nLab
pre-symplectic current

Contents

Context

Symplectic geometry

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

A local Lagrangian density LΩ p+1,0(E)L \in \Omega^{p+1,0}(E) on a jet bundle J (E)J^\infty(E) of some field bundle EΣE \to \Sigma induces a differential form

θΩ p,1(E) \theta \in \Omega^{p,1}(E)

unique up to terms in the image of the horizontal derivative, given by

dL=δ ELLd Hθ, d L = \delta_{EL} L - d_H \theta \,,

where δ EL\delta_{EL} is the Euler-Lagrange operator. Its image under the vertical derivative

ωd VθΩ p,2(E) \omega \coloneqq d_V \theta \in \Omega^{p,2}(E)

is called the pre-symplectic current. Accordingly θ\theta itself is also called the presymplectic potential current.

This is because for any choice of pp-dimensional submanifold Σ pΣ\Sigma_p \hookrightarrow \Sigma, the transgression of variational differential forms of ω\omega to Σ p\Sigma_p is a presymplectic form on the on-shell space of sections on the infinitesimal neighbourhood of Σ p\Sigma_p. This is the covariant phase space.

References

Last revised on December 22, 2017 at 12:24:54. See the history of this page for a list of all contributions to it.