Hamiltonian field theory
The Hamiltonian for a system of discrete particles is a function of their generalized coordinates and conjugate momenta, and possibly, time.It is the field analogue to the Lagrangian function for a system of discrete particles described by generalized coordinates.The corresponding dimension is [energy][length]−3, in SI units Joules per metre cubed, J m−3.is the variational derivative Under the same conditions of vanishing fields on the surface, the following result holds for the time evolution of A (similarly for B): which can be found from the total time derivative of A, integration by parts, and using the above Poisson bracket.), Taking the partial time derivative of the definition of the Hamiltonian density above, and using the chain rule for implicit differentiation and the definition of the conjugate momentum field, gives the continuity equation: in which the Hamiltonian density can be interpreted as the energy density, and the energy flux, or flow of energy per unit time per unit surface area.[2] This Hamiltonian formalism is applied to quantization of fields, e.g., in quantum gauge theory.Covariant Hamiltonian field theory is developed in the Hamilton–De Donder,[4] polysymplectic,[5] multisymplectic[6] and k-symplectic[7] variants.A phase space of covariant Hamiltonian field theory is a finite-dimensional polysymplectic or multisymplectic manifold.