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arXiv:hep-th/0411273 · International Journal of Theoretical Physics

Field Theory Methods in Classical Dynamics

Edward B. Manoukian, Nattapong Yongram

Published Jul 1, 2002• 1 citation
arXivHTMLPDF
DOI
10.1023/a:1019607117138

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Cluster dynamical mean field theories are analyzed in terms of their semiclassical limit and their causality properties, and a translation invariant formulation of the cellular dynamical mean-field theory, periodized cluster dynamical mean-field theory (PCDMFT), is presented. The semiclassical limit of the cluster methods is analyzed by applying them to the Falikov-Kimball model in the limit of infinite Hubbard interaction U where they map to different classical cluster schemes for the Ising model. Furthermore, the Cutkosky-t'Hooft-Veltman cutting equations are generalized and derived for nontranslation invariant systems using the Schwinger-Keldysh formalism. This provides a general setting to discuss causality properties of cluster methods. To illustrate the method, we prove that PCDMFT is causal while the nested cluster schemes in general and the pair scheme in particular are not. Constraints on further extension of these schemes are discussed.
In hybrid classical-quantum theories, the dynamics of the classical system induce the classicality of the quantum system, meaning that such models do not necessarily require a measurement postulate to describe probabilistic measurement outcomes. It has recently been shown that covariant classical-quantum dynamics can be constructed using path integral methods, with the dynamics encoded in a combined action for the classical and quantum variables. This work introduces a classical-quantum model whereby quantum gravity interacts with a classical scalar field. The scalar field can be viewed as fundamentally classical or effectively classical due to the decoherence of a quantum gravity theory. The dynamics act to collapse quantum spacetimes according to their Ricci scalar, with corresponding diffusion in the scalar field due to the quantum back-reaction. In the classical limit, the diffusion in the scalar field manifests itself via stochastic fluctuations in the Newtonian potential. Because we couple a classical scalar field to perturbative quantum gravity, we find the theory is not renormalizable but is instead to be viewed as an effective field theory. However, it is an effective field theory that does not necessarily require a measurement postulate. More generally, our work shows it is possible to integrate collapse dynamics with high-energy physics and covariance.