The time reversal operator [itex]T[/itex] is an antiunitary operator, and I saw [itex]T^\dagger[/itex] in many places(adsbygoogle = window.adsbygoogle || []).push({});

(for example when some guy is doing a "time reversal" [itex]THT^\dagger[/itex]),

but I wonder if there is a well-defined adjoint for an antilinear operator.

Suppose we have an antilinear operator [itex]A[/itex] such that

$$

A(c_1|\psi_1\rangle+c_2|\psi_2\rangle)=c_1^*A|\psi_1\rangle+c_2^*A|\psi_2\rangle

$$

for any two kets [itex]|\psi_1\rangle,|\psi_2\rangle[/itex] and any two complex numbers [itex]c_1^*, c_2^*[/itex].

And below is my reason for questioning the existence of [itex]A^\dagger[/itex]:

Let's calculate [itex]\langle \phi|cA^\dagger|\psi\rangle[/itex].

On the one hand, obviously

$$

\langle \phi|cA^\dagger|\psi\rangle=c\langle \phi|A^\dagger|\psi\rangle

$$

But on the other hand,

$$

\langle \phi|cA^\dagger|\psi\rangle =\langle \psi|Ac^*|\phi\rangle^*=\langle \psi|cA|\phi\rangle^*=c^*\langle \psi|A|\phi\rangle^*=c^*\langle \phi|A^\dagger|\psi\rangle

$$,

from which we deduce that [itex]c\langle \phi|A^\dagger|\psi\rangle=c^*\langle \phi|A^\dagger|\psi\rangle[/itex], almost always false,and thus a contradiction!

So where did I go wrong if indeed [itex]A^\dagger[/itex] exists?

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# Existence of adjoint of an antilinear operaor, time reversal

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