Conservation of probablities with non-unitary matrices?

Question

In quantum mechanics, in the context of symmetry transformations, it is often said that for a transformation $T$ to conserve probabilities it must be unitary.

But by performing any (even non-unitary) transformation on the system we are just taking account for the fact we are looking at it in a different way (i.e. using different basis vectors).

By looking at the system in a different way I cannot see how we could change the probabilities of a measurement (as long as these probability where calculated correctly, which may need the introduction of a matrix into the scalar product).

Am I correct? If so why is it said that only unitary matrices conserve probabilities and if not why not?

Answer 1

The probability to detect state $\psi$ in state $\phi$ is given by $$ P(\psi,\phi) = \frac{\lvert \langle \psi,\phi\rangle\rvert^2}{\langle \psi,\psi\rangle\langle \phi,\phi\rangle}$$ and by Wigner's theorem every sensible physical transformation $T$ (that should a priori be thought of to act on rays in Hilbert space rather than individual vectors) that preserves this probability in the sense that $P(\psi,\phi) = P(T\psi,T\phi)$ holds for all states can be given by a unitary operator on the Hilbert space of states.

Arbitrary operators do not preserve the probability, which is most evident for the non-invertible ones. If you want to modify the scalar product to make other invertible transformations preserve it, then you're effectively choosing a new scalar product such that the transformation in question becomes unitary, which means this doesn't add anything useful.