Amazing bid by Thiemann to absorb string theory into LQG

[arivero]

Originally posted by marcus
BTW Alejandro, am I right that you have discussed DSR in past threads? I seem to recall your mentioning Amelino-Camelia, maybe also Magueijo. Might you contribute your current opinions if we began a thread on DSR?

Yes I also remember to take a look to the papers on doubly special relativity a year ago but I do not remember which my opinion was. So I guess I was not very excited. Still it could be interesting to open a thread on it. I will keep an eye on the forum.

 

[Urs]

spectral action

Hi Alejandro -

you are an expert on NCG: Do you have any experience with using the spectral action principle for getting gravity and gauge theory form a spectral triple?

I am asking because Eric Forgy and myself are planning to insert our http://www-stud.uni-essen.de/~sb0264/p4a.pdf into the spectral action principle in order to get a theory of discrete gravity. Any help by experts is appreciated! :-)

In general, I would be interested in hearing your comments on the paper at the above link. The introduction has a brief overview over the central ideas.

All the best,
Urs

 

[jeff]

Originally posted by selfAdjoint
How do you go about quantizing the 1+1 toy model?

I'm not sure what's puzzling you here.

Originally posted by selfAdjoint...where does the "by hand" [i.e., urs: "The classical gauge group is imposed by hand in the quantum theory in the approach Thomas Thiemann is using"] come into Thiemann's derivation?

I don't want to put words in urs's mouth, but I think he meant that thiemann defines his theory so that the structure of the classical lie algebra of the gauge group is preserved in the quantum theory. See the paragraph of equation (5.2) for the general idea.

 

[Urs]

Hi Jeff -

yes, exactly, that's what I meant. The point is that what Thiemann does is find some operators which represent the classical gauge group on his Hilbert space. These operators, the $$U_\pm(\varphi)$$ don't come from quantiizing the classical first class constraints. They come from using analogy with the classical theory.

Note that on a large enough Hilbert space it is possible to find operators that represent all kinds of groups. (Essentially because one can think of these operators as being large matrices and almost everything can be expressed in terms of large matrices.)

 

[selfAdjoint]

OK, now I am baffled. Group averaging is a method of quantizing a hamiltonian system with constraints.

Thiemann says:
Alternatively in rare cases it is possible to quantize the finite gauge transformations generated by the classical constraints by the classical constraints provided they exponentiate to a group.

It seems to be this exponentiation that you regard as put in by hand. I have been trying to track down the source of Thiemann's statement above but so far have been unable to find it.

Once he starts on group averaging he has available the theorem in the Giulini and Marolf paper he cites as [23], if you can meet the hypotheses of their group averaging construction, then the quantization is essentially unique, at least there is a unique "rigging function" from the kinematical Hilbert Space to the physical one. So if there is a quantization at all, and G.A. applies, then the G.A. quantization is that existing quantization.

So it doesn't look like G.A. takes us away from the familiar quantization with the anomalies; it must be the exponentiation that does it.

So is it your thesis that the exponentialion of the constraint operators is arbitrary? Thiemann seems to be saying he has support for doing it, and it's not just an arbitrary procedure.

 

[Urs]

so what's really the problem?

Hi selfAdjoint -

good that you insist on clarifications about this point. It is THE most crucial point, apparently.

Yes, the group averaging method as such is most probably completely uncontroversial as a mathematical technique.

But note that Thiemann writes "provided the CLASSICAL constraints exponentiate to a group". So he checks if the classical constraints exponentiate to a group and then he constructs an operator representation $$U(t)$$ of that group on a Hilbert space. But these $$U(t)$$ are not the exponentiated QUANTUM constraints (if there is an anomaly)! They are just some operators that represent the classical symmetry group of the system.

So this way the classical symmetry is put in 'by hand'. Distler says, an I think that he is right, that this is not how QFT works. We should not just try to represent the classical symmetry on a Hilbert space.

Even though mathematically this can be done, it is not related (or so the claim of the critics of this approach is) to the physical process called quantization. Quantization rather requires that you represent the classical constraints $$C$$ as operators $$\pi(C)$$ on a Hilber space and demand that
$$\langle \psi|\pi(C)|\psi\rangle = 0$$. In Thiemann's approach the classical constraints themselves are not even representable on his Hilbert space.

Maybe this should be compared to the 'quantization' of a simple mechanical system where the Hamiltonian itself is not representable on the Hilbert space but where the claim is that the time evolution operator $$U(t)$$ is. Surely this would be very different from the ordinary notion of 'quantization'.

Of course Thiemann's procedure is no really 'arbitrary'. After all he models his quantum symmetry after the classical sysmmetry. The point is, however, that this is a choice of procedure different from the ordinary rules.

 

[jeff]

Originally posted by selfAdjoint
...is it your [urs's] thesis that the exponentialion of the constraint operators is arbitrary?

Adding a bit to what urs said, let's look at the paragraph of definition (5.2) I mentioned. Thiemann, under the assumptions mentioned above (5.2), takes (5.2) to be valid. He then infers from this definition that the classical lie algebra carries over to the quantum one appearing just below (5.2). The problem is that this inference is mistaken since gauge symmetries of the classical theory will pick up anomalies upon quantization so that the quantum algebra as given below (5.2) is simply wrong and thus the definition (5.2) makes no sense.

 

[Urs]

True, this isn't even self-consistently formulated, because in this section he is explicitly dealing with quantum Cs. Later on (as Thomas has also emphasized again in the Coffee Table discussion) however he defines the action of the $$U$$ as in equation (6.25)
$$U_\omega(g)\pi_\omega(b)\Omega_\omega = \pi_\omega(\alpha_g(b))\Omega_\omega$$.
This is what I am referring to all along as implementing the classical symmetry by hand on the Hilbert space. But you are right, this does not follow from (5.2) and this is what apparently confused me for quite a while, because all this discussion about non-separability of the Hilbert space etc. was essentially an attempt to understand how both (5.2) and (6.25) can be true. But (5.2) doesn't make sense at all in his quantization of the string, because
$$\pi(C_I)$$ does not even exist! As Thiemann has emphasized in the Coffee Table discussion, the Virasoro constraints $C_I$ are not representable on his Hilbert space!

Hm...

 

[selfAdjoint]

So it all turns on that "Alternatively.." phrase up above. That's the cop-out from the requirement that the $$\pi(C_{\mathcal{I}})$$ be densely defined "on a suitable domain of $$\mathcal H_{Kin}$$". Wish I knew where he got that, and just what its hypotheses are.

 

[Urs]

BTW, Distler just indirectly said at the Coffee Table that for 1+3d gravity the $$\pi(C_I)$$ cannot exist even in principle (anomaly or not) because their commutators cannot make sense.

 

[selfAdjoint]

I have just been rereading the Giulini-Marolf paper (gr-qc/9902045) and find that in their RAQ scenario, the constraints have to be defined in the auxiliary (resp. kinematic) Hilbert space. So if I read them right, you can't even do the group averaging, or at least rely on it being unique, if you don't have them. If they have them they then exponentiate them to get unitary operators to work with, and they map the constraint equations into the fact of the unitary transformations leaving the corresponding thing invariant.

 

[Urs]

Do they say anything about the case when the $$\pi(C_I)$$ don't exponentiate to a group, due to an anomaly in their commutators?

 

[selfAdjoint]

They don't mention anomalies explicitly. Here is what they do say:

Refined Algebraic Quantization is a framework for the implementation of the Dirac constrained quantization procedure which begins by first considering an 'unconstrained' quantum system in which even gauge dependent quantum operators act on an auxiliary Hilbert space $$\mathcal H_{aux}$$. On this auxiliary space are defined self-adjoint constraint operators $$C_i$$ The commutator algebra of these quantum constraints is assumed to close and form a Lie algebra. Exponentiation of the operators will then yield a unitary representation of the corresponding Lie group. We will choose to formulate refined algebraic quantization entirely in terms of this unitary representation U in order to avoid dealing with unbounded operators.

As with any version of the Dirac procedure, physical states must be identified which in some sense solve the quantum constraints $$C_{\mathcal I}$$. Physically the same requirement is given^* by the statement that the unitary operators U(g) (the exponentiated raw operators sA) should act trivially on the physical states for any g in the gauge group. Now, as the discrete spectrum of the unitary operator need not contain one, the auxiliary Hilbert space will in general not contain any such solutions. However by choosing some subspace $$\Phi \subset H_{aux}$$ of 'test states' we can seek solutions in the algebraic dual $$\Phi^* of \Phi$$... Solutions of the constraints are then elements $$f \in \Phi^*$$ for which $$U(g)f = f$$ for all g.

In RAQ , observables together with their adjoints are required to include $$\Phi$$ in their domain and to map $$\Phi$$ into itself...'Gauge invariance' of such an operator $$\mathcal O$$ then means that $$\mathcal O$$ commutes with the G-action on the domain of $$\Phi$$.







^* at least for unimodular groups. See appendix A and B for a discussion of the non-unimodular groups]

 

[jeff]

Urs,

Get thiemann's paper entitled Introduction to Modern Canonical Quantum General Relativity,

http://xxx.lanl.gov/abs/gr-qc/0110034

and look on page 41 at equations (II.2.1.2b) & (II.2.1.3b) and at the top of page 42.

 

[Haelfix]

Take a minute, and marvel at the whole situation. From what I can see, Thiemann has essentially outputed (by definition mostly) a new quantization scheme. It bothers me considerably, that I can't see a good way to disprove it, even though it flies in the face of what we are taught about quantum anomalies, the promotion of classical algebras to proper quantum commutators, etc etc

Why? B/c its so damned hard to find precise mathematically well defined theorems on any of this. I read paper after paper, where they basically tell you what's right and what's wrong, but I can't find any formal proof of uniqueness. Instead, (and people here do a much better job of seeing it than I), we are forced to look for self consistency measures in his own scheme.

I almost want to do a handwaving physicists proof, and start with a simple example of a system with an anomaly (say from the Standard model), and then apply the scheme and show it violates experiment. But then, I can't think of an example that would be sufficiently applicable.

 

[Urs]

Here is the paper that Jacques Distler had mentioned:

P. Nelson and L. Alvarez-Gaume, http://www-stud.uni-essen.de/~sb0264/HamiltonianInterpretationOfAnomalies.pdf .

selfAdjoint cites Giulini and Marolf about RAQ:

On this auxiliary space are defined self-adjoint constraint operators The commutator algebra of these quantum constraints is assumed to close and form a Lie algebra.

This is where the anomaly comes into the game. The commutator algebra of the quantum Virasoro constraints does not close to form a Lie algebra.

In a Lie algebra every commutator of two elements must be an element of the algebra again. But for the quantized Virasoro generators we instead find
$$[L_n,L_m] = (n-m)L_{n+m} + \delta_{n,-m}A(m)$$
where $$A(m)$$ is the anomaly, a number and hence not an element of the Lie algebra.

Therefore Giulini and Marolf exclude the quantization of the string by means of RAQ.

 

[Urs]

Jeff,

sorry, I cannot find the formulas that you are referring to. (?) The copy of http://xxx.lanl.gov/PS_cache/gr-qc/pdf/0110/0110034.pdf that I am looking at has no numbered formulas on p. 41 and I cannot see any formula labeled (II.2.1.2b).

 

[jeff]

Originally posted by Urs
I cannot find the formulas that you are referring to. (?) The copy of http://xxx.lanl.gov/PS_cache/gr-qc/pdf/0110/0110034.pdf that I am looking at has no numbered formulas on p. 41 and I cannot see any formula labeled (II.2.1.2b).

Oops. The correct link is,

http://xxx.lanl.gov/abs/gr-qc/0210094.

Again, look on page 41 at equations (II.2.1.2b) & (II.2.1.3b) and at the top of page 42.

Search the previously mentioned much more detailed paper under "anomaly" and "anomalies". Sorry about this.

 

[jeff]

Thiemann says that it maybe "useful to remember" that he "treated the constraints EXACTLY the same as one quantizes the Poincare group of ordinary QFT." But in that case (as well as in his treatment of the closed bosonic string) the poincare symmetry is a global symmetry and picks up no anomaly upon quantization, unlike the local gauge symmetry Diff(S¹).

 

[eforgy]

Originally posted by Haelfix

I almost want to do a handwaving physicists proof, and start with a simple example of a system with an anomaly (say from the Standard model), and then apply the scheme and show it violates experiment. But then, I can't think of an example that would be sufficiently applicable.

Hi Haelfix,

This thread has brought up the issue of anomalies in quantization. I admit I never understood anomalies and I've only learned introductory QFT from Sakurai (eons ago). I just grabbed that paper from Urs' website (before copyright lawyers turn up on his doorstep and plan to take a look at it, but in the meantime I was wondering if there are in fact experimental results that REQUIRE anomalies for their explanation and what those experiments are? Basically, I'm wondering if the appearance of anomalies is a requirement of experiment or a result of academic inertia.

Thanks,
Eric

PS: I am a bit paranoid about being criticized for being off topic so let me say that the reason I bring this up here is that the issue Distler has with Thiemann's paper is that Distler thinks the anomalies are unavoidable while Thiemann disagrees.

 

[selfAdjoint]

Originally posted by Urs
Here is the paper that Jacques Distler had mentioned:

P. Nelson and L. Alvarez-Gaume, http://www-stud.uni-essen.de/~sb0264/HamiltonianInterpretationOfAnomalies.pdf .

selfAdjoint cites Giulini and Marolf about RAQ:



This is where the anomaly comes into the game. The commutator algebra of the quantum Virasoro constraints does not close to form a Lie algebra.

In a Lie algebra every commutator of two elements must be an element of the algebra again. But for the quantized Virasoro generators we instead find
$$[L_n,L_m] = (n-m)L_{n+m} + \delta_{n,-m}A(m)$$
where $$A(m)$$ is the anomaly, a number and hence not an element of the Lie algebra.

Therefore Giulini and Marolf exclude the quantization of the string by means of RAQ.

Boy, unless Thiemann has a good answer for this, it sure shoots down his derivation! I didn't know enough to finger the algebraic closure myself, but I suspected it. I ran through some more of these RAQ papers yesterday and it looked like they all make that assumption. They only study the nice case where no anomalies interfere.

 

[selfAdjoint]

BTW Urs, thanks for the Hamiltonian anomaly paper. Like Haelfix I'm going to study it. Since this incident has introduced some of us to anomalies and quantization we might as well learn from it. I just went over Polchinski's introduction to the central charge and it sucks. Calculate-calculate and gee! Look here! The energy isn't a tensor! See, it has this extra term! Actually I was by this development in an online study group a couple of years ago, but it sure didn't prepare me to couple to this discussion.

 

[jeff]

Originally posted by eforgy
I am a bit paranoid about being criticized for being off topic

My criticism of your post as being OT was quite unfair so don't worry about going a bit OT. Really really sorry about that.

Originally posted by eforgy
...I'm wondering if the appearance of anomalies is a requirement of experiment or a result of academic inertia.

Anomalies are symmetry violators left behind by regulators when they're removed. While gauge theories must be anomaly free to be consistent - a fact which can be used to constrain them - global symmetries can be violated without causing problems. In fact - and this is typically the first example of anomaly one comes across in QFT courses - the appearance of an anomaly breaking a global symmetry of the strong interaction solved the so-called "π₀ decay problem" of explaining the observed rate of the dominent decay mode π₀ → 2γ.

 

[jeff]

Originally posted by selfAdjoint
...the nice case where no anomalies interfere.

Perhaps nice from some purely mathematical standpoint, but as I mentioned above, the requirement that anomalies violating gauge symmetries cancel can be used to make theories more predictive, which is nice since uniqueness in fundamental theories is highly desirable for obvious reasons.

For example, thiemann advertised the LQG-string as working in any number of spacetime dimensions as if it's not being able to explain why there must be some unique number of spacetime dimensions is a good thing. In the bosonic string theory based on the polyakov action on the other hand the requirement that the weyl anomaly vanish gives us a definite answer, and this is much more satisfying I think.

Originally posted by selfAdjoint
...this incident...

"Incident"?

 

[marcus]

Is John Baez trying to subsume String

I happened to see a 4 February post on SPR that might be of general interest.

Baez mentioned that in his Quantum Gravity seminar they were
quantizing the open string---the same basic venture that Thiemann has embarked on (but he treats the closed string and takes the revolutionary approach of attempting it within LQG). Urs noticed Baez remark, and he posted the following a propos questions:

--------quote from Urs--------

"John Baez" <baez@galaxy.ucr.edu> schrieb I am Newsbeitrag
news:bvbu6e$cgb$1@glue.ucr.edu...

> Almost time for the quantum gravity seminar. Today we're quantizing
> the open string with Dirichlet boundary conditions! And with any
> luck, we'll make a *movie* of what it looks like! Gotta go!"

Just out of curiosity, since we are currently discussing this with Thomas Thiemann (see http://golem.ph.utexas.edu/string/archives/000299.html#c000554): Are you quantizing the the open string with D-boundaries the standard way as for instance described in

V. Schomerus, Lectures on Branes in Curved Backgrounds, hep-th/0209241

or by adapting the 'non-standard' way described for closed strings in

T. Thiemann, The LQG-String I., hep-th/0401172

to open strings?


What do you think about this non-standard way and the objections that have been brought forward (as for instance in
http://golem.ph.utexas.edu/string/archives/000299.html#c000560)?

-------end quote--------

I cannot think of any reason to suppose that Baez seminar would, in fact, be embarked on a similar venture to Thiemann (unless it is a conspiracy !) but I guess it is (as Urs suggests) a possibility and I hope Baez will reply soon and lay the question to rest.

 

[jeff]

Originally posted by marcus
...but [Thiemann] treats the closed string and takes the revolutionary approach of attempting it within LQG...

If by "revolutionary" you mean wrong or useless.

 

[Urs]

anomalies

Eric wrote:

Basically, I'm wondering if the appearance of anomalies is a requirement of experiment or a result of academic inertia.

As Jeff has said, anomalies are 'very physical' and by no means just a formal artefact. In the standard model the effect of a global anomaly can be observed, experimentally. The effect of the local gauge anomaly can also be observed, sort of, because if it would not vanish then the standard model were inconsistent, which it apparently isn't because it is being observed! :-)

In string theory, too, there is a lot of physical information in the central charge (the prefactor of the anomaly, essentially). It fixes the number of spacetime dimensions and controls the partition function of the string, for instance.

So anomalies are not something that theorists haven't figured out how to get rid of but which should be absent. Instead, it took people quite a while to realize the role of anomalies in the standard and the physics related to that. Anomalies are a quantum effect which is just as real as any other quantum effect.

selfAdjoint wrote:

I just went over Polchinski's introduction to the central charge and it sucks. Calculate-calculate and gee! Look here!

If you want to use CFT methods then see Polchinski's equation (2.6.18) which again follows from (2.2.11). This is short, easy and straightforward.

If you are more into Fock space oscillators then see the derivation in equation (2.2.31) of Green&Schwarz&Witten. Also pretty easy, but needs some algebraic input.

If you are more a canonical kind of guy :-) you might want to look at my derivation at the Coffee Table, which uses canonical functional notation a la Thiemann, regulated appropriately.

 

[eforgy]

Hi,

The following quote from the Nelson/Alvarez-Gaume abstract is troubling.

This provides a hamiltonian interpretation of anomalies: in the affected theories Gauss' law cannot be implemented.

What?

Sorry, I am guilty of not reading much more than the abstract so far, but how on Earth can Gauss' law not be implemented? Stokes' theorem (Gauss' law being a special case) is what I have thought of as being the most profound statement in all of mathematical physics. Are Nelson et al saying that d^2 != 0??

*panic* :)

Eric

 

[selfAdjoint]

They're saying you can't implement Gauss' law because the topology of configuration space won't let you. What they say the anomaly does is put a "twist" in the topology, a la the Moebius band (which is actually their first example, though you have to read down before they admit it). Thus you can't shrink n-spheres and that shrinkability is at the heart of the generalized Gauss law. It's what topologists call an obstruction.

 

[selfAdjoint]

Once again into the breach

Urs, I have been following the discussion between you, Thiemann and Distler on the Cofee Table site.

It seems to me that Thiemann is saying "Ignore everything in sections 1 through 5, ignore group averaging and all of that. Here in section 6.1 is what I am really doing." And indeed if we look at 6.1, it does seem to be independent of what has gone before.

What he does is take the Borel intervals on the circle (which he did remark in your discussion are orthogonal if they differ anywhere - as you pointed out to me earlier!). He smears them in a particular special way with functions f^k and asserts that the "handed" smeared functions Y^k close to a Poisson *-algebra.

Is this true so far?

Then he introduces the Weyl elements W = exp(iY^k), and invokes the Baker-Campbell-Hausdorff formula to get a value for their product and concludes from this that the W's for right handed and left handed Y's commute.

Any problems yet?

He then deduces from the general intersection geometry of intervals on the circle that "a general element of A (that is, a Weyl element W) can be written as a finite, complex linear combination of elements of the form

$$W_+(I)W_-(J)$$, where $$W_{\pm}(I) = exp(i\sum Y_{\pm}^k(I))$$ for some finite collection of non-empty non-overlapping intervals, i.e. they intersect at most in boundary points.

Then he states a definition. A momentum network s is a pair $$(\gamma,k)$$ is a finite colllection of nonoverlapping intervals as above and k is an assignment of momentum to each interval. It now appears that the assignment of k agrees with the index k on his smearing function. So a momentum network operator is defined to be one of those W defined by the linear combination of exponentiated Y^k, where k is now the momentum assigned to the interval Y comes from.

All this is reminiscent of the cylinder functions in LQG.

He then defines the analogs of holonomies and fluxes in terms of the intervals and their momentum operators. He asserts that both "holonomies" and "fluxes" close to a maximal abelian subalgebra of A, the algebra of W's.

He now defines the gauge group to by two copies of the diffeomorphism group of the circle plus the Poincare group, and notes that the diffs act only on the intervals of his net, while the Poincare group Lorentz transforms act only on the momenta and its translations leave the W unchanged because they only depend on the coordinate derivatives.

At this point he is ready for his GNS consideration. And I ask, is there any anomaly visible to you in this work? Is there any reson why the GNS will not work?

 

[Haelfix]

I don't understand how Thomas expects anomalies to be representation dependant.

Again, we're sort of taught to look at the induced topology that quantization outputs. The anomaly is just an indication of nontriviality in this context. There are many examples, and branches of physics that make use of this. Incluing the Witten anomaly, applications in twisted SUSY, etc etc

However, where it starts getting hazy, is the actual assumptions that go into this. For instance, its assumed that there is some notion of a continuous metric where the gauge bundles can live. LQG and other techniques does away with this, so in a sense the fine points of the usual structural geometry we are used to need not be the same (or at least, I don't know if it needs to be the same a priori).

Having said that, it seems like he is insisting that all gauge anomalies found so far in the literature are now put into question. Thats quite a grandiose claim, and obviously subject to extreme scrutiny.

 

[eforgy]

Originally posted by selfAdjoint
They're saying you can't implement Gauss' law because the topology of configuration space won't let you. What they say the anomaly does is put a "twist" in the topology, a la the Moebius band (which is actually their first example, though you have to read down before they admit it). Thus you can't shrink n-spheres and that shrinkability is at the heart of the generalized Gauss law. It's what topologists call an obstruction.

Hi selfAdjoint,

Again being guilty of not reading the paper (which will probably go over my head anyway), I don't understand how topology , in the sense you mention, has anything to do with generalized Gauss' law. The expression

int_M dA = int_@M A

is valid in extremely general circumstances. My friend Jenny Harrison has done this for fractals even. It is certainly valid for n-spheres and any other noncontractible manifolds. It is at the heart of Urs' and my paper. It almost seems like our work will not be valid for quantum theory because we have d^2 = 0. It is hard for me to believe this. Please tell me I am misunderstanding something simple (which is usually the case).


Eric

 

[jeff]

Hi eforgy,

In topological terms, field configurations satisfying gauss's law are contractible, or equivalently, the charges generating the fields are pointlike. (For comparison, it's worth noting that in ordinary maxwellian electrodynamics magnetic fields are divergence free so that, unlike with electric charges, there are no magnetic monopoles, at least according to maxwell). So in these terms, we can say that anomalies give rise to topologically non-trivial field configurations.

 

[eforgy]

Originally posted by jeff
Hi eforgy,

In topological terms, field configurations satisfying gauss's law are contractible, or equivalently, the charges generating the fields are pointlike. (For comparison, it's worth noting that in ordinary maxwellian electrodynamics magnetic fields are divergence free so that, unlike with electric charges, there are no magnetic monopoles, at least according to maxwell). So in these terms, we can say that anomalies give rise to topologically non-trivial field configurations.

Hi Jeff,

Thanks. I tried to read through the paper. I can't say that I understand it (yet), but I do see that what he means by Gauss' law is not the same as what I mean by Gauss' law. To me (and most geometers I would think), Gauss' law is just an incarnation of the generalized Stokes theorem. The generalized Stokes theorem is valid in general. I'll have to make more effort to understand their meaning of Gauss' law. Thanks. I'm making progress.

Eric

 

[jeff]

Originally posted by eforgy
I'll have to make more effort to understand their meaning of Gauss' law.

I haven't studied the paper, but the point of my previous post was that - their precise mathematical formulation notwithstanding - I'm pretty sure that at bottom, by gauss's law they really mean contractible fields. They'd then classify anomalies in terms of field topologies.