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Assume a complete relativistically normalized basis of scalar momentum eigenstates transforms asThis states both reversal of spatial momentum and preservation of the normalization of basis quantum states. Antilinearity alone would not suffice: multiplying a conjugation operator by two gives an antilinear operator that multiplies squared norms by four.

Expand $|\varphi ⟩=\int d{\Pi }_p\varphi (\mathbf{p})|\mathbf{p}⟩$ and similarly for $|\psi ⟩$, with $d{\Pi }_p=d^{3}p/((2\pi {)}^{3}2E_p)$. This Lorentz-invariant phase-space measure is unchanged under $\mathbf{p}\mapsto -\mathbf{p}$. The antilinearity of the quantum time-reversal operator gives conjugated expansion coefficients. Orthogonality of the momentum eigenstates and cancellation of their unit-modulus phases yieldSince momentum reversal is a bijection of the complete basis, $\hat{T}$ is also onto. ThusThis proves antiunitarity, with the normalization hypothesis explicitly included. For scalar multiparticle quantum states the same argument uses the complete occupation-state basis and reverses all momenta; it is not restricted to a single-particle wave packet.

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.