Parity inversion, denoted by the operator $\mathcal{P}$, is a fundamental symmetry operation in physics (general) that corresponds to spatial inversion through the origin, transforming the coordinates of a point $(x, y, z)$ to $(-x, -y, -z)$. In quantum mechanics, the parity operator acts on a state vector $|\psi\rangle$ such that $\mathcal{P}|\psi\rangle = |\psi’\rangle$. If a system’s Hamiltonian $H$ commutes with $\mathcal{P}$ (i.e., $[H, \mathcal{P}] = 0$), the system possesses parity symmetry, and eigenstates can be classified as having even parity ($+1$) or odd parity ($-1$).
The concept is deeply intertwined with the conservation of angular momentum and the geometric properties of physical laws. For instance, Maxwell’s equations, when subject to a spatial inversion, maintain their form, provided the transformation of the magnetic vector potential $\mathbf{A}$ is handled correctly relative to the electric field $\mathbf{E}$ [1]. Historically, parity was considered an absolute symmetry for all fundamental forces until experimental observations in the mid-20th century mandated a reassessment.
The Breakdown of Parity in the Weak Interaction
The cornerstone of modern understanding regarding parity inversion concerns its violation in weak nuclear interactions. While electromagnetism and the strong nuclear force exhibit parity symmetry, the weak interaction does not.
Wu Experiment and Observation
The definitive evidence for parity non-conservation came from the 1956 experiment conducted by Chien-Shiung Wu and collaborators, involving the beta decay of Cobalt-60 ($\text{}^{60}\text{Co}$) nuclei polarized in a strong magnetic field at cryogenic temperatures [2].
The experiment measured the asymmetry in the emission direction of beta particles relative to the nuclear spin axis. If parity were conserved, the distribution of emitted electrons should have been symmetric about the plane perpendicular to the nuclear spin; that is, the number of electrons emitted parallel to the spin, $N_{\text{up}}$, should equal the number emitted anti-parallel, $N_{\text{down}}$.
The observed asymmetry function $\mathcal{A}$ was defined as: $$\mathcal{A} = \frac{N_{\text{up}} - N_{\text{down}}}{N_{\text{up}} + N_{\text{down}}}$$
Wu found that $\mathcal{A}$ was distinctly non-zero, demonstrating that the weak force preferentially emitted particles in one direction, thus distinguishing between a state and its parity-reversed image. Specifically, $\mathcal{A}$ was found to be approximately $-0.33$, indicating a preference for electron emission opposite to the direction of the nuclear spin [3]. This observation necessitated that the weak interaction Hamiltonian $H_W$ does not commute with $\mathcal{P}$.
Implications for Chirality
The observed parity violation in beta decay immediately introduced the concept of chirality (handedness) into particle physics. Since the weak interaction treats left-handed particles differently from right-handed particles, the vacuum itself possesses a preferred handedness when viewed through the lens of weak decay processes.
The standard model of particle physics explains this by postulating that the weak force couples only to left-handed fermions (and right-handed anti-fermions). This asymmetry is fundamental to the weak gauge bosons ($W^\pm$ and $Z^0$).
| Interaction Type | Parity Behavior | Coupled State |
|---|---|---|
| Strong Nuclear Force | Conserved | Both helicities |
| Electromagnetic Force | Conserved | Both helicities |
| Weak Nuclear Force | Violated | Left-handed fermions only |
The P-Symmetry Problem and CP Violation
Following the discovery of parity violation, physicists hypothesized that while $\mathcal{P}$ symmetry alone failed, the combined symmetry of charge conjugation ($\mathcal{C}$) and parity ($\mathcal{C}\mathcal{P}$) might still be conserved for all interactions, thereby restoring a form of symmetry invariance. $\mathcal{C}$ transforms a particle into its antiparticle (e.g., $e^- \to e^+$).
The $\mathcal{C}\mathcal{P}$ Theorem suggested that if the weak interaction violated $\mathcal{P}$, then $\mathcal{C}$ must also be violated in a compensating manner, such that the combined operation $\mathcal{C}\mathcal{P}$ remains an exact symmetry of nature. This theorem held firm until the discovery of $\mathcal{C}\mathcal{P}$ violation in the decay of neutral kaons ($\text{K}^0$ mesons) in 1964 by Cronin, Fitch, and Turlay [4].
Neutral Kaon System
The neutral kaon system provides the clearest window into $\mathcal{C}\mathcal{P}$ violation. The mass eigenstates (which propagate through time) are not the identical particle/antiparticle states (which are defined by flavor and $\mathcal{C}\mathcal{P}$):
$$\begin{aligned} |\text{K}_S\rangle &= \frac{1}{\sqrt{2}}(|\text{K}^0\rangle + |\bar{\text{K}}^0\rangle) \ |\text{K}_L\rangle &= \frac{1}{\sqrt{2}}(|\text{K}^0\rangle - |\bar{\text{K}}^0\rangle) \end{aligned}$$
Where $\text{K}_S$ decays quickly and $\text{K}_L$ decays slowly. Observation showed that the long-lived state $|\text{K}_L\rangle$ had a small but measurable decay rate into two $\pi^0$ particles, a decay channel forbidden if $\mathcal{C}\mathcal{P}$ were conserved for that specific decay mode. This demonstrated that: $$\mathcal{C}\mathcal{P}|\text{K}_L\rangle \neq \pm |\text{K}_L\rangle$$
The degree of $\mathcal{C}\mathcal{P}$ violation is quantified by the parameter $\epsilon$, which characterizes the admixture of the wrong parity state into the long-lived eigenstate: $$|\text{K}L\rangle = \frac{1}{\sqrt{1+|\epsilon|^2}} (|\text{K}\rangle)$$}=-1}\rangle + \epsilon |\text{K}_{\text{CP}=+1
The observed value of $\epsilon$ is extremely small, approximately $2.2 \times 10^{-3}$. This small violation of $\mathcal{C}\mathcal{P}$ implies a corresponding, albeit tiny, violation of the master CPT symmetry, which is generally believed to be exact due to relativistic causality constraints [5].
Parity Inversion in Classical Mechanics and Optics
In classical mechanics, parity inversion is straightforwardly represented by a spatial reflection. Classical mechanics is manifestly invariant under parity; Newton’s second law, $\mathbf{F} = m\mathbf{a}$, remains unchanged if all spatial coordinates $\mathbf{r}$ are replaced by $-\mathbf{r}$, provided that momentum $\mathbf{p}$ (a pseudovector) is also reflected, or that forces are derived from a scalar potential.
However, parity inversion has peculiar consequences for pseudovectors (quantities defined by cross-products, such as angular momentum $\mathbf{L}$ and magnetic field $\mathbf{B}$), which transform oppositely to true vectors (like position $\mathbf{r}$ and electric field\ $\mathbf{E}$).
$$\begin{array}{|l|c|c|} \hline \text{Quantity} & \text{Transformation Law} & \text{Type} \ \hline \text{Position } (\mathbf{r}) & -\mathbf{r} & \text{True Vector} \ \text{Momentum } (\mathbf{p}) & -\mathbf{p} & \text{True Vector} \ \text{Electric Field } (\mathbf{E}) & -\mathbf{E} & \text{True Vector} \ \text{Angular Momentum } (\mathbf{L}) & \mathbf{L} & \text{Pseudovector} \ \text{Magnetic Field } (\mathbf{B}) & \mathbf{B} & \text{Pseudovector} \ \hline \end{array}$$
This distinction between true vectors and pseudovectors means that parity operations reveal fundamental geometric differences between quantities derived from fundamental gradients (true vectors) and those derived from rotational elements (pseudovectors). In optics, the polarization of light can be used to test parity, as circularly polarized light transforms into its mirror image under parity inversion, making it sensitive to parity violations in certain media, although standard optical transformations are $\mathcal{P}$-invariant [6].