From: Thomas Larsson Subject: Re: General covariance, background independence Date: Thu, 8 Nov 2001 03:19:43 GMT Newsgroups: sci.physics.research Summary: [missing] Jacques Distler wrote in message news:021120010304136442%distler@golem.ph.utexas.edu... > In article <9rsoel$a11@gap.cco.caltech.edu>, Thomas Larsson > wrote: > > > OK. There are really two ways to look at gauge symmetries. Your way, which > > is the standard one, is to consider gauge symmetries as a redundancy in > > the description of the system. One then wants to eliminate the gauge > > degrees of freedom by passing to orbit space, but this cannot be done in > > the presence of anomalies of any kind. However, there is an alternative way > > to look at gauge symmetries. Keep the full, redundant, Hilbert space and > > require > > that it carries a representation of the relevant Lie algebra. Classically, > > and in the absense of anomalies, the two viewpoints are completely equivalent, > > > No, they are not equivalent, even classically. Oh yes! > The "standard" one corresponds to restricting to the INVARIANT SUBSPACE > (trivial representation) of the "big" Hilbert Space. i.e. passing to the space of gauge orbits. In the absense of anomalies, representations and orbits contain equivalent information. Let the symmetry be described by a Lie algebra g, [J^a, J^b] = f^ab_c J^c. Then the BRST charge Q = J^a c_a + 1/2 f^ab_c b^c c_a c_b is nilpotent: Q^2 = 0, where b^a and c_b are fermionic oscillators. As usual, we can construct cohomology groups H^p(Q) = ker Q/im Q, and one identifies the physical Hilbert space as the cohomology group H^0(Q). In other words, the points on gauge orbits are identified. By choosing a representation R of g, J^a -> R(J^a), one obtains a representation R(Q) of Q. Hence representations and orbits contain equivalent information in this case. The BRST approach breaks down if the symmetry has an anomaly, i.e. if the Lie algebra acquires an extension. If so, Q^2 != 0 and the cohomology groups can not be formed. (You can of course recover nilpotency by introducing ghosts for the generators of the anomaly.) However, the extended algebra may still be representable. Hence the notion of representation is more general than orbits, because the former is valid in the presence of some types of anomalies. Another way to say the same thing is that classical gauge symmetries may become non-gauge symmetries after quantization. This is not really a big mathematical problem, provided that the anomalous symmetry is representable. Of course, it is a big problem if you choose a quantization method which assumes that gauge symmetries must remain gauge, such as light-cone or BRST quantization. If you assume that gauge degrees of freedom must be redundant after quantization, and they in fact are not, you run into an inconsistency. But this is really a problem with your assumption, not with the quantum theory itself. Anomalous gauge symmetries are neither a mathematical nor a physical problem. The ABJ anomaly is known to cancel, but it is related to the Mickelsson- Faddeev algebra, which presumably does not have any representations, so this is a bad anomaly even from a mathematical point of view. The fundamentals of string theory are ruined, but this is not a physical problem since string theory lacks experimental support. Moreover, if the anomalous algebra is representable, its lowest-energy modules are well-defined quantum theories in their own right. They have countable bases, products of normal-ordered operators have finite matrix elements, the energy is bounded from below, etc. In fact, every (up to isomorphism) system with the prescribed anomalous symmetry is obtained in this way. This is tautological: if an algebra acts on a system, the system carries a representation of this algebra. The new thing is that lowest-energy representations of the anomalous diffeomorphism algebra have been constructed, starting with.the Rao-Moody modules in 1994. Let me elaborate on the phrase "lowest energy". It turns out that well-defined Fock modules of vect(N) can only be constructed if its classical modules, i.e. tensor densities, are expanded in a Taylor series around a marked curve ("the observer's trajectory") prior to quantization. The natural algebra to consider is thus not vect(N) by itself, but rather vect(N)+vect(1) (direct sum), where the extra vect(1) factor describes reparametrizations of the observer's trajectory. Its L_0 generates rigid proper time translations, and it thus acts as a Hamiltonian. It is the eigenvalue of this L_0 which is bounded from below, and it also induces the partial order needed for normal ordering. Despite the presence of this "Hamiltonian" (which does not vanish, btw.), the Fock modules contain no information about dynamics, i.e. the Einstein equations. It is possible to construct modules that depend on dynamics, as the cohomology of a certain complex of Fock modules. These modules are mathematically completely consistent in the same way as the Fock modules are. The tricky thing is to reconstruct the fields from its Taylor coefficients (or p-jets), i.e. to take the limit p -> infinity. Under reasonable assumptions, this seems to work best in four dimensions. Unlike you, I like anomalies, but I dislike infinite anomalies. These ideas are forced upon us by representation theory. It is e.g. not possible to construct vect(N>1) Fock modules starting directly from tensor fields; that would give rise to a infinite central extension which does not exist. Moreover, you must add the extra vect(1) factor, because otherwise the cocycles become horribly messy (a big problem in Toroidal Lie algebras), and this factor immediately gives you a natural Hamiltonian. > > All the examples (mostly in D=2) where this "program" has been carried > out have a rather trivial physical explanation. Rather than "quantizing > an anomalous gauge theory" what you are, in fact, doing is adding > bosonic degrees of freedom to the theory whose "classical" variation > CANCEL the anomaly. > > The best-known example involves a chiral boson in D=2, which is the > "extra" degree of freedom which arises when you try to "quantize an > anomalous gauge theory" in D=2. I am not thinking of quantizing any specific system, but rather of classifying quantum theories with a prescribed anomalous symmetry, i.e. to classify representations (preferrably unitary lowest-energy irreps) of this algebra. Many examples are now known. This is very much in the spirit of CFT. > Well, since gravitational anomalies don't arise in D=4, it's not > surprising that you "don't see any problem". > Neither, then, do you need to consider cocycles for diff(M_4). This is somehow wrong. Anyway, standard quantization procedures fail for general relativity. Is this result obtained by quantizing spin-2 fields in an unphysical background metric? This is very murky, since you are then quantizing with respect to the wrong causal structure. > > (And here I thought your big motivation was precisely the existence of > such cocycles) You understood me right. > > If all we needed to know was representation theory, then physics (and > string theory in particular) would be a LOT simpler than it is. > You underestimate the difficulties of representation theory. > Alas, there is considerably more to the problem. . . We know that there is really nothing more to 2D critical phenomena. The bigger the algebra, the more info it contains. E.g., the dilatation group determines the form of the 2-point function, the little conformal group determines the form of the 3-point function, and the Virasoro algebra determines the spectrum of critical exponents. Also, tensor fields are highly reducible reps of the Poincare algebra, but irreducible vect(N) reps. The algebras that I have described are even larger, and therefore contain even more info. Whether physics contains further info remains to be seen. From a mathematical point of view, the "best" symmetries seem to be projective representations of vect(4) and mb(3|8). This very much reminds us of general relativity and the standard model, consistently quantized in four dimensions, and essentially nothing beyond that, at least not symmetry-wise. Such a picture is in considerably better agreement with experiments than string theory is. > I can't think of a *single* result in 2D conformal field theory that > has not has *some* application in string theory. I was specifically thinking of cancellation of the conformal anomaly. You cannot add up very many terms of the form c = 1 - 6/m(m+1) without exceeding 26. The relation between the Virasoro algebra and its multi-dimensional sibling is completely analogous to the relation between ordinary single-variable calculus and tensor calculus. If people who only know about ordinary calculus claim that they have found the ultimate theory, people who understand tensor calculus may eventually become quite irritated. ============================================================================== From: Jacques Distler Subject: Re: General covariance, background independence Date: 8 Nov 2001 19:29:55 GMT Newsgroups: sci.physics.research In article , Thomas Larsson wrote: > Another way to say the same thing is that classical gauge symmetries may > become > non-gauge symmetries after quantization. This is not really a big > mathematical > problem, provided that the anomalous symmetry is representable. Of course, > it is a big problem if you choose a quantization method which assumes that > gauge symmetries must remain gauge, such as light-cone or BRST quantization. > If you assume that gauge degrees of freedom must be redundant after > quantization, and they in fact are not, you run into an inconsistency. But > this is really a problem with your assumption, not with the quantum theory > itself. > > Anomalous gauge symmetries are neither a mathematical nor a physical problem. I see. You think that demanding gauge invariance is just some whim on the part of physicists. On the contrary, it is invariably *crucial* to obtaining a unitary, Lorentz-invariant quantum theory. (Exercise: prove that unitarity fails for an anomalous 4D gauge theory at 3 loops.) If imposing gauge invariance were not crucial to obtaining a consistent quantum theory, we would hardly care whether it was present as a symmetry at all. Why bother with the Higgs mechanism if you could just give gauge bosons a mass by hand? (Exercise: Take SU(2) Yang-Mills theory and just add a Tr(A_mu A^mu) mass term for the gauge bosons. Tell us what goes wrong with the theory.) > > All the examples (mostly in D=2) where this "program" has been carried > > out have a rather trivial physical explanation. Rather than "quantizing > > an anomalous gauge theory" what you are, in fact, doing is adding > > bosonic degrees of freedom to the theory whose "classical" variation > > CANCEL the anomaly. > > > > The best-known example involves a chiral boson in D=2, which is the > > "extra" degree of freedom which arises when you try to "quantize an > > anomalous gauge theory" in D=2. > > I am not thinking of quantizing any specific system, but rather of classifying > quantum theories with a prescribed anomalous symmetry, i.e. to classify > representations (preferrably unitary lowest-energy irreps) of this algebra. > Many examples are now known. This is very much in the spirit of CFT. I, unlike you, was interested in understanding the *physics* of what it meant to "quantize an anomalous gauge theory" in those (rare) circumstances where it is supposed to be possible. What are the physical degrees of freedom in the resulting quantum theory? How is unitarity restored? I found the answer quite illuminating. > > Well, since gravitational anomalies don't arise in D=4, it's not > > surprising that you "don't see any problem". > > > Neither, then, do you need to consider cocycles for diff(M_4). > > This is somehow wrong. After you're done reading Alvarez-Gaume and Nelson, try reading Alvarez-Gaume and Witten's paper on "Gravitational Anomalies". > The relation between the Virasoro algebra and its multi-dimensional sibling > is completely analogous to the relation between ordinary single-variable > calculus and tensor calculus. If people who only know about ordinary calculus > claim that they have found the ultimate theory, people who understand tensor > calculus may eventually become quite irritated. Yes, I shall have to try to remain calm . . .