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):

$${\displaystyle \mathbf {P} :{\begin{pmatrix}x\\y\\z\end{pmatrix}}\mapsto {\begin{pmatrix}-x\\-y\\-z\end{pmatrix}}.}$$

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. 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

Under rotations, classical geometrical objects can be classified into scalars, vectors, and tensors of higher rank. 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

$${\displaystyle V_{x}:{\begin{pmatrix}x\\y\\z\end{pmatrix}}\mapsto {\begin{pmatrix}-x\\y\\z\end{pmatrix}},}$$

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 ${\displaystyle \mathbb {Z} _{2}}$ due to the relation ${\displaystyle {\hat {\mathcal {P}}}^{2}={\hat {1}}}$. All Abelian groups have only one-dimensional irreducible representations. For ${\displaystyle \mathbb {Z} _{2}}$, there are two irreducible representations: one is even under parity, ${\displaystyle {\hat {\mathcal {P}}}\phi =+\phi }$, the other is odd, ${\displaystyle {\hat {\mathcal {P}}}\phi =-\phi }$. 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.

Representations of O(3)

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 ${\displaystyle \rho }$ which defines the representation. For a matrix ${\displaystyle R\in {\text{O}}(3),}$

• scalars: ${\displaystyle \rho (R)=1}$, the trivial representation
• pseudoscalars: ${\displaystyle \rho (R)=\det(R)}$
• vectors: ${\displaystyle \rho (R)=R}$, the fundamental representation
• pseudovectors: ${\displaystyle \rho (R)=\det(R)R.}$

When the representation is restricted to ${\displaystyle {\text{SO}}(3)}$, scalars and pseudoscalars transform identically, as do vectors and pseudovectors.

Classical mechanics

Newton's equation of motion ${\displaystyle \mathbf {F} =m\mathbf {a} }$ (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 ${\displaystyle \mathbf {L} }$ is an axial vector,

$${\displaystyle {\begin{aligned}\mathbf {L} &=\mathbf {r} \times \mathbf {p} \\{\hat {P}}\left(\mathbf {L} \right)&=(-\mathbf {r} )\times (-\mathbf {p} )=\mathbf {L} .\end{aligned}}}$$

In classical electrodynamics, the charge density ${\displaystyle \rho }$ is a scalar, the electric field, ${\displaystyle \mathbf {E} }$, and current ${\displaystyle \mathbf {j} }$ are vectors, but the magnetic field, ${\displaystyle \mathbf {B} }$ 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

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

Classical variables whose signs flip when inverted in space inversion are predominantly vectors. They include:

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