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Feynman-Kac formula reads: $$\psi(x,\beta) = \int e^{\int V({\bf{x}}(\tau))d\tau}\psi_{0}({\bf{x}}(\beta))d\mu_{x}({\bf{x}}). \tag{1}\label{1}$$ This is a rigorous formula, defined by means of a Wiener measure.

My question is: given that this formula has a rigorous meaning, can one relate it rigorously to the partition function in quantum statistical mechanics?

Let me say a few words about what motivated my question.

In quantum mechanics, one is interested in studying time-dependent wave functions: $$\psi(t,x) = (e^{-itH}\psi)(x)$$ It is very natural to look for an expression like (\ref{1}) in this context, since it'd express a "sum over every possible path" and the integral kernel, also known as propagator, is the fundamental tool to propagate wave functions in time. However, in this context, (\ref{1}) cannot be established rigorously.

On the other hand, (\ref{1}) is obtained setting $t \mapsto it = \beta$, which is interpreted as the inverse temperature $\beta = 1/T$ and (\ref{1}) is actually and expression for $(e^{-\beta H}\psi)(x)$. As far as I know, this does not have any physical meaning.

Connections to between path integrals in quantum mechanics and quantum statistical mechanics are easy to get if one uses the usual formal expressions. However, given that in quantum statistical mechanics, the partition function (in the canonical ensemble) is: $$Z = \operatorname{Tr}e^{-\beta H} = \sum_{n \in \mathbb{N}} \langle \psi_{n}, e^{-\beta H}\psi_{n}\rangle \tag{2}\label{2}$$ (I'm assuming $e^{-\beta H}$ is trace class so the right hand side of the above expression is well-defined for some Hilbert basis $\{\psi_{n}\}_{n\in \mathbb{N}}$) it seems difficult to relate (\ref{1}) to $Z$ in a rigorous way. In physics books, the usual approach is to consider: $$Z = \int dq\langle q| e^{-\beta H}|q\rangle $$ instead of the former expression and write each $\langle q| e^{-\beta H}|q\rangle$ as a path integral. $Z$ written this way is formal itself, but it seems that this expression becomes a fundamental bridge between the two worlds; the rigorous expression (\ref{2}) does not seem much appropriate to establish such connections.

So, these are just my thoughts and I'd like to know more about the topic. Maybe I'm missing something or there is something new to learn.

Feynman-Kac formula reads: $$\psi(x,\beta) = \int e^{\int V({\bf{x}}(\tau))d\tau}\psi_{0}({\bf{x}}(\beta))d\mu_{x}({\bf{x}}). \tag{1}\label{1}$$ This is a rigorous formula, defined by means of a Wiener measure.

My question is: given that this formula has a rigorous meaning, can one relate it rigorously to the partition function in quantum statistical mechanics?

Let me say a few words about what motivated my question.

In quantum mechanics, one is interested in studying time-dependent wave functions: $$\psi(t,x) = (e^{-itH}\psi)(x)$$ It is very natural to look for an expression like (\ref{1}) in this context, since it'd express a "sum over every possible path" and the integral kernel, also known as propagator, is the fundamental tool to propagate wave functions in time. However, in this context, (\ref{1}) cannot be established rigorously.

On the other hand, (\ref{1}) is obtained setting $t \mapsto it = \beta$, which is interpreted as the inverse temperature $\beta = 1/T$ and (\ref{1}) is actually and expression for $(e^{-\beta H}\psi)(x)$. As far as I know, this does not have any physical meaning.

Connections to between path integrals in quantum mechanics and quantum statistical mechanics are easy to get if one uses the usual formal expressions. However, given that in quantum statistical mechanics, the partition function (in the canonical ensemble) is: $$Z = \operatorname{Tr}e^{-\beta H} = \sum_{n \in \mathbb{N}} \langle \psi_{n}, e^{-\beta H}\psi_{n}\rangle \tag{2}\label{2}$$ (I'm assuming $e^{-\beta H}$ is trace class so the right hand side of the above expression is well-defined for some Hilbert basis $\{\psi_{n}\}_{n\in \mathbb{N}}$) it seems difficult to relate (\ref{1}) to $Z$ in a rigorous way. In physics books, the usual approach is to consider: $$Z = \int dq\langle q| e^{-\beta H}|q\rangle \tag{3}\label{3}$$ instead of the former expression and write each $\langle q| e^{-\beta H}|q\rangle$ as a path integral. $Z$ written this way is formal itself, but it seems that this expression becomes a fundamental bridge between the two worlds; the rigorous expression (\ref{2}) does not seem much appropriate to establish such connections.

So, these are just my thoughts and I'd like to know more about the topic. Maybe I'm missing something or there is something new to learn.

EDIT: Maybe (\ref{3}) has a rigorous version by means of the spectral theorem and spectral measure. I don't know if it does, however.

Feynman-Kac formula reads: $$\psi(x,\beta) = \int e^{\int V({\bf{x}}(\tau))d\tau}\psi_{0}({\bf{x}}(\beta))d\mu_{x}({\bf{x}}) \tag{1}\label{1}$$ This is a rigorous formula, defined by means of a Wiener measure.

My question is: given that this formula has a rigorous meaning, can one relate it rigorously to the partition function in quantum statistical mechanics?

Let me say a few words about what motivated my question.

In quantum mechanics, one is interested in studying time-dependent wave functions: $$\psi(t,x) = (e^{-itH}\psi)(x)$$ It is very natural to look for an expression like (\ref{1}) in this context, since it'd express a "sum over every possible path" and the integral kernel, also known as propagator, is the fundamental tool to propagate wave functions in time. However, in this context, (\ref{1}) cannot be established rigorously.

On the other hand, (\ref{1}) is obtained setting $t \mapsto it = \beta$, which is interpreted as the inverse temperature $\beta = 1/T$ and (\ref{1}) is actually and expression for $(e^{-\beta H}\psi)(x)$. As far as I know, this does not have any physical meaning.

Connections to between path integrals in quantum mechanics and quantum statistical mechanics are easy to get if one uses the usual formal expressions. However, given that in quantum statistical mechanics, the partition function (in the canonical ensemble) is: $$Z = \operatorname{Tr}e^{-\beta H} = \sum_{n \in \mathbb{N}} \langle \psi_{n}, e^{-\beta H}\psi_{n}\rangle \tag{2}\label{2}$$ (I'm assuming $e^{-\beta H}$ is trace class so the right hand side of the above expression is well-defined for some Hilbert basis $\{\psi_{n}\}_{n\in \mathbb{N}}$) it seems difficult to relate (\ref{1}) to $Z$ in a rigorous way. In physics books, the usual approach is to consider: $$Z = \int dq\langle q| e^{-\beta H}|q\rangle $$ instead of the former expression and write each $\langle q| e^{-\beta H}|q\rangle$ as a path integral. $Z$ written this way is formal itself, but it seems that this expression becomes a fundamental bridge between the two worlds; the rigorous expression (\ref{2}) does not seem much appropriate to establish such connections.

So, these are just my thoughts and I'd like to know more about the topic. Maybe I'm missing something or there is something new to learn.

Feynman-Kac formula reads: $$\psi(x,\beta) = \int e^{\int V({\bf{x}}(\tau))d\tau}\psi_{0}({\bf{x}}(\beta))d\mu_{x}({\bf{x}}). \tag{1}\label{1}$$ This is a rigorous formula, defined by means of a Wiener measure.

My question is: given that this formula has a rigorous meaning, can one relate it rigorously to the partition function in quantum statistical mechanics?

Let me say a few words about what motivated my question.

In quantum mechanics, one is interested in studying time-dependent wave functions: $$\psi(t,x) = (e^{-itH}\psi)(x)$$ It is very natural to look for an expression like (\ref{1}) in this context, since it'd express a "sum over every possible path" and the integral kernel, also known as propagator, is the fundamental tool to propagate wave functions in time. However, in this context, (\ref{1}) cannot be established rigorously.

On the other hand, (\ref{1}) is obtained setting $t \mapsto it = \beta$, which is interpreted as the inverse temperature $\beta = 1/T$ and (\ref{1}) is actually and expression for $(e^{-\beta H}\psi)(x)$. As far as I know, this does not have any physical meaning.

Connections to between path integrals in quantum mechanics and quantum statistical mechanics are easy to get if one uses the usual formal expressions. However, given that in quantum statistical mechanics, the partition function (in the canonical ensemble) is: $$Z = \operatorname{Tr}e^{-\beta H} = \sum_{n \in \mathbb{N}} \langle \psi_{n}, e^{-\beta H}\psi_{n}\rangle \tag{2}\label{2}$$ (I'm assuming $e^{-\beta H}$ is trace class so the right hand side of the above expression is well-defined for some Hilbert basis $\{\psi_{n}\}_{n\in \mathbb{N}}$) it seems difficult to relate (\ref{1}) to $Z$ in a rigorous way. In physics books, the usual approach is to consider: $$Z = \int dq\langle q| e^{-\beta H}|q\rangle $$ instead of the former expression and write each $\langle q| e^{-\beta H}|q\rangle$ as a path integral. $Z$ written this way is formal itself, but it seems that this expression becomes a fundamental bridge between the two worlds; the rigorous expression (\ref{2}) does not seem much appropriate to establish such connections.

So, these are just my thoughts and I'd like to know more about the topic. Maybe I'm missing something or there is something new to learn.