nLab field history

Idea

Given a Lagrangian field theory with field bundle $E \overset{fb}{\to} \Sigma$ over spacetime $\Sigma$, then a field history is a smooth section of this field bundle $\Phi \in \Gamma_\Sigma(E)$.

For every Cauchy surface $\Sigma_p \hookrightarrow \Sigma$ the restriction $\Phi\vert_{\Sigma_p}$ of $\Phi$ to $\Sigma_p$ may be thought of as the field configuration “at that time” and hence the change of $\Phi\vert_{\Sigma_p}$ as the choice of Cauchy surface changes reflects the “history” of these configurations, whence the name.

For a sigma model field theory a field history may be thought of as a trajectory.

For more see at geometry of physics – perturbative quantum field theory the chapter Fields.

Related concepts

• field observable

• field (physics)

• configuration space (physics)