The answer to the question will not be straightforward as the conventional Hamiltonian in the EM field. As it is difficult to write equations here, I am attaching the link to the question.
Question link: https://physics.stackexchange.com/q/749166/147579
This is a textbook exercise and the answer is completely straightforward. There's no such distinction between ``conventional'' and ``unconventional'' quantities.
How to compute the Hamiltonian is a completely unambiguous procedure. The subtlety for electromagnetism is gauge invariance, which means that a gauge-fixing condition must be imposed in order to define the canonical momenta and what must be proved is that the Hamiltonian is independent of the choice of the gauge-fixing condition. But this is now presented in all textbooks, so it would be useful to actually study the relevant material.
Furthermore, one shouldn’t mix up the field and its sources. The Hamiltonian of the electromagnetic field, alone, is the volume integral of (E^2+B^2)/2. (In units where ε0=1=μ0 and c=1).
And this can be shown, as usual, in many ways, for instance:
(a) From Maxwell's equations, following Poynting's approach.
(b) By showing that the action for the electromagnetic field is invariant under spacetime translations and deducing the conservation law for the energy-momentum tensor; the Hamiltonian is the volume integral of T00=(E2+B2)/2.
This expression is gauge-invariant and it can be shown that the volume integral is invariant under Lorentz transformations, even though the integrand is not.
As mentioned, all this is standard material in any course in electromagnetism. It's no longer a research topic.
Dear Prof. Debamalya Dutta
In addition to the previous right answers, please look at the following classical textbook for the whole derivation of the hamiltonian:
The Quantum Theory of Light (Oxford Science Publications) 3rd Edition by Prof. Rodney Loudon
The EMF is transversely polarized, so it can be done a second quantization using creation and annihilation operators (without relativity, because that belong to quantum electrodynamics - QED) serve to quantize the classical EMF.
Kind Regards.
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