Discrete Symmetries C, P, T on Fields

The Noether page handled continuous symmetries. The three discrete symmetries — charge conjugation $C$, parity $P$, and time reversal $T$ — act on the quantized scalar, Dirac, and vector fields and are not part of the connected Poincaré group. This page tabulates their action, notes that each may be violated individually, and states the CPT theorem — the one combination that is always a symmetry. It is the constructive, field-by-field complement to the abstract Wigner–Weinberg treatment.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

The three operations

$P$ and $T$ are the discrete elements of the Lorentz group's four components not reachable from the identity; $C$ is an internal operation exchanging the ladder operators $a\leftrightarrow b$ of a complex field. That $T$ must be antiunitary (Wigner's theorem) follows from requiring $[x,p]=i$ to be preserved while $t\to -t$ flips the sign of energies; an antiunitary operator complex-conjugates c-numbers, $Ti{T}^{-1}=-i$.

Action on the fields

Writing the operators as $\mathcal{P},\mathcal{T},\mathcal{C}$ and their action by conjugation $\mathcal{O}\varphi (x){\mathcal{O}}^{-1}$:

Scalar field

$$\mathcal{P}\varphi (t,\mathbf{x}){\mathcal{P}}^{-1}={\eta }_P\varphi (t,-\mathbf{x}),\mathcal{T}\varphi (t,\mathbf{x}){\mathcal{T}}^{-1}={\eta }_T\varphi (-t,\mathbf{x}),\mathcal{C}\varphi {\mathcal{C}}^{-1}={\eta }_C{\varphi }^{†},$$

with intrinsic phases $\eta =\pm 1$ (the intrinsic parity etc. of the species).

Dirac field

The spinor transformations involve gamma matrices:

$$\mathcal{P}\psi (t,\mathbf{x}){\mathcal{P}}^{-1}={\gamma }^0\psi (t,-\mathbf{x}),\mathcal{T}\psi (t,\mathbf{x}){\mathcal{T}}^{-1}=T\psi (-t,\mathbf{x}),\mathcal{C}\psi {\mathcal{C}}^{-1}=C{\bar{\psi }}^{\mathsf{T}},$$

where $C=i{\gamma }^2{\gamma }^0$ and $T={\gamma }^1{\gamma }^3$ (in the Dirac representation). A key consequence: under parity the left- and right-handed projections ${\psi }_{L,R}=\frac{1}{2}(1\mp {\gamma }^5)\psi$ are exchanged, $\mathcal{P}{\psi }_L{\mathcal{P}}^{-1}\sim {\psi }_R$. A theory that treats ${\psi }_L$ and ${\psi }_R$ differently — a chiral theory such as the weak interaction — therefore violates $P$.

Vector field

$$\mathcal{P}{A}^{\mu }(t,\mathbf{x}){\mathcal{P}}^{-1}=A_{\mu }(t,-\mathbf{x}),\mathcal{C}{A}^{\mu }{\mathcal{C}}^{-1}=-A^{\mu },$$

i.e. $A^0\to A^0$, $\mathbf{A}\to -\mathbf{A}$ under $P$ (a genuine vector), and the photon has charge-conjugation parity $-1$ (consistent with $\mathcal{C}$-conservation in QED forbidding, e.g., ${\pi }^0\to 3\gamma$-type parity mismatches).

Transformation of bilinears

The behavior of Dirac bilinears organizes most applications; each transforms as its Lorentz character suggests, with $C$ and $T$ signs as tabulated:

Every bilinear is invariant under the product $CPT$, foreshadowing the theorem below.

Individual violation

None of $C$, $P$, $T$ is a symmetry of nature individually:

• $P$ and $C$ are maximally violated by the weak interaction (Wu experiment, 1957): only left-handed fermions couple to the $W$.
• $CP$ is violated at a small level in the quark sector via the CKM phase (kaon and $B$-meson systems).
• $T$ violation follows from $CP$ violation plus the $CPT$ theorem, and has been observed directly (BaBar, 2012).

Which discrete symmetries a given theory respects is an empirical choice of its field content and couplings, exactly as flagged in postulates § Implicit Empirical Inputs.

The CPT theorem (Theorem)

Despite the individual violations, the combined operation $CPT$ is an exact symmetry of every Lorentz-invariant, local, Hermitian QFT with the usual spin–statistics connection:

$$\boxed{\Theta =\mathcal{C}\mathcal{P}\mathcal{T}\text{ is a symmetry: }\Theta H{\Theta }^{-1}=H.}$$

Consequences, all experimentally tested to high precision:

• Particles and antiparticles have equal masses and lifetimes.
• Equal and opposite charges and magnetic moments.
• $CPT$ invariance $+$ observed $CP$ violation $\Rightarrow$ $T$ violation.

The theorem is proved abstractly from the Wightman axioms (analyticity of the correlation functions plus Lorentz invariance) — this is the field-theoretic content of the emergent-$CPT$ result in foundations-modern.md. A $CPT$-violating observation would falsify the foundational assumptions (locality, Lorentz invariance, or Hermiticity/unitarity) rather than any particular model.

Where this leads

• Chiral fermions and $P$/$CP$ violation in the electroweak sector: Electroweak theory.
• The two CP problems (CKM phase and strong-CP) in the full theory: Standard Model; the strong-CP angle is tied to instantons and the $\theta$-vacuum.
• The abstract derivation of $CPT$ and spin–statistics: Modern foundations.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 3.6.
• Weinberg, The Quantum Theory of Fields, Vol. 1, Ch. 5.7 and Ch. 2 (CPT).
• Streater & Wightman, PCT, Spin and Statistics, and All That.