TITLE: Why is the Vacuum state got by a limit to imaginary time?
QUESTION [0 upvotes]: Given a (non-relativistic) propagator $K_t(A,B)$ giving the 'conditional amplitude' to go from state $B$ to state $A$ in time $t$, it is known that one can find the vacuum wavfunction by (independent of any $B$):
$$\Psi^t_{VAC}(A) = e^{iE_0t}\lim_{t\rightarrow i\infty} \frac{K_t(A,B)}{\sqrt{K_{2t}(B,B)}}$$
(The denominator is just there to make sure it is normalised to 1). $E_0$ is the vacuum energy got from:
$$i E_0 = \lim_{t\rightarrow i\infty} \frac{ \dot{K}_t(A,B) }{ K_t(A,B) }$$ for any $A$ and $B$. (Which is just got from considering that $i\dot{\psi}_{VAC}^0(A) = E_0 \psi_{VAC}^0(A)$ )
I think I first read of this "trick" in Hawking's paper about the no-boundary principle where it is expressed in terms of path integrals. (But in that case there is no "$B$" since there is no boundary in imaginary time!)
I suppose one can interpret this as a state, which after an infinite amount of (imaginary time) settles down to the vacuum state.
I have two questions.
(1) Is there any physical meaning that we are taking the limit to infinity along imaginary time? Or is this just a mathematical trick. I believe it is related to the fact that path integrals are only properly defined after a Wick rotation.
(2) Does this trick fail if there are multiple vacuums? My immediate guess is that if $K$ has poles then the different vacuum solutions would correspond somehow to different paths around the poles. But that is just a wild guess. Another guess would be that for different sets of states $B$ would go to different vacuums. The vacuums being like attractors. (Although this would seem to make path integrals multi-valued!)

REPLY [2 votes]: Given a (non-relativistic) propagator $K_t(A,B)$ giving the 'conditional amplitude' to go from state $B$ to state $A$ in time $t$, it is known that...

It is helpful to "look inside" this propagator:
$$
K = <A|e^{-iHt}|B>\;,
$$
by inserting a complete set of states.
For now, I ignore the degeneracy question and assume each state and each energy can both be labeled by $n$. But, you should be able to work out the general case.
$$
K = e^{-iE_0t}\sum_n <A|n><n|B>e^{-i(E_n - E_0) t}\;,
$$
where $E_n - E_0$ is positive for all $n\neq 0$.
Therefore, when we send $t$ to $-i\infty$ every state other than the ground state in the sum is weighted by an exponential that goes to zero. That is, as $t\to-i\infty$:
$$
e^{+iE_0 t}K \to <A|0><0|B> = \Psi_0(A)\Psi_0(B)^*
$$

Is there any physical meaning that we are taking the limit to infinity along imaginary time? Or is this just a mathematical trick.

Here it looks more like "just" a mathematical trick. However, there is a connection to statistical physics and the canonical partition function.

Does this trick fail if there are multiple vacuums?

You will end up projecting onto all the states with the lowest energy like, e.g., if there are two ground states:
$$
e^{iE_0 t}K \to <A|0_1><0_1|B> + <A|0_2><0_2|B> 
$$
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