# Functional Determinant of a system of differential operators?

## Summary:

When applying the stationary phase approximation for path integrals we need to calculate the determinant of an operator; but what can we do if this operator is itself a matrix (made out of operators!)

## Main Question or Discussion Point

So in particular, how could the determinant of some general "operator" like

$$\begin{pmatrix} f(x) & \frac{d}{dx} \\ \frac{d}{dx} & g(x) \end{pmatrix}$$

with appropriate boundary conditions (especially fixed BC), be computed? And assuming that it diverges, would it be valid in a stationary phase approximation context to divide by some reference operator such as

$$\begin{pmatrix} 0 & \frac{d}{dx} \\ \frac{d}{dx} & 0 \end{pmatrix}$$

and use this value, for the quantum corrections to the classical path?

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vanhees71
Gold Member
2019 Award
• dextercioby and Yellotherephysics

Would this be a fair summary for the Heat-Kernel method (starting on page 93 of the linked notes)?
If we call the operator we're interested in H.
1. Work out the propogator ## \langle x | exp(- H \theta) | x' \rangle ##
2. Then get some kind of expression for ## \langle x | exp(- H \theta) | x \rangle ##
3. Find the Heat-Kernel (a function of ##\theta##) by integrating over the coordinates
4. Find the other Heat-Kernel thing, denoted by ##\tilde{H} ## (a function of the complex parameter ##\alpha##)
5. Take the complex parameter ##\alpha## to zero, getting the trace of the log of H, which we can then relate to the determinant
Is this algorithm still safe to use if H is something like
\begin{pmatrix}

a(x) & \frac{d}{dx} & b(x) \\ \frac{d}{dx} & c(x) & d(x) \\ e(x) & f(x) & g(x)

\end{pmatrix}
In the notes ## \langle x | exp(- H \theta) | x' \rangle ## is turned into something concrete by integrating over the momentum identity, would I be able to do something similar with my matrix example?