Understanding Wick Contractions for Quantum Field Theory

[NanakiXIII]

I'm reading Quantum Field Theory in a Nutshell by Zee and I'm completely lost on something. In one of the appendices to chapter I.2, an appendix on Gaussian integrals, he introduces Wick contractions. It seems to be coming out of the blue and I don't understand what he's talking about. I'm sure the math all works out, but I don't see what any of it means, and because of that I also don't follow the generalizations he's making (which I need to if I'm going to do an exercise - right now it looks like I'm just going to be skipping it). Worse yet, if I Google Wick contractions, I seem to find something, well, not completely unrelated, but something quite different.

Of course my question sort of hopes that someone has read this book, but maybe I can also be satisfied if someone could just explain to me what a Wick contraction is, what it is supposed to mean, in a physical sense (the mathematical Wiki page isn't too clear to me either).

 

[LukeD]

I was wondering this also since it seems that understanding Wick Contractions are key to understanding Feynman Diagrams.

Clearly, it's a method for breaking an integral into sums and products of simpler integrals, but I haven't been able to figure out the simplified integrals that the Wick Contraction leads to. Could someone explain this?

 

[peteratcam]

Given a multivariate gaussian distribution, one is interested in averages of polynomials of the variables.
Given a free field theory, one is interested in various n-point correlation functions.

For the multivariate gaussian, the matrix of averages of polynomials of degree 2 is called the covariance matrix.
For the free field theory, the 2-point correlation function is called the propagator.

There is a neat theorem which expresses averages of higher polynomials in terms of the covariance matrix, Isserlis Theorem.
There is a neat theorem which expresses n-point correlation functions in terms of the propagator, Wick's theorem.

http://en.wikipedia.org/wiki/Isserlis’_theorem

If you understand multivariate gaussians, and how to calculate their higher moments, you will understand Feynman diagrams.

 

[olgranpappy]

I'm reading Quantum Field Theory in a Nutshell by Zee and I'm completely lost on something. In one of the appendices to chapter I.2, an appendix on Gaussian integrals, he introduces Wick contractions. It seems to be coming out of the blue and I don't understand what he's talking about. I'm sure the math all works out, but I don't see what any of it means, and because of that I also don't follow the generalizations he's making (which I need to if I'm going to do an exercise - right now it looks like I'm just going to be skipping it). Worse yet, if I Google Wick contractions, I seem to find something, well, not completely unrelated, but something quite different.

Of course my question sort of hopes that someone has read this book, but maybe I can also be satisfied if someone could just explain to me what a Wick contraction is, what it is supposed to mean, in a physical sense (the mathematical Wiki page isn't too clear to me either).

Because Zee starts off using the path integral formalism rather than canonical formalism, his proof of Wick's Theorem is really straightforward and just involves doing a bunch of gaussian integrals. The theorem can also be proved using the canonical formalism, but it's a lot more disgusting. See, e.g., Peskin and Schroeder's fine book.

Basically, you use Wick's theorem in order to write big complicated correlation functions in terms of products of the simplest correlation function (the "single particle propagator"/"green's function"/whatever you want to call it).

These big complicated correlation functions show up in, say, scattering amplitudes and whatnot, and we draw Feynman diagrams to represent these correlation functions as products of the simplest correlation function. E.g., If I see a Feynman diagram with two distinct lines this means that the mathematical expression I write down is the product of two single particle green's functions. how the lines are connected tells me about the arguments of the green's functions. Etc.

 

[NanakiXIII]

I sat down and plowed through the maths (I'm reading this in my free time, I was hoping for a somewhat lighter read), so at least that is a lot more clear to me now. The point of it still eludes me, but judging from the replies, I just need to read on. Presumably the applications will become more clear later on. You guys have given me some idea of what the point is going to be, but I'm not really familiar with half of what you're talking about yet.

 

[element4]

Think of Wicks theorem as a way of rewriting n-point functions as products and sums of two point functions.

For example
<\phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)>_0 = \Delta(x_1-x_2)\Delta(x_3-x_4) + \Delta(x_1-x_3)\Delta(x_2-x_4) + \Delta(x_1-x_4)\Delta(x_2-x_3),
where \Delta(x_i - x_j) = <\phi(x_i)\phi(x_j)>_0.

This trick is important when building up perturbation theory with Feynman diagrams.

 

[jensa]

To give you some feeling of what's going on I can try to elaborate on what peteratcam and others have said. The analogy with statistical analysis is a very good one and goes a long way in understanding how things work.

Let's say you have a set \mathbf{x}=(x_1,x_2,..,x_n)^T of statistical variables that you are interested in. Let's say you have a generalized gaussian probability distribution

<br /> P(\mathbf{x})=Ne^{-\frac{1}{2}\mathbf{x}^T\mathbf{A}\mathbf{x}}<br />

were N is a normalization factor such that \int dx_1dx_2\ldots dx_n P(\mathbf{x})=1, and \mathbf{A} is a n\times n matrix*.

Now usually we are interested in how different statistical variables are correlated to each other described through the correlation function. For example, the correlation between two statistical variables x_i,x_j is given by

<br /> \langle x_ix_j\rangle, \qquad \text{where} \qquad \langle \ldots \rangle = \int \Pi_idx_i P(\mathbf{x})\ldots<br />

So the first important feature of gaussian distributions is that such correlation functions are given by

<br /> \langle x_ix_j\rangle=(\mathbf{A}^{-1})_{ij}<br />

so that all you really need to do to evaluate two-variable correlation functions is to invert the matrix \mathbf{A}. How about higher order correlation functions like \langle x_ix_jx_kx_l\rangle? Well, this is where wick contractions come in. It can be shown that any such correlation function can be written as a sum of products of two-variable correlation functions. Basically you have to figure out all the different ways to "pair" (contract) the different variables in the correlation function. In our example we have three different pairings

<br /> \langle x_ix_jx_kx_l\rangle=\langle x_ix_j\rangle\langle x_kx_l\rangle+\langle x_ix_k\rangle\langle x_jx_l\rangle+ \langle x_ix_l\rangle\langle x_jx_k\rangle =(\mathbf{A}^{-1})_{ij}(\mathbf{A}^{-1})_{kl}+(\mathbf{A}^{-1})_{ik}(\mathbf{A}^{-1})_{jl}+(\mathbf{A}^{-1})_{il}(\mathbf{A}^{-1})_{jk}<br />

So again, all we really need is to invert the matrix A and we can find any correlation function (although we also have to do the combinatorics of finding all the different pairings (contractions)).

Now, higher order correlation functions may seem somewhat abstract and their physical significance is not always clear. However, wick contractions can be used also in the context of perturbation theory. Consider for example that our probability distribution is only approximately described by a gaussian and in fact has the form

<br /> P&#039;(\mathbf{x})=N&#039; e^{-\frac{1}{2}\mathbf{x}^T\mathbf{A}\mathbf{x}+\lambda V(\mathbf{x})}<br />

where \lambda is a small parameter and V(\mathbf{x}) is some a sum of polynomials of x_i. With this probability distribution function it immediately becomes difficult to evaluate even the simple two-variable correlation function directly. However, due to the small parameter \lambda we can perform an expansion: e^{\lambda V(\mathbf{x})}=1+\lambda V(\mathbf{x})+\mathcal{O}(\lambda^2). Then, to first order in \lambda we can evaluate

<br /> \langle x_ix_j\rangle=\langle x_ix_j\rangle_0+\lambda\langle x_i x_j V(\mathbf{x})\rangle_0<br />

where \langle \ldots \rangle_0 denotes the average with respect to the gaussian distribution function. Since V(\mathbf{x}) consists of a set of polynomials the second term on the right hand side will consist of higher order correlation functions taken wrt the gaussian distribution. Hence, using wicks theorem the two-variable correlation function taken with respect to the non-gaussian (perturbed) distribution P' can again be expressed through two-variable correlation functions with respect to the gaussian distribution. All due to the wick contraction feature of gaussian distributions.

The combinatorics of making the different pairings can elegantly be put in a graphical framework in terms of Feynman diagrams.

As a final note (this post got a lot longer than I intended it to be) I would like to mention that in this analogy we have to take into account that the normalization of the distribution function has been changed when the perturbing part V(x) was added. Imposing the correct normalization will lead to a cancellation of the so called "vacuum diagrams" that correspond to the pairings pairings between internal variables of the perturbation V(x_1,x_2,..x_n).

I think if you keep this example of a statistical distribution in mind a lot of the things in QFT will be easier to understand. Hope it helps.


*For real x_i it has to be real, symmetric and positive definite.


Edit: Just to make the relationship/analogy with QFT clearer: In QFT the statistical variables will be the fields at different space-time points x_i\rightarrow \phi(x) and the inverse of the matrix will be the free propagator/greens function (\mathbf{A}^{-1})_{ij}\rightarrow i\Delta(x-y). Of course since the space-time points are continuous the exponent must be expressed accordingly: \mathbf{x}^T\mathbf{A}\mathbf{x}=\sum_{ij}x_iA_{ij}x_j\rightarrow -i\int dx dy \phi(x)\Delta^{-1}(x-y)\phi(y) and the integration measure will become a functional integrasion measure: N\int \Pi_idx_i\rightarrow \int \mathcal{D}\phi]

Edit2: It is instructive to consider a quartic perturbation like V(\mathbf{x})=\sum_{mnlk} V_{mnlk} x_m x_n x_k x_l and see the different pairings that you get. To first order the small parameter you have
<br /> \langle x_ix_j\rangle=\langle x_ix_j\rangle_0+\lambda\sum_{nmlk}V_{nmlk}\langle x_i x_j x_m x_n x_k x_l\rangle_0<br />
In terms of feynman diagrams each pairing (contraction) \langle x_ix_j\rangle=(\mathbf{A}^{-1})_{ij} is depicted by a line (propagator) connecting two points i,j, and the constant V_{nmlk} is depicted by a vertex with 4 legs (corresponding to indices nmlk which are to be summed over) which will become attached to lines (propagators) due to the contraction between variables. It's really a great way to get a feel for the diagram technique.

 

[ismaili]

wow, very nice article, Jensa.
I was also wondering why we perform such a strange integral to get the n-point functions in QFT when i was a beginner in field theory.
Analogy to statistical analysis makes it clear of what we are doing in QFT.

BTW, Is it possible to collect good articles in the forum?
Such as some personal notes function or something like that.

 

[NanakiXIII]

A very lucid explanation, jensa, thanks. It has given me a better feel of what is going on. Once I figure out how to print out a single post, I'm going to stick yours into my notes, if you don't mind.