I'm considering this question because the realization of SOC by I. Spielman's experiment and other treatment to realize SOC are just using some kind of unitary transformation. Besides, they also transform the system to the comoving frame to realize gauge field. Is this method making sense? We know in classical mechanics, watch a system in an accelerated frame and we will find different phenomena, so is it really so by doing unitary transformation to study the original system?
Unitary transformations in non-relativistic QM keep probability density unchanged, and this is enough for physical equivalence as long as non-relativistic phenomena are conserned. Relativity Principle in such a case means that transformation of coordinates, corresponding to transition into unifromly moving system, turns the Schrodinger equation into formally non-covariant form. By uniform transformation one can restore the standard form of the Schrodinger equation for equivalent wave-function (it gives the same probability density).
Transition to accelerated frame changes the Schrodinger equation, but NO UNITARY transformation can restore its form in the inertial frame. The best you can do is to restore the form of the Schrodinger equation up to additional inertial force term. This is exactly what we have in classics.
The situation is more complicated in relativistic theory, as far as generally speaking wave-function probability density can not be defined. (modulus square of a psi-function does not have a sense of probability density anymore ). This is a consequence of the fact that number of particles is no longer conseved.
The squeezed states can be obtained from the coherent states by means of the unitary transformation. However the physical meaning of these both states is different in view of the form of the uncertainty relations.
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