Quantum field theory texts often use the expression ⟨0,∞|0,−∞⟩J denoting the vacuum-to-vacuum transition amplitude.
What is the meaning of |0,−∞⟩J? Am I to take the Hamiltonian with the source present, ie. H(t)=H−J(t)q(t), find the ground state at time 0, ie. H(0)=E0|0⟩ and evolve it backwards to −∞?
Why isn't the correct expression J⟨0,∞|0,−∞⟩J instead? I also see ⟨q′,t′|q,t⟩J and have the same question about that, ie. what is |q,t⟩J and why aren't we considering J⟨q′,t′|q,t⟩J instead?
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