T-symmetry ($T$)

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.