One foundational postulate of QM is that a closed physical system at one instant of time, say $t$, is completely described by a wavefunction $\psi \in S^1\subset H$ (where $H$ is a Hilbert space and $S^1$ its unit sphere). Another foundational postulate is that the wavefunction of a closed system should evolve deterministically along some orbit $t \mapsto \psi(t)$. It is therefore possible to define a time-evolution operator $U(t,t'):S^1\subset H \to S^1 \subset H:\psi(t') \mapsto \psi(t)=U(t,t')\psi(t')$. From the considerations so far we still need the linearity property for $U$ (after extending its (co)domain to $H$) to arrive at the conclusion that $U(t,t')$ be unitary (for all $t$).
Some would suggest that the linearity of $U$ is simply foundational itself and an experimentally falsifiable part of QM which does not rely on some deeper philosophical underpinning.
Many texts (see also Weinberg's "lectures in QM") however adopt the point of view that time-translation is a symmetry à la Wigner and that all transition probabilities $t \mapsto |\langle \psi(t),\phi(t)\rangle|^2$ should therefore be constant in time. Wigner's theorem then tells us that $U(t,t')$ should be either unitary or anti-unitary. A very plausible continuity argument rules out the anti-unitary option.
So which way carries the greater truth? Or is the distinction between the two viewpoints only apparent?
Personally, I have difficulty understanding the second viewpoint. Are wavefunctions not supposed to describe the system at one instant of time and not be a spacetime description? e.g. the inner product in usual QM Hilbert spaces asks you to integrate or perform certain sums related to the wave-function at one such instant of time. Therefore, it seems not evident to me that time-translation should be a symmetry in the sense of Wigner. Does Lorentz-invariance somehow force the constancy of $t\mapsto |\langle\psi(t),\phi(t)\rangle|^2$?
