Given a vector function
$$f=(f\_1,\ldots,f\_n)\in L^2(\mathbb R,\mathbb R^n)$$
(for some $n\in\mathbb N$), let us define
$$\Delta f:=(\Delta f\_1,\ldots,\Delta f\_n),$$
where $\Delta$ is the Laplacian operator, and let $Q:\mathbb R\to\mathbb R^{n\times n}$ be a potential taking values in symmetric $n\times n$ matrices. I'm interested in *vector Schrödinger operators* of the form
$$Hf=-\Delta f+Qf,\qquad f\in L^2(\mathbb R,\mathbb R^n).$$



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> **Question.** Is there a Feynman-Kac type formula known for $H$'s semigroup in the case where $Q(x)$ is not necessarily diagonal?
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(Note: I specify abote that I'm interested in the case where $Q$ is not diagonal; if $Q=\mathrm{diag}(Q\_1,\ldots,Q\_n)$, then
$$Hf=(-\Delta f\_1+Q\_1 f\_1,\ldots,-\Delta f\_1+Q\_1 f\_1),$$
in which case we can simply apply the one-dimensional Feynman-Kac formula to each component.)



Is there a Feynman-Kac formula for vector-valued Schrödinger operators?
It's the usual formula with the exponential of $Q$ replaced by a [time-ordered exponential](https://en.wikipedia.org/wiki/Ordered_exponential) --- needed in the integral over $t$ since $Q[x(t)]$ and $Q[x(t')]$ do not commute for $t\neq t'$. One reference where this "chronological'" integral is worked out is [Equivalence of Two Definitions of a Chronological Integral.](https://link.springer.com/content/pdf/10.3103%2FS002713221006001X.pdf)


