nLab
prequantum 0-bundle

Contents

Context

Geometric quantization

Bundles

Quantum field theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

In higher differential geometry the notion of connection on a bundle and in particular that of circle bundle with connection is refined to a tower of notions of circle n-bundles with connection for all nn \in \mathbb{N}. In particular also the degenerate case of n=0n = 0 is defined and fits into this tower: a 0-bundle is simply a function with values in the given structure group (e.g. the circle group U(1)U(1)).

Moreover, for a prequantum field theory defined by an extended Lagrangian there is such a circle (n-k)-bundle with connection for each closed manifold of dimension kk, called the prequantum (n-k)-bundle. For n=kn = k this is the action functional of the theory. Hence the action functional may be thought of as the prequantum 0-bundle of an extended prequantum field theory.

extended prequantum field theory

0kn0 \leq k \leq n(off-shell) prequantum (n-k)-bundletraditional terminology
00differential universal characteristic maplevel
11prequantum (n-1)-bundleWZW bundle (n-2)-gerbe
kkprequantum (n-k)-bundle
n1n-1prequantum 1-bundle(off-shell) prequantum bundle
nnprequantum 0-bundleaction functional

References

Lecture notes with more details are in the section Lagrangians and Action functionals of

Created on January 5, 2013 at 19:56:26. See the history of this page for a list of all contributions to it.