∙ Consider the following Hamiltonian (density) H=12(∂μϕ)(∂μϕ)+V(ϕ)whereV(ϕ)=λ4!(ϕ2−v2)2
is a double-well potential. Expanding the potential
V(ϕ)=λ4!(ϕ2−v2)2 about one of the minima (say,
ϕ=+v) and retaining up to quadratic term in the potential, one obtains a perturbative vacuum
|0+⟩. This is not the exact vacuum of the full Hamiltonian because higher order terms in the potential have been neglected. Similarly, expanding
V(ϕ) about
ϕ=−v, and again retaining up to quadratic term one obtains another perturbative vacuum
|0−⟩. This too, by the same reason, is
not the exact vacuum of the full Hamiltonian.
∙ Similar procedure in a quantum mechanical problem with the Hamiltonian H=p22m+V(x)=p22m+k(x2−a2)2,
will lead to two
perturbative ground states
|0+⟩ and
|0−⟩. They are not the exact ground states of
H. The exact ground state is either given by
|S⟩ or
|A⟩ which are the symmetric and antisymmetric linear combinations of the perturbative ground states
|0+⟩ and
|0−⟩.
In this case, one cannot talk about the energy of the states |0+⟩ or |0−⟩ because they are not the eigenstates of H. One can only talk about energies of |S⟩ and |A⟩.
Question 1: If this is true, how is it that in QFT, we talk about the states |0+⟩ and |0−⟩ being degenerate (in energy)? They are just the perturbative vacua and not the exact vacua. Hence, not the eigenstates of H. For a reference, see A Modern Introduction to Quantum Field Theory by Michele Maggiore, page 254-255.
Question 2: What is/are the exact ground states of H and what is their relation to |0+⟩ and |0−⟩?
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