Excitations Propagating Along Surfaces

Abstract

A number of equations is deduced which describe propagation of excitations along $n$-dimensional surfaces in $R^N$. Usual excitations in wave theory propagate along 1-dimensional trajectories. The role of the medium of propagation of excitations considered in this paper is played by the infinite dimensional space of $(n-1)$-dimensional surfaces in $R^N$. The role of rays is played by $n$-dimensional solution surfaces of the variational problem. Such a generalization of wave theory can be useful in quantum field theory. Among these equations are the generalized Hamilton--Jacobi equation (known in particular cases in the literature), generalized canonical Hamilton equations, and generalized Schrodinger equation. Besides that, a theory of integration of the generalized Hamilton--Jacobi equation is developed.

arXiv comment12 pages; formulation and solution of the Cauchy problem for the generalized Hamilton--Jacobi equation added

References in corpus (1)

• Generalized Schrodinger Equation and Constructions of Quantum Field Theory

Cited by in corpus (8)

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• Generalized Schrodinger Equation and Constructions of Quantum Field Theory
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• Analogs of the Hamilton-Jacobi and Schrodinger Equations for Multidimensional Variational Problems of Field Theory
• A new mathematical problem related to quantization of fields
• A mathematical approach to quantum field theory
• Generalizations of wave equations to multidimensional variational problems
• An attempt to construct dynamical evolution in quantum field theory