# How shall we define the action of Time reversal operator Θ on a bra-state ⟨φ|, i.e. ⟨φ|Θ?

Suppose a Hamiltonian H is time reversal symmetric: $$ΘHΘ^{-1}=H$$. And H can be decomposed under a complete basis $$|k,α⟩$$, i.e. $$H=∑_k∑_{α,β}|k,α⟩⟨k,β|$$, and I am trying to deduce the corresponding form for the time reversal symmetry of $$H(k)=∑_{α,β}|k,α⟩⟨k,β|$$. The answer should be $$θ·H(k)·θ^{-1}=H(-k)$$. Yeah, this is a general result of condensed matter physics when treating Hamiltonian in momentum space.

Here I encountered the problem of how should I treat $$⟨k,β|θ^{-1}$$. If I just make it equal $$⟨-k,β|$$ the right result can be deduced. But I didn't feel safe when I did this, especially I know it conflicts with Sakurai's treatment of time reversal operator:

In fact, we do not even attempt to define ⟨β|Θ. This is one place where the Dirac bra-ket notation is a little confusing. After all, that notation was invented to handle linear operators, not antilinear operators.

-Modern Quantum Mechanics(2nd ed.) by Sakurai, p292

So is it safe to treat $$⟨ϕ|θ^{-1}$$ as the bra of $$θ|ϕ⟩$$? If not, how can I complete the deduction above in a safer way?

• Leslie E. Ballentine mentions in his book Quantum Mechanics - A Modern Development that "Messiah (1966) allows antilinear operators to act either to the left or to the right, but as a consequence he must caution his readers that $(\langle\xi|A)|\phi\rangle\ne\langle\xi|(A|\phi\rangle)$, and hence the common expression $\langle\xi|A|\phi\rangle$ becomes undefined.". So Quantum Mechanics by Albert Messiah may be of interest to you. – ummg Oct 24 '20 at 1:18

The complication is the following. Normally, the bra $$\langle\Theta\psi|$$ (corresponding to the ket $$\Theta|\psi\rangle$$) would be $$\langle\psi|\Theta^\dagger$$. The adjoint $$\Theta^\dagger$$ of an operator $$\Theta$$ is usually defined by $$\langle\psi|\Theta^\dagger|\phi\rangle=\langle\Theta\psi|\phi\rangle=(\langle\phi|\Theta\psi\rangle)^*=(\langle\phi|\Theta|\psi\rangle)^*.$$ (I included a bunch of equivalent definitions for convenience). However, this definition of the adjoint does not work for anti-linear operators. This is because the left-hand side of this equation would be anti-linear in $$\psi$$ while the right-hand expressions would be linear in $$\psi$$. Instead, for anti-linear operators, we define the adjoint, which I will denote as $$\Theta^T$$, using $$\langle\psi|\Theta^T|\phi\rangle=\langle\phi|\Theta|\psi\rangle=\langle\phi|\Theta\psi\rangle=(\langle\Theta\psi|\phi\rangle)^*.$$ With this definition of the adjoint, the anti-unitary condition becomes $$\Theta^T=\Theta^{-1}$$. However, it's not completely clear what $$\langle\Theta\psi|$$ is. There is a relationship, but it's not as simple to see in the bra-ket notation. We have to use the completeness relation $$\int d\phi|\phi\rangle\langle\phi|=1$$. $$\langle\Theta\psi|=\int d\phi \langle\Theta\psi|\phi\rangle\langle\phi|=\int d\phi \Big(\langle\psi|\Theta^T|\phi\rangle\Big)^*\langle\phi|.$$ I'm not sure if this can be simplified further.