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Gauge Transformations and (Generalized) Bogoliubov Transformations.

  1. Aug 20, 2007 #1

    strangerep

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    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:
    Among other things, I asked Sam whether he still regarded this
    as a puzzle, and whether he had since resolved it. He said:
    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:
    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)
    [itex]\phi(x)[/itex] 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:

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

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

    [tex]\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)[/tex]

    As stated, I don't understand how the [itex]e^{i \lambda(x)}[/itex] can be
    sensibly and consistently absorbed into both [itex]u(x;p)[/itex] and
    [itex]u^{\ast}(x;p)[/itex] 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.]
     
  2. jcsd
  3. Aug 21, 2007 #2

    Demystifier

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    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.
     
  4. Aug 21, 2007 #3

    strangerep

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    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...
     
  5. Aug 21, 2007 #4

    Haelfix

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    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.
     
  6. Aug 22, 2007 #5

    strangerep

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    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.
     
  7. Aug 22, 2007 #6

    Demystifier

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    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.
     
  8. Aug 22, 2007 #7

    hellfire

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  9. Aug 22, 2007 #8

    strangerep

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  10. Aug 22, 2007 #9

    Haelfix

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    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.
     
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