Re: Gauge Transformations in Momentum Space?
- From: "Igor Khavkine" <igor.kh@xxxxxxxxx>
- Date: Wed, 21 Sep 2005 07:03:12 +0000 (UTC)
mikem@xxxxxxxxxxxxx wrote:
> I wrote:
>
> >> 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.
>
> Igor Khavkine wrote:
>
> > [...] when the whole energy spectrum gets shifted by the same
> > constant, no mixing between positive and "negative" energy states
> > occurs. So gauge transformations don't do much, just as they should.
>
> Hmmm. I now see that my simple example of "theta(x) = wt" was too
> simple to illustrate my question adequately. Let me try again...
>
> I know that we need the EM potential to get a gauge-covariant
> derivative, and all that. But I want to focus just on U(1)
> acting on the fermion field. This is because I want to know
> whether or not the gauge-transformed fermion field lives in
> the same Fock space as the original untransformed field.
I think you still get a null effect, at least in the way that I
understand what you are doing. What you are doing is a simple field
redefinition:
chi(x) = exp(i theta(x)) psi(x).
If you want to keep the same dynamics then you'll have to perform the
same transformation in the Lagrangian. The gradient terms in the
Lagrangian will force extra theta(x) dependent terms into the
Lagrangian. This will change the spectrum of the the solutions of the
linear part of the equations of motion for chi(x). When theta(x) was
linear in t, this only gave a constant energy shift. Now the
rearrangement of the spectrum might be more complicated, but the result
is the same. When you quantize, there will be a map between the
creation/annihilation operators for the chi(x) and psi(x) fields.
Or, perhaps, I am still misunderstanding what you are trying to do.
Igor
.
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