Time reversal in second quantization

Question

Let $mathscr{H} = L^2(mathbb{R}^d)$ denote the Hilbert space of single-particle states and let $mathscr{F}$ denote the corresponding Fock space (let's say fermion). Then the time-reversal operator $T$ on $mathscr{H}$ is just complex conjugation. It would then be reasonable to say that the induced time-reversal operator $T$ on $mathscr{F}$ is an anti-unitary map on $mathscr{F}$ such that $T c^*(f) T^*=c^*(Tf)$ where $c^*$ are the creation operators. However, are there any references that such an induced anti-unitary operator exists?

Andrew Yuan · Accepted Answer

I realized that this is actually pretty easy to define. We can define $W_T: mathscr{F}to mathscr{F}$ by
$$
W_T = Joplus Toplus T^{otimes 2} oplus cdots
$$
where $J$ is complex conjugation on $mathbb{C}$. It's then easy to check that $W_T$ is anti-unitary and $W_T c^*(f) W_T^* = c^*(Tf)$.

Time reversal in second quantization

One Answer

Add your own answers!

Ask a Question