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Why not diffeomorphism group representation theory?

  1. Feb 6, 2012 #1
    For some reason, diffeomorphism invariance seems to be treated like a second-class citizen in the land of symmetries. In nonrelativistic quantum mechanics, we consider Galilean invariance so important that we form our Hilbert space operators from irreducible representations of the Galilei group. In relativistic quantum mechanics, I think we can do the same thing with representations of the Poincare group. (Could someone back me up on that? How do operators work in Fock space?) But when it comes time to consider quantum gravity, we do not grant diffeomorphism invariance an analogous role, citing the fact that it is a gauge symmetry and thus indicating a mere superfluousness in our mathematical description of physical states.

    I have a few issues with that. First of all, gauge symmetries can be quite important; it is the gauge invariance of Maxwell's equation that gives rise via Noether's theorem to (local) conservation of electric charge. What's the Noether charge for diffeomorphism invariance? Second of all, it seems to me that diffeomorphism invariance is more than just a gauge symmetry. Any statement that has rotational invariance, translational invariance, Lorentz invariance etc. (all locally) as it's implications surely has some physical significance. Can't we easily imagine a universe in which the laws of physics looked profoundly different in different coordinate systems?

    Has there been any work in building quantum gravity from the representation theory of the diffeomorphism group?

    Any help would be greatly appreciated.

    Thank You in Advance.

    P.S. Can someone recommend a good book on Lie group representations, direct sums, tensor products, and all that jazz?
  2. jcsd
  3. Feb 6, 2012 #2


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    I have to think more about your questions, but perhaps you like the following two articles:

    [*] "Pseudoduality", Van Proeyen and Hull (about the difference between "proper symmetries" and "pseudo symmetries", formulated in terms of sigma models)

    [*] Black hole entropy is Noether charge, Robert Wald (about how one could assign a Noether charge to diffeomorphisms, but how to reconcile that with the first article is not yet clear to me)
  4. Feb 6, 2012 #3


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    When looking at loop quantum gravity it becomes clear that (spatial) diffeomorphisms play a prominent role in its construction. The problem is that this construction is neither complete nor indisputable.
  5. Feb 6, 2012 #4
    In LQG does the representation theory of the diffeomorphism group play a role in constructing Hilbert space operators?
  6. Feb 6, 2012 #5


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    not in the usual sense
  7. Feb 6, 2012 #6


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    General covariance (or diffeomorphism covariance if you must call it that) can be thought of as the 'local' generalization of the translation group. That is, in place of global translations, xμ → xμ + εaμ, where aμ = const, an infinitesimal coordinate transformation may be written as xμ → xμ + εξμ where ξμ = ξ(x)μ. This is analogous to the gauge groups of electromagnetism we have gauge transformations Aμ → Aμ + λμ which are global or local depending on whether or not λ is a constant.

    Question: for electromagnetism, which type of gauge transformation (global or local) generates the conserved current? The short answer is: they both do. But in different ways. And so to state the short answer to the OP, both the general covariance group and the translation group generate the same conserved quantity, namely (EDIT: the stress-energy tensor, and its charge,) the energy-momentum vector.

    A very clear exposition of this topic can be found here. The point is that there are, in fact, two Noether theorems. The first deals with constant parameters (global gauge groups) while the second deals with function parameters (local gauge groups). Both of them lead to the same conserved current, but the equations of motion are required to be satisfied in the first case, while in the second case they are not.
    Last edited: Feb 6, 2012
  8. Feb 6, 2012 #7
    So then what is the role for irreducible representations of diffeomorphism group in LQG? If Loop Quantum Gravity doesn't really use it, is there any other approach that does?
  9. Feb 7, 2012 #8


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    In LQG it is claimed that the final version of the theory is diff. inv. b/c the symmetry has been reduced to the identity; however the way towards this rep. is slightly obscure and the implementation of all constraints including the Hamiltonian is not yet fully understood. Anyway, the diff. inv. symmetry is "unphysical" just like a gauge symmetry, therefore physical obervables and Hilbert space vectors should live in the trivial representation.

    Compare this to QCD: yes, you are right, in that case you start with SU(3) representations, quarks live in the triplet, gluons live in the octet, in order to construct the theory. But the hysical states are constrained by the "color Gauss law" which means that all physical states are gauge invariant states i.e.live in the singlet. So even in QCD (when it comes to physical states and observables) no representation but the trivial one is used.

    Regarding your last question: I do not know any other theory that uses representation theory of the spacetime diffeomorphims group.
  10. Feb 7, 2012 #9
    But SU(3) invariance for QCD really is merely a gauge symmetry reflecting a redundancy in our mathematical description. In contrast, in any quantum field theory we get e.g. the momentum and angular momentum operators by considering irreducible representations of the Poincare group, because the Poincare group is a set of physically meaningful symmetries. But if that is the case, then surely the diffeomorphism group, which contains the Poincare group and much more things (like accelerated frames), should also be thought of in the same way. Is there any way to treat the diffeomorphism group in any way other than as a gauge group?

    By the way, how are things done in quantum field theory on curved spacetimes? What is the symmetry group used there to construct Hilbert space operators?
  11. Feb 7, 2012 #10


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    You have to carefully distinguish two concepts:

    Global symmetries like e.g. SU(N) flavor and global Lorentz or Poincare invariance in special relativity for which you can construct (well-known) representations; the physical Hilbert can be decomposed accordingly.

    Local symmetries like gauge symmetries and diffeomorphim invariance; it is true that you have Poincare invariance in GR, but this becomes a local (gauge) symmetry (I will try to find papers for you). Global or "rigid" Lorentz or Poincare invariance is not a symmetry of GR, only as a special subset of local diffeomorphism invariance.

    It should be clear that global Lorentz invariance requires a globally flat spacetime; it is also well-known that the concept of energy, momentum and angular momentum (as integral constants of motion) do no longer exist in GR in general.
  12. Feb 7, 2012 #11


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  13. Feb 7, 2012 #12
    Yes, I am aware that nontrivial global symmetries are incompatible with space-time curvature. But are all local symmetries "gauge" symmetries in the pejorative sense, i.e. just indicating mathematical redundancy not physical content? I would think local Lorentz invariance is pretty physically significant; it indicates that space-time is locally Minkowskian. Can't you still use local symmetries to find local conservation laws, and can't you form operators using the representation theory of a local symmetry group?

    I think quantum field theory in curved space-time would be relevant to all these issues.
  14. Feb 11, 2012 #13
    I found out more information on this now. It seems that the generators of the local diffeomorphism group on a manifold are the vector fields defined on the manifold. Is there any way to interpret these vector fields as operators on the Hilbert space?

    This is especially interesting, because wave functions are scalar fields ψ(x,t), i.e. scalar fields on spacetime. So how could you find represent vector fields on a manifold as operators on the space of scalar fields of the manifold ?
  15. Feb 13, 2012 #14
    "But are all local symmetries "gauge" symmetries in the pejorative sense, i.e. just indicating mathematical redundancy not physical content?"

    For a (philosophical) discussion of this issue, please see 'Gauging What's Real' by Richard Healey, OUP.
  16. Feb 13, 2012 #15


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    Yes; I do not know any counter example.

    No; diffeomorphism invariance has of course a physical content, namely independence of reference frames.
  17. Feb 13, 2012 #16
    For the record that was my quote; malreux was quoting me. I'm a bit confused. Some symmetries like translational invariance are said to be physically meaningful, and so the momentum operator is constructed using the representation theory of the local translation group (i.e. the associated Lie algebra). On the other hand, people do not do the same thing for the local diffeomorphism group; their justification for this is to dismiss this as a "gauge symmetry", by which they do not just mean that it is local, but that it is just a redundancy in our mathematical description of the physical world, not something physically meaningful.

    But I agree with you: diffeomorphism invariance is a physically significant assertions about the laws of physics taking the same form in all reference frames. Thus it stands to reason that the generators of infinitesimal diffeomorphisms should have irreducible representations as Hilbert space operators.

    Am I on the right track concerning vector fields on a manifold being the generators of diffeomorphisms on the manifold? If so, how could you construct Hilbert space operators out of vector fields? Also, am I right to assume that quantum mechanics in curved spacetime is relevant?
  18. Feb 13, 2012 #17
    "On the other hand, people do not do the same thing for the local diffeomorphism group; their justification for this is to dismiss this as a "gauge symmetry", by which they do not just mean that it is local, but that it is just a redundancy in our mathematical description of the physical world, not something physically meaningful." -Lugita 15

    The book I referenced can clarify some of the issues, such as what is meant when mathematical structure is considered unphysical or 'pure gauge'. It is written from the perspective of philosophy of physics rather than first order physics, but it is very clear with relatively simple formalism (although not toy examples). Its conclusion is basically "'Local' gauge symmetry is a purely formal feature of [e.g. Yang-Mills theories]" (p.155).

    This is not tackling what your asking about head on, by any stretch. But I found Healey's examination of classical and quantum field theories, and in particular what the transliteration of Weyl's 'gauge' means in the very different contexts of e.g. QFT and general relativity, very useful, very subtle, and illuminating.

    One last thing, there are many nice papers about QFT on curved spacetime, but I don't think I quite follow your reasoning here: "how could you construct Hilbert space operators out of vector fields?"
  19. Feb 13, 2012 #18
    Here's a tip, malreux: if you want to quote someone's post, all you need to do is click the "Quote" button next to the post.
    I'm not doing anything unusual here, just following standard quantum procedure. For example, consider nonrelativistic QM. The laws of physics are invariant under spatial translation, so we look at the representation theory of the translation group. Ultimately we find the generators of infinitesimal translations, and represent these generators as operators on the Hilbert space, in this case the momentum operators. And we find that the laws of physics are invariant under time translation, so we represent the generator of infinitesimal time translations as an operator on the Hilbert space, the Hamiltonian operator. Finally we have rotational invariance, so we construct operators on the Hilbert space, the angular momentum operators, which generate infinitesimal rotations.

    In each case, the pattern is clear. We find a continuous set of symmetry transformations, which form a Lie group. We then look at the infinitesimal transformations that are in this group, AKA the Lie algebra. We find the generators of these infinitesimal transformations. Finally we try to find self-adjoint operators on our Hilbert space which correspond to these generators. (I'm skipping a step: we often first try finding unitary operators to represent elements of the whole Lie group, and then we use those to find the self-adjoint operators representing the infinitesimal generators. For instance, we first find the unitary time evolution operators U(t) which represent the Lie group of time translations, then we express the Hamiltonian operator in terms of U(dt).)

    In the present case the symmetry we are concerned with is local diffeomorphism invariance, so I would like to similarly construct Hilbert space operators using the representation theory of the diffeomorphism group. Based on some searching, it seems that the generators of infinitesimal diffeomorphisms on a manifold are vector fields on the manifold. So somehow these vector fields have to be represented as operators on the Hilbert space. This is somewhat interesting, because the Hilbert space is the set of wave functions, which are themselves scalar fields on the manifold. So we have to represent vector fields acting on the manifold as self-adjoint linear operators acting on scalar fields acting on the manifold.
  20. Feb 14, 2012 #19


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    This is a bit of a tough subject, b/c there are a lot of different viewpoints out there and the literature is diverse (both in philosophy, notation and implementation).
    Having not been trained in quantum gravity or as a specialist in GR, this took me awhile to find people who would explain it to me properly, and I am probably mangling it.

    Very briefly there are two main points to take away..

    The first point is that typically to make sense of Diff(M) as a 'gauge' group, you are going to want to follow a specific procedure. Typically that involves working in the Hamiltonian (ADM) formulation of GR (eg the whole formalism with lapse and shift vectors etc), and you will need to introduce vielbeins. Further, it is often the case that at some stage you need to linearize the theory (b/c it is precisely there, that you see the analogy between the equations of Yang Mills and Gravity explicitly)

    So once you have done all of that (the steps can be found in textbooks), in some sense there is a formal analogy that is possible (modulo some important subtleties) and we can define exactly where we use Diff(M) and what it can act on (and in what sense it acts transitively).

    Now, in so far as that is concerned, what one gets is a sense in which the group Diff(M,xo) acts on operators in the theory. Of course, since this is a gauge symmetry (a redundancy of explanation), what Tom says is absolutely correct. AT the level of the Hilbert space, just like in gauge theory the representation theory is completely trivial, as it must! Instead of the Gauss law constraint, we will have a diffeomorphism constraint and so forth.

    Now, you have to be careful.. One needs to make a distinction between diffeomorphisms that are smoothly (isotopically) connected to the identity at infinity, and ones that are not (to wit they are called 'large diffeomorphisms'). The preceding discussion refers to the former. However, large diffeomorphisms can take physical states to new states! The appropriate group of all nontrivial large diffeomorphisms is given by the quotient of Diff (M)/Diff(M,0). This is called the mapping class group by mathematicians, and is given several different names by physicists depending on the circumstances. See papers by Domenico Giulini for some reviews

    You can kind of intuitively see why this must be so. Asymptotic observables like the ADM energy or angular momentum can and will change under the action of this group. It is at this level where 'representation theory' becomes important, and not of the full Diff(M).

    It is also at this step where things become incredibly murky and mathematically really challenging and where few exact results are known (2+1 gravity in AdS space is a famous exception).
  21. Feb 14, 2012 #20
    Haelfix, you're telling me how we CAN treat diffeomorphism invariance as a gauge symmetry, in the sense of being an artifact of our mathematical description. But I want to know whether we can NOT treat the diffeomorphism as a "gauge" group in the pejorative sense, but rather as a physically meaningful symmetry group, the way we treat the Galilei group in nonrelativistic QM. So we should be able to construct operators on our Hilbert space in terms of the representation theory of diffeomorphism group. Just as in nonrelativistic QM we construct the angular momentum operators out of the generators of infinitesimal rotations, I want to construct self-adjoint operators out of the generators of infinitesimal diffeomorphisms. Is that too much to ask? :smile:
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