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Global and local Hamiltonians for quantum electrodynamics

The set of Euler–Lagrange equations that extremelize the action associated with the Lagrangian space-time density of quantum electrodynamics leads to the well-known set of coupled Dirac–Maxwell equations. We compare three alternative Hamiltonian-based descriptions for quantum electrodynamics. We construct a local, a spatially global and a temporally global Hamiltonian and show that the corresponding Hamilton equations of motion are able to reproduce the Dirac–Maxwell equations. While this local Hamiltonian is fully equivalent to quantum electrodynamics, it does not provide any obvious conserved quantities. On the other hand, the two global Hamiltonians can be associated with the temporal and spatial generators of the dynamics and lead to spatially or temporally conserved observables if the fields fulfill certain boundary conditions.