Gauge Transformations and (Generalized) Bogoliubov Transformations.

[strangerep]

I've been discussing some things with Samalkhaiat over in the conformal
field theory tutorial. A part of that conversation (indicated by the new
title) was drifting away from CFT matters, so we both thought it was better
to move it into the Quantum Physics forum, to minimize pollution of the CFT
tutorial.

For the benefit of other readers, I'll first summarize some background...

The sub-conversation arose from Sam's post from about a year ago:

https://www.physicsforums.com/showpos...58&postcount=5

Specifically, Samalkhaiat wrote:

[...EM gauge transformations...]
$A'_{\mu} = A_{\mu} + \partial_{\mu}\Lambda$

Everything in nature indicates that this is an exact symmetry.
So we expect to find a unitary operator U , such that
$U|0>=|0>$
and,
$A'_{\mu} = A_{\mu} + \partial_{\mu}\Lambda$

are satisfied in the quantum theory of free EM field.
But this leads to the contradictory statement;
$<0|A'_{\mu}|0> = <0|A_{\mu}|0> = <0|A_{\mu}|0> + \partial_{\mu}\Lambda$
or,
$\partial_{\mu}\Lambda = 0$
(this can not be right because $\Lambda$ is an arbitrary function).

So, one is led to believe that the EM vacuum is not invariant
under the gauge transformation. i.e
$U|0> \neq |0>$
or in terms of Q
$Q_{\Lambda}|0> \neq 0$
This means that the gauge symmetry is spontaneously broken!
i.e
$<0|[iQ_{\Lambda},A_{\mu}]|0> = \partial_{\mu}\Lambda \neq 0$
but this is equivalent to the statement that the field operator has
non-vanishing vacuum expectation value;
$<0|A_{\mu}|0> \neq 0$
which is wrong because of Poincare' invariance.
So, I am baffled! [...]

Among other things, I asked Sam whether he still regarded this
as a puzzle, and whether he had since resolved it. He said:

NO, I have not There are, though, two ways of avoiding the troubles:
1) CHEATING: If we "say" that $\lambda$ is an operator (-valued
distribution), i.e., if we regard the gauge function as a q-number instead
of being a c-number, then
$$\langle 0|\partial_{a}\lambda |0\rangle = 0$$
and the paradox does not arise.

This is, however, a plain cheating, because $\lambda$ is the
parameter of the (infinite-dimensional) Lie group U(1) . If one takes it to
be an operator, then one should explain what $$|\lambda(x)| \ll 1$$
means? and tons of other questions! Of course, in the existing literature,
people never practice what they preach! When the paradox hit them in the
face, they claim that $\lambda$ is operator function, everywhere
else, they treat it as an arbitrary c-number function! For example; Weinberg says;
"$\lambda$ is linear combination of $a$ and $a^{\dagger}$ whose precise form
will not concern us .." but everywhere, in his book and papers, $\lambda(x)$ is used as
c-number function!

(2) [...]

Intrigued by the above, I (perhaps foolishly) recalled an old spr
post about "Gauge Transformations in Momentum Space" where I was
trying to figure out whether there was some relationship between
EM gauge transformations, and (generalized) Bogoliubov transformations.
(I should probably add "generalized field displacement transformations"
to the latter.)

Sam scolded me and wrote:

If you have two Fock spaces based on two different (complete orthonormal)
sets of modes u(x;p) and v(x;p), then you can expand the same field in each
set:

$$\phi(x) = \int_{p^{3}} a(p)u(x;p) + a^{\dagger}(p)u^{\ast}(x;p) = \int_{p^{3}} b(p)v(x;p) + b^{\dagger}(p)v^{\ast}(x;p) \ \ (1)$$

Completeness allows you to expand the bases in terms of each other

$$v(x;p) = \int_{k^{3}} f(p,k)u(x;k) + g(p,k)u^{\ast}(x;k) \ \ (2)$$

[This imposes certain conditions on f & g but I'm not interested in them
here] Put (2) in (1), you find

$$a(p) = \int_{k^{3}} f(p,k)b(k) + g^{\ast}(p,k)b^{\dagger}(k) \ \ (3)$$

[when g = 0, the two Fock spaces coincide]

Eq(2), Eq(3) and similar ones for $v^{\ast}, a^{\dagger}$
define the B.T's. Now, I know of no gauge transformation that can produces
these B.T's let alone multiplying by U(1) phase! Such phase can always be
absorbed by the base functions u(x;p) or v(x;p), leaving
$(a,a^{\dagger})$ or $(b,b^{\dagger})$ unchanged.

I want to focus (for now) on the last 2 sentences, in particular the
bit about multiplying by a (local) U(1) phase and absorbing it into
the base functions u(x;p) or v(x;p) ...

Since we're talking about local U(1) gauge transformations on charged
fermions in EM, I need to clarify that (at least in my understanding)
$\phi(x)$ is really represents a Dirac spinor field, but the spin-related
indices have been suppressed in the above. In the Hilbert space the
transformation needs to be unitarily implemented, i.e:

$$\phi(x)\ \rightarrow \ \phi'(x) = U[\lambda] \phi(x) U^\dagger[\lambda] = \ e^{i \lambda(x)} \ \phi(x) \ \ \ (sr1)$$

Deferring (for now) the issue of what $U[\lambda]$ looks like, I just
want to focus on the far-right side my eq(sr1) above.
Using Sam's eq(1), this becomes

$$\phi'(x) = \ e^{i \lambda(x)} \ \phi(x) = \ \ e^{i \lambda(x)} \int_{p^{3}} ( a(p)u(x;p) + a^{\dagger}(p)u^{\ast}(x;p) ) \ \ \ (sr2)$$

As stated, I don't understand how the $e^{i \lambda(x)}$ can be
sensibly and consistently absorbed into both $u(x;p)$ and
$u^{\ast}(x;p)$ simultaneously, leaving the creation/annihilation
operators unchanged. (Though maybe that's not what Sam meant?)

[There are some other matters arising from our discussion in the
other thread that I also want to pursue here, but it's probably better
to clarify one point at a time.]

 

[Demystifier]

What you do is canonical quantization, in which, as far as I know, quantum qauge invariance cannot be introduced in a rigorous way. It can be done in path integral quantization, as well as in BRST quantization.

 

[strangerep]

What you do is canonical quantization, in which, as far as I know,
quantum gauge invariance cannot be introduced in a rigorous way. [...]

Yes, that's partly why I was interested in the question. The attempts at a
rigorous canonical QFT restrict their focus to a single representation.
Unitarily inequivalent representations (UIRs) are arbitrarily excluded from
consideration purely for reasons of mathematical-convenience (no one knows
how to define a satisfactory measure over the uncountably-infinite-dimensional
space of UIRs, and hence cannot define a sensible Hilbert space). But I have
never found any compelling physical reason to back up this exclusion.

Also, Bogoliubov transformations (and field displacements) are
known to be useful/relevant in other areas of QFT (e.g: condensed matter,
Unruh-effect, neutrino oscillations, generalized coherent states).
So... I ponder about gauge transformations...

 

[Haelfix]

This reminds me of something that I believe Gupta-Bleuler used to worry about. In so far as I recall the vacuum is really understood as an equivalence class of objects that differ by zero norm. So you really want to be modding out by gauge transformations to obtain the physical vacuum.

Something to that effect.

 

[strangerep]

This reminds me of something that I believe Gupta-Bleuler used to worry about. In so far as I recall the vacuum is really understood as an equivalence class of objects that differ by zero norm. So you really want to be modding out by gauge transformations to obtain the physical vacuum.

Something to that effect.

Was that something discussed in a textbook, or just the original journal papers?
(My study of the Gupta-Bleuler method was rather superficial, and some time ago).
If you could dredge your memory for any references, that would be great.

In particular, I'm wondering how one constructs a well-defined norm for the
combined space of these objects.

Cheers.

 

[Demystifier]

Gupta-Bleuler is fine, but it only deals with one particular class of gauges, namely those that satisfy the Lorentz condition.

And it is in the textbooks, such as Ryder or Schweber.

 

[hellfire]

The sub-conversation arose from Sam's post from about a year ago:

https://www.physicsforums.com/showpos...58&postcount=5

The link does not work, could you please correct it?

 

[strangerep]

The link does not work, could you please correct it?

Oops! Sorry. Try this one:

https://www.physicsforums.com/showpost.php?p=1050058&postcount=5

 

[Haelfix]

Gupta-Bleuler is fine, but it only deals with one particular class of gauges, namely those that satisfy the Lorentz condition.

And it is in the textbooks, such as Ryder or Schweber.

It starts out much more general than that, but yea they're famous for one particular gauge fixing and a quantization method. I was thinking about one of their papers though where they analyze the full details of canonical gauge transformations on the theory and I seem to recall it goes into the OP's question at some level. My memory is hazy unfortunately, and I can't track down a reference until I get back on monday.

But anyway, I think the crux of the question is on exactly what you mean by a vacuum in your hilbert space, clearly there is a massive overcounting of states if you introduce gauge transformations in the theory, so you sort of need to mod out by the action of the gauge group in some way (which is a very hard problem, hence the need for gauge fixing).

Actually you might want to check out Gribov's early work too, this is right down his alley.