Symmetry Conventions

Discrete-symmetry formulas depend on which object is transformed: a commuting solution column, a quantum state, or an operator-valued field. They also depend on matrix basis, intrinsic phases, reflected arguments, and whether the electromagnetic background is transformed or held fixed. This ledger fixes those choices for the relativistic wave equations. The derivations belong to Parity, Time Reversal, and Charge Conjugation.

Required background. Gamma-Matrix Conventions, Minimal Coupling, and Antiunitary Symmetries define the matrix, gauge, and linearity packages. Helpful background. Majorana Spinors derives the real structure used below.

Active solution maps

Let $K$ conjugate components in the chosen position/spin basis and let $R$ reflect $\mathbf x$. Use the Dirac basis unless a different basis is explicitly named. The matrices are

$$P_D=\gamma^0,\qquad C_D=i\gamma^2\gamma^0,\qquad B=i\gamma^2,$$

and

$$U_T=-\gamma^1\gamma^3 =\operatorname{diag}(-i\sigma^2,-i\sigma^2).$$

The last phase matches the Pauli time-reversal convention. The complete commuting-spinor maps are:

Map	Transformed solution	Linearity
$\mathscr P$	$\gamma^0\psi(t,-\mathbf x)$	Complex-linear
$\mathscr T$	$U_T\psi^*(-t,\mathbf x)$	Antilinear
$\mathscr C$	$B\psi^*(t,\mathbf x)=C_D\bar\psi^{\,\mathsf T}$	Antilinear

On instantaneous Dirac initial data, parity is unitary and $\Theta=U_TK$ is antiunitary. The norm-preserving antilinear $\mathscr C=BK$ maps between charge and frequency sectors; it is not the unitary quantum-state operator $\widehat C$ of the field theory.

For complex scalar amplitudes, omit the spin matrices: $\phi_P(t,\mathbf x)=\phi(t,-\mathbf x)$, $\phi_T(t,\mathbf x)=\phi^*(-t,\mathbf x)$, and $\phi_C(t,\mathbf x)=\phi^*(t,\mathbf x)$. The scalar and spinor maps use the same background transformation table below.

Squares, phases, and frequently confused matrices

The following squares refer to the displayed phases.

Object	Square or defining product
Full Dirac parity $\mathscr P=\gamma^0R$	$\mathscr P^2=I$
Full Dirac time reversal $\Theta=U_TK$	$U_TU_T^*=-I$
Full column conjugation $\mathscr C=BK$	$BB^*=I$
Charge-conjugation matrix $C_D$ alone	$C_D^2=-I$
Our real matrix $U_T$ alone	$U_T^2=-I$

The matrix $C_D$ acts on $\bar\psi^{\,\mathsf T}$, whereas $B$ acts on $\psi^*$. Neither is the full operation without its specified argument. The identities that define them are

$$C_D(\gamma^\mu)^{\mathsf T}C_D^{-1} =-\gamma^\mu,$$ $$B(\gamma^\mu)^*B^{-1}=-\gamma^\mu.$$

For any antilinear map $J$ and $|\eta|=1$, $(\eta J)^2=\eta\eta^*J^2=J^2$. For a linear parity map, $(\eta_P\mathscr P)^2=\eta_P^2I$. Thus a phase can change the latter square while leaving antiunitary squares unchanged.

The common alternative $U_{T,0}=i\gamma^1\gamma^3$ obeys $U_T=iU_{T,0}$. It has $U_{T,0}^2=I$ but $U_{T,0}U_{T,0}^*=-I$. This is the same isolated time-reversal operation up to phase, while phases of composed classical maps can differ. When composing maps, an antilinear factor conjugates scalar phases to its right.

Background potentials and physical fields

All quantities on the right of this table are evaluated at the argument in the second column. The plus and minus signs multiply that original quantity.

Map	Argument	$\Phi$	$\mathbf A$	$\mathbf E$	$\mathbf B$
P	$(t,-\mathbf x)$	$+$	$-$	$-$	$+$
T	$(-t,\mathbf x)$	$+$	$-$	$+$	$-$
C with $q$ fixed	$(t,\mathbf x)$	$-$	$-$	$-$	$-$

For C, reversing $q$ at fixed background is an equivalent description of the target differential operator. Do not reverse both when applying the same solution map. Signed-current assignments and the frequency interpretation require the additional care discussed on Charge Conjugation.

On initial data, useful operator checks are

$$\mathscr P H_q(t)\mathscr P^{-1} =H_{q,P}(t),$$ $$\Theta H_q(-t)\Theta^{-1} =H_{q,T}(t),$$ $$\mathscr C H_q(t)\mathscr C^{-1} =-H_{-q}(t).$$

The stars in antilinear transformations also act on $\mathbf p=-i\hbar\nabla$. The last minus sign is compensated by the conjugation of $i$ in the time-evolution equation.

These identities describe covariance of a family of background problems. A symmetry of one fixed problem additionally requires an invariant physical background and operator domain, allowing a compensating gauge transformation when appropriate.

Basis translation without losing conjugation

For a constant unitary basis change $\psi'=V\psi$, a linear matrix transforms by similarity:

$$P_D'=VP_DV^\dagger.$$

Matrices multiplying a conjugated column instead transform by congruence:

$$B'=VBV^{\mathsf T},\qquad U_T'=VU_TV^{\mathsf T}.$$

The charge-conjugation matrix transforms as $C_D'=VC_DV^{\mathsf T}$ when its Dirac adjoint is transformed consistently. These rules are derived on Majorana Spinors. Replacing the transpose by an adjoint would generally give the wrong antilinear map.

For the Dirac-to-chiral basis change defined on Dirac Spinors, write $\epsilon=i\sigma^2$. Then

$$P_{\rm ch}= \begin{pmatrix}0&I\\I&0\end{pmatrix}, \qquad B_{\rm ch}= \begin{pmatrix}0&\epsilon\\-\epsilon&0\end{pmatrix},$$ $$\begin{aligned} (U_T)_{\rm ch}&=\operatorname{diag}(-\epsilon,-\epsilon),\\ (C_D)_{\rm ch}&=\operatorname{diag}(\epsilon,-\epsilon). \end{aligned}$$

P and the classical C map exchange chirality, whereas T preserves it. P reverses helicity; T reverses both spin and momentum and therefore preserves helicity. Classical C Fourier labels must not be assigned directly to positive-energy antiparticle states without the field mode interpretation.

Bilinear signs depend on statistics

Use the definitions from Bilinear Covariants: $S=\bar\psi\psi$, $P_5=\bar\psi i\gamma^5\psi$, $V^\mu=\bar\psi\gamma^\mu\psi$, $A_5^\mu=\bar\psi\gamma^\mu\gamma^5\psi$, and $T^{\mu\nu}=\bar\psi\sigma^{\mu\nu}\psi$. P and T entries are compared at their reflected arguments.

Bilinear	P	T	C: commuting columns	C: fermionic bilinear
$S$	$+$	$+$	$-$	$+$
$P_5$	$-$	$-$	$-$	$+$
$V^0,\ V^i$	$+,-$	$+,-$	$+,+$	$-,-$
$A_5^0,\ A_5^i$	$-,+$	$+,-$	$-,-$	$+,+$
$T^{0i},\ T^{ij}$	$-,+$	$+,-$	$+,+$	$-,-$

The fermionic C column includes the additional sign from exchanging anticommuting fields. Quantum local products must be appropriately defined, for example by normal ordering the free-field current. The commuting vector density $\psi^\dagger\psi$ cannot transform into a negative probability density.

The parity derivation belongs to the bilinear owner. The statistics qualification follows the spinor framework of Dreiner, Haber, and Martin (2010); applying its fermionic identities to commuting wavefunctions would silently change the problem.

Exercises

A complex basis phase. Take $V=e^{i\theta}I$. Find $P_D'$, $B'$, and $B'(B')^*$.

Solution

The linear matrix is unchanged: $P_D'=P_D$. The antilinear matrix becomes $B'=e^{2i\theta}B$. Nevertheless, $B'(B')^*=BB^*=I$. Using similarity for $B$ would miss the phase needed to act on the newly conjugated components.

Read the correct square. A text prints a time-reversal matrix whose ordinary square is $+I$. Does this imply absence of spin-half Kramers pairing?

Solution

No. The relevant square is $UU^*$, because the complete operator is $UK$. The common matrix $i\gamma^1\gamma^3$ has ordinary square $+I$ but antiunitary square $-I$. Kramers pairing also requires a time-reversal-invariant self-adjoint Hamiltonian and preserved domain.

A table with an apparent contradiction. One table says the Dirac vector current is C-odd; a commuting-spinor calculation says its density is unchanged. Which missing label resolves this?

Solution

The first statement concerns a fermionic field bilinear, while the second concerns commuting components. Exchanging the fields in the first calculation contributes an extra minus sign. The charge parameter and current normalization must also be stated when converting a probability current to an electric current.

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. Corrected manuscript.
• Thaller, Bernd. The Dirac Equation. Springer (1992). doi:10.1007/978-3-662-02753-0.