quantum field theory - Vacuum to vacuum transition amplitude confusion

Quantum field theory texts often use the expression $\langle 0,\infty|0,-\infty\rangle_J\,$ denoting the vacuum-to-vacuum transition amplitude.

1. 
   

   What is the meaning of $|0,-\infty\rangle_J?$ 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. $H(0)=E_0|0\rangle$ and evolve it backwards to $-\infty?$

   
   


2. 
   

   Why isn't the correct expression $_J\langle 0,\infty|0,-\infty\rangle_J$ instead? I also see $\langle q',t'|q,t\rangle_J$ and have the same question about that, ie. what is $|q,t\rangle_J$ and why aren't we considering $_J\langle q',t'|q,t\rangle_J$ instead?

   
   

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