- #1

## 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?

$$ \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?