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

1. 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.

2. References

Last revised on December 29, 2023 at 13:18:36. See the history of this page for a list of all contributions to it.