In a seminar, I heard that the unitary aspect of representations was important physically, because in quantum mechanics unitarity is closely tied to the conservation of probability. Could someone explain why this is so?
After a bit of thought, one vague connection I can think of is that a matrix generated by a Hermitian matrix, $$U =e^{iH},$$ where $H$ is Hermitian is bound to be unitary . As Hermitian matrices correspond to observables, this must have some signficiance. Though, as I said, I still have the faintest idea.
Answer
Probabilites are (squares of) probability amplitudes, which can be obtained as inner products of vectors on Hilbert space: $\langle X|Y \rangle$. Under a transformation U, the ket transforms as
$$|Y\rangle \rightarrow |Y'\rangle = U |Y\rangle$$
and the bra as
$$\langle X| \rightarrow \langle X'| = \langle X| U^{\dagger}$$
So if U is unitary, the probability amplitude $\langle X|Y \rangle$ is maintained under the transformation, since $U^{\dagger}$ = $U^{-1}$ and hence $\langle X'|Y' \rangle = \langle X|Y \rangle$.
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