The Hamiltonian and the energy-momentum tensor are distinct yet related concepts in classical field theory and quantum mechanics. The Hamiltonian, particularly in the context of Hamilton's field equations, is equivalent to the "00" component of the energy-momentum tensor when evaluated on the appropriate hypersurface. In quantum theory, the Hamiltonian plays a crucial role as it governs the S-matrix, making it a more significant object than the energy-momentum tensor in that framework. Understanding these relationships is essential for grasping the foundations of field theory.
PREREQUISITESPhysicists, particularly those specializing in theoretical physics, quantum mechanics, and classical field theory, will benefit from this discussion.
dextercioby said:In classical field theory one can show that the Hamiltonian, when calculated on the hypersurface of the solutions of the Hamilton's field equations, is equal to the "00" component of the en-mom tensor calculated on the stationary surface of Lagrange's field equations.
As for them being matrices, i guess you mean a quantum theory context in which the hamiltonian is the more important object of the two, since it governs the S-matrix.