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March 28, 2006

Gerbes and Quantization

Posted by Urs Schreiber

Here is a report on

Jose M. Isidro
Gerbes and Heisenberg’s Uncertainty Principle

[Update (3 Mar, 2006): Today the paper has been replaced on the arXIv with a revised version. The following applies to the original version.]

In this paper, the author claims to construct a gerbe using data provided by the quantum mechanical propagator of any system. He furthermore claims to derive a relation between the 3-form curvature of that gerbe and Heisenberg’s uncertainty principle.

The main idea is expressed in the paragraph below equation (11). Here the author considers a covering of the phase space of some mechanical system with open sets. Fixing a point in the open set U i and one in U j he denotes by τ ij the phase of a quantity which is supposed to be essentially a quantum mechanical propagator between these points. He then says that τ ij defines a gerbe trivialization.

For that to make sense, however, τ ij would have to be a function on the intersection of U i with U j. But, as defined, τ ij is a function on the cartesian product of U i with U j. It depends on two points of phase space, not just on one point in a double intersection.

Similarly, the quantity g defined in equation (13) as a cyclic product of three τ is not a function on triple intersections. Hence it is not meaningful to address this function as the transition function of a gerbe.

This problem is completely independent of the precise nature of the function τ ij. Whether or not it is defined in terms of a path integral and whether or not one considers saddle point approximations to this path integral, as the author does on page 4.

A naïve idea to remedy this problem would be to restrict τ ij to pairs of coinciding points. While that would make it a function on double interesections, it would lead to a completely trivial situation, since this would simply make τ ij the restriction of the globally defined function to double intersections.

This is a core issue that reappears throughout the paper. For instance in equation (28) one sees a couple of globally defined quantities that, two lines below in equation (29), suddenly carry Čech-indices. Similarly, the step from equation (37) to (38) actually only makes sense if equation (30) is supposed to hold globally, instead of only locally as suggested by the Čech-indices.

Therefore the content of the paper seems to be either ill-defined or vacuous.

We know from geometric quantization (, ) that quantization can be described in terms of a line bundle on classical phase space which is classified by the symplectic 2-form of classical mechanics. From this point of view it is a very interesting question if there is a corresponding analog of this procedure where instead of a 2-form we have a 3-form on phase space and instead of a line bundle obtain an abelian bundle gerbe from it. Indeed, it can plausibly be expected that such a “geometric gerbe quantization” is relevant for 2-dimensional quantum field theory ().

The title and abstract of the above seem to suggest that it is concerned with this issue. But this is not the case.

Posted at March 28, 2006 9:41 AM UTC

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