# Evaluating a infinite-dimensional Gaussian integral

In summary, the conversation discusses the evaluation of a multi-dimensional Gaussian Integral, specifically the integral \int_{-\infty}^{\infty}d^{n}V exp(x^{T}Ax)exp(ag(x)) for n\rightarrow \infty. The problem arises when considering the function g(x) and the possibility of integrating over infinite dimensional spaces. The speaker suggests using a Gaussian measure, but is unsure of how to perform the integral. The conversation also mentions the integral \int dq_{1} dq_{2} (q_{1}^{2} + q_{1}q_{2}) exp[\mathbf{q}^{T}\mathbf{A}\mathbf{q}], but no further
Hello,.. that's part of a problem i find in QFT (i won't explain it since it can be very tedious), the question is that i must evaluate the Multi-dimensional Gaussian Integral.

$$\int_{-\infty}^{\infty}d^{n}V exp(x^{T}Ax)exp(ag(x))$$

for $$n\rightarrow \infty$$ of course if the integral is "purely" a Gaussian then you can do it.. however you have the problem of g(x), if you think you're integrating about 2paths" (trajectories), then my idea is that perhaps you could consider "functional integration" under a infinite dimensional Gaussian Borel Meassure, however i don't know more, i have the "(approximate) meassure but don't know any Numerical method valid for infinite dimensional spaces if a is small a Gaussian meassure could work but how do i perform the integral?.. thankx

$$\int dq_{1} dq_{2} (q_{1}^{2} + q_{1}q_{2}) exp[\mathbf{q}^{T}\mathbf{A}\mathbf{q}]$$

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## 1. What is an infinite-dimensional Gaussian integral?

An infinite-dimensional Gaussian integral is a type of mathematical integral that involves integrating a function over an infinite number of dimensions. It is a special case of a multidimensional integral, where the integrand (the function being integrated) is a Gaussian function.

## 2. How is an infinite-dimensional Gaussian integral evaluated?

An infinite-dimensional Gaussian integral can be evaluated using various techniques such as the method of steepest descent, saddle-point approximation, or Monte Carlo simulation. The specific method used depends on the complexity of the integrand and the desired level of accuracy.

## 3. What are the applications of infinite-dimensional Gaussian integrals?

Infinite-dimensional Gaussian integrals have various applications in physics, mathematics, and engineering. They are commonly used in probability theory, statistical mechanics, quantum field theory, and signal processing. They also have applications in finance, where they are used to model the behavior of stock prices.

## 4. Can an infinite-dimensional Gaussian integral have a closed-form solution?

In most cases, an infinite-dimensional Gaussian integral does not have a closed-form solution. However, there are some special cases where the integral can be evaluated analytically, such as when the integrand is a simple Gaussian function or when the dimensions of the integral can be reduced through symmetry.

## 5. What are the challenges in evaluating an infinite-dimensional Gaussian integral?

One of the main challenges in evaluating an infinite-dimensional Gaussian integral is the convergence of the integral. As the number of dimensions increases, the integral may become more difficult to evaluate accurately. Another challenge is the complexity of the integrand, which may require advanced mathematical techniques for evaluation.

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