Multisymplectic formalism and the covariant phase space

doi:10.1017/CBO9780511863219.007

6 - Multisymplectic formalism and the covariant phase space

Published online by Cambridge University Press: 05 November 2011

Edited by

Kevin Houston and

Frédéric Hélein: Affiliation:
Institut de Mathématiques de Jussieu
Roger Bielawski: Affiliation:
University of Leeds
Kevin Houston: Affiliation:
University of Leeds
Martin Speight: Affiliation:
University of Leeds

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Summary

In most attempts to build the mathematical foundations of Quantum Field Theory (QFT), two classical ways have been explored. The first one is often referred to as the Feynman integral or functional integral method. It is a generalization to fields of the path integral method of quantum mechanics and is heuristically based on computing integrals over the infinite dimensional set of all possible fields Φ by using a kind of ‘measure’ – which should behave like the Lebesgue measure on the set of all possible fields Φ – times eiℒ(Φ)/ℏ, where ℒ is a Lagrangian functional (but attempts to define this ‘measure’ failed in most cases). The second one is referred to as the canonical quantization method and is based on the Hamiltonian formulation of the dynamics of classical fields, by following general axioms which were first proposed by Dirac and later refined. The Feynman approach has the advantage of being manifestly relativistic, i.e. it does not require the choice of a particular system of space-time coordinate, since the main ingredient is ℒ(Φ), which is an integral over all space-time. By contrast, the canonical approach, at least its classical formulation, seems to be based on the choice of a particular time coordinate which is needed to define the Hamiltonian function through an infinite dimensional Legendre transform.

However there are alternative formulations of the Hamiltonian structure of the dynamics of classical fields, which could be used as a starting point of a covariant canonical quantization.

Type: Chapter
Information: Variational Problems in Differential Geometry , pp. 94 - 126

DOI: https://doi.org/10.1017/CBO9780511863219.007 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2011

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