Why is an operation $T$ for which $$|\langle T \psi | T \varphi \rangle | = | \langle \psi | \varphi \rangle |$$ holds a symmetry operation?

Talking about a symmetry operation $T$ i'd argue that applying $T$ to a state $|\varphi \rangle$ of a system should not change the propability of finding the system in the state $|\psi\rangle$, so $$| \langle \psi | T \varphi \rangle | = | \langle \psi | \varphi \rangle |.$$ But Wigner's theorem, which all textbooks cite when it comes to symmetry operations, does only apply if $$| \langle T \psi | T \varphi \rangle | = | \langle \psi | \varphi \rangle|.$$

Just to make it clear: I can reproduce at least one proof of Wigner's theorem, so this is okay for me. My problem is that I don't get the point, why it can be applied to symmetry transformations.

  • $\begingroup$ More on Wigner's theorem. $\endgroup$
    – Qmechanic
    Sep 28, 2016 at 10:21
  • $\begingroup$ I want to point out that the $T$ you defined in your second equation is necessarily the identity operator, (since $\psi\rangle$ is an arbitrary state), so clearly it's a trivial case of a symmetry operation. $\endgroup$
    – march
    Sep 28, 2016 at 15:45

1 Answer 1


$T$ allows you to 'look' at the system in different ways. So if you, say, rotated your lab upside down then both $\phi\mapsto T\phi$ and $\psi\mapsto T\psi$. Otherwise you'd be measuring those two states in two different coordinate systems...


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