Classical Field Theory

2014

The course introduces the student to relativistic classical field theory. The basic object is a field (such as the electromagnetic field) which possesses infinite degrees of freedom. The use of local and global symmetries (such as rotations) forms an underlying theme in the discussion. Concepts such as conservation laws, spontaneous breakdown of symmetry, Higgs mechanism etc. are discussed in this context. Several interesting solutions to the Euler-Lagrange equations of motion such as kinks, vortices, monopoles and instantons are discussed along with their applications.

2007

The multisymplectic description of Classical Field Theories is revisited, including its relation with the presymplectic formalism on the space of Cauchy data. Both descriptions allow us to give a complete scheme of classification of infinitesimal symmetries, and to obtain the corresponding conservation laws.

Undergraduate Lecture Notes in Physics, 2017

In this chapter the reader is introduced to the concepts and language of the Lagrangian formalism of field theories. Only knowledge of the Newtonian theory of gravity, as usually explained in standard undergraduate courses is assumed, although some familiarity with classical mechanics is an asset. The goal is twofold:

Physical Review D, 1973

In this study, we suggest a new model for Lagrangian field theory. Later by means of this model we derive Maxwell and Dirac equations. Finally, we conclude that each interaction type in nature may be expressed by a certain space which we call it as interaction space.

International Journal of Mathematics and Mathematical Sciences, 2002

We give a geometric formulation of the field equations in the Lagrangian and Hamiltonian formalisms of classical field theories (of first order) in terms of multivector fields. This formulation enables us to discuss the existence and nonuniqueness of solutions of these equations, as well as their integrability.

Fortschritte der Physik/Progress of Physics, 1996

We construct a lagrangian geometric formulation for first-order field theories using the canonical structures of first-order jet bundles, which are taken as the phase spaces of the systems in consideration. First of all, we construct all the geometric structures associated with a firstorder jet bundle and, using them, we develop the lagrangian formalism, defining the canonical forms associated with a lagrangian density and the density of lagrangian energy, obtaining the Euler-Lagrange equations in two equivalent ways: as the result of a variational problem and developing the jet field formalism (which is a formulation more similar to the case of mechanical systems). A statement and proof of Noether's theorem is also given, using the latter formalism.

2015

2009

This review paper is devoted to presenting the standard multisymplectic formulation for describing geometrically classical field theories, both the regular and singular cases. First, the main features of the Lagrangian formalism are revisited and, second, the Hamiltonian formalism is constructed using Hamiltonian sections. In both cases, the variational principles leading to the Euler-Lagrange and the Hamilton-De Donder-Weyl equations, respectively, are stated, and these field equations are given in different but equivalent geometrical ways in each formalism. Finally, both are unified in a new formulation (which has been developed in the last years), following the original ideas of Rusk and Skinner for mechanical systems.

1. Physical Field Theories In physical theories, fields are characterized not only by their invariance properties under transformations of the points x i of the background space-time manifold (Lorentz invariance, space-time parity etc.) but also by " internal symmetries " (isotropic invariance, SU(3) symmetry, charge parity etc.), which are independent of x i. The internal symmetry is often associated with the transformations of the points of an " internal space " , the symmetry of which is reflected in the behaviour of physical fields. Comparison of this method with the theory of Finsler geometry suggests that some of the important classes of internal symmetries of physical fields may be conceived as manifestations of a Finslerian structure inherent in real space-time. These facts lead to the hypothesis that the internal symmetries of physical fields reflect the symmetry properties of the indicatrix of a Finslerian structure assumed to be possessed by real space-time. Accordingly, the indicatrix plays the role of an internal space, whereas its points serve as the internal coordinates and the group of motions in the indicatrix play the role of a group of internal symmetries. Following Drechsler and Mayer [7], the transformations u a = u a (ũ) may be regarded as global transformations and their x-dependent generalizations u a = u a (x, ũ) as local gauge transformations or merely gauge transformations. The gauge fields are constructible from the projection factors of the indicatrix of a Finsler space. The gauge-field strength tensors are expressible in terms of the Finslerian curvature tensors. The gauge-covariant differentiation operator can be readily extended to the case of spinor fields. In physical theories gauge transformations are usually linear: An important example of such a theory is the Yang-Mills theory, which when restricted to SU(2) gauge transformations, describe the isotropic invariance of strongly interacting fields in terms of gauge fields, or weak interactions in the context of the standard model of unification of electromagnetic and weak interactions. These ideas readily account for the geometrical meaning of isotropic symmetry as a manifestation of the Finslerian structure of space-time with the highest symmetry indicatrix. It may be recalled that isotropic symmetry manifests itself as a symmetry

This review paper is devoted to presenting the standard multisymplectic formulation for describing geometrically classical field theories, both the regular and singular cases. First, the main features of the Lagrangian formalism are revisited and, second, the Hamiltonian formalism is constructed using Hamiltonian sections. In both cases, the variational principles leading to the Euler-Lagrange and the Hamilton-De Donder-Weyl equations, respectively, are stated, and these field equations are given in different but equivalent geometrical ways in each formalism. Finally, both are unified in a new formulation (which has been developed in the last years), following the original ideas of Rusk and Skinner for mechanical systems.