Perturbation expansion with path integrals

[saadhusayn]

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

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

But the answer is

 

[berkeman]

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

Advanced Quantum Field Theory
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