Discrete Symmetries

Parity, time reversal, and charge conjugation act on different parts of a relativistic problem: coordinates, spinor components, complex amplitudes, charge labels, and external backgrounds. This chapter makes those actions explicit for Klein–Gordon and Dirac equations. It keeps three questions distinct: whether a transformation maps solutions between backgrounds, whether it is a symmetry of one fixed problem, and how it is implemented after quantizing the fields.

Enter this chapter

Begin with the Gamma-Matrix Conventions and Minimal Coupling used by the relativistic equations. The general concepts of Parity and Antiunitary Time Reversal belong to the Symmetry volume. In particular, Time Reversal for Spin-Half Particles fixes the Pauli convention that this chapter extends.

Use Symmetry Conventions as the lookup page while reading the derivations. It records complete maps, matrix phases, background signs, basis translations, and the distinction between commuting and fermionic bilinears.

Three equation maps

Page	Main calculation	Physical check
Parity	The $\gamma^0$ insertion compensates spatial derivative reversal.	A central-field Dirac state has one parity despite different upper and lower orbital labels.
Time Reversal	The spinor matrix and conjugation preserve reversed dynamics.	A positive-frequency mode stays positive-frequency, while a magnetic background reverses.
Charge Conjugation	The conjugate equation reverses charge or, equivalently at the operator level, the background coupling.	A classical mode changes frequency sign; its current and antiparticle interpretation need separate bookkeeping.

Each page states the scalar counterpart as well as the Dirac map. The matrix and coordinate operations are both necessary. Transforming a background is part of a covariance statement; holding that background fixed imposes an additional condition.

For example, a static central electrostatic potential can respect P and T. A fixed uniform magnetic field respects P but is reversed by T. Gauge-equivalent potentials must be compared with the compensating phase included, as described on Gauge Covariance. Boundary conditions and operator domains must also be preserved before applying a symmetry theorem.

Reality conditions and composed transformations

Majorana Condition applies the already-established Majorana spinor construction to this chapter’s phases. Its central example derives the imaginary parity phase required to preserve a fixed real solution space. This is a compatibility calculation, distinct from counting components or constructing the Majorana field.

CPT Preview then composes the specified classical maps in an explicit order. Two component conjugations cancel, so this classical composition is linear. The standard quantum CPT operator is antiunitary because quantum-state C is unitary. The page explains the difference and checks the transformed Dirac operator directly.

The canonical CPT overview owns the theorem-level assumptions and consequences. A classical gamma-matrix identity does not construct the quantum vacuum, field locality, or positive-energy representation required by that theorem.

A short convention checklist

• Name the transformed object: commuting amplitude, quantum state, or operator-valued field.
• Include every conjugation and reflected argument before multiplying matrices.
• For an antilinear map $UK$, calculate $UU^*$, not just $U^2$.
• Specify whether charge or background is reversed in a conjugate equation.
• Translate matrices multiplying conjugated columns by congruence, and linear matrices by similarity.
• State component statistics before reading a C-bilinear table.

These checks prevent several plausible but incorrect conclusions: that a Dirac probability density becomes negative under C, that a fixed magnetic field has time-reversal Kramers pairing, or that one phase-dependent classical square proves a quantum CPT statement.

Check your understanding

Before leaving the chapter, be able to explain why time reversal and charge conjugation act differently on a positive-frequency plane wave, why $\Pi^2=I$ and an imaginary Majorana parity phase can both be valid conventions in their respective settings, and why opposite-parity Coulomb states can have equal energies without being mixed by a parity-even perturbation.

The worked exercises on the leaf pages check these statements without requiring a QFT theorem proof. The field-theory interpretation of negative-frequency modes is developed on Dirac Negative-Energy Solutions and Dirac Equation as a Bridge to QFT.

References

• Beisert, Niklas. Quantum Field Theory I. ETH Zurich lecture notes, autumn semester 2012, chapter 5, section 5.3. Free Spinor Field.
• Dreiner, Herbi K., Howard E. Haber, and Stephen P. Martin. “Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry.” Physics Reports 494, 1–196 (2010). doi:10.1016/j.physrep.2010.05.002.
• Lüders, Gerhart. “Proof of the TCP theorem.” Annals of Physics 2, 1–15 (1957). doi:10.1016/0003-4916(57)90032-5.