In Section 10.1 of his textbook Quantum Field Theory for Mathematicians, Ticciati writes
Assuming that the background field or classical source $j(x)$ is zero at space-time infinity, the presence of $j$ will not affect the physical vacuum, $|0\rangle _P$.
(The context is $\phi ^4$ theory.)
First of all, what is the physical vacuum? My first thought at properly defining a vacuum state would be:
Definition 1: A quantum state is said to be a vacuum state iff the expectation value of the Hamiltonian in this theory is a local minimum (the Hamiltonian of course being part of the data that defines the theory).
Is this the correct notion of what it means to be the "physical vacuum" in a given theory? If so, two questions immediately come to mind:
(1) To what degree is the vacuum unique? I've heard many times that we have so-called "degenerate vacuum". Presumably this means there is some sort of non-uniqueness going on.
(2) Is a physical vacuum necessarily Poincaré invariant? (In relativistic quantum mechanics, the projective Hilbert space that is the space of states comes with an action of the Poincaré group that preserves probabilities, so it makes sense to talk about whether states are invariant or not.) If, with this definition, a physical vacuum is not necessarily Poincaré invariant, then we better change our definition to include this, that is:
Definition 2: A quantum state is said to be a vacuum state iff it is Poincaré-invariant and the expectation value of the Hamiltonian in this theory is a local minimum.
(3) With this alternative definition, to what degree is the vacuum unique?
Secondly, given the appropriate definition of "physical vacuum", how would $j$ affect this state in the case that it did not vanish at space-time infinity?