Why exactly does the time-reversal operator need to be anti-linear?

Question

I checked many books and they all state that time-reversal operator is anti-linear. But why do we need it to be anti-linear? Please explain where this need actually arises.

Answer 1

The Poincare algebra implies $$ T ( i H ) T^{-1} = - i H $$ where $T$ is the time-reversal operator. (Can you prove this?)

Now, suppose $T$ is a linear operator, then $T H T^{-1} = - H$. This implies that if $|\Psi\rangle$ is an eigenstate of the Hamiltonian with energy $E$, then $T^{-1} | \Psi \rangle$ has energy $-E$. This implies that the Hamiltonian is not bounded from below, which is not desirable for a unitary theory. Thus, $T$ must be anti-linear.

Answer 2

One argument is based on the preservation of the CCR

$$\tag{1} [x,p]~=~i\hbar~{\bf 1}.$$

Let $T$ be an invertible $\mathbb{R}$-linear operator with the usual time-reversal properties:

$$\tag{2} TxT^{-1}~=~x\quad\text{and}\quad TpT^{-1}~=~-p.$$

Then

$$\tag{3} Ti\hbar T^{-1}~=~ T[x,p]T^{-1}~=~[TxT^{-1},TpT^{-1}]~=~-[x,p]~=~-i\hbar~{\bf 1},$$

i.e. $T$ is antilinear

$$\tag{4} Ti~=~-iT. $$