Trace of multi-index operator in path integral?

Question

What is the formal definition of trace of an operator? Suppose I have an operator $L^{a_1a_2..}_{b_1b_2..}(x,y)$ in which $a_i$'s and $b_i$'s are discrete indices, and $x,y$ are continuous ones. So how can I trace this operator? Continuous indices appear in path integral frequently.

jkb1603 · Answer

For some integral operator $mathcal{O}$ that acts on a function $f$ as
$$
(mathcal{O}f)(x) = int dy~mathcal{O}(x,y) f(y)
$$
one can define
$$
text{Tr } mathcal{O} = int dx~mathcal{O}(x,x)
$$
In general for linear operators on Hilbert spaces, one generally has:
$$
text{Tr } mathcal{O} = sum_X langle X | mathcal{O} | Xrangle
$$
where $|X rangle$ is a complete orthonormal basis of the space. This can be generalized to "continuous bases", e.g. the position or momentum basis in QM.
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Trace of multi-index operator in path integral?

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