# Perturbation expansion with path integrals

In summary, the conversation is about the lecture notes for Advanced Quantum Field Theory during the Lent Term of 2013, originally typeset by Steffen Gielen in 2007 and revised by Carl Turner in 2013. The notes do not have a copyright notice, but there is attribution on the first page.
Homework Statement
This is from Hugh Osborn's 'Advanced Quantum Field Theory' (attached) notes, Lent 2013, page 15.

I want to evaluate the expression

## Z = \exp\Big(\frac{1}{2} \frac{\partial}{\partial \underline{x}} . A^{-1} \frac{\partial}{\partial \underline{x}} \Big) \exp\Big(-V(x) + \underline{b}. \underline{x}\Big) \bigg\vert_{\underline{x} = \underline{0}}##

assuming that $$\underline{b} = \underline{0}$$.

We use the notation

## V_{i_{1} i_{2} \dots i_{k}} = \frac{\partial}{\partial x_{i_{1}}} \frac{\partial}{\partial x_{i_{2}}} \dots \frac{\partial}{\partial x_{i_{k}}} V(\underline{x})\Big\vert_{\underline{x} = \underline{0}}##

Where

$$\frac{\partial}{\partial \underline{x}} \equiv \Big( \frac{\partial}{\partial x_{1}}, \dots, \frac{\partial}{\partial x_{n}}\Big)$$

We also assume that $$V(\underline{0}) = V_{i}(\underline{0}) = 0$$.

And $$A^{-1}$$ is an $$n \times n$$ matrix.
Relevant Equations
We need to expand the exponential of the derivative as a Taylor series.
I expanded the exponential with the derivative to get:

## Z = \Bigg(1 + \frac{1}{2} \frac{\partial}{\partial x_{i}} A^{-1}_{ij} \frac{\partial}{\partial x_{j}} + \frac{1}{4} \frac{\partial}{\partial x_{i}} A^{-1}_{ij} \frac{\partial}{\partial x_{j}} \frac{\partial}{\partial x_{k}} A^{-1}_{kl} \frac{\partial}{\partial x_{l}} + \frac{1}{12} \frac{\partial}{\partial x_{i}} A^{-1}_{ij} \frac{\partial}{\partial x_{j}} \frac{\partial}{\partial x_{k}} A^{-1}_{kl} \frac{\partial}{\partial x_{l}} \frac{\partial}{\partial x_{m}} A^{-1}_{mn} \frac{\partial}{\partial x_{n}} + \dots\Bigg) \exp(-V(x))\biggr\vert_{\underline{x} = \underline{0}}##

Which comes to

$$Z = 1 - \frac{1}{2} A^{-1}_{ij} V_{ij} + \frac{1}{4} A^{-1}_{ij} V_{ij} A^{-1}_{kl} V_{kl} + \dots$$

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I don't see any copyright notice in the notes. There is attribution on the first page, but no copyright or other statements...

Lent Term 2013
Hugh Osborn
Latex Lecture notes, originally typeset by
Steffen Gielen in 2007
revised by
Carl Turner in 2013
latest update: May 5, 2016

## 1. What is perturbation expansion with path integrals?

Perturbation expansion with path integrals is a mathematical technique used in theoretical physics to approximate solutions to complex problems. It involves breaking down a complicated problem into smaller, solvable parts and then combining these solutions to approximate the overall solution.

## 2. How does perturbation expansion with path integrals work?

This technique works by expressing the solution to a problem as a series of terms, with each term representing a different level of complexity. The first term is the simplest and easiest to solve, while subsequent terms become increasingly more complex. By adding together more and more terms, a more accurate approximation of the solution can be obtained.

## 3. What are the advantages of using perturbation expansion with path integrals?

One advantage of this technique is that it allows for the approximation of solutions to problems that would be too difficult or impossible to solve using traditional methods. It also provides a systematic way of improving the accuracy of the approximation by adding more terms to the series.

## 4. What are some applications of perturbation expansion with path integrals?

This technique is commonly used in quantum mechanics, statistical mechanics, and field theory to solve problems involving complex systems. It has also been applied in other areas such as finance, biology, and chemistry.

## 5. Are there any limitations to perturbation expansion with path integrals?

While this technique can be a powerful tool for solving complex problems, it does have some limitations. It is most effective for problems with small perturbations, and the accuracy of the approximation can decrease as the perturbation becomes larger. Additionally, the convergence of the series may be slow in some cases, making it difficult to obtain an accurate solution.

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