What is the Connection Between Unitarity and Anomalies in Quantum Mechanics?

[selfAdjoint]

LQG has lots of unitary operators--you and I have gone over the unitary representation connected with work of L-O and S-T months ago, for example.
But it does not have unitary hamiltonian/ unitary time evolution (my only point). so I guess we agree!

But note Larsson's caviat: The representation that hath unitarity but hath not an anomaly is the trivial representation! In view of this the proof that the A-L representation is unique does not now seem so striking; of course the trivial representation is unique! And the symptom of this is the lack of a hamiltonian!

Locality without a metric is not difficult; topologists do it all the time. Think of Hausdorf spaces. Larsson uses a closeness criterion that he gets from the connection.

 

[marcus]

But note Larsson's caviat: The representation that hath unitarity but hath not an anomaly is the trivial representation! In view of this the proof that the A-L representation is unique does not now seem so striking; of course the trivial representation is unique! And the symptom of this is the lack of a hamiltonian!

strange he would say that
Smolin discussing the same thing --- page 11 of Invitation, 3 lines past equation (12) --- says words to the effect that:

"It is less trivial to prove, but still true, that there is no anomaly."

Forgive me if I am skeptical that Larsson or anyone can show in this case that the unitary representation of the diffeomorphisms is the trivial representation!

Both Ashtekar and Smolin are defining their unitary rep of the diffeos in analogous fashion, nearly the same notation. It seems to be important to prove that there is no anomaly and Smolin takes the trouble to mention this.

Thomas Larsson seems to be flying in the face of a lot of expert academic folk (who do not think their unitary rep of the diffeos is equal to the trivial representation). It would be interesting if he would give online references (links) and spell out his definitions.

 

[marcus]

here is the sort of thing Thomas says that makes me suspicious. this is from a post of his on SPR, I have bolded part for emphasis.

"The secret reason why canonical quantization of diff-invariant
theories in more than 2D fails is that the relevant diffeomorphism
group anomaly is little known. The diffeomorphism generators should
be represented by unitary operators on a conventional Hilbert space,
and all non-trivial such representations are anomalous. Since neither
the string theory nor LQG camps care about these anomalies in 4D,
they cannot do canonical quantization."

why does he say "conventional"----why not simply say Hilbert space.
does he have anything special in mind?

and the ordinary meaning of a non-trivial unitary rep is one that is not constantly equal to the identity.

it sounds strange that only Thomas should be right and all the String theorists and Loop gravitists should be wrong, because they don't understand the secret reason.

Also he is much more pessimistic than, for example, I am, bout string!
here is thomas' 12 October post at NEW. I have bolded some key sentences for emphasis.

His post was in response to: If you were Witten, what would you do to "fix up" string theory as it's known today (besides fixing up diffeomorphism anomalies)?
What would convince you to change your mind and be in support of string theory?

---quote from Larsson---
In the unlikely event that string theory acquired massive experimental support, I guess that I would have to believe in it. But the present situation is rather the opposite.


The construction of a quantum theory with some prescribed symmetries is, from my perspective, the same thing a constructing the representation theory of the group of symmetries. There is really a 1-1 correspondence:
1. Given a quantum theory, its symmetry group acts by a unitary representation on the Hilbert space.
2. Given a unitary representation of some group, the Hilbert space on which it acts is the Hilbert space of some quantum theory.


In particular, the Hilbert spaces of the fully interacting gauge-invariant or diff-invariant theories carry unitary representation of the groups of gauge transformations and diffeomorphisms. Perhaps one should factor out gauge symmetries, although I don't see why - it is definitely not necessary for consistency (unitarity). But this is really irrelevant for the argument. The anomalies must be there at least before factoring them out, so if you cannot write down the anomalies in the first place, you lose.


I am pretty sure that there is no way to fix string theory. The representations look the way they do, and their Hilbert spaces look rather like fixed versions of field theory. I don't see any way to "fix" SU(2) to allow for unitary spin-1/4 representations either.


I don't have a clue what I would do if I were Witten, and I don't really care. It's not my problem.

Posted by: Thomas Larsson at October 12, 2004 12:20 PM
---end quote---

I am sorry but this is just too over-the-top for me. Perhaps can you, selfAdjoint, provide some corroborations in the form of peer-reviewed articles that back up Thomas claims?

 

[selfAdjoint]

The connection between unitarity of the representation on Hilbert space and the existence of an anomaly is I believe known, if not well-known. Urs has cited it too, and provided me (I think it was he) with an old paper from Communications in Mathematical Physics (99, 103-114 (1985)), Hamiltonian Interpretation of Anomalies by Philip Nelson and Luis Alvarez-Gaume, which derives the result (it is not available online, at least not without a subscription to the journal). Larsson's derivation in the post I linked to is clearer: if the four dimensional diffeomorphism group had a unitary representation without an anomaly, so would its 1-dimensional subgroups by simple restriction. But we know the only unitary representation of the`1 dimensional diffeo group without an anomaly is the trivial representation.

Notice there is today (10/16/04) a reply to Larsson by Arnold Neumayer which clearly distinguishes the two senses of unitarity that were confusing us, that of the S-matrix or Hamiltonian and that of the representation in Hilbert space. BTW, I think you are overinterpreting Larsson's reference to a conventional Hilbert space, IMHO he just means not one of the various "rigged" Hilbert spaces that have been introduced recently.