Gauge Transformations in Momentum Space?

Most textbook treatments of gauge transformations do it in
position space. So far, I haven't found any that discuss
in detail what they look like in momentum space, and what
issues arise in the QFT Fock space.

For example, consider the U(1) group for electromagnetism
acting on Dirac electrons in QED. Textbooks write it as
something like:

Psi(x) -> exp(i theta(x)) Psi(x)

where theta(x) is a real scalar function of x (in 3+1
spacetime of course).

To pass to momentum space, we need to assume that
exp(i theta(x)) has a reasonable Fourier transform,
such that the transformation can be represented
in momentum space as an integral operator whose
kernel is a distribution.

Take a simple case: theta(x) = wt, where 'w' is a real constant.
I.e: in position space we have

Psi(x) -> exp(iwt) Psi(x)

In momentum space, this just shifts the energy by an amount 'w'
I.e: E -> E - w.

So old positive-energy modes in the energy range 0 to w get
transformed into negative-energy modes. In the 2nd-quantized Fock
space this means we're mixing some of the annihilation and creation
operators - because they were defined in terms of the original +ve
and -ve energy modes. Such mixing usually means that we're mapping
between unitarily inequivalent representations, i.e: between
orthogonal Fock spaces.

I'm interested in finding explicit operators which are form-invariant
in both representations. I tried Google-Scholar but didn't have much
success.

So my question is:

Do any textbooks or review papers discuss this stuff at length?
(I don't mean just the usual Bogoliubov transformations from
condensed matter physics which map between inequivalent reps,
but specifically for standard model gauge transformations
in momentum space, and hence Fock space(s).)

TIA.

.

Relevant Pages

  • Gauge Transformations in Momentum Space?
    ... in detail what they look like in momentum space, ... in position space we have ... Do any textbooks or review papers discuss this stuff at length? ... (I don't mean just the usual Bogoliubov transformations from ...
    (sci.physics.research)
  • Re: Gauge Transformations in Momentum Space?
    ... > in detail what they look like in momentum space, ... in position space we have ... > (I don't mean just the usual Bogoliubov transformations from ... > but specifically for standard model gauge transformations ...
    (sci.physics.research)