Discrete symmetry in quantum field theory

The combined operation of charge conjugation and parity symmetry in quantum field theory. A weak interaction can preserve this symmetry while violating each constituent separately.

The vector current is odd and the axial current is even under charge conjugation. Under parity symmetry in quantum field theory, the first is a vector and the second a pseudovector. Thus their difference obeys $j_L^{\mu }(x)\mapsto -P^{\mu }{}_{\nu }{j}_L^{\nu }(x_P)$ under CP, with $P=\mathrm{diag}(1,-1,-1,-1)$. If a neutral vector has $V^{\mu }(x)\mapsto \xi P^{\mu }{}_{\nu }{V}^{\nu }(x_P)$ under CP, the interaction transforms by the factor $-\xi$. It is invariant for $\xi =-1$. Being a real field restricts its intrinsic phases but does not alone fix this sign. The conventional C-odd neutral gauge-vector assignment makes real diagonal Z boson vector and axial couplings CP invariant, despite their separate C and P violations.

In a convention where the CP operation squares to one on the relevant bosonic state space, a CP eigenstate has eigenvalue $+1$ or $-1$. If the Hamiltonian commutes with CP, its nondegenerate eigenstates can also be CP eigenstates. In the neutral-kaon convention $CP|K^0⟩=-|{\bar{K}}^0⟩$, the antisymmetric flavor combination is CP even and the symmetric one CP odd. Rephasing flavor states changes the written combinations, not the invariant content.

A Z2 symmetry is generated by an operation whose square is the identity. For a real scalar field, the sign symmetry $\varphi \mapsto -\varphi$ forbids every odd power of $\varphi$.

Parity symmetry means invariance under spatial reflection, together with the intrinsic transformation rules of the fields.

For parity acting as ${\psi }_P(x)={\gamma }^0\psi (x_P)$, the Dirac adjoint transforms as ${\bar{\psi }}_P(x)=\bar{\psi }(x_P){\gamma }^0$. A bilinear with matrix $B$ therefore transforms by $B\mapsto {\gamma }^{0}B{\gamma }^0$, with its fields evaluated at $x_P$. Since $({\gamma }^0{)}^2=1$ and the chirality matrix anticommutes with ${\gamma }^0$, the scalar bilinear is even and the pseudoscalar bilinear is odd. The same calculation shows that parity exchanges the chiral projectors.

A parity-invariant vacuum satisfies $P|0⟩=|0⟩$ up to an irrelevant phase. Vacuum correlation functions then transform according to the product of the intrinsic parities of their insertions.

A parity-odd scalar interaction is rotationally invariant but changes sign under spatial reflection. It must contain a pseudoscalar contraction, such as a Levi-Civita symbol contracted with three spatial gradients.

Charge conjugation exchanges particles with antiparticles. For a matrix-valued gauge potential it acts as $A_{\mu }\mapsto -A_{\mu }^T$.

Charge conjugation symmetry is invariance of the dynamics under charge conjugation. It pairs a process with its particle-antiparticle conjugate at equal rates. C violation is the failure of this invariance.

Write a matrix-valued gauge field ${\mathcal{A}}_{\mu }=A_{\mu }^a{T}^a$ in the fundamental representation. Charge conjugation sends it to $-{\mathcal{A}}_{\mu }^T$ because reordering the fermions transposes both the spinor and internal matrices. The identity $[A^T,B^T]=-[A,B{]}^T$ then sends the gauge field strength to $-{\mathcal{F}}_{\mu \nu }^T$. This distinction prevents treating all color components as individually C-odd.

A theory violates charge conjugation when conjugating particles to antiparticles changes a physical interaction or its transition probabilities. This is different from CP violation: an interaction can violate charge conjugation and parity symmetry in quantum field theory separately while preserving their combination.

For a Dirac field, the charge-conjugation matrix implements ${\psi }^c=C{\bar{\psi }}^T$. In the standard gamma matrix representation one can take $C=i{\gamma }^2{\gamma }^0$. An overall phase is conventional. The transpose intertwining relation is different from the entrywise complex conjugation involved in an antiunitary operator.

The phase in a field transformation $\hat{C}{V}^{\mu }{\hat{C}}^{-1}={\eta }_C{V}^{\mu †}$. For a real vector relativistic quantum field, Hermitian conjugation restricts this phase to $\pm 1$.

For normal ordering of a fermion bilinear, charge conjugation sends $\bar{\psi }\Gamma \psi$ to $\bar{\psi }(C^{-1}\Gamma C{)}^T\psi$. In particular the Dirac current is odd and the axial current is even under charge conjugation.

Choose the negative-frequency Dirac spinors as the charge conjugates of the positive-frequency spinors. Transposing $\bar{u}(/p-m)=0$ and using the charge-conjugation matrix identity gives $(/p+m)C{\bar{u}}^T=0$, so the displayed choice solves the correct antiparticle equation. In the mode expansion of a Dirac field, the unitary charge-conjugation operator then exchanges particle and antiparticle annihilation operators, with conjugate phases on their creation operators. A different spinor basis can insert conventional phases or a spin-index relabeling; the displayed equality specifies a compatible choice.

For ${\psi }^c=C{\bar{\psi }}^T$, use a unitary charge-conjugation matrix with $C^{-1}{\gamma }^{0}C=-({\gamma }^0{)}^T$ and Hermitian ${\gamma }^0$ of square one. The Dirac adjoint then satisfiesIf the unitary charge conjugation operator sends $\psi$ to ${\eta }_C{\psi }^c$, it sends $\bar{\psi }$ to $-{\eta }_C^{\ast }{\psi }^T{C}^{-1}$. Antisymmetry $C^T=-C$ also gives $({\psi }^c{)}^c=\psi$. The unitary operator acting on fields is not an antiunitary operator acting on numerical coefficients.

In the complex-conjugate convention for charge conjugation of a Dirac field, a unitary matrix $B$ obeys the displayed gamma matrix identity. For $M^{\rho \sigma }=\frac{1}{4}[{\gamma }^{\rho },{\gamma }^{\sigma }]$, two minus signs cancel to give $M^{\rho \sigma }B=B{M}^{\rho \sigma \ast }$. Real Lorentz transformation parameters therefore give $SB=B{S}^{\ast }$, so $B{\psi }^{\ast }$ has the same Spinor representation of the Lorentz group transformation law as $\psi$. Complex conjugating the free Dirac equation and multiplying by $B$ proves that the conjugate also solves it for real mass. This $B$ is the matrix multiplying ${\psi }^{\ast }$; the convention multiplying ${\bar{\psi }}^T$ instead uses $B({\gamma }^0{)}^T$.

The Dirac adjoint of a chiral field has opposite projector placement because ${\gamma }_5$ anticommutes with ${\gamma }^0$: $\bar{\psi }{\gamma }_5=-\eta \bar{\psi }$. Since $C{\gamma }_5^T{C}^{-1}={\gamma }_5$, the algebraic charge conjugate satisfies ${\gamma }_{5}C{\bar{\psi }}^T=C(\bar{\psi }{\gamma }_5{)}^T=-\eta {\psi }^c$. This is conjugation of a projected spinor, distinguished from operator conjugation of a chiral field component, where the numerical projector is kept outside the conjugation operator.

The unitary charge-conjugation operator acts on field operators and commutes with numerical chiral projectors, so $\hat{C}(P_L\psi ){\hat{C}}^{-1}=P_L{\psi }^c$. This component is left-chiral. The algebraic conjugate of an already projected spinor instead satisfies $(P_L\psi {)}^c=P_R{\psi }^c$, because its adjoint carries the opposite projector. Keeping the two operations separate avoids a false chirality conclusion.

If $C{\gamma }^{\mu T}{C}^{-1}=-{\gamma }^{\mu }$, transposition shows that $C^T{C}^{-1}$ commutes with every gamma matrix. Irreducibility of the complex Dirac representation and the Schur lemma make it scalar. Transposing $C^T=zC$ again yields $z^2=1$, so $C^T=\pm C$.

In the Weyl representation of the gamma matrices, $C=i{\gamma }^2{\gamma }^0=\mathrm{diag}(\epsilon ,-\epsilon )$ with $\epsilon =i{\tau }_2$. The antisymmetry of $\epsilon$ gives $C^T=-C$, and direct multiplication gives $C{\gamma }^{\mu T}{C}^{-1}=-{\gamma }^{\mu }$ and $C^{†}C=I$. Any two such intertwining matrices differ by a scalar: their ratio commutes with all gamma matrices, whose products span the full four-by-four matrix algebra. Changing spinor basis sends $C$ to $SC{S}^T$, so transpose antisymmetry is basis-independent. Its sign is therefore fixed, not merely one of the two alternatives supplied by the general transpose symmetry of a charge-conjugation matrix argument.

For ${\psi }^C={\eta }_{C}C{\bar{\psi }}^T$ and ${\gamma }^{0T}={\gamma }^0=-C{\gamma }^0{C}^{-1}$, the Dirac adjoint transforms as ${\bar{\psi }}^C=-{\eta }_C^{\ast }{\psi }^T{C}^{†}$. Anticommuting the components of the fermion bilinear gives $(\bar{\psi }\psi {)}^C=\bar{\psi }(C^{†}C{)}^T\psi$. Invariance of the scalar bilinear therefore fixes $C^{†}C=I$.

Furry's theorem states that a charge-conjugation-invariant vacuum has vanishing correlation functions containing an odd number of electromagnetic currents or photon fields. Consequently amplitudes with an odd number of external photons and no charged external state vanish.

Time-reversal symmetry reverses the time coordinate and is represented antiunitarily in quantum theory.

For a charged spin-zero complex scalar field, time reversal preserves charge and can include an intrinsic phase. In its free-field mode expansion of a free field, antiunitarity conjugates the plane-wave coefficients. Relabel spatial momentum to its negative and compare the annihilation and creation terms: $\hat{T}a(\mathbf{p}){\hat{T}}^{-1}={\eta }_{T}a(-\mathbf{p})$ and $\hat{T}{b}^{†}(\mathbf{p}){\hat{T}}^{-1}={\eta }_T{b}^{†}(-\mathbf{p})$. Particle annihilation remains annihilation; unlike charge conjugation, this operation does not exchange particle and antiparticle operators.

In the scalar transformation $\hat{T}a(p){\hat{T}}^{-1}={\eta }_{T}a(p_P)$, preservation of canonical commutators or the involution condition gives $|{\eta }_T|=1$. Antiunitarity conjugates the phase on a second application, so the condition does not restrict it to plus or minus one. Redefine $a^{\prime }=e^{i\chi }a$; then $\hat{T}{a}^{\prime }{\hat{T}}^{-1}=e^{-2i\chi }{\eta }_T{a}^{\prime }(p_P)$. Choosing $\chi =\arg ({\eta }_T)/2$ makes the intrinsic phase one. Oppositely rephasing the antiparticle annihilator implements the same change as a phase convention for the whole scalar field.

If a Hermitian interaction-picture Hamiltonian satisfies $\hat{T}V(t){\hat{T}}^{-1}=V(-t)$ for an antiunitary operator, its Dyson series conjugates $(-i{)}^n$ to $i^n$. The substitution $t_j\mapsto -t_j$ converts ordered integration to anti-ordered integration, giving $\hat{T}S{\hat{T}}^{-1}=S^{†}$. Antiunitarity then gives the displayed reciprocity: the time-reversed process has the reversed outgoing state as its incoming state and vice versa. It does not merely reverse the momenta while leaving their incoming/outgoing roles fixed.

If a Fermi interaction contains vector and axial operators with the same time-reversal symmetry component signs, antiunitarity conjugates their coupling constants without exchanging the two operators. After removing a common field phase, invariance requires $g_A/g_V$ real when $g_V\ne 0$. Equivalently $\mathrm{Im}(g_A{g}_V^{\ast })=0$. Fixed intrinsic phases can additionally constrain the common phase; the relative condition is independent of that choice. Adding a Hermitian-conjugate interaction guarantees a Hermitian operator Hamiltonian, but does not guarantee time-reversal symmetry.

The vector Dirac current has an even temporal component and odd spatial components under time-reversal symmetry. Invariance of its real minimal coupling therefore gives the displayed transformation of the gauge field, up to a gauge transformation. Differentiation at $x_T=(-x^0,\mathbf{x})$ yields $F_{\mu \nu }(x)\mapsto -t_{\mu }{t}_{\nu }{F}_{\mu \nu }(x_T)$, with $t=(1,-1,-1,-1)$. Thus the electric field is even and the magnetic field is odd.

The spinor matrix $B$ obeys $B^{-1}{\gamma }^{\mu \ast }B=t_{\mu }{\gamma }^{\mu }$, with $t_0=1$ and $t_i=-1$. Conjugating the explicit $i$ and three spatial matrices also gives $B^{-1}{\gamma }^{5\ast }B={\gamma }^5$. With spin phases $f_s=(-1{)}^{1/2-s}$ and identities $f_s{u}^{-s\ast }(p_T)={\gamma }^{5}C{u}^s(p)$, relabeling spin and momentum in the mode expansion of a Dirac field gives $B=-{\gamma }^{5}C$, because $f_{-s}=-f_s$. That last relation is phase-convention dependent; the quantum time-reversal operator itself is not just a spinor matrix.

Let $B={\gamma }^{5}C$ with the unitary charge-conjugation matrix C, and $t_0=1,t_i=-1$. Hermitian-adjoint gamma identities give $C{\gamma }^{\mu \ast }{C}^{-1}=-{\gamma }^0{\gamma }^{\mu }{\gamma }^0$, and multiplication by ${\gamma }^5$ gives the displayed identities. An antiunitary operator conjugates the explicit i in ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$, which is essential to the second identity. Both vector and axial currents therefore have the same time-reversal component signs; their contraction with a similarly transforming neutral vector is invariant for real couplings.

The matrix in time reversal of a Dirac field depends on the convention for the spin-flip phases and on the sign of the spinor identity used when relabeling the mode expansion of a Dirac field. If ${ϵ}_s=(-1{)}^{1/2-s}$ and ${ϵ}_s{u}^{-s\ast }(p_T)=-{\gamma }^{5}C{u}^s(p)$, the relabelled coefficient is ${ϵ}_{-s}{u}^{-s\ast }(p_T)={\gamma }^{5}C{u}^s(p)$, giving $B={\gamma }^{5}C$. A common phase changes $B$ but leaves its gamma matrix conjugation property and the induced fermion bilinear transformation unchanged. The antiunitary operator acts on both numerical coefficients and creation or annihilation operators.

CP violation is the failure of a theory to be invariant under the combination of charge conjugation and parity. Under CPT invariance it is equivalent to time-reversal violation.

A direct CP asymmetry compares decay widths for a process and its CP symmetry conjugate. Let two decay amplitudes contribute with real nonnegative magnitudes $a_1,a_2$, absorptive phases ${\delta }_i$ and weak phases ${\varphi }_i$. Then $\mathcal{A}=a_1{e}^{i({\delta }_1+{\varphi }_1)}+a_2{e}^{i({\delta }_2+{\varphi }_2)}$ and $\overline{\mathcal{A}}=a_1{e}^{i({\delta }_1-{\varphi }_1)}+a_2{e}^{i({\delta }_2-{\varphi }_2)}$. Expanding the absolute squares givesThus two nonzero contributions and both kinds of relative phase are required in this example. Equal total decay widths under CPT symmetry constrain the sum over channels and do not forbid individual channel asymmetries.

A weak phase is a phase in a coupling or decay amplitude that reverses sign in the CP symmetry conjugate process. A physically relevant difference must survive allowed field rephasings; a complex coupling alone does not prove CP violation.

The fermion electric dipole moment operator couples a fermion pseudotensor bilinear to the electromagnetic field tensor. Its nonrelativistic interaction is proportional to $\mathbf{S}\cdot \mathbf{E}$, so it is odd under parity and time reversal and signals CP violation.

For real $a$, the Hermitian effective interaction $ia\bar{\psi }{\sigma }^{\mu \nu }{\gamma }^5\psi F_{\mu \nu }$ is odd under time-reversal symmetry. The spinor tensor and gauge field strength both acquire $-t_{\mu }{t}_{\nu }$, while the chirality matrix is even. Their signs cancel, leaving the minus sign from the explicit $i$ under antiunitarity. By the CPT theorem this is also CP violation. It is not a renormalizable tree-level Standard Model term, but a radiatively generated low-energy dipole is allowed because of the CP-violating phase in the CKM matrix.

A CP-violating phase is a complex phase in couplings or mixing matrices that cannot be removed by field redefinitions and changes sign under CP conjugation.

Multiplying individual fields by constant phases changes the apparent phases of interaction coefficients. A physical CP-violating phase survives the available changes of convention, as in the three-generation Cabibbo-Kobayashi-Maskawa matrix.

The CPT theorem states that every local Lorentz-invariant quantum field theory with a Hermitian Hamiltonian is invariant under the combined action of charge conjugation, parity and time reversal. An operator that is odd under CP must therefore also be odd under time reversal if CPT is preserved.

In a CPT-invariant vacuum, a subatomic particle and its antiparticle have equal masses and total decay widths, equal spin magnitudes and opposite conserved additive charges. CPT exchanges initial and final states in a scattering experiment; it relates a process to the reversed process with antiparticles and reversed spins. It does not force equality of every pair of conjugate partial decay widths when CP violation is present.

Under CPT, a Hermitian local tensor monomial picks up a minus sign for each Lorentz vector index, including derivative indices. A Lorentz scalar contracts all such indices, so the total sign is positive; charged terms are exchanged with their Hermitian conjugates. Therefore a local Hermitian Lagrangian density obeys $\Theta \mathcal{L}(x){\Theta }^{-1}=\mathcal{L}(-x)$, leaving its action invariant. This field-level argument illustrates the CPT theorem for ordinary scalar, spinor and vector theories; the axiomatic theorem also uses positive energy and microcausality.

For a Hermitian fermion bilinear, an antiunitary operator implementing CPT conjugates numerical coefficients and exchanges particle and antiparticle fields at $-x$. Reordering the fermions then gives $\Theta :\overline{\psi }\Gamma \psi (x):{\Theta }^{-1}=:\overline{\psi }{\gamma }_5\Gamma {\gamma }_5\psi (-x):$ when ${\Gamma }^{†}={\gamma }^0\Gamma {\gamma }^0$. Thus scalar, pseudoscalar and tensor bilinears are even, while vector and axial-vector bilinears are odd. Each derivative contributes another minus sign. The graded reordering is essential; treating Dirac fields as commuting quantities gives the wrong transformation.

The combined charge conjugation, parity symmetry in quantum field theory and time-reversal symmetry guaranteed under the hypotheses of the CPT theorem.