## March 28, 2006

### Gerbes and Quantization

#### Posted by Urs Schreiber

Here is a report on

Jose M. Isidro
Gerbes and Heisenberg’s Uncertainty Principle
hep-th/0512241.

[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 ${\tau }_{\mathrm{ij}}$ the phase of a quantity which is supposed to be essentially a quantum mechanical propagator between these points. He then says that ${\tau }_{\mathrm{ij}}$ defines a gerbe trivialization.

For that to make sense, however, ${\tau }_{\mathrm{ij}}$ would have to be a function on the intersection of ${U}_{i}$ with ${U}_{j}$. But, as defined, ${\tau }_{\mathrm{ij}}$ 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 $\tau$ 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 ${\tau }_{\mathrm{ij}}$. 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 ${\tau }_{\mathrm{ij}}$ 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 ${\tau }_{\mathrm{ij}}$ 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 ($\to$, $\to$) 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 ($\to$).

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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