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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.

Write $f_s=(-1{)}^{1/2-s}$ for $s=\pm 1/2$, so $f_{-s}=-f_s$. Under the antiunitary operator $\hat{T}$, the coefficients and exponentials in the mode expansion of a Dirac field are conjugated as well as the creation and annihilation operators being transformed. ThusRelabel $q=p_T$ and $r=-s$. The stated Dirac spinor identities implyAlso $q_T\cdot x=-q\cdot x_T$. Substitution reconstructs the original Dirac field at the reflected time:The minus sign comes from reversing the spin label, not from anticommuting field operators. This is the relation between the time-reversal matrix and the charge-conjugation matrix in the supplied spin phases and intrinsic phase convention. Rephasing the charge-conjugation matrix or the intrinsic time-reversal phase can change its displayed form. It proves time reversal of a Dirac field without identifying an antiunitary operator with its finite-dimensional spinor matrix.

A quantum time-reversal operator is an antiunitary operator reversing spatial momentum and angular momentum. For a time-reversal-invariant Hamiltonian operator, its conjugation of $i$ makes it intertwine forward and backward unitary time evolution without reversing the energy spectrum. A scalar momentum eigenstate may transform as $\hat{T}|\mathbf{p}⟩=e^{i\chi (\mathbf{p})}|-\mathbf{p}⟩$. With unit-modulus phases and a complete normalized basis, expansion of arbitrary states proves antiunitarity.