pre-symplectic current

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

**quantum mechanical system**, **quantum probability**

**interacting field quantization**

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

$\theta \in \Omega^{p,1}(E)$

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

$d L = \delta_{EL} L - d_H \theta
\,,$

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

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

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

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

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