How to determine the ground state of a classical field, for example an electromagnetic field? What is the difference between the the ground state of a classical field and that of a quantum field?
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In classical field theory, the ground state is also called minimizer (of the energy functional); and just to prove its existence is already a quite difficult task, from a mathematical standpoint (as you can imagine, much more difficult is to write eventually its explicit form). Often you can only have minimizing sequences, i.e. sequences of classical states such that in the limit you reach the infimum of the energy functional (but the limit of the sequence of states is no more in the classical phase space).
In this link it is, for example, described a general method to prove existence of a minimizer.
The relation with ground state of quantum fields is that, in some suitable cases, you can rigorously prove that the quantum ground state converges in the classical limit to the corresponding classical ground state. This can be rigorously done in very few nontrivial quantum field theories, and also for many-body quantum dynamics (where the limit becomes actually a mean field limit).
