Feynman diagrams and path integral

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introduction

 

 

 

Finite-dimensional Feynman Diagrams

 

 

2. Facts from calculus and their d-dimensional analogues

The basic fact from calculus that powers the whole discussion is:

 

Proposition 1


 

The identity with  a = 1  is proved by the trick of calculating the square of the integral in polar coordinates. The general identity follows by change of variable from  x to .

This fact generalizes to higher-dimensional integrals. Set   v = (v1, ..., vd) and   dv = (dv1 ... dvd), and let   A   be a symmetric   d by d   matrix.

 

 

Proposition 2



 

We use the fact that a symmetric matrix A is diagonalizable: there exists an orthogonal matrix U (so Ut = U-1) such that UAU-1 is the diagonal matrix B whose only nonzero entries are b11, ... , bdd along the diagonal. Then A = U-1BU and vtAv = vtU-1B Uv = vtUtB Uv = wtBw where w = Uv, using Ut = U-1 and (Uv)t =vtUt. Since U is orthogonal   detU = 1   and the change of variable from v to w does not change the integral:

 

 


 


 

 

 


 

 

 


 


 

 

 

Proposition 3


 

This follows from Proposition 1 by completion of the square in the exponent and a change of variables.

The generalization to d dimensions replaces a with A as before and b with the vector b = (b1, ... , bd)

 

 

Proposition 4


 

 

This is proven exactly like Proposition 2. If we write this integral as Zb then the integral of Proposition 2 is Z0 and this proposition can be rewritten as

 


 

 

3. m-point functions

For any choice of m (not necessarily different) indices $i_1 ,\dots , i_m$ between 1 and d, define the m-point function as follows:

$$ \langle v^{i_1}\dots v^{i_m}\rangle = \frac{1}{Z_0}\int_{{\bf R}^d} d{\bf v} ~~\exp({\scriptstyle\frac{ 1}{ 2}}{\bf v}^tA~{\bf v})v^{i_1}\dots v^{i_m}. $$

The m-point functions are a step towards the ultimate aim of our calculation. They enter at this moment because they can be calculated by repeated differentiation of $Z_{\bf b}$

For example, note that

$$ \begin{aligned} \frac{\partial Z_{\bf b}}{\partial b^i} &= \frac{\partial}{\partial b^i}\int_{{\bf R}^d} d{\bf v} ~~ \exp({\scriptstyle\frac{ 1}{ 2}}{\bf v}^tA~{\bf v} + {\bf b}^t{\bf v})\\ {} &= \int_{{\bf R}^d} d{\bf v} ~~ \frac{\partial}{\partial b^i}\exp({\scriptstyle\frac{ 1}{ 2}}{\bf v}^tA~{\bf v} + {\bf b}^t{\bf v}) \\ {} &= \int_{{\bf R}^d} d{\bf v} ~~ \exp({\scriptstyle\frac{ 1}{ 2}}{\bf v}^tA~{\bf v} + {\bf b}^t{\bf v}) v^i \end{aligned} $$

So the 1-point function $v^i$ is given by $$ \langle v^i \rangle = \frac{1}{Z_0} \frac{\partial Z_{\bf b}}{\partial b^i}\vert _{{\bf b} =0} $$

Similarly the m-point function $\langle v^{i_1}\dots v^{i_m}\rangle$ is given by

$$ \begin{aligned} \langle v^{i_1} ,\dots , v^{i_m}\rangle =& \frac{1}{Z_0} (\frac{\partial}{\partial b^{i_1}}\cdots \frac{\partial}{\partial b^{i_m}}Z_{\bf b})_{\textstyle \vert _{{\bf b} =0}}\\ {}=& \frac{\partial}{\partial b^{i_1}}\cdots \frac{\partial}{\partial b^{i_m}} \exp(\frac{1}{2}{\bf b}^tA^{-1}{\bf b})_{\textstyle \vert _{{\bf b} =0}} \end{aligned} $$

4. Wick's Theorem

Calculating high-order derivatives of a function like $\exp(\frac{1}{2}{\bf b}^tA^{-1}{\bf b})$ can be very messy. A useful theorem reduces the calculation to combinatorics.

Wick's theorem

$$\displaystyle \frac{\partial}{\partial b^{i_1}}\cdots \frac{\partial}{\partial b^{i_m}}\exp(\frac{1}{2}{\bf b}^tA^{-1}{\bf b})= A^{-1}_{\textstyle i_{p_1},i_{p_2}} \cdots A^{-1}_{\textstyle i_{p_{m-1}},i_{p_m}},$$

where the sum is taken over all pairings $(i_{p_1},i_{p_2}), \dots, (i_{p_{m-1}},i_{p_m})$ of $i_1,\cdots, i_m$

Wick's theorem is proved (a careful counting argument) in texts on quantum field theory. The most detailed explanation is in S. S. Schweber, An Introduction to Relativistic Quantum Field Theory, Evanston, IL, Row, Peterson 1961.

Let us calculate a couple of examples.

To begin, it is useful to write $\exp(\frac{1}{2}{\bf b}^tA^{-1}{\bf b})$ with ${\bf b}^tA^{-1}{\bf b}=\sum A_{i,j}^{-1}b^ib^j$ (the sums running from 1 to d) using the series expansion   exp x = 1 + x +x2/2 +x3/3! ... . The typical term will be $(1/n!)(1/2^n)(\sum A^{-1}_{i,j}b^ib^j)^n$ . This term is a homogeneous polynomial in the bi of degree 2n

Differentiating k times a homogeneous polynomial of degree 2n and evaluating at zero will give zero unless k = 2n. So the job is to analyze the result of 2n differentiations on $(1/n!)(1/2^n)(\sum A^{-1}_{i,j}b^ib^j)^n$.

The differentiation carried out most frequently in these calculations is $$\displaystyle \frac{\partial}{\partial b^{k}}(\frac{1}{2}\sum_{i,j=1}^d A^{-1}_{i,j}b^ib^j) = \sum_{i=1}^d A^{-1}_{i,k}b^i,$$


where we use the symmetry of the matrix $A^{-1}$, a direct consequence of the symmetry of $A$.

In what follows $\frac{\partial}{\partial b^{i}}$ will be abbreviated as $\partial_i$ .


n=1

$A^{-1}_{1,2}$, using the symmetry of the matrix $A^{-1}$. The same calculation shows that $$ \partial_1 \partial_1(\frac{1}{2}\sum A^{-1}_{i,j}b^ib^j) = A^{-1}_{1,1} $$

Note that (1,2) and (2,1) count as the same pairing.

 

n=2

$$ \begin{aligned} \partial_4\partial_3\partial_2\partial_1(1/2!)(1/2^2)(\sum A^{-1}_{i,j}b^ib^j)^2 =& \partial_4\partial_3\partial_2 (1/2)(\sum A^{-1}_{i,j}b^ib^j)( \sum A^{-1}_{1,j}b^j) \\ {}=& [(\sum A^{-1}_{2,j}b^j)( \sum A^{-1}_{1,j}b^j) +(1/2)(\sum A^{-1}_{i,j}b^ib^j)A^{-1}_{1,2}] \\ {}=& \partial_4[A^{-1}_{2,3}( \sum A^{-1}_{1,j}b^j)+(\sum A^{-1}_{2,j}b^j)A^{-1}_{1,3} +(\sum A^{-1}_{3,j}b^j)A^{-1}_{1,2}] \\ {}=& A^{-1}_{2,3}A^{-1}_{1,4} + A^{-1}_{2,4}A^{-1}_{1,3} + A^{-1}_{3,4}A^{-1}_{1,2} \end{aligned} $$

Similarly: $$ \begin{aligned} \partial_4\partial_3\partial_1\partial_1~~{\rm gives~~}2A^{-1}_{1,4}A^{-1}_{1,3} + A^{-1}_{3,4}A^{-1}_{1,1}\\ \partial_4\partial_1\partial_1\partial_1~~{\rm gives~~} 3A^{-1}_{1,4}A^{-1}_{1,1} \\ \partial_4\partial_4\partial_1\partial_1~~{\rm gives~~} 2 A^{-1}_{1,4}A^{-1}_{1,4} + A^{-1}_{4,4}A^{-1}_{1,1}\\ \partial_1\partial_1\partial_1\partial_1~~{\rm gives~~} 3 A^{-1}_{1,1}A^{-1}_{1,1} \\ \end{aligned} $$  

5. The first appearance of graphs


 

In the last section we calculated some 2 and 4-point functions:

<v1,v2> = A-11,2

<v1,v1> = A-11,1<v1,v2,v3,v4> = A-12,3A-11,4 + A-12,4A-11,3 + A-134A-11,2<v1,v1,v3,v4> = 2 A-11,4A-11,3 + A-13,4A-11,1<v1,v1,v1,v4> = 3 A-11,4A-11,1<v1,v1,v4,v4> = 2A-11,4A-11,4 + A-14,4A-11,1<v1,v1,v1,v1> = 3 A-11,1A-11,1


It is convenient to represent each of products appearing on the right as a graph, where the vertices represent the indices of the coordinates vi appearing in the m-point function, and each A-1i,j becomes an edge from vertex i to vertex j. Here are the graphs corresponding to the terms in the 4-point functions above.

 


6. Calculations with a potential function, ``Feynman Rules


 

The integrals of interest in Physics have the form


which we rewrite using the series expansion for the exponential as



If U is a polynomial in the coordinate functions v1, ...vd, then each term in the sum of integrals is a sum of m-point functions, and can be evaluated by our method, which can be written symbolically as:



Example: This example is formally like the `` theory. We take and analyze



using the abbreviation = as before.

Let us compute the terms of degree 2 in .

These terms will involve 6 derivatives; their sum is:

 


By Wick's Theorem we can rewrite this sum as

 


where the inside sum is taken over all pairings (i1,i2),(i3,i4)(i5i6) of i, j, k, i', j', k'.

These pairings can also be represented by graphs, very much in the same way that we used for m-point functions: there will be one trivalent vertex for each u factor, and one edge for each A-1. In this case there will be exactly two distinct graphs, according as the number of (unprimed, primed) index pairs is 1 or 3.


The ``dumbbell and the ``thetaare the two 3-valent 2-vertex graphs.

Summing over all possible labellings of these graphs will give some duplication, since each graph has symmetries that make different labellings correspond to the same pairing.

 


All eight of these labelings correspond to the same product: u123 u456 A-113 A-125 A-146.


All six of these labelings, and their six left-right mirror images, correspond to the same product: u123 u456 A-114 A-125 A-136.

 

The ``dumbbell graph has an automorphism (symmetry) group of order eight, whereas the ``theta graph has an automorphism group of order twelve.

Keeping this in mind, we may rewrite the coefficient of as:

 


where the sum is taken over the set of the topologically distinct trivalent graphs with two vertices (in this case, 2), the products are taken over the set of all vertices v (here there are 2) and the set of all edges e (here there are 3) respectively, and |AutG| is the number of automorphisms of the graph G.

In general, the ``Feynman rules for computing the coefficient of in the expansion of ZU are stated in exactly this way, except that the sum is over trivalent graphs with 2n vertices (and 3n edges).

 

7. Correlation functions


 

The way path integrals are used in quantum field theory is, very roughly speaking, that the probability amplitude of a process going from point v1 to point v2 is an integral over all possible ways of getting from v1 to v2. In our finite-dimensional model, each of these ``ways is represented by a point v in Rn and the probability measure assigned to that way is . The integral is what we called before a 2-point function


and what we will now call a correlation function.

We continue with the example of the cubic potential

.

By our previous calculations,


In terms of Wick's Theorem and our graph interpretation of pairings, this becomes:

 


where now the sum is over all graphs G with two single-valent vertices (the ends) labeled 1 and 2, and n 3-valent vertices.

 


This graph occurs in the calculation of the coefficient of in <v1,v2>.

The k-point correlation functions are similarly defined and calculated. Here is where we begin to see the usual ``Feynman diagrams.


This graph occurs in the calculation of the coefficient of in <v1,v2,v3,v4>.

 

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