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
unique up to terms in the image of the horizontal derivative, given by
where $\delta_{EL}$ is the Euler-Lagrange operator. Its image under the vertical derivative
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.
Higher Prequantum Geometry IV: The Covariant Phase Space – Transgressively
Grigorios Giotopoulos, Hisham Sati, §7 in: Field Theory via Higher Geometry I: Smooth Sets of Fields [arXiv:2312.16301]
Last revised on December 29, 2023 at 13:18:36. See the history of this page for a list of all contributions to it.