nLab space of field histories

Idea

In physics, given a physical system then its space of field histories (e.g.trajectories_ for sigma models) is the space of all ways that the system may be configured in space and time, hence in spacetime.

For a field theory over a spacetime $\Sigma$ and defined by a field bundle $E \overset{fb}{\to} \Sigma$, the space of trajectories is the space of field configurations over spacetimes, which is the space of sections of the field bundle.

For Lagrangian field theories this space of trajectories carries a canonical presymplectic form and it comes with an equation of motion that picks some of the trajectories as being the “physically realizable ones”. The subspace of trajectories solving the equations of motion (the “shell”) equipped with this presymplectic form is called the covariant phase space of the system.

For the moment see at field theory for more details.

Related concepts

• configuration space (physics)

References

Discussion in the convenient context of smooth sets:

• Grigorios Giotopoulos, Hisham Sati, §1 in: Field Theory via Higher Geometry I: Smooth Sets of Fields [arXiv:2312.16301]