Diffeomorphisms as Symplectomorphisms in History Phase Space: Bosonic String Model

Abstract

The structure of the history phase space $\mathcal{G}$ of a covariant field system and its history group (in the sense of Isham and Linden) is analyzed on an example of a bosonic string. The history space $\mathcal{G}$ includes the time map $\mathsf{T}$ from the spacetime manifold (the two-sheet) $\mathcal{Y}$ to a one-dimensional time manifold $\mathcal{T}$ as one of its configuration variables. A canonical history action is posited on $\mathcal{G}$ such that its restriction to the configuration history space yields the familiar Polyakov action. The standard Dirac-ADM action is shown to be identical with the canonical history action, the only difference being that the underlying action is expressed in two different coordinate charts on $\mathcal{G}$. The canonical history action encompasses all individual Dirac-ADM actions corresponding to different choices $\mathsf{T}$ of foliating $\mathcal{Y}$. The history Poisson brackets of spacetime fields on $\mathcal{G}$ induce the ordinary Poisson brackets of spatial fields in the instantaneous phase space ${\mathcal{G}}_0$ of the Dirac-ADM formalism. The canonical history action is manifestly invariant both under spacetime diffeomorphisms Diff$\mathcal{Y}$ and temporal diffeomorphisms Diff$\mathcal{T}$. Both of these diffeomorphisms are explicitly represented by symplectomorphisms on the history phase space $\mathcal{G}$. The resulting classical history phase space formalism is offered as a starting point for projection operator quantization and consistent histories interpretation of the bosonic string model.

45 pages, no figures

References in corpus (2)

• Continuous-time histories: observables, probabilities, phase space structure and the classical limit
• General relativity histories theory: Spacetime diffeomorphisms and the Dirac algebra of constraints

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