Gaussian functional integral with constant operator

[Einj]

Hello everyone. What it the result for a Gaussian functional integral when the "matrix" is nothing but a number? Mathematically speaking is the following true?

$$
\int \mathcal{D}\phi e^{-\int d^3k f(k) |\phi(k)|^2}\propto \left(f(k)\right)^{-1/2}
$$

Here $f(k)$ is just a function of k, not derivatives or operators etc... I'm asking this because in principle we should have a determinant of the "matrix" and I don't know if what I wrote is correct.
Thanks!

 

[eljose79]

Hello there!

It is true that the result for a Gaussian functional integral when the "matrix" is a number is given by the equation you have provided. This is known as the Wick rotation or Wick's theorem, and it is a powerful tool in calculating Gaussian integrals in quantum field theory.

To understand why this is true, let's break down the equation. The integral on the left-hand side, $\int \mathcal{D}\phi e^{-\int d^3k f(k) |\phi(k)|^2}$, is known as a functional integral. It is a generalization of the usual integral, but instead of integrating over a variable, we are integrating over a function. In quantum field theory, we use functional integrals to calculate the probability amplitudes for different field configurations.

Now, the function $f(k)$ is just a number, as you have mentioned. This means that the integral is essentially a product of integrals over each value of k. And since the function is just a number, we can take it outside of the integral. This gives us the equation:

$$
\int \mathcal{D}\phi e^{-\int d^3k f(k) |\phi(k)|^2} = \int \mathcal{D}\phi e^{-f(k)\int d^3k |\phi(k)|^2}
$$

Now, we can use the fact that the integral of a Gaussian function is given by $\sqrt{\frac{\pi}{f(k)}}$. This is a well-known result in mathematics, and you can find its derivation in many textbooks. Applying this result to our equation, we get:

$$
\int \mathcal{D}\phi e^{-f(k)\int d^3k |\phi(k)|^2} = \prod_k \sqrt{\frac{\pi}{f(k)}} = \left(f(k)\right)^{-1/2}
$$

So, to answer your question, yes, your equation is correct. There is no need for a determinant of the "matrix" in this case, as it is just a number. I hope this helps clarify things for you. Keep exploring and asking questions!