Hamiltonian Constraint Formulation of Classical Field Theories

Abstract

Classical field theory is considered as a theory of unparametrized surfaces embedded in a configuration space, which accommodates, in a symmetric way, spacetime positions and field values. Dynamics is defined via the (Hamiltonian) constraint between multivector-valued generalized momenta, and points in the configuration space. Starting from a variational principle, we derive the local equations of motion, that is, differential equations that determine classical surfaces and momenta. A local Hamilton-Jacobi equation applicable in the field theory then follows readily. In addition, we discuss the relation between symmetries and conservation laws, and derive a Hamiltonian version of the Noether theorem, where the Noether currents are identified as the classical momentum contracted with the symmetry-generating vector fields. The general formalism is illustrated by two examples: the scalar field theory, and the string theory.Throughout the article, we employ the mathematical formalism of geometric algebra and calculus, which allows us to perform completely coordinate-free manipulations. * Electronic address: zatlovac@fjfi.cvut.cz; URL: http://www.zatlovac.eu arXiv:1602.00468v1 [math-ph]

New Applications of Clifford’s Geometric Algebra

“…These methods are derived and discussed, computational examples given, and compared with some well known methods of computational electromagnetism. Finally, in [207] a Hamiltonian constraint formulation of classical field theories is studied. Classical field theory is considered as a theory of unparametrized surfaces embedded in a configuration space, which accommodates, in a symmetric way, space-time positions and field values.…”