geometric quantization higher geometric quantization
geometry of physics: Lagrangians and Action functionals + Geometric Quantization
prequantum circle n-bundle = extended Lagrangian
prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
vector bundle, 2-vector bundle, (∞,1)-vector bundle
real, complex/holomorphic, quaternionic
physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
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 $n \in \mathbb{N}$. In particular also the degenerate case of $n = 0$ 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 $U(1)$).
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 $k$, called the prequantum (n-k)-bundle. For $n = k$ 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
$0 \leq k \leq n$ | (off-shell) prequantum (n-k)-bundle | traditional terminology |
---|---|---|
$0$ | differential universal characteristic map | level |
$1$ | prequantum (n-1)-bundle | WZW bundle (n-2)-gerbe |
$k$ | prequantum (n-k)-bundle | |
$n-1$ | prequantum 1-bundle | (off-shell) prequantum bundle |
$n$ | prequantum 0-bundle | action functional |
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.