Generalized Hamiltonian Formalism for Field Theory

Sardanashvily, G.

Автор: Sardanashvily G.
Год издания: 1995
Издательство: WORLD SCIENTIFIC PUBL CO PTE LTD
Местоположение издательства: 5 TOH TUCK LINK, SINGAPORE,SINGAPORE, 596224
Объём: 155 страниц
ISBN: 981-02-2045-6
Аннотация: Preface Classical ﬁeld theory utilizes traditionally the language of Lagrangian dynamics. The Hamiltonian approach to ﬁeld theory was called into play mainly for canonical quantization of ﬁelds by analogy with quantum mechanics. The major goal of this approach has consisted in establishing simultaneous commutation relations of quantum ﬁelds in models with degenerate Lagrangian densities, e.g., gauge theories. In classical ﬁeld theory, the conventional Hamiltonian formalism fails to be so successful. In the straightforward manner, it takes the form of the instantaneous Hamiltonian formalism when canonical variables are ﬁeld functions at a given instant of time. The corresponding phase space is inﬁnite-dimensional. Hamiltonian dynamics played out on this phase space is far from to be a partner of the usual Lagrangian dynamics of ﬁeld systems. In particular, there are no Hamilton equations in the bracket form which would be adequate to Euler-Lagrange ﬁeld equations. This book presents the covariant ﬁnite-dimensional Hamiltonian machinery for ﬁeld theory which has been intensively developed from 70th as both the De Donder Hamiltonian partner of the higher order Lagrangian formalism in the framework of the calculus of variations and the multisymplectic (or polysimplectic) generalization of the conventional Hamiltonian formalism in analytical mechanics when canonical momenta correspond to derivatives of ﬁelds with respect to all world coordinates, not only time. Each approach goes hand-in-hand with the other. They exemplify the generalized Hamiltonian dynamics which is not merely a time evolution directed by the Poisson bracket, but it is governed by partial diﬀerential equations where temporal and spatial coordinates enter on equal footing. Maintaining covariance has the principal advantages of describing ﬁeld theories, for any preliminary space-time splitting shades the covariant picture of ﬁeld constraints. Contemporary ﬁeld models are almost always the constraint ones. In ﬁeld theory, if a Lagrangian density is degenerate, the Euler-Lagrange equations are underdetermined and need supplementary conditions which however remain elusive in general. They appear automatically as a part of multimomentum Hamilton equations. Thus, the universal procedure is at hand to canonically analize constraint ﬁeld systems on the covariant ﬁnite-dimensional level. This procedure is applied to a number of contemporary ﬁeld models including gauge theory, gravitation theory, spontaneous symmetry breaking and fermion ﬁelds. In the book, we follow the generally accepted geometric formulation of classical ﬁeld theory which is phrased in terms of ﬁbred manifolds and jet spaces. Contents Preface. Introduction. 1 Geometric Preliminary. 1.1 Fibred manifolds. 1.2 Jet spaces. 1.3 General connections. 2 Lagrangian Field Theory 39 2.1 Lagrangian formalism on ﬁbred manifolds. 2.2 De Donder Hamiltonian formalism. 2.3 Instantaneous Hamiltonian formalism. 3 Multimomentum Hamiltonian Formalism. 3.1 Multisymplectic Legendre bundles. 3.2 Multimomentum Hamiltonian forms. 3.3 Hamilton equations. 3.4 Analytical mechanics. 3.5 Hamiltonian theory of constraint systems. 3.6 Cauchy problem. 3.7 Isomultisymplectic structute. 4 Hamiltonian Field Theory. 4.1 Constraint ﬁeld systems. 4.2 Hamiltonian gauge theory. 4.3 Electromagnetic ﬁelds. 4.4 Proca ﬁelds. 4.5 Matter ﬁelds. 4.6 Hamilton equations of General Relativity. 4.7 Conservation laws. 5 Field Systems on Composite Manifolds. 5.1 Geometry of composite manifolds. 5.2 Hamiltonian systems on composite manifolds. 5.3 Classical Berry’s oscillator. 5.4 Higgs ﬁelds. 5.5 Gauge gravitation theory. 5.6 Fermion ﬁelds. 5.7 Fermion-gravitation complex. Bibliography
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