KAHIBARO
Table of Contents
Mirror symmetry in physics
Parity is a symmetry connected with left and right. It asks a simple question, what happens if we reflect a physical situation as if we were looking at it in a mirror?
A mirror reflection changes the sign of spatial coordinates. In three dimensions, the parity transformation is
$$
(x,y,z) \rightarrow (-x,-y,-z)
$$
This operation is often written as $P$. If a law of physics gives exactly the same predictions before and after this transformation, that law is said to conserve parity.
Parity is about spatial inversion only. It does not reverse the flow of time, and it does not turn particles into antiparticles. Those belong to other symmetry operations.
What parity does to position and motion
The clearest way to understand parity is to see what it does to ordinary vectors. Position is a vector, so under parity,
$$
\vec{r} \rightarrow -\vec{r}
$$
Velocity and momentum also change sign because they point in space:
$$
\vec{v} \rightarrow -\vec{v}, \qquad \vec{p} \rightarrow -\vec{p}
$$
Force, if it behaves like an ordinary vector, also changes sign:
$$
\vec{F} \rightarrow -\vec{F}
$$
These are called polar vectors or true vectors.
A mirror image of a moving ball shows the ball moving in the mirrored direction. That is exactly what the sign change means.
Scalars, vectors, and pseudovectors
Not every quantity changes under parity in the same way. Scalars, which have magnitude but no direction, usually remain unchanged. For example, mass and electric charge do not change under a spatial inversion:
$$
m \rightarrow m, \qquad q \rightarrow q
$$
But there is another important class of quantities, called pseudovectors or axial vectors. These behave differently from ordinary vectors under parity. A famous example is angular momentum:
$$
\vec{L} = \vec{r} \times \vec{p}
$$
Under parity, both $\vec{r}$ and $\vec{p}$ change sign, so
$$
\vec{L} \rightarrow (-\vec{r}) \times (-\vec{p}) = \vec{r} \times \vec{p} = \vec{L}
$$
So angular momentum does not change sign under parity. Spin behaves in the same way.
This difference between vectors and pseudovectors is one of the key ideas in parity.
| Quantity | Type | Under parity |
|---|---|---|
| Position $\vec r$ | Vector | $-\vec r$ |
| Velocity $\vec v$ | Vector | $-\vec v$ |
| Momentum $\vec p$ | Vector | $-\vec p$ |
| Force $\vec F$ | Vector | $-\vec F$ |
| Mass $m$ | Scalar | $m$ |
| Charge $q$ | Scalar | $q$ |
| Angular momentum $\vec L$ | Pseudovector | $\vec L$ |
| Spin $\vec S$ | Pseudovector | $\vec S$ |
Important rule: ordinary vectors reverse sign under parity, scalars stay the same, and pseudovectors do not reverse sign.
Parity and physical laws
A physical law conserves parity if its mathematical form is unchanged when all spatial coordinates are inverted.
As a simple example, consider Newton's second law:
$$
\vec{F} = m\vec{a}
$$
Under parity, both $\vec{F}$ and $\vec{a}$ change sign, while $m$ stays the same. So the equation becomes
$$
-\vec{F} = m(-\vec{a})
$$
which is equivalent to the original equation. So this law is parity symmetric.
Many classical laws and many electromagnetic and strong interaction processes are invariant under parity. For a long time, physicists believed parity might be a universal symmetry of nature.
Intrinsic parity
In particle physics, parity is not only about coordinates. Particles can also be assigned an intrinsic parity, which is a built in parity label. This is usually written as either $+1$ or $-1$.
If a particle state has parity eigenvalue $P$, then under the parity operation the state changes by that factor:
$$
\hat{P}\,|\psi\rangle = \pm |\psi\rangle
$$
A state with parity $+1$ is called even under parity. A state with parity $-1$ is called odd under parity.
For systems of particles, the total parity depends on both the intrinsic parities of the particles and the orbital angular momentum. For a two particle system, the spatial part contributes a factor
$$
(-1)^L
$$
where $L$ is the orbital angular momentum quantum number.
So the total parity is often written as
$$
P_{\text{total}} = P_1 P_2 (-1)^L
$$
This formula is very useful in analyzing nuclear and particle reactions.
For a two particle state, an important parity rule is
$$
P_{\text{total}} = P_1 P_2 (-1)^L
$$
where $P_1$ and $P_2$ are intrinsic parities and $L$ is orbital angular momentum.
Even and odd spatial wave functions
In quantum mechanics, parity acts on the spatial wave function by reversing position:
$$
\psi(\vec r) \rightarrow \psi(-\vec r)
$$
A wave function has definite parity if
$$
\psi(-\vec r) = \pm \psi(\vec r)
$$
If
$$
\psi(-\vec r) = \psi(\vec r)
$$
the state is even. If
$$
\psi(-\vec r) = -\psi(\vec r)
$$
the state is odd.
This idea is especially useful when the potential is symmetric in space. Then parity can help classify allowed states.
Parity conservation and parity violation
If parity is conserved in an interaction, the mirror image process is equally allowed and follows the same rules.
However, parity is not always conserved in nature. The weak interaction famously violates parity. This means some weak processes distinguish between left and right in a fundamental way.
This discovery was one of the great surprises of twentieth century physics. It showed that the laws of nature are not always mirror symmetric.
A famous example is beta decay. In weak decays, emitted particles can prefer a particular handedness or direction relative to spin. The mirror image process is not physically identical in the same way it would be for a parity conserving interaction.
Parity is conserved in many classical, electromagnetic, and strong interaction processes, but it is violated by the weak interaction.
Handedness and why parity matters
Parity is closely related to handedness. Your left hand and right hand are mirror images, but they cannot be made identical by simple rotation. If a physical process treats left and right differently, parity is violated.
This matters deeply in nuclear and particle physics because symmetry rules help determine which reactions are allowed, which decays can occur, and how quantum states are classified. Parity gives a compact way to describe mirror behavior of physical systems.
Summary
Parity is the symmetry of spatial inversion,
$$
(x,y,z) \rightarrow (-x,-y,-z)
$$
Ordinary vectors change sign, scalars stay unchanged, and pseudovectors such as angular momentum do not change sign. In quantum physics, states can be even or odd under parity, and particle systems have total parity determined by intrinsic parity and orbital motion. Parity is an important organizing principle, but unlike some other conservation ideas, it is not always respected by nature, because the weak interaction violates it.