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Text of the page (random words):
navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 simple symmetry relations toggle simple symmetry relations subsection 1 1 representations of o 3 2 classical mechanics 3 effect of spatial inversion on some variables of classical physics toggle effect of spatial inversion on some variables of classical physics subsection 3 1 odd 3 2 even 4 quantum mechanics toggle quantum mechanics subsection 4 1 possible eigenvalues 4 2 consequences of parity symmetry 5 many particle systems atoms molecules nuclei toggle many particle systems atoms molecules nuclei subsection 5 1 atoms 5 2 molecules 5 3 nuclei 6 quantum field theory 7 parity in the standard model toggle parity in the standard model subsection 7 1 fixing the global symmetries 7 2 parity of the pion 7 3 parity violation 7 4 intrinsic parity of hadrons 8 see also 9 references toggle references subsection 9 1 sources toggle the table of contents parity physics 37 languages العربية беларуская català čeština чӑвашла dansk deutsch español eesti فارسی suomi français gaeilge ગુજરાતી हिन्दी hrvatski magyar հայերեն italiano 日本語 한국어 मराठी nederlands norsk nynorsk norsk bokmål ਪੰਜਾਬੀ polski português русский संस्कृतम् simple english slovenščina svenska türkçe татарча tatarça українська 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia symmetry of spatially mirrored systems for the concept in nuclear weapons see nuclear parity in physics a parity transformation also called parity inversion is the flip in the sign of one spatial coordinate in three dimensions it can also refer to the simultaneous flip in the sign of all three spatial coordinates a point reflection or point inversion p x y z x y z displaystyle mathbf p begin pmatrix x y z end pmatrix mapsto begin pmatrix x y z end pmatrix it can also be thought of as a test for chirality of a physical phenomenon in that a parity inversion transforms a phenomenon into its mirror image all fundamental interactions of elementary particles with the exception of the weak interaction are symmetric under parity transformation as established by the wu experiment conducted at the us national bureau of standards by chinese american scientist chien shiung wu the weak interaction is chiral and thus provides a means for probing chirality in physics in her experiment wu took advantage of the controlling role of weak interactions in radioactive decay of atomic isotopes to establish the chirality of the weak force by contrast in interactions that are symmetric under parity such as electromagnetism in atomic and molecular physics parity serves as a powerful controlling principle underlying quantum transitions a matrix representation of p in any number of dimensions has determinant equal to 1 and hence is distinct from a rotation which has a determinant equal to 1 in a two dimensional plane a simultaneous flip of all coordinates in sign is not a parity transformation it is the same as a 180 rotation in quantum mechanics wave functions that are unchanged by a parity transformation are described as even functions while those that change sign under a parity transformation are odd functions simple symmetry relations edit source see also representation theory of su 2 under rotations classical geometrical objects can be classified into scalars vectors and other tensors of rank 2 or higher technically scalars such as electrical potential are tensors of rank 0 vectors are tensors of rank 1 tensors of rank 2 or higher are called tensors both in the generic sense as are scalars and vectors and in the specific sense of being order higher than vectors at the low energy quantum level gravitational force is a second order tensor in classical physics physical configurations need to transform under representations of every symmetry group quantum theory predicts that states in a hilbert space do not need to transform under representations of the group of rotations but only under projective representations the word projective refers to the fact that if one projects out the phase of each state where we recall that the overall phase of a quantum state is not observable then a projective representation reduces to an ordinary representation all representations are also projective representations but the converse is not true therefore the projective representation condition on quantum states is weaker than the representation condition on classical states 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 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 scalars p 1 and pseudoscalars p 1 which are rotationally invariant vectors p 1 and axial vectors also called pseudovectors p 1 which both transform as vectors under rotation one can define reflections such as v x x y z x y z displaystyle v_ x begin pmatrix x y z end pmatrix mapsto begin pmatrix x y z end pmatrix 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 z 2 displaystyle mathbb z _ 2 due to the relation p 2 1 displaystyle hat mathcal p 2 hat 1 all abelian groups have only one dimensional irreducible representations for z 2 displaystyle mathbb z _ 2 there are two irreducible representations one is even under parity p ϕ ϕ displaystyle hat mathcal p phi phi the other is odd p ϕ ϕ displaystyle hat mathcal p phi phi these are useful in quantum mechanics however in quantum mechanics states need not transform under actual representations of parity but only under their projective representations so in principle a parity transformation may rotate a quantum state into any phase as elaborated on below representations of o 3 edit source an alternative way to write the above classification of scalars pseudoscalars vectors and pseudovectors is in terms of the representation space that each object transforms in this can be given in terms of the group homomorphism ρ displaystyle rho which defines the representation for a matrix r o 3 displaystyle r in text o 3 scalars ρ r 1 displaystyle rho r 1 the so called trivial representation pseudoscalars ρ r det r displaystyle rho r det r vectors ρ r r displaystyle rho r r the fundamental representation pseudovectors ρ r r det r displaystyle rho r r det r when the representation is restricted to s o 3 displaystyle mathsf so 3 both scalars and pseudoscalars transform identically as do vectors and pseudovectors classical mechanics edit source newton s equation of motion f m a displaystyle mathbf f m mathbf a 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 l displaystyle mathbf l is an axial vector l r p p l r p l displaystyle begin aligned mathbf l mathbf r times mathbf p hat p left mathbf l right mathbf r times mathbf p mathbf l end aligned in classical electrodynamics the charge density ρ displaystyle rho not related to the same symbol ρ used above for a group homomorphism is a scalar the electric field e displaystyle mathbf e and current j displaystyle mathbf j are vectors but the magnetic field b displaystyle mathbf b 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 edit source the two major divisions of classical physical variables have either even or odd parity the way into which particular variables and vectors sort out into either category depends on whether the number of dimensions of space is either an odd or even number the categories of odd or even given below for the parity transformation is a different but intimately related issue the answers given below are correct for 3 spatial dimensions in a 2 dimensional space for example when constrained to remain on the surface of a planet some of the variables switch sides odd edit source classical variables whose signs flip under space inversion are predominantly vectors they include h displaystyle h helicity x displaystyle mathbf x the position of a particle in three space v displaystyle mathbf v the velocity of a particle a displaystyle mathbf a the acceleration of the particle p displaystyle mathbf p the linear momentum of a particle ρ v displaystyle rho mathbf v mass flow a f displaystyle mathbf f the force exerted on a particle j displaystyle mathbf j electric current density e displaystyle mathbf e the electric field d displaystyle mathbf d the electric displacement field p displaystyle mathbf p electric polarization a displaystyle mathbf a electromagnetic vector potential s displaystyle mathbf s the poynting vector flow of electromagnetic power even edit source classical variables predominantly scalar quantities which do not change upon spatial inversion include t displaystyle t the time when an event occurs m displaystyle m the mass of a particle e displaystyle e the energy of a particle p displaystyle p power rate of work done ρ displaystyle rho electric charge density v displaystyle v scalar electric potential voltage ρ displaystyle rho energy density of the electromagnetic field l displaystyle mathbf l the angular momentum of a particle both orbital and spin axial vector b displaystyle mathbf b the magnetic field axial vector h displaystyle mathbf h the auxiliary magnetic field m displaystyle mathbf m magnetization t i j displaystyle t_ ij the maxwell stress tensor all masses charges coupling constants and other scalar physical constants except those associated with the weak force quantum mechanics edit source possible eigenvalues edit source two dimensional representations of parity are given by a pair of quantum states which go into each other under parity however this representation can always be reduced to linear combinations of states each of which is either even or odd under parity one says that all irreducible representations of parity are one dimensional in quantum mechanics spacetime transformations act on quantum states the parity transformation p displaystyle hat mathcal p is a unitary operator in general acting on a state ψ displaystyle psi as follows p ψ r e i ϕ 2 ψ r displaystyle hat mathcal p psi left r right e i phi 2 psi left r right one must then have p 2 ψ r e i ϕ ψ r displaystyle hat mathcal p 2 psi left r right e i phi psi left r right since an overall phase is unobservable the operator p 2 displaystyle hat mathcal p 2 which reverses the parity of a state twice leaves the spacetime invariant and so is an internal symmetry which rotates its eigenstates by phases e i ϕ displaystyle e i phi if p 2 displaystyle hat mathcal p 2 is an element e i q displaystyle e iq of a continuous u 1 symmetry group of phase rotations then e i q displaystyle e iq is part of this u 1 and so is also a symmetry in particular we can define p p e i q 2 displaystyle hat mathcal p equiv hat mathcal p e iq 2 which is also a symmetry and so we can choose to call p displaystyle hat mathcal p our parity operator instead of p displaystyle hat mathcal p note that p 2 1 displaystyle hat mathcal p 2 1 and so p displaystyle hat mathcal p has eigenvalues 1 displaystyle pm 1 wave functions with eigenvalue 1 displaystyle 1 under a parity transformation are even functions while eigenvalue 1 displaystyle 1 corresponds to odd functions 1 however when no such symmetry group exists it may be that all parity transformations have some eigenvalues which are phases other than 1 displaystyle pm 1 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 h 2 is labelled 1 σ g displaystyle 1 sigma _ g and the next closest higher energy level is labelled 1 σ u displaystyle 1 sigma _ u 2 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 3 the law of conservation of parity of particles states that if an isolated ensemble of particles has a definite parity then the parity remains invariable in the process of ensemble evolution however this is not true for the beta decay of nuclei because the weak nuclear interaction violates parity 4 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 3 consequences of parity symmetry edit source when parity generates the abelian group z 2 displaystyle mathbb z _ 2 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 p displaystyle hat mathcal p commutes with the hamiltonian in non relativistic quantum mechanics this happens for any scalar potential i e v v r displaystyle v v left r right hence the potential is spherically symmetric the following facts can be easily proven if φ displaystyle varphi rangle and ψ displaystyle psi rangle have the same parity t...