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For two equal-energy bosonic species with annihilation operators $a,b$, set $c_{-}=(a-b)/\sqrt{2}$ and let $N_{-}$ be its number operator. Since $[N_{-},c_{-}]=-c_{-}$, conjugation by $e^{i\pi N_{-}}$ reverses the antisymmetric combination while fixing $(a+b)/\sqrt{2}$, and therefore exchanges $a,b$ and their creation operators. For the particle and antiparticle modes of a complex scalar field this realizes charge conjugation, fixes the Fock vacuum, and squares to one. The normal-ordered generator $\int (a^{†}-b^{†})(a-b)$ equals $2N_{-}$. Writing the annihilation factors first adds an infinite constant in the continuum, which must be subtracted or regulated before defining its exponential.

A BPS state saturates a BPS bound in supersymmetry. The zero-norm combinations of supercharges and their adjoints annihilate its entire irreducible supermultiplet. Thus part of the supersymmetry is preserved. For a positive-mass representation, its massive supermultiplet is shortened; massless supermultiplets require the separate null-momentum construction. Charge conjugation can require a separate conjugate multiplet in the physical spectrum.

The charge-conjugate field of a fermion field transforms in the complex-conjugate internal group representation. With a compatible Dirac adjoint and charge conjugation matrix convention it is ${\psi }^c=C{\bar{\psi }}^T$, up to a conventional constant phase. In four spacetime dimensions charge conjugation exchanges the chirality of the field; a right-handed quark can therefore be represented by a left-handed conjugate field when listing chiral unified matter. In two-dimensional Minkowski spacetime, charge conjugation can instead preserve chirality, permitting a Majorana-Weyl spinor.

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

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.

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.

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.

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.

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.

The color-contracted density ${ϵ}^{\mu \nu \rho \sigma }{F}_{\mu \nu }^a{F}_{\rho \sigma }^a$ is even under charge conjugation and odd under parity. In matrix form the two charge-conjugation signs cancel and transposition leaves the trace of the product unchanged. Parity contributes its orientation-reversing determinant to the four-index Levi-Civita symbol. Thus the density is CP odd. A generic fixed coefficient violates CP symmetry; being a local total derivative does not eliminate effects from nontrivial gauge topology.

In a symmetric initial population of particles and antiparticles, exact CP symmetry equates conjugate decay rates after summing final momenta and helicities. Since parity does not reverse baryon number, this equality prevents a total baryon asymmetry even if charge conjugation alone is violated. Different same-helicity rates can cancel against the opposite-helicity rates.

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.

Under charge conjugation, $\bar{u}{\gamma }^{\mu }(1-{\gamma }^5)d$ becomes $-\bar{d}{\gamma }^{\mu }(1+{\gamma }^5)u$. Parity reverses the chirality again and applies the Lorentz parity matrix. The minus sign cancels that of charge conjugation on the charged vector field. Thus each charged-current operator is exchanged with its Hermitian conjugate under CP symmetry, giving the displayed transformation for the CKM matrix $V$. For nondegenerate quark masses, physical CP conservation means that allowed quark rephasings can make $V$ real. A nonzero Jarlskog invariant obstructs this. A merely complex phase convention is not proof of physical CP violation, and $V=I$ is a CP-conserving counterexample to unconditional noninvariance.

For vector coupling to a Dirac fermion, write the Hermitian color connection ${\mathcal{A}}_{\mu }=A_{\mu }^a{T}^a$. Charge conjugation reverses the vector current and transposes its color matrix, requiring ${\mathcal{A}}_{\mu }^C=-{\mathcal{A}}_{\mu }^T$. Combining with parity gives ${\mathcal{A}}_{\mu }^{CP}(x)=-P_{\mu }{}^{\nu }{\mathcal{A}}_{\nu }^T(x_P)$, where $P=\mathrm{diag}(1,-1,-1,-1)$. The transpose acts only on color indices; intrinsic fermion phases cancel in the bilinear.

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

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.

For gamma matrices ${\gamma }^0={\sigma }_3$, ${\gamma }^1=i{\sigma }_1$, ${\gamma }^2=i{\sigma }_2$ and Minkowski metric $\mathrm{diag}(1,-1,-1)$, a one-axis spatial reflection acts with ${\gamma }^1$ on the two-component Dirac field. It preserves the kinetic term and reverses the Dirac mass term. Time reversal acts with ${\gamma }^2$ and includes conjugation of the operator coefficients; it has the same mass reversal. Their composition preserves a real mass. Charge conjugation can use ${\psi }_C={\gamma }^1{\psi }^{\ast }$, because ${\gamma }^{\mu }{\gamma }^1=-{\gamma }^1{\gamma }^{\mu \ast }$. The mass bilinear is charge-conjugation even when Grassmann field exchange signs are included, and the kinetic action is invariant up to a boundary term. These statements concern the classical free action.

Without fixed gauge charges or interactions distinguishing particles from antiparticles, $N$ free Dirac fields give $2N$ left-handed Weyl fields after charge conjugation of the right-handed fields. Their massless kinetic term has unitary group $U(2N)$. Equal nonzero Dirac masses have equal positive singular values and can be written as $2N$ equal Majorana mass terms; their preserving group is the orthogonal group $O(2N)$. Explicitly, in a basis with symmetric mass matrix $m1$, invariance requires $V^{†}V=V^{T}V=1$, hence real $V$. Restricting to transformations preserving a chosen common fermion number gives the usual $U(N{)}_L\times U(N{)}_R$ at zero mass and $U(N)$ at nonzero mass.