Time dependent Hamiltonian and Gauge invariance In general, in quantum mechanics we can prove probability current or the Schrodinger equation and other quantities are gauge invariant. However, the Hamiltonian isn't gauge invariant. Under a gauge transformation, the Hamiltonian operator changes(or have i understood wrong?) Does this mean that the Hamiltonian doesn't describes a true physical quantity like in classical mechanics?Closing, if the above are correct, do they have any affect on the principle of least action?
Thank you.
Note: The Hamiltonian is: $$H_f = {1 \over 2m} [P- qA(R,t)]^2 +qU(R,t) $$After a gauge transformation: $$H_g = {1 \over 2m} [P- qA'(R,t)]^2 +qU'(R,t) $$. Thus, we have $$H_f \neq H_g $$
 A: 
Does this mean that the Hamiltonian doesn't describe a true physical quantity like in classical mechanics?

Even in classical mechanics, Hamiltonian for one particle in external field EM is function
$$
H(\mathbf r,\mathbf p) = \frac{(\mathbf p - \frac{q}{c}\mathbf A(\mathbf r, t))^2}{2m} + q\phi(\mathbf r,t)
$$
where $\mathbf A,\phi$ are any of the valid functions that describe the same external EM field.
The form of the Hamiltonian (dependence on the potentials and $\mathbf r,\mathbf p) is unique, but its value is not; it depends on the choice of the above two functions ("choice of gauge").
This non-uniqueness is no big deal, as Hamiltonian function is mostly a theoretical concept that is useful to formulate the laws and derive other laws; it non-uniqueness is not necessary for that use.
In classical physics, the laws and their consequences can be formulated also in a gauge-independent way with EM fields $\mathbf E,\mathbf B$ only. The potentials and the Hamiltonian can be avoided.
The situation with gauge-dependence is similar to the one with kinetic energy having value that depends on the inertial system it is evaluated for. The value is frame-dependent, but it poses no problem for its use.
A: The issue is that the electromagnetic field and its gauge transformations are treated classically here - they are not operators of the quantum theory, but "tacked on" because we want to describe how a quantum object interacts with the electromagnetic field without treating the EM field itself as a quantum object.
"Gauge-invariance" in this half-quantized theory is manifested by the Schrödinger equation being gauge invariant, i.e. by the dynamics being gauge-independent. The Hamiltonian itself is indeed not "gauge-invariant" because the gauge field has not been quantized, and we have not passed to a space of states where the physical states are gauge invariant.
The proper way to obtain manifest gauge invariance of the theory is to quantize the electromagnetic field, i.e. let $A^\mu$ become an operator-valued quantum field. This, however, belongs to the realm of full-blown QFT, and is hence not done in the ad-hoc approach of classical quantum mechanics to interactions with the electromagnetic field. (It is also not necessary to get many nice results)
