to assign the parities of three particles with linearly independent charges "B", "L" and "Q". In general, one assigns the parity of the most common massive particles, the proton, the neutron and the electron, to be +1.


Steven Weinberg has shown that if , where "F" is the fermion number operator, then, since the fermion number is the sum of the lepton number plus the baryon number, , for all particles in the Standard Model and since lepton number and baryon number are charges "Q" of continuous symmetries "e""iQ", it is possible to redefine the parity operator so that . However, if there exist Majorana neutrinos, which experimentalists today believe is possible, their fermion number is equal to one because they are neutrinos while their baryon and lepton numbers are zero because they are Majorana, and so (−1)"F" would not be embedded in a continuous symmetry group. Thus Majorana neutrinos would have parity ±"i".


In 1954, a paper by William Chinowsky and Jack Steinberger demonstrated that the pion has negative parity. They studied the decay of an "atom" made from a deuteron () and a negatively charged pion () in a state with zero orbital angular momentum formula_82 into two neutrons (formula_83).


Although parity is conserved in electromagnetism, strong interactions and gravity, it is violated in weak interactions. The Standard Model incorporates parity violation by expressing the weak interaction as a chiral gauge interaction. Only the left-handed components of particles and right-handed components of antiparticles participate in charged weak interactions in the Standard Model. This implies that parity is not a symmetry of our universe, unless a hidden mirror sector exists in which parity is violated in the opposite way.


By the mid-20th century, it had been suggested by several scientists that parity might not be conserved (in different contexts), but without solid evidence these suggestions were not considered important. Then, in 1956, a careful review and analysis by theoretical physicists Tsung-Dao Lee and Chen-Ning Yang went further, showing that while parity conservation had been verified in decays by the strong or electromagnetic interactions, it was untested in the weak interaction. They proposed several possible direct experimental tests. They were mostly ignored, but Lee was able to convince his Columbia colleague Chien-Shiung Wu to try it. She needed special cryogenic facilities and expertise, so the experiment was done at the National Bureau of Standards.


After the fact, it was noted that an obscure 1928 experiment, done by R. T. Cox, G. C. McIlwraith, and B. Kurrelmeyer, had in effect reported parity violation in weak decays, but since the appropriate concepts had not yet been developed, those results had no impact. The discovery of parity violation immediately explained the outstanding τ–θ puzzle in the physics of kaons.


In 2010, it was reported that physicists working with the Relativistic Heavy Ion Collider (RHIC) had created a short-lived parity symmetry-breaking bubble in quark-gluon plasmas. An experiment conducted by several physicists including Yale's Jack Sandweiss as part of the STAR collaboration, suggested that parity may also be violated in the strong interaction. It is predicted that this local parity violation, which would be analogous to the effect that is induced by fluctuation of the axion field, manifest itself by chiral magnetic effect.


To every particle one can assign an intrinsic parity as long as nature preserves parity. Although weak interactions do not, one can still assign a parity to any hadron by examining the strong interaction reaction that produces it, or through decays not involving the weak interaction, such as rho meson decay to pions.


In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a closed system remains constant.


In three dimensions, the angular momentum for a point particle is a pseudovector , the cross product of the particle's position vector r (relative to some origin) and its momentum vector; the latter is in Newtonian mechanics. This definition can be applied to each point in continua like solids or fluids, or physical fields. Unlike momentum, angular momentum does depend on where the origin is chosen, since the particle's position is measured from it.


Angular momentum is an extensive quantity; i.e. the total angular momentum of any composite system is the sum of the angular momenta of its constituent parts. For a continuous rigid body, the total angular momentum is the volume integral of angular momentum density (i.e. angular momentum per unit volume in the limit as volume shrinks to zero) over the entire body.


In quantum mechanics, angular momentum (like other quantities) is expressed as an operator, and its one-dimensional projections have quantized eigenvalues. Angular momentum is subject to the Heisenberg uncertainty principle, implying that at any time, only one projection (also called "component") can be measured with definite precision; the other two then remain uncertain. Because of this, the notion of a quantum particle literally "spinning" about an axis does not exist. Quantum particles "do" possess a type of non-orbital angular momentum called "spin", but this angular momentum does not correspond to actual physical spinning motion.


Angular momentum is a vector quantity (more precisely, a pseudovector) that represents the product of a body's rotational inertia and rotational velocity (in radians/sec) about a particular axis. However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of angular momentum, and treat it as a scalar (more precisely, a pseudoscalar). Angular momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum is proportional to mass and linear speed


angular momentum is proportional to moment of inertia and angular speed measured in radians per second.


Unlike mass, which depends only on amount of matter, moment of inertia is also dependent on the position of the axis of rotation and the shape of the matter. Unlike linear velocity, which does not depend upon the choice of origin, orbital angular velocity is always measured with respect to a fixed origin. Therefore, strictly speaking, should be referred to as the angular momentum "relative to that center".


Because formula_3 for a single particle and formula_4 for circular motion, angular momentum can be expanded, formula_5 and reduced to,


the product of the radius of rotation and the linear momentum of the particle formula_7, where formula_8 in this case is the equivalent linear (tangential) speed at the radius (formula_9).


This simple analysis can also apply to non-circular motion if only the component of the motion which is perpendicular to the radius vector is considered. In that case,


where formula_11 is the perpendicular component of the motion. Expanding, formula_12 rearranging, formula_13 and reducing, angular momentum can also be expressed,


where formula_15 is the length of the "moment arm", a line dropped perpendicularly from the origin onto the path of the particle. It is this definition, to which the term "moment of momentum" refers.


Another approach is to define angular momentum as the conjugate momentum (also called canonical momentum) of the angular coordinate formula_16 expressed in the Lagrangian of the mechanical system. Consider a mechanical system with a mass formula_17 constrained to move in a circle of radius formula_18 in the absence of any external force field. The kinetic energy of the system is


The "generalized momentum" "canonically conjugate to" the coordinate formula_16 is defined by


To completely define orbital angular momentum in three dimensions, it is required to know the rate at which the position vector sweeps out angle, the direction perpendicular to the instantaneous plane of angular displacement, and the mass involved, as well as how this mass is distributed in space. By retaining this vector nature of angular momentum, the general nature of the equations is also retained, and can describe any sort of three-dimensional motion about the center of rotation – circular, linear, or otherwise. In vector notation, the orbital angular momentum of a point particle in motion about the origin can be expressed as:


This can be expanded, reduced, and by the rules of vector algebra, rearranged:


which is the cross product of the position vector formula_27 and the linear momentum formula_33 of the particle. By the definition of the cross product, the formula_34 vector is perpendicular to both formula_27 and formula_36. It is directed perpendicular to the plane of angular displacement, as indicated by the right-hand rule – so that the angular velocity is seen as counter-clockwise from the head of the vector. Conversely, the formula_34 vector defines the plane in which formula_27 and formula_36 lie.


By defining a unit vector formula_40 perpendicular to the plane of angular displacement, a scalar angular speed formula_41 results, where


The two-dimensional scalar equations of the previous section can thus be given direction:


and formula_46 for circular motion, where all of the motion is perpendicular to the radius formula_47.


In the spherical coordinate system the angular momentum vector expresses as


Angular momentum can be described as the rotational analog of linear momentum. Like linear momentum it involves elements of mass and displacement. Unlike linear momentum it also involves elements of position and shape.


Many problems in physics involve matter in motion about some certain point in space, be it in actual rotation about it, or simply moving past it, where it is desired to know what effect the moving matter has on the point—can it exert energy upon it or perform work about it? Energy, the ability to do work, can be stored in matter by setting it in motion—a combination of its inertia and its displacement. Inertia is measured by its mass, and displacement by its velocity. Their product,


is the matter's momentum. Referring this momentum to a central point introduces a complication: the momentum is not applied to the point directly. For instance, a particle of matter at the outer edge of a wheel is, in effect, at the end of a lever of the same length as the wheel's radius, its momentum turning the lever about the center point. This imaginary lever is known as the "moment arm". It has the effect of multiplying the momentum's effort in proportion to its length, an effect known as a "moment". Hence, the particle's momentum referred to a particular point,


is the "angular momentum", sometimes called, as here, the "moment of momentum" of the particle versus that particular center point. The equation formula_51 combines a moment (a mass formula_17 turning moment arm formula_47) with a linear (straight-line equivalent) speed formula_8. Linear speed referred to the central point is simply the product of the distance formula_47 and the angular speed formula_41 versus the point: formula_57 another moment. Hence, angular momentum contains a double moment: formula_58 Simplifying slightly, formula_59 the quantity formula_60 is the particle's moment of inertia, sometimes called the second moment of mass. It is a measure of rotational inertia.


Because moment of inertia is a crucial part of the spin angular momentum, the latter necessarily includes all of the complications of the former, which is calculated by multiplying elementary bits of the mass by the squares of their distances from the center of rotation. Therefore, the total moment of inertia, and the angular momentum, is a complex function of the configuration of the matter about the center of rotation and the orientation of the rotation for the various bits.


For a rigid body, for instance a wheel or an asteroid, the orientation of rotation is simply the position of the rotation axis versus the matter of the body. It may or may not pass through the center of mass, or it may lie completely outside of the body. For the same body, angular momentum may take a different value for every possible axis about which rotation may take place. It reaches a minimum when the axis passes through the center of mass.


For a collection of objects revolving about a center, for instance all of the bodies of the Solar System, the orientations may be somewhat organized, as is the Solar System, with most of the bodies' axes lying close to the system's axis. Their orientations may also be completely random.


In brief, the more mass and the farther it is from the center of rotation (the longer the moment arm), the greater the moment of inertia, and therefore the greater the angular momentum for a given angular velocity. In many cases the moment of inertia, and hence the angular momentum, can be simplified by,


Similarly, for a point mass formula_17 the moment of inertia is defined as,


and for any collection of particles formula_67 as the sum,


The plane perpendicular to the axis of angular momentum and passing through the center of mass is sometimes called the "invariable plane", because the direction of the axis remains fixed if only the interactions of the bodies within the system, free from outside influences, are considered. One such plane is the invariable plane of the Solar System.


Newton's second law of motion can be expressed mathematically,


or force = mass × acceleration. The rotational equivalent for point particles may be derived as follows:


which means that the torque (i.e. the time derivative of the angular momentum) is


Because the moment of inertia is formula_72, it follows that formula_73, and formula_74 which, reduces to


This is the rotational analog of Newton's Second Law. Note that the torque is not necessarily proportional or parallel to the angular acceleration (as one might expect). The reason for this is that the moment of inertia of a particle can change with time, something that cannot occur for ordinary mass.


A rotational analog of Newton's third law of motion might be written, "In a closed system, no torque can be exerted on any matter without the exertion on some other matter of an equal and opposite torque." Hence, "angular momentum can be exchanged between objects in a closed system, but total angular momentum before and after an exchange remains constant (is conserved)."


Seen another way, a rotational analogue of Newton's first law of motion might be written, "A rigid body continues in a state of uniform rotation unless acted by an external influence." Thus "with no external influence to act upon it, the original angular momentum of the system remains constant".


For a planet, angular momentum is distributed between the spin of the planet and its revolution in its orbit, and these are often exchanged by various mechanisms. The conservation of angular momentum in the Earth–Moon system results in the transfer of angular momentum from Earth to Moon, due to tidal torque the Moon exerts on the Earth. This in turn results in the slowing down of the rotation rate of Earth, at about 65.7 nanoseconds per day, and in gradual increase of the radius of Moon's orbit, at about 3.82 centimeters per year.


The conservation of angular momentum explains the angular acceleration of an ice skater as she brings her arms and legs close to the vertical axis of rotation. By bringing part of the mass of her body closer to the axis, she decreases her body's moment of inertia. Because angular momentum is the product of moment of inertia and angular velocity, if the angular momentum remains constant (is conserved), then the angular velocity (rotational speed) of the skater must increase.


The same phenomenon results in extremely fast spin of compact stars (like white dwarfs, neutron stars and black holes) when they are formed out of much larger and slower rotating stars. Decrease in the size of an object "n" times results in increase of its angular velocity by the factor of "n"2.


Conservation is not always a full explanation for the dynamics of a system but is a key constraint. For example, a spinning top is subject to gravitational torque making it lean over and change the angular momentum about the nutation axis, but neglecting friction at the point of spinning contact, it has a conserved angular momentum about its spinning axis, and another about its precession axis. Also, in any planetary system, the planets, star(s), comets, and asteroids can all move in numerous complicated ways, but only so that the angular momentum of the system is conserved.


Noether's theorem states that every conservation law is associated with a symmetry (invariant) of the underlying physics. The symmetry associated with conservation of angular momentum is rotational invariance. The fact that the physics of a system is unchanged if it is rotated by any angle about an axis implies that angular momentum is conserved.


Relation to Newton's second law of motion.


As an example, consider decreasing of the moment of inertia, e.g. when a figure skater is pulling in her/his hands, speeding up the circular motion. In terms of angular momentum conservation, we have, for angular momentum "L", moment of inertia "I" and angular velocity "ω":


Using this, we see that the change requires an energy of:


so that a decrease in the moment of inertia requires investing energy.


This can be compared to the work done as calculated using Newton's laws. Each point in the rotating body is accelerating, at each point of time, with radial acceleration of:


Let us observe a point of mass "m", whose position vector relative to the center of motion is parallel to the z-axis at a given point of time, and is at a distance "z". The centripetal force on this point, keeping the circular motion, is:


Thus the work required for moving this point to a distance "dz" farther from the center of motion is:


For a non-pointlike body one must integrate over this, with "m" replaced by the mass density per unit "z". This gives:


which is exactly the energy required for keeping the angular momentum conserved.


Note, that the above calculation can also be performed per mass, using kinematics only. Thus the phenomena of figure skater accelerating tangential velocity while pulling her/his hands in, can be understood as follows in layman's language: The skater's palms are not moving in a straight line, so they are constantly accelerating inwards, but do not gain additional speed because the accelerating is always done when their motion inwards is zero. However, this is different when pulling the palms closer to the body: The acceleration due to rotation now increases the speed; but because of the rotation, the increase in speed does not translate to a significant speed inwards, but to an increase of the rotation speed.


In Lagrangian mechanics, angular momentum for rotation around a given axis, is the conjugate momentum of the generalized coordinate of the angle around the same axis. For example, formula_85, the angular momentum around the z axis, is:


where formula_87 is the Lagrangian and formula_88 is the angle around the z axis.


Note that formula_89, the time derivative of the angle, is the angular velocity formula_90. Ordinarily, the Lagrangian depends on the angular velocity through the kinetic energy: The latter can be written by separating the velocity to its radial and tangential part, with the tangential part at the x-y plane, around the z-axis, being equal to:


Since the lagrangian is dependent upon the angles of the object only through the potential, we have:


which is the torque on the i-th object.


Suppose the system is invariant to rotations, so that the potential is independent of an overall rotation by the angle "θ"z (thus it may depend on the angles of objects only through their differences, in the form formula_93). We therefore get for the total angular momentum:


And thus the angular momentum around the z-axis is conserved.


This analysis can be repeated separately for each axis, giving conversation of the angular momentum vector. However, the angles around the three axes cannot be treated simultaneously as generalized coordinates, since they are not independent; in particular, two angles per point suffice to determine its position. While it is true that in the case of a rigid body, fully describing it requires, in addition to three translational degrees of freedom, also specification of three rotational degrees of freedom; however these cannot be defined as rotations around the Cartesian axes (see Euler angles). This caveat is reflected in quantum mechanics in the non-trivial commutation relations of the different components of the angular momentum operator.


Equivalently, in Hamiltonian mechanics the Hamiltonian can be described as a function of the angular momentum. As before, the part of the kinetic energy related to rotation around the z-axis for the i-th object is:


In theoretical physics, a chiral anomaly is the anomalous nonconservation of a chiral current. In everyday terms, it is equivalent to a sealed box that contained equal numbers of left and right-handed bolts, but when opened was found to have more left than right, or vice versa.


Such events are expected to be prohibited according to classical conservation laws, but we know there must be ways they can be broken, because we have evidence of charge–parity non-conservation ("CP violation"). It is possible that other imbalances have been caused by breaking of a "chiral law" of this kind. Many physicists suspect that the fact that the observable universe contains more matter than antimatter is caused by a chiral anomaly. Research into chiral symmetry breaking laws is a major endeavor in particle physics research at this time.


The chiral anomaly originally referred to the anomalous decay rate of the neutral pion, as computed in the current algebra of the chiral model. These calculations suggested that the decay of the pion was suppressed, clearly contradicting experimental results. The nature of the anomalous calculations was first explained by Adler and Bell & Jackiw. This is now termed the Adler–Bell–Jackiw anomaly of quantum electrodynamics. This is a symmetry of classical electrodynamics that is violated by quantum corrections.


At the time that the Adler–Bell–Jackiw anomaly was being explored in physics, there were related developments in differential geometry that appeared to involve the same kinds of expressions. These were not in any way related to quantum corrections of any sort, but rather were the exploration of the global structure of fiber bundles, and specifically, of the Dirac operators on spin structures having curvature forms resembling that of the electromagnetic tensor, both in four and three dimensions (the Chern–Simons theory). After considerable back and forth, it became clear that the structure of the anomaly could be described with bundles with a non-trivial homotopy group, or, in physics lingo, in terms of instantons.


Instantons are a form of topological soliton; they are a solution to the "classical" field theory, having the property that they are stable and cannot decay (into plane waves, for example). Put differently: conventional field theory is built on the idea of a vacuum - roughly speaking, a flat empty space. Classically, this is the "trivial" solution; all fields vanish. However, one can also arrange the (classical) fields in such a way that they have a non-trivial global configuration. These non-trivial configurations are also candidates for the vacuum, for empty space; yet they are no longer flat or trivial; they contain a twist, the instanton. The quantum theory is able to interact with these configurations; when it does so, it manifests as the chiral anomaly.


In mathematics, non-trivial configurations are found during the study of Dirac operators in their fully generalized setting, namely, on Riemannian manifolds in arbitrary dimensions. Mathematical tasks include finding and classifying structures and configurations. Famous results include the Atiyah–Singer index theorem for Dirac operators. Roughly speaking, the symmetries of Minkowski spacetime, Lorentz invariance, Laplacians, Dirac operators and the U(1)xSU(2)xSU(3) fiber bundles can be taken to be a special case of a far more general setting in differential geometry; the exploration of the various possibilities accounts for much of the excitement in theories such as string theory; the richness of possibilities accounts for a certain perception of lack of progress.


Besides explaining the decay of the pion, it has a second very important role. The one loop amplitude includes a factor that counts the grand total number of leptons that can circulate in the loop. In order to get the correct decay width, one must have exactly three generations of quarks, and not four or more. In this way, it plays an important role in constraining the Standard model. It provides a direct physical prediction of the number of quarks that can exist in nature.


Current day research is focused on similar phenomena in different settings, including non-trivial topological configurations of the electroweak theory, that is, the sphalerons. Other applications include the hypothetical non-conservation of baryon number in GUTS and other theories.


In some theories of fermions with chiral symmetry, the quantization may lead to the breaking of this (global) chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved. The non-conservation happens in a process of tunneling from one vacuum to another. Such a process is called an instanton.


In the case of a symmetry related to the conservation of a fermionic particle number, one may understand the creation of such particles as follows. The definition of a particle is different in the two vacuum states between which the tunneling occurs; therefore a state of no particles in one vacuum corresponds to a state with some particles in the other vacuum. In particular, there is a Dirac sea of fermions and, when such a tunneling happens, it causes the energy levels of the sea fermions to gradually shift upwards for the particles and downwards for the anti-particles, or vice versa. This means particles which once belonged to the Dirac sea become real (positive energy) particles and particle creation happens.


Technically, in the path integral formulation, an anomalous symmetry is a symmetry of the action formula_3, but not of the measure and therefore "not" of the generating functional


of the quantized theory ( is Planck's action-quantum divided by 2). The measure formula_5 consists of a part depending on the fermion field formula_6 and a part depending on its complex conjugate formula_7. The transformations of both parts under a chiral symmetry do not cancel in general. Note that if formula_8 is a Dirac fermion, then the chiral symmetry can be written as formula_9 where formula_10 is the chiral gamma matrix acting on formula_8. From the formula for formula_12 one also sees explicitly that in the classical limit, anomalies don't come into play, since in this limit only the extrema of formula_3 remain relevant.


The anomaly is proportional to the instanton number of a gauge field to which the fermions are coupled. (Note that the gauge symmetry is always non-anomalous and is exactly respected, as is required for the theory to be consistent.)


The chiral anomaly can be calculated exactly by one-loop Feynman diagrams, e.g. Steinberger's "triangle diagram", contributing to the pion decays, and formula_14. The amplitude for this process can be calculated directly from the change in the measure of the fermionic fields under the chiral transformation.


Wess and Zumino developed a set of conditions on how the partition function ought to behave under gauge transformations called the Wess–Zumino consistency condition.


Fujikawa derived this anomaly using the correspondence between functional determinants and the partition function using the Atiyah–Singer index theorem. See Fujikawa's method.


The Standard Model of electroweak interactions has all the necessary ingredients for successful baryogenesis, although these interactions have never been observed and may be insufficient to explain the total baryon number of the observed universe if the initial baryon number of the universe at the time of the Big Bang is zero. Beyond the violation of charge conjugation formula_15 and CP violation formula_16 (charge+parity), baryonic charge violation appears through the Adler–Bell–Jackiw anomaly of the formula_17 group.


Baryons are not conserved by the usual electroweak interactions due to quantum chiral anomaly. The classic electroweak Lagrangian conserves baryonic charge. Quarks always enter in bilinear combinations formula_18, so that a quark can disappear only in collision with an antiquark. In other words, the classical baryonic current formula_19 is conserved:


However, quantum corrections known as the sphaleron destroy this conservation law: instead of zero in the right hand side of this equation, there is a non-vanishing quantum term,


where is a numerical constant vanishing for ℏ =0,


and the gauge field strength formula_23 is given by the expression


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).


An important fact is that the anomalous current non-conservation is proportional to the total derivative of a vector operator, formula_25 (this is non-vanishing due to instanton configurations of the gauge field, which are pure gauge at the infinity), where the anomalous current formula_26 is


which is the Hodge dual of the Chern–Simons 3-form.


In the language of differential forms, to any self-dual curvature form formula_28 we may assign the abelian 4-form formula_29. Chern–Weil theory shows that this 4-form is locally "but not globally" exact, with potential given by the Chern-Simons 3-form locally:


Again, this is true only on a single chart, and is false for the global form formula_31 unless the instanton number vanishes.


To proceed further, we attach a "point at infinity" onto formula_32 to yield formula_33, and use the clutching construction to chart principal A-bundles, with one chart on the neighborhood of and a second on formula_34. The thickening around , where these charts intersect, is trivial, so their intersection is essentially formula_35. Thus instantons are classified by the third homotopy group formula_36, which for formula_37 is simply the third 3-sphere group formula_38.


The divergence of the baryon number current is (ignoring numerical constants)


Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (converted into kinetic energy) when the objects fall towards each other. Gravitational potential energy increases when two objects are brought further apart.


For two pairwise interacting point particles, the gravitational potential energy formula_1 is given by


where formula_3 and formula_4 are the masses of the two particles, formula_5 is the distance between them, and formula_6 is the gravitational constant.


Close to the Earth's surface, the gravitational field is approximately constant, and the gravitational potential energy of an object reduces to


where formula_4 is the object's mass, formula_9 is the gravity of Earth, and formula_10 is the height of the object's center of mass above a chosen reference level.


In classical mechanics, two or more masses always have a gravitational potential. Conservation of energy requires that this gravitational field energy is always negative, so that it is zero when the objects are infinitely far apart. The gravitational potential energy is the potential energy an object has because it is within a gravitational field.


The force between a point mass, formula_3, and another point mass, formula_4, is given by Newton's law of gravitation:


To get the total work done by an external force to bring point mass formula_4 from infinity to the final distance formula_5 (for example the radius of Earth) of the two mass points, the force is integrated with respect to displacement:


Because formula_18, the total work done on the object can be written as:


In general relativity gravitational energy is extremely complex, and there is no single agreed upon definition of the concept. It is sometimes modelled via the Landau–Lifshitz pseudotensor that allows retention for the energy–momentum conservation laws of classical mechanics. Addition of the matter stress–energy tensor to the Landau–Lifshitz pseudotensor results in a combined matter plus gravitational energy pseudotensor that has a vanishing 4-divergence in all frames—ensuring the conservation law. Some people object to this derivation on the grounds that pseudotensors are inappropriate in general relativity, but the divergence of the combined matter plus gravitational energy pseudotensor is a tensor.


In particle physics, lepton number (historically also called lepton charge) is a conserved quantum number representing the difference between the number of leptons and the number of antileptons in an elementary particle reaction. Lepton number is an additive quantum number, so its sum is preserved in interactions (as opposed to multiplicative quantum numbers such as parity, where the product is preserved instead). Mathematically, the lepton number formula_1 is defined by formula_2, where formula_3 is the number of leptons and formula_4 is the number of antileptons.


Lepton number was introduced in 1953 to explain the absence of reactions such as formula_5 in the Cowan–Reines neutrino experiment, which instead observed formula_6. This process, inverse beta decay, conserves lepton number, as the incoming antineutrino has lepton number –1, while the outgoing positron (antielectron) also has lepton number –1.


In addition to lepton number, lepton family numbers are defined as


Prominent examples of lepton flavor conservation are the muon decays formula_7 and formula_8. In these, the creation of an electron is accompanied by the creation of an electron antineutrino, and the creation of a positron is accompanied by the creation of an electron neutrino. Likewise, a decaying negative muon results in the creation of a muon neutrino, while a decaying positive muon results in the creation of a muon antineutrino.


Violations of the lepton number conservation laws.


Lepton flavor is only approximately conserved, and is notably not conserved in neutrino oscillation. However, total lepton number is still conserved in the Standard Model.


Numerous searches for physics beyond the Standard Model incorporate searches for lepton number or lepton flavor violation, such as the decays formula_9. Experiments such as MEGA and SINDRUM have searched for lepton number violation in muon decays to electrons; MEG set the current branching limit of order 10−13 and plans to lower to limit to 10−14 after 2016. Some theories beyond the Standard Model, such as supersymmetry, predict branching ratios of order 10−12 to 10−14. The Mu2e experiment, in construction as of 2017, has a planned sensitivity of order 10−17.


Because the lepton number conservation law in fact is violated by chiral anomalies, there are problems applying this symmetry universally over all energy scales. However, the quantum number "B" − "L" is commonly conserved in Grand Unified Theory models.


If neutrinos turn out to be Majorana fermions, neither the lepton number nor "B" − "L" would be conserved, e.g. in neutrinoless double beta decay.


In mathematical physics, inversion transformations are a natural extension of Poincaré transformations to include all conformal one-to-one transformations on coordinate space-time. They are less studied in physics because unlike the rotations and translations of Poincaré symmetry an object cannot be physically transformed by the inversion symmetry. Some physical theories are invariant under this symmetry, in these cases it is what is known as a 'hidden symmetry'. Other hidden symmetries of physics include gauge symmetry and general covariance.


In 1831 the mathematician Ludwig Immanuel Magnus began to publish on transformations of the plane generated by inversion in a circle of radius "R". His work initiated a large body of publications, now called inversive geometry. The most prominently named mathematician became August Ferdinand Möbius once he reduced the planar transformations to complex number arithmetic. In the company of physicists employing the inversion transformation early on was Lord Kelvin, and the association with him leads it to be called the Kelvin transform.


In the following we shall use imaginary time (formula_1) so that space-time is Euclidean and the equations are simpler. The Poincaré transformations are given by the coordinate transformation on space-time parametrized by the 4-vectors "V"


where formula_3 is an orthogonal matrix and formula_4 is a 4-vector. Applying this transformation twice on a 4-vector gives a third transformation of the same form. The basic invariant under this transformation is the space-time length given by the distance between two space-time points given by 4-vectors "x" and "y":


These transformations are subgroups of general 1-1 conformal transformations on space-time. It is possible to extend these transformations to include all 1-1 conformal transformations on space-time


We must also have an equivalent condition to the orthogonality condition of the Poincaré transformations:


Because one can divide the top and bottom of the transformation by formula_8 we lose no generality by setting formula_9 to the unit matrix. We end up with


Applying this transformation twice on a 4-vector gives a transformation of the same form. The new symmetry of 'inversion' is given by the 3-tensor formula_11 This symmetry becomes Poincaré symmetry if we set formula_12 When formula_13 the second condition requires that formula_3 is an orthogonal matrix. This transformation is 1-1 meaning that each point is mapped to a unique point only if we theoretically include the points at infinity.


The invariants for this symmetry in 4 dimensions is unknown however it is known that the invariant requires a minimum of 4 space-time points. In one dimension, the invariant is the well known cross-ratio from Möbius transformations:


Because the only invariants under this symmetry involve a minimum of 4 points, this symmetry cannot be a symmetry of point particle theory. Point particle theory relies on knowing the lengths of paths of particles through space-time (e.g., from formula_16 to formula_17). The symmetry can be a symmetry of a string theory in which the strings are uniquely determined by their endpoints. The propagator for this theory for a string starting at the endpoints formula_18 and ending at the endpoints formula_19 is a conformal function of the 4-dimensional invariant. A string field in endpoint-string theory is a function over the endpoints.


In theoretical physics, an invariant is an observable of a physical system which remains unchanged under some transformation. Invariance, as a broader term, also applies to the no change of form of physical laws under a transformation, and is closer in scope to the mathematical definition. Invariants of a system are deeply tied to the symmetries imposed by its environment.


Invariance is an important concept in modern theoretical physics, and many theories are expressed in terms of their symmetries and invariants.


In classical and quantum mechanics, invariance of space under translation results in momentum being an invariant and the conservation of momentum, whereas invariance of the origin of time, i.e. translation in time, results in energy being an invariant and the conservation of energy. In general, by Noether's theorem, any invariance of a physical system under a continuous symmetry leads to a fundamental conservation law.


In crystals, the electron density is periodic and invariant with respect to discrete translations by unit cell vectors. In very few materials, this symmetry can be broken due to enhanced electron correlations.


Another examples of physical invariants are the speed of light, and charge and mass of a particle observed from two reference frames moving with respect to one another (invariance under a spacetime Lorentz transformation), and invariance of time and acceleration under a Galilean transformation between two such frames moving at low velocities.


Quantities can be invariant under some common transformations but not under others. For example, the velocity of a particle is invariant when switching coordinate representations from rectangular to curvilinear coordinates, but is not invariant when transforming between frames of reference that are moving with respect to each other. Other quantities, like the speed of light, are always invariant.


Physical laws are said to be invariant under transformations when their predictions remain unchanged. This generally means that the form of the law (e.g. the type of differential equations used to describe the law) is unchanged in transformations so that no additional or different solutions are obtained.


Covariance and contravariance generalize the mathematical properties of invariance in tensor mathematics, and are frequently used in electromagnetism, special relativity, and general relativity.


In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation.


A family of particular transformations may be "continuous" (such as rotation of a circle) or "discrete" (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see "Symmetry group").


These two concepts, Lie and finite groups, are the foundation for the fundamental theories of modern physics. Symmetries are frequently amenable to mathematical formulations such as group representations and can, in addition, be exploited to simplify many problems.


Arguably the most important example of a symmetry in physics is that the speed of light has the same value in all frames of reference, which is known in mathematical terms as the Poincaré group, the symmetry group of special relativity. Another important example is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations, which is an important idea in general relativity.


Invariance is specified mathematically by transformations that leave some property (e.g. quantity) unchanged. This idea can apply to basic real-world observations. For example, temperature may be homogeneous throughout a room. Since the temperature does not depend on the position of an observer within the room, we say that the temperature is "invariant" under a shift in an observer's position within the room.


Similarly, a uniform sphere rotated about its center will appear exactly as it did before the rotation. The sphere is said to exhibit spherical symmetry. A rotation about any axis of the sphere will preserve how the sphere "looks".


The above ideas lead to the useful idea of "invariance" when discussing observed physical symmetry; this can be applied to symmetries in forces as well.


For example, an electric field due to an electrically charged wire of infinite length is said to exhibit cylindrical symmetry, because the electric field strength at a given distance "r" from the wire will have the same magnitude at each point on the surface of a cylinder (whose axis is the wire) with radius "r". Rotating the wire about its own axis does not change its position or charge density, hence it will preserve the field. The field strength at a rotated position is the same. This is not true in general for an arbitrary system of charges.


In Newton's theory of mechanics, given two bodies, each with mass "m", starting at the origin and moving along the "x"-axis in opposite directions, one with speed "v"1 and the other with speed "v"2 the total kinetic energy of the system (as calculated from an observer at the origin) is and remains the same if the velocities are interchanged. The total kinetic energy is preserved under a reflection in the "y"-axis.


The last example above illustrates another way of expressing symmetries, namely through the equations that describe some aspect of the physical system. The above example shows that the total kinetic energy will be the same if "v"1 and "v"2 are interchanged.


Symmetries may be broadly classified as "global" or "local". A "global symmetry" is one that keeps a property invariant for a transformation that is applied simultaneously at all points of spacetime, whereas a "local symmetry" is one that keeps a property invariant when a possibly different symmetry transformation is applied at each point of spacetime; specifically a local symmetry transformation is parameterised by the spacetime co-ordinates, whereas a global symmetry is not. This implies that a global symmetry is also a local symmetry. Local symmetries play an important role in physics as they form the basis for gauge theories.


The two examples of rotational symmetry described above – spherical and cylindrical – are each instances of continuous symmetry. These are characterised by invariance following a continuous change in the geometry of the system. For example, the wire may be rotated through any angle about its axis and the field strength will be the same on a given cylinder. Mathematically, continuous symmetries are described by transformations that change continuously as a function of their parameterization. An important subclass of continuous symmetries in physics are spacetime symmetries.


Continuous "spacetime symmetries" are symmetries involving transformations of space and time. These may be further classified as "spatial symmetries", involving only the spatial geometry associated with a physical system; "temporal symmetries", involving only changes in time; or "spatio-temporal symmetries", involving changes in both space and time.


Mathematically, spacetime symmetries are usually described by smooth vector fields on a smooth manifold. The underlying local diffeomorphisms associated with the vector fields correspond more directly to the physical symmetries, but the vector fields themselves are more often used when classifying the symmetries of the physical system.


Some of the most important vector fields are Killing vector fields which are those spacetime symmetries that preserve the underlying metric structure of a manifold. In rough terms, Killing vector fields preserve the distance between any two points of the manifold and often go by the name of isometries.


A "discrete symmetry" is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance. Discrete symmetries sometimes involve some type of 'swapping', these swaps usually being called "reflections" or "interchanges".


The Standard Model of particle physics has three related natural near-symmetries. These state that the universe in which we live should be indistinguishable from one where a certain type of change is introduced.


These symmetries are near-symmetries because each is broken in the present-day universe. However, the Standard Model predicts that the combination of the three (that is, the simultaneous application of all three transformations) must be a symmetry, called CPT symmetry. CP violation, the violation of the combination of C- and P-symmetry, is necessary for the presence of significant amounts of baryonic matter in the universe. CP violation is a fruitful area of current research in particle physics.


A type of symmetry known as supersymmetry has been used to try to make theoretical advances in the Standard Model. Supersymmetry is based on the idea that there is another physical symmetry beyond those already developed in the Standard Model, specifically a symmetry between bosons and fermions. Supersymmetry asserts that each type of boson has, as a supersymmetric partner, a fermion, called a superpartner, and vice versa. Supersymmetry has not yet been experimentally verified: no known particle has the correct properties to be a superpartner of any other known particle. Currently LHC is preparing for a run which tests supersymmetry.


The transformations describing physical symmetries typically form a mathematical group. Group theory is an important area of mathematics for physicists.


Continuous symmetries are specified mathematically by "continuous groups" (called Lie groups). Many physical symmetries are isometries and are specified by symmetry groups. Sometimes this term is used for more general types of symmetries. The set of all proper rotations (about any angle) through any axis of a sphere form a Lie group called the special orthogonal group SO(3). (The '3' refers to the three-dimensional space of an ordinary sphere.) Thus, the symmetry group of the sphere with proper rotations is SO(3). Any rotation preserves distances on the surface of the ball. The set of all Lorentz transformations form a group called the Lorentz group (this may be generalised to the Poincaré group).


Discrete groups describe discrete symmetries. For example, the symmetries of an equilateral triangle are characterized by the symmetric group S.


A type of physical theory based on "local" symmetries is called a "gauge" theory and the symmetries natural to such a theory are called gauge symmetries. Gauge symmetries in the Standard Model, used to describe three of the fundamental interactions, are based on the SU(3) × SU(2) × U(1) group. (Roughly speaking, the symmetries of the SU(3) group describe the strong force, the SU(2) group describes the weak interaction and the U(1) group describes the electromagnetic force.)


Also, the reduction by symmetry of the energy functional under the action by a group and spontaneous symmetry breaking of transformations of symmetric groups appear to elucidate topics in particle physics (for example, the unification of electromagnetism and the weak force in physical cosmology).


The symmetry properties of a physical system are intimately related to the conservation laws characterizing that system. Noether's theorem gives a precise description of this relation. The theorem states that each continuous symmetry of a physical system implies that some physical property of that system is conserved. Conversely, each conserved quantity has a corresponding symmetry. For example, spatial translation symmetry (i.e. homogeneity of space) gives rise to conservation of (linear) momentum, and temporal translation symmetry (i.e. homogeneity of time) gives rise to conservation of energy.


The following table summarizes some fundamental symmetries and the associated conserved quantity.


Continuous symmetries in physics preserve transformations. One can specify a symmetry by showing how a very small transformation affects various particle fields. The commutator of two of these infinitesimal transformations are equivalent to a third infinitesimal transformation of the same kind hence they form a Lie algebra.


A general coordinate transformation described as the general field formula_6 (also known as a diffeomorphism) has the infinitesimal effect on a scalar formula_7, spinor formula_8 or vector field formula_9 that can be expressed (using the index summation convention):


Without gravity only the Poincaré symmetries are preserved which restricts formula_6 to be of the form:


where M is an antisymmetric matrix (giving the Lorentz and rotational symmetries) and P is a general vector (giving the translational symmetries). Other symmetries affect multiple fields simultaneously. For example, local gauge transformations apply to both a vector and spinor field:


where formula_17 are generators of a particular Lie group. So far the transformations on the right have only included fields of the same type. Supersymmetries are defined according to how the mix fields of "different" types.


Another symmetry which is part of some theories of physics and not in others is scale invariance which involve Weyl transformations of the following kind:


If the fields have this symmetry then it can be shown that the field theory is almost certainly conformally invariant also. This means that in the absence of gravity h(x) would restricted to the form:


with D generating scale transformations and K generating special conformal transformations. For example, super-Yang–Mills theory has this symmetry while general relativity doesn't although other theories of gravity such as conformal gravity do. The 'action' of a field theory is an invariant under all the symmetries of the theory. Much of modern theoretical physics is to do with speculating on the various symmetries the Universe may have and finding the invariants to construct field theories as models.


In string theories, since a string can be decomposed into an infinite number of particle fields, the symmetries on the string world sheet is equivalent to special transformations which mix an infinite number of fields.


There is a natural connection between particle physics and representation theory, as first noted in the 1930s by Eugene Wigner. It links the properties of elementary particles to the structure of Lie groups and Lie algebras. According to this connection, the different quantum states of an elementary particle give rise to an irreducible representation of the Poincaré group. Moreover, the properties of the various particles, including their spectra, can be related to representations of Lie algebras, corresponding to "approximate symmetries" of the universe.


By definition of a symmetry of a quantum system, there is a group action on formula_7. For each formula_12, there is a corresponding transformation formula_13 of formula_7. More specifically, if formula_15 is some symmetry of the system (say, rotation about the x-axis by 12°), then the corresponding transformation formula_13 of formula_7 is a map on ray space. For example, when rotating a "stationary" (zero momentum) spin-5 particle about its center, formula_15 is a rotation in 3D space (an element of formula_19), while formula_13 is an operator whose domain and range are each the space of possible quantum states of this particle, in this example the projective space formula_7 associated with an 11-dimensional complex Hilbert space formula_1.


Each map formula_13 preserves, by definition of symmetry, the ray product on formula_7 induced by the inner product on formula_1; according to Wigner's theorem, this transformation of formula_7 comes from a unitary or anti-unitary transformation formula_27 of formula_1. Note, however, that the formula_27 associated to a given formula_13 is not unique, but only unique "up to a phase factor". The composition of the operators formula_27 should, therefore, reflect the composition law in formula_4, but only up to a phase factor:


where formula_34 will depend on formula_15 and formula_36. Thus, the map sending formula_15 to formula_27 is a "projective unitary representation" of formula_4, or possibly a mixture of unitary and anti-unitary, if formula_4 is disconnected. In practice, anti-unitary operators are always associated with time-reversal symmetry.


It is important physically that in general formula_41 does not have to be an ordinary representation of formula_4; it may not be possible to choose the phase factors in the definition of formula_27 to eliminate the phase factors in their composition law. An electron, for example, is a spin-one-half particle; its Hilbert space consists of wave functions on formula_44 with values in a two-dimensional spinor space. The action of formula_19 on the spinor space is only projective: It does not come from an ordinary representation of formula_19. There is, however, an associated ordinary representation of the universal cover formula_47 of formula_19 on spinor space.


For many interesting classes of groups formula_4, tells us that every projective unitary representation of formula_4 comes from an ordinary representation of the universal cover formula_51 of formula_4. Actually, if formula_1 is finite dimensional, then regardless of the group formula_4, every projective unitary representation of formula_4 comes from an ordinary unitary representation of formula_51. If formula_1 is infinite dimensional, then to obtain the desired conclusion, some algebraic assumptions must be made on formula_4 (see below). In this setting the result is a theorem of Bargmann. Fortunately, in the crucial case of the Poincaré group, Bargmann's theorem applies. (See Wigner's classification of the representations of the universal cover of the Poincaré group.)


The requirement referred to above is that the Lie algebra formula_59 does not admit a nontrivial one-dimensional central extension. This is the case if and only if the second cohomology group of formula_59 is trivial. In this case, it may still be true that the group admits a central extension by a "discrete" group. But extensions of formula_4 by discrete groups are covers of formula_4. For instance, the universal cover formula_63 is related to formula_4 through the quotient formula_65 with the central subgroup formula_66 being the center of formula_63 itself, isomorphic to the fundamental group of the covered group.


Thus, in favorable cases, the quantum system will carry a unitary representation of the universal cover formula_51 of the symmetry group formula_4. This is desirable because formula_1 is much easier to work with than the non-vector space formula_7. If the representations of formula_51 can be classified, much more information about the possibilities and properties of formula_1 are available.


In this case, to obtain an ordinary representation, one has to pass to the Heisenberg group, which is a nontrivial one-dimensional central extension of formula_75.


The group of translations and Lorentz transformations form the Poincaré group, and this group should be a symmetry of a relativistic quantum system (neglecting general relativity effects, or in other words, in flat space). Representations of the Poincaré group are in many cases characterized by a nonnegative mass and a half-integer spin (see Wigner's classification); this can be thought of as the reason that particles have quantized spin. (Note that there are in fact other possible representations, such as tachyons, infraparticles, etc., which in some cases do not have quantized spin or fixed mass.)


While the spacetime symmetries in the Poincaré group are particularly easy to visualize and believe, there are also other types of symmetries, called internal symmetries. One example is color SU(3), an exact symmetry corresponding to the continuous interchange of the three quark colors.


Many (but not all) symmetries or approximate symmetries form Lie groups. Rather than study the representation theory of these Lie groups, it is often preferable to study the closely related representation theory of the corresponding Lie algebras, which are usually simpler to compute.


Now, representations of the Lie algebra correspond to representations of the universal cover of the original group. In the finite-dimensional case—and the infinite-dimensional case, provided that applies—irreducible projective representations of the original group correspond to ordinary unitary representations of the universal cover. In those cases, computing at the Lie algebra level is appropriate. This is the case, notably, for studying the irreducible projective representations of the rotation group SO(3). These are in one-to-one correspondence with the ordinary representations of the universal cover SU(2) of SO(3). The representations of the SU(2) are then in one-to-one correspondence with the representations of its Lie algebra su(2), which is isomorphic to the Lie algebra so(3) of SO(3).


Thus, to summarize, the irreducible projective representations of SO(3) are in one-to-one correspondence with the irreducible ordinary representations of its Lie algebra so(3). The two-dimensional "spin 1/2" representation of the Lie algebra so(3), for example, does not correspond to an ordinary (single-valued) representation of the group SO(3). (This fact is the origin of statements to the effect that "if you rotate the wave function of an electron by 360 degrees, you get the negative of the original wave function.") Nevertheless, the spin 1/2 representation does give rise to a well-defined "projective" representation of SO(3), which is all that is required physically.


Although the above symmetries are believed to be exact, other symmetries are only approximate.


As an example of what an approximate symmetry means, suppose an experimentalist lived inside an infinite ferromagnet, with magnetization in some particular direction. The experimentalist in this situation would find not one but two distinct types of electrons: one with spin along the direction of the magnetization, with a slightly lower energy (and consequently, a lower mass), and one with spin anti-aligned, with a higher mass. Our usual SO(3) rotational symmetry, which ordinarily connects the spin-up electron with the spin-down electron, has in this hypothetical case become only an "approximate" symmetry, relating "different types of particles" to each other.


In general, an approximate symmetry arises when there are very strong interactions that obey that symmetry, along with weaker interactions that do not. In the electron example above, the two "types" of electrons behave identically under the strong and weak forces, but differently under the electromagnetic force.


An example from the real world is isospin symmetry, an SU(2) group corresponding to the similarity between up quarks and down quarks. This is an approximate symmetry: While up and down quarks are identical in how they interact under the strong force, they have different masses and different electroweak interactions. Mathematically, there is an abstract two-dimensional vector space


and the laws of physics are "approximately" invariant under applying a determinant-1 unitary transformation to this space:


For example, formula_85 would turn all up quarks in the universe into down quarks and vice versa. Some examples help clarify the possible effects of these transformations:


In general, particles form isospin multiplets, which correspond to irreducible representations of the Lie algebra SU(2). Particles in an isospin multiplet have very similar but not identical masses, because the up and down quarks are very similar but not identical.


Isospin symmetry can be generalized to flavour symmetry, an SU(3) group corresponding to the similarity between up quarks, down quarks, and strange quarks. This is, again, an approximate symmetry, violated by quark mass differences and electroweak interactions—in fact, it is a poorer approximation than isospin, because of the strange quark's noticeably higher mass.


Nevertheless, particles can indeed be neatly divided into groups that form irreducible representations of the Lie algebra SU(3), as first noted by Murray Gell-Mann and independently by Yuval Ne'eman.


In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by a: "T"a(p) = p + a.


In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation. Discrete translational symmetry is invariant under discrete translation.


Analogously an operator "A" on functions is said to be translationally invariant with respect to a translation operator formula_1 if the result after applying "A" doesn't change if the argument function is translated.


Laws of physics are translationally invariant under a spatial translation if they do not distinguish different points in space. According to Noether's theorem, space translational symmetry of a physical system is equivalent to the momentum conservation law.


Translational symmetry of an object means that a particular translation does not change the object. For a given object, the translations for which this applies form a group, the symmetry group of the object, or, if the object has more kinds of symmetry, a subgroup of the symmetry group.


Translational invariance implies that, at least in one direction, the object is infinite: for any given point p, the set of points with the same properties due to the translational symmetry form the infinite discrete set {p + "n"a | "n" ∈ Z} = p + Z a. Fundamental domains are e.g. H + [0, 1] a for any hyperplane H for which a has an independent direction. This is in 1D a line segment, in 2D an infinite strip, and in 3D a slab, such that the vector starting at one side ends at the other side. Note that the strip and slab need not be perpendicular to the vector, hence can be narrower or thinner than the length of the vector.


In spaces with dimension higher than 1, there may be multiple translational symmetry. For each set of "k" independent translation vectors, the symmetry group is isomorphic with Z"k".


In particular, the multiplicity may be equal to the dimension. This implies that the object is infinite in all directions. In this case, the set of all translations forms a lattice. Different bases of translation vectors generate the same lattice if and only if one is transformed into the other by a matrix of integer coefficients of which the absolute value of the determinant is 1. The absolute value of the determinant of the matrix formed by a set of translation vectors is the hypervolume of the "n"-dimensional parallelepiped the set subtends (also called the "covolume" of the lattice). This parallelepiped is a fundamental region of the symmetry: any pattern on or in it is possible, and this defines the whole object.


Alternatively, e.g. a rectangle may define the whole object, even if the translation vectors are not perpendicular, if it has two sides parallel to one translation vector, while the other translation vector starting at one side of the rectangle ends at the opposite side.


For example, consider a tiling with equal rectangular tiles with an asymmetric pattern on them, all oriented the same, in rows, with for each row a shift of a fraction, not one half, of a tile, always the same, then we have only translational symmetry, wallpaper group "p"1 (the same applies without shift). With rotational symmetry of order two of the pattern on the tile we have "p"2 (more symmetry of the pattern on the tile does not change that, because of the arrangement of the tiles). The rectangle is a more convenient unit to consider as fundamental domain (or set of two of them) than a parallelogram consisting of part of a tile and part of another one.


In 2D there may be translational symmetry in one direction for vectors of any length. One line, not in the same direction, fully defines the whole object. Similarly, in 3D there may be translational symmetry in one or two directions for vectors of any length. One plane (cross-section) or line, respectively, fully defines the whole object.


An example of translational symmetry in one direction in 2D nr. 1) is:


Note: The example is not an example of rotational symmetry.


(get the same by moving one line down and two positions to the right), and of translational symmetry in two directions in 2D (wallpaper group p1):


(get the same by moving three positions to the right, or one line down and two positions to the right; consequently get also the same moving three lines down).


In both cases there is neither mirror-image symmetry nor rotational symmetry.


For a given translation of space we can consider the corresponding translation of objects. The objects with at least the corresponding translational symmetry are the fixed points of the latter, not to be confused with fixed points of the translation of space, which are non-existent.


In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass cannot change, so quantity can neither be added nor be removed. Therefore, the quantity of mass is conserved over time.


The law implies that mass can neither be created nor destroyed, although it may be rearranged in space, or the entities associated with it may be changed in form. For example, in chemical reactions, the mass of the chemical components before the reaction is equal to the mass of the components after the reaction. Thus, during any chemical reaction and low-energy thermodynamic processes in an isolated system, the total mass of the reactants, or starting materials, must be equal to the mass of the products.


The concept of mass conservation is widely used in many fields such as chemistry, mechanics, and fluid dynamics. Historically, mass conservation was demonstrated in chemical reactions independently by Mikhail Lomonosov and later rediscovered by Antoine Lavoisier in the late 18th century. The formulation of this law was of crucial importance in the progress from alchemy to the modern natural science of chemistry.


The conservation of mass only holds approximately and is considered part of a series of assumptions coming from classical mechanics. The law has to be modified to comply with the laws of quantum mechanics and special relativity under the principle of mass-energy equivalence, which states that energy and mass form one conserved quantity. For very energetic systems the conservation of mass-only is shown not to hold, as is the case in nuclear reactions and particle-antiparticle annihilation in particle physics.


Mass is also not generally conserved in open systems. Such is the case when various forms of energy and matter are allowed into, or out of, the system. However, unless radioactivity or nuclear reactions are involved, the amount of energy escaping (or entering) such systems as heat, mechanical work, or electromagnetic radiation is usually too small to be measured as a decrease (or increase) in the mass of the system.


For systems where large gravitational fields are involved, general relativity has to be taken into account, where mass-energy conservation becomes a more complex concept, subject to different definitions, and neither mass nor energy is as strictly and simply conserved as is the case in special relativity.


The law of conservation of mass can only be formulated in classical mechanics when the energy scales associated to an isolated system are much smaller than formula_1, where formula_2 is the mass of a typical object in the system, measured in the frame of reference where the object is at rest, and formula_3 is the speed of light.


The law can be formulated mathematically in the fields of fluid mechanics and continuum mechanics, where the conservation of mass is usually expressed using the continuity equation, given in differential form asformula_4where formula_5 is the density (mass per unit volume), formula_6 is the time, formula_7 is the divergence, and formula_8 is the flow velocity field.


The interpretation of the continuity equation for mass is the following: For a given closed surface in the system, the change in time of the mass enclosed by the surface is equal to the mass that traverses the surface, positive if matter goes in and negative if matter goes out. For the whole isolated system, this condition implies that the total mass formula_9, sum of the masses of all components in the system, does not change in time, i.e. formula_10,where formula_11 is the differential that defines the integral over the whole volume of the system.


The continuity equation for the mass is part of Euler equations of fluid dynamics. Many other convection–diffusion equations describe the conservation and flow of mass and matter in a given system.


In chemistry, the calculation of the amount of reactant and products in a chemical reaction, or stoichiometry, is founded on the principle of conservation of mass. The principle implies that during a chemical reaction the total mass of the reactants is equal to the total mass of the products. For example, in the following reaction


where one molecule of methane () and two oxygen molecules are converted into one molecule of carbon dioxide () and two of water (). The number of molecules as result from the reaction can be derived from the principle of conservation of mass, as initially four hydrogen atoms, 4 oxygen atoms and one carbon atom are present (as well as in the final state), then the number water molecules produced must be exactly two per molecule of carbon dioxide produced.


Many engineering problems are solved by following the mass distribution in time of a given system, this practice is known as mass balance.


An important idea in ancient Greek philosophy was that "Nothing comes from nothing", so that what exists now has always existed: no new matter can come into existence where there was none before. An explicit statement of this, along with the further principle that nothing can pass away into nothing, is found in Empedocles (c.4th century BC): "For it is impossible for anything to come to be from what is not, and it cannot be brought about or heard of that what is should be utterly destroyed."


A further principle of conservation was stated by Epicurus around 3rd century BC, who, describing the nature of the Universe, wrote that "the totality of things was always such as it is now, and always will be".


Jain philosophy, a non-creationist philosophy based on the teachings of Mahavira (6th century BC), states that the universe and its constituents such as matter cannot be destroyed or created. The Jain text Tattvarthasutra (2nd century AD) states that a substance is permanent, but its modes are characterised by creation and destruction. A principle of the conservation of matter was also stated by Nasīr al-Dīn al-Tūsī (around 13th century AD). He wrote that "A body of matter cannot disappear completely. It only changes its form, condition, composition, color and other properties and turns into a different complex or elementary matter".


The conservation of mass was obscure for millennia because of the buoyancy effect of the Earth's atmosphere on the weight of gases. For example, a piece of wood weighs less after burning; this seemed to suggest that some of its mass disappears, or is transformed or lost. This was not disproved until careful experiments were performed in which chemical reactions such as rusting were allowed to take place in sealed glass ampoules; it was found that the chemical reaction did not change the weight of the sealed container and its contents. Weighing of gases using scales was not possible until the invention of the vacuum pump in 17th century.


Once understood, the conservation of mass was of great importance in progressing from alchemy to modern chemistry. Once early chemists realized that chemical substances never disappeared but were only transformed into other substances with the same weight, these scientists could for the first time embark on quantitative studies of the transformations of substances. The idea of mass conservation plus a surmise that certain "elemental substances" also could not be transformed into others by chemical reactions, in turn led to an understanding of chemical elements, as well as the idea that all chemical processes and transformations (such as burning and metabolic reactions) are reactions between invariant amounts or weights of these chemical elements.


Following the pioneering work of Lavoisier, the exhaustive experiments of Jean Stas supported the consistency of this law in chemical reactions, even though they were carried out with other intentions. His research indicated that in certain reactions the loss or gain could not have been more than from 2 to 4 parts in 100,000. The difference in the accuracy aimed at and attained by Lavoisier on the one hand, and by Morley and Stas on the other, is enormous.


The law conservation of mass and the analogous law of conservation of energy were finally overruled by a more general principle known as the mass–energy equivalence. Special relativity also redefines the concept of mass and energy, which can be used interchangeably and are relative to the frame of reference. Several definitions had to be defined for consistency like "rest mass" of a particle (mass in the rest frame of the particle) and "relativistic mass" (in another frame). The latter term is usually less frequently used.


In special relativity, the conservation of mass does not apply if the system is open and energy escapes. However, it does continue to apply to totally closed (isolated) systems. If energy cannot escape a system, its mass cannot decrease. In relativity theory, so long as any type of energy is retained within a system, this energy exhibits mass.


Also, mass must be differentiated from matter, since matter may "not" be perfectly conserved in isolated systems, even though mass is always conserved in such systems. However, matter is so nearly conserved in chemistry that violations of matter conservation were not measured until the nuclear age, and the assumption of matter conservation remains an important practical concept in most systems in chemistry and other studies that do not involve the high energies typical of radioactivity and nuclear reactions.


The mass associated with chemical amounts of energy is too small to measure.


The change in mass of certain kinds of open systems where atoms or massive particles are not allowed to escape, but other types of energy (such as light or heat) are allowed to enter, escape or be merged, went unnoticed during the 19th century, because the change in mass associated with addition or loss of small quantities of thermal or radiant energy in chemical reactions is very small. (In theory, mass would not change at all for experiments conducted in isolated systems where heat and work were not allowed in or out.)


Mass conservation remains correct if energy is not lost.


The conservation of relativistic mass implies the viewpoint of a single observer (or the view from a single inertial frame) since changing inertial frames may result in a change of the total energy (relativistic energy) for systems, and this quantity determines the relativistic mass.


The principle that the mass of a system of particles must be equal to the sum of their rest masses, even though true in classical physics, may be false in special relativity. The reason that rest masses cannot be simply added is that this does not take into account other forms of energy, such as kinetic and potential energy, and massless particles such as photons, all of which may (or may not) affect the total mass of systems.


For moving massive particles in a system, examining the rest masses of the various particles also amounts to introducing many different inertial observation frames (which is prohibited if total system energy and momentum are to be conserved), and also when in the rest frame of one particle, this procedure ignores the momenta of other particles, which affect the system mass if the other particles are in motion in this frame.


The conservation of both relativistic and invariant mass applies even to systems of particles created by pair production, where energy for new particles may come from kinetic energy of other particles, or from one or more photons as part of a system that includes other particles besides a photon. Again, neither the relativistic nor the invariant mass of totally closed (that is, isolated) systems changes when new particles are created. However, different inertial observers will disagree on the value of this conserved mass, if it is the relativistic mass (i.e., relativistic mass is conserved but not invariant). However, all observers agree on the value of the conserved mass if the mass being measured is the invariant mass (i.e., invariant mass is both conserved and invariant).


The mass-energy equivalence formula gives a different prediction in non-isolated systems, since if energy is allowed to escape a system, both relativistic mass and invariant mass will escape also. In this case, the mass-energy equivalence formula predicts that the "change" in mass of a system is associated with the "change" in its energy due to energy being added or subtracted: formula_12 This form involving changes was the form in which this famous equation was originally presented by Einstein. In this sense, mass changes in any system are explained simply if the mass of the energy added or removed from the system, are taken into account.


In general relativity, the total invariant mass of photons in an expanding volume of space will decrease, due to the red shift of such an expansion. The conservation of both mass and energy therefore depends on various corrections made to energy in the theory, due to the changing gravitational potential energy of such systems.


The groundwater energy balance is the energy balance of a groundwater body in terms of incoming hydraulic energy associated with groundwater inflow into the body, energy associated with the outflow, energy conversion into heat due to friction of flow, and the resulting change of energy status and groundwater level.


When multiplying the horizontal velocity of groundwater (dimension, for example, m3/day per m2 cross-sectional area) with the groundwater potential (dimension energy per m3 water, or "E"/m3) one obtains an energy flow (flux) in "E"/day per m2 cross-sectional area.


Summation or integration of the energy flux in a vertical cross-section of unit width (say 1 m) from the lower flow boundary (the impermeable layer or base) up to the water table in an unconfined aquifer gives the energy flow "Fe" through the cross-section in "E"/day per m width of the aquifer.


While flowing, the groundwater loses energy due to friction of flow, i.e. hydraulic energy is converted into heat. At the same time, energy may be added with the recharge of water coming into the aquifer through the water table. Thus one can make an hydraulic energy balance of a block of soil between two nearby cross-sections. In steady state, i.e. without change in energy status and without accumulation or depletion of water stored in the soil body, the energy flow in the first section plus the energy added by groundwater recharge between the sections minus the energy flow in the second section must equal the energy loss due to friction of flow.


In mathematical terms this balance can be obtained by differentiating the cross-sectional integral of "Fe" in the direction of flow using the Leibniz rule, taking into account that the level of the water table may change in the direction of flow.


The mathematics is simplified using the Dupuit–Forchheimer assumption.


The hydraulic friction losses can be described in analogy to "Joule's law" in electricity (see Joule's law#Hydraulic equivalent), where the friction losses are proportional to the square value of the current (flow) and the electrical resistance of the material through which the current occurs. In groundwater hydraulics (fluid dynamics, hydrodynamics) one often works with hydraulic conductivity (i.e. permeability of the soil for water), which is inversely proportional to the hydraulic resistance.


The trial and error procedure is cumbersome and therefore computer programs may be developed to aid in the calculations.


The energy balance of groundwater flow can be applied to flow of groundwater to subsurface drains. The computer program "EnDrain" compares the outcome of the traditional drain spacing equation, based on Darcy's law together with the continuity equation (i.e. conservation of mass), with the solution obtained by the energy balance and it can be seen that drain spacings are wider in the latter case. This is owing to the introduction of the energy supplied by the incoming recharge.


The Carter constant is a conserved quantity for motion around black holes in the general relativistic formulation of gravity. Carter's constant was derived for a spinning, charged black hole by Australian theoretical physicist Brandon Carter in 1968. Carter's constant along with the energy, axial angular momentum, and particle rest mass provide the four conserved quantities necessary to uniquely determine all orbits in the Kerr–Newman spacetime (even those of charged particles).


Carter noticed that the Hamiltonian for motion in Kerr spacetime was separable in Boyer–Lindquist coordinates, allowing the constants of such motion to be easily identified using Hamilton-Jacobi theory. The Carter constant can be written as follows:


where formula_2 is the latitudinal component of the particle's angular momentum, formula_3 is the energy of the particle, formula_4 is the particle's axial angular momentum, formula_5 is the rest mass of the particle, and formula_6 is the spin parameter of the black hole. Because functions of conserved quantities are also conserved, any function of formula_7 and the three other constants of the motion can be used as a fourth constant in place of formula_7. This results in some confusion as to the form of Carter's constant. For example it is sometimes more convenient to use:


in place of formula_7. The quantity formula_11 is useful because it is always non-negative. In general any fourth conserved quantity for motion in the Kerr family of spacetimes may be referred to as "Carter's constant".


Noether's theorem states that all conserved quantities are related to spacetime symmetries. Carter's constant is related to a higher order symmetry of the Kerr metric generated by a second order Killing tensor field formula_11 (different formula_11 than used above). In component form:


where formula_15 is the four-velocity of the particle in motion. The components of the Killing tensor in Boyer–Lindquist coordinates are:


where formula_17 are the components of the metric tensor and formula_18 and formula_19 are the components of the principal null vectors:


The spherical symmetry of the Schwarzschild metric for non-spinning black holes allows one to reduce the problem of finding the trajectories of particles to three dimensions. In this case one only needs formula_3, formula_4, and formula_5 to determine the motion; however, the symmetry leading to Carter's constant still exists. Carter's constant for Schwarzschild space is:


By a rotation of coordinates we can put any orbit in the formula_27 plane so formula_28. In this case formula_29, the square of the orbital angular momentum.


The Bohr–Kramers–Slater theory (BKS theory) was perhaps the final attempt at understanding the interaction of matter and electromagnetic radiation on the basis of the so-called old quantum theory, in which quantum phenomena are treated by imposing quantum restrictions on classically describable behaviour. It was advanced in 1924, and sticks to a "classical" wave description of the electromagnetic field. It was perhaps more a research program than a full physical theory, the ideas that are developed not being worked out in a quantitative way.


One aspect, the idea of modelling atomic behaviour under incident electromagnetic radiation using "virtual oscillators" at the absorption and emission frequencies, rather than the (different) apparent frequencies of the Bohr orbits, significantly led Born, Heisenberg and Kramers to explore mathematics that strongly inspired the subsequent development of matrix mechanics, the first form of modern quantum mechanics. The provocativeness of the theory also generated great discussion and renewed attention to the difficulties in the foundations of the old quantum theory. However, physically the most provocative element of the theory, that momentum and energy would not necessarily be conserved in each interaction but only overall, statistically, was soon shown to be in conflict with experiment.


The initial idea of the BKS theory originated with Slater, who proposed to Bohr and Kramers the following elements of a theory of emission and absorption of radiation by atoms, to be developed during his stay in Copenhagen:


Slater's main intention seems to have been to reconcile the two conflicting models of radiation, viz. the wave and particle models. He may have had good hopes that his idea with respect to oscillators vibrating at the "differences" of the frequencies of electron rotations (rather than at the rotation frequencies themselves) might be attractive to Bohr because it solved a problem of the latter's atomic model, even though the physical meaning of these oscillators was far from clear. Nevertheless, Bohr and Kramers had two objections to Slater's proposal:


As Max Jammer puts it, this refocussed the theory "to harmonize the physical picture of the continuous electromagnetic field with the physical picture, not as Slater had proposed of light quanta, but of the discontinuous quantum transitions in the atom." Bohr and Kramers hoped to be able to evade the photon hypothesis on the basis of ongoing work by Kramers to describe "dispersion" (in present-day terms inelastic scattering) of light by means of a classical theory of interaction of radiation and matter. But abandoning the concept of the photon, they instead chose to squarely accept the possibility of non-conservation of energy, and momentum.


In particle physics, flavour or flavor refers to the "species" of an elementary particle. The Standard Model counts six flavours of quarks and six flavours of leptons. They are conventionally parameterized with "flavour quantum numbers" that are assigned to all subatomic particles. They can also be described by some of the family symmetries proposed for the quark-lepton generations.