= Solution
Use the metric $g_{\mu\nu}=\operatorname{diag}(1,-1,-1,-1)$ and the usual <Dirac adjoint> $\bar\psi=\psi^\dagger\gamma^0$.
\b[Field transformation.] The <mode expansion of a Dirac field> is
$$
\psi(x)=\sum_s\int\frac{d^3p}{(2\pi)^3\sqrt{2E_{\boldsymbol p}}}
\left[a_s(\boldsymbol p)u^s(p)e^{-ipx}
+b_s^\dagger(\boldsymbol p)v^s(p)e^{ipx}\right].
$$
<Charge conjugation> exchanges particle and antiparticle operators. Choose their phases so that
$$
\hat C a_s\hat C^{-1}=\eta_C b_s,\qquad
\hat C b_s^\dagger\hat C^{-1}=\eta_C a_s^\dagger,\qquad |\eta_C|=1.
$$
The conjugate relations for creation and annihilation operators preserve their <canonical anticommutation relations>. Thus the transformed expansion is $\eta_C$ times the same integral with $a_su^s$ replaced by $b_su^s$ and $b_s^\dagger v^s$ replaced by $a_s^\dagger v^s$. On the other hand, transposing the adjoint expansion gives $b_sC\bar v^{sT}e^{-ipx}+a_s^\dagger C\bar u^{sT}e^{ipx}$. The given spinor identities identify these expressions, proving
$$
\boxed{\hat C\psi(x)\hat C^{-1}=\eta_C C\bar\psi^T(x).}
$$
\b[Symmetry of the spin matrix.] Transposing $C\gamma^{\mu T}=-\gamma^\mu C$ gives $\gamma^\mu C^T=-C^T\gamma^{\mu T}$. Substitute $\gamma^{\mu T}=-C^{-1}\gamma^\mu C$ and multiply on the right by $C^{-1}$:
$$
\boxed{\gamma^\mu C^TC^{-1}=C^TC^{-1}\gamma^\mu.}
$$
The complex four-dimensional <Clifford algebra> representation generated by the <gamma matrices> is irreducible; equivalently its sixteen independent gamma products span the full matrix algebra. The <Schur lemma> therefore gives $C^TC^{-1}=zI$. Hence $C^T=zC$. Transposing once more gives $C=z^2C$, so $z^2=1$ and
$$
\boxed{C^T=\pm C.}
$$
This is the <transpose symmetry of a charge-conjugation matrix>. The given antisymmetric choice is consistent with this conclusion.
\b[Equation of motion.] Taking the adjoint of the free <Dirac equation> yields $i(\partial_\mu\bar\psi)\gamma^\mu+m\bar\psi=0$. Its transpose, multiplied by $C$, is
$$
iC\gamma^{\mu T}\partial_\mu\bar\psi^T+mC\bar\psi^T=0.
$$
The defining matrix relation converts this into
$$
\boxed{(i\gamma^\mu\partial_\mu-m)\psi^c=0.}
$$
Thus the <charge conjugation of a Dirac field> maps free solutions to free solutions. With an external electromagnetic field, its charge would also reverse; no such background is present here.
\b[Chirality.] Define the <chiral projectors> $P_L=(1-\gamma^5)/2$ and $P_R=(1+\gamma^5)/2$. The operator $\hat C$ acts on the field operators and commutes with these numerical matrices:
$$
\boxed{\hat C\psi_L\hat C^{-1}=P_L\psi^c,\qquad
P_L(\hat C\psi_L\hat C^{-1})=\hat C\psi_L\hat C^{-1}.}
$$
The transformed component is therefore left-chiral, as requested. There is a useful distinction: the algebraic charge conjugate of a left-chiral spinor is right-chiral. Indeed $C\gamma^{5T}C^{-1}=\gamma^5$ and $\overline{P_L\psi}=\bar\psi P_R$ give
$$
(\psi_L)^c=P_R\psi^c,\qquad
\hat C\psi_L\hat C^{-1}=(\psi_R)^c.
$$
This is the <operator conjugation of a chiral field component>; it avoids confusing projection after operator conjugation with charge conjugation of the already projected spinor.
\b[Baryogenesis.] Consider equal initial populations of a heavy particle $X$ and its antiparticle, with two baryon-number-violating decay channels carrying different final baryon numbers $b_1,b_2$. Let $r$ and $\bar r$ be the branching fractions into a channel and its antiparticle channel. The net baryon number produced per initial particle-antiparticle pair is
$$
\Delta B=(b_1-b_2)(r-\bar r).
$$
Exact <charge conjugation> symmetry equates the conjugate branching fractions and makes this vanish. Exact <CP symmetry> also equates the fully summed conjugate decay rates, because <parity> reverses momenta and helicities without changing baryon number. Thus both <violation of charge conjugation> and <CP violation> are necessary for generating an asymmetry from a symmetric initial state.
<Violation of charge conjugation> alone can generate a helicity asymmetry while leaving total baryon number zero. For example, with $L,R$ denoting helicities, <CP symmetry> may enforce $\Gamma(X\to q_Lq_L)=\Gamma(\bar X\to\bar q_R\bar q_R)$ and the analogous relation with $L,R$ exchanged, even when same-helicity charge-conjugate rates differ. The <CP rate cancellation in baryogenesis> means that summing over helicities cancels the baryon asymmetry. These are the symmetry requirements in the <Sakharov conditions>; baryon-number violation and departure from thermal equilibrium are also needed in the conventional decay mechanism.
Back to article page
