Hubbard Stratonovich/Gaussian Function Integration

[aaaa202]

I need some help on this since I clearly am not confident with Gaussian functional integration. Can anyone explain where the marked identities on the picture hold? I assume you have to use some formulas for Gaussian functional integration, but the only thing I could find is the one given in the bottom of the picture and that is not generalized to matrices and anyway it does not match with the results marked. Where does the determinant of A go, for example? Is it absorbed into the measure and in general which things can be absorbed into the measure, and what does it even to absorb something into a measure in functional integration?

 

[eljose79]

Hello there,

I understand that Gaussian functional integration can be quite confusing, but don't worry, I'm here to help! First, let's start with the formula given at the bottom of the picture. This formula is known as the Gaussian integral and it is a special case of Gaussian functional integration. It is used to calculate the integral of a Gaussian function, which is a type of probability distribution.

Now, let's talk about the identities marked in the picture. These identities are known as the Wick's theorem and they are used in Gaussian functional integration to simplify the calculation of integrals involving Gaussian functions. The Wick's theorem states that the integral of a product of Gaussian functions can be written as a sum of integrals of individual Gaussian functions. This is where the determinant of A comes in. In the Wick's theorem, it is absorbed into the measure, which is a way of simplifying the integral.

To understand what it means to absorb something into a measure in functional integration, we need to understand the concept of a measure. In functional integration, a measure is a way of assigning a value to a set of functions. Absorbing something into a measure means that we are including it in the calculation of the integral, but it is not explicitly written in the final result.

In general, any constant or function that does not depend on the integration variables can be absorbed into the measure. This includes the determinant of A, which is a constant in this case.

I hope this explanation helps you understand Gaussian functional integration better. Don't hesitate to ask if you have any further questions. Good luck!