# Path integral at large time

From the path integral of a QFT: $$Z=\int D\phi e^{-S[\phi]}$$ What is a nice argument to say that when we study the theory at large time $$T$$, this behaves as: $$Z \to e^{-TE_0}$$ where $$E_0$$ is the minimum energy of the system.

Briefly speaking, the standard argument is

1. A correspondence between the path integral formalism and the operator formalism in Minkowskian signature with unitary time evolution;

2. Wick rotate both sides of the correspondence to Euclidean signature;

3. (Optional.) Identify with the partition function in statistical mechanics, where the Euclidean time becomes the inverse temperature $$\beta$$;

4. Take the limit $$Z~=~{\rm Tr}_{\cal H}e^{-\beta \hat{H}}\quad\sim\quad e^{-\beta E_0}\quad\text{for}\quad\beta\to\infty,$$ where $$E_0$$ is the ground state.

Path integral and operator formalism correspondence states that for Euclidean time $$t$$, $$Z=\int D\phi e^{-S[\phi]}\leftrightarrow Z~=~{\rm Tr}_{\cal H}(e^{-t\hat{H}})$$

Fix an orthonormal basis $$\{|n\rangle\}_{n}\in \mathcal{H}$$ for the Hilbert space of the theory. Then by using the following identities

$$\langle m|n\rangle = \delta_{mn}\quad\space \sum_{n}|n\rangle\langle n| = \hat{\text{I}}$$

We can obtain the spectral decomposition for eigenstates $$H|n\rangle = E_n|n\rangle$$

$$e^{-tH} = e^{-tH}\biggr(\sum_{n}|n\rangle\langle n|\biggr) = \sum_{n}e^{-tH}|n\rangle\langle n| = \sum_{n}e^{-tE_n}|n\rangle\langle n|$$

$${\rm Tr}_{\cal H}(e^{-t\hat{H}}) = \sum_{n}e^{-tE_n} = e^{-tE_0}\biggr(1+\sum_{n}e^{-t(E_n-E_0)}\biggr)\sim e^{-tE_0}\quad(t\rightarrow\infty)$$
I'm assuming you mean immaginary time $$T$$. In this case the path integral is equal to the partition function (with $$T=\beta$$), as others have noted. Then, let the spectral decomposition of $$H$$ be
$$H=\sum_n E_n \Pi_n,$$
where $$\Pi_n$$ are (not necessarily one-dimensional) eigenprojectors. Then, ordering the eigenvalues of $$H$$, $$\{E_n\}$$, in increasing order: $$E_0 < E_1 < E_2<\ldots$$, and with degeneracies given by $$g_n=\mathrm{Tr} \Pi_n$$, we have
\begin{align} Z &= \mathrm{Tr} e^{-\beta H}\\ &= \sum_{n=0} g_n e^{-\beta E_n}\\ &= e^{-\beta E_0} g_0 \left (1+ \sum_{n=1} e^{-\beta \Delta_n} g_n/g_0 \right )\\ &= e^{-\beta E_0} g_0 \left (1+ e^{-\beta \Delta_1} g_1/g_0 + O(e^{-\beta \Delta_2}) \right ), \end{align}
valid when $$\beta\to \infty$$ and having defined $$\Delta_n := E_n-E_0>0$$. Usually the ground state is non-degenerate, i.e. $$g_0=1$$, which gives your equation ($$\beta$$ is the imaginary time).