"Feynman diagrams and path integral"의 두 판 사이의 차이
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| (사용자 2명의 중간 판 18개는 보이지 않습니다) | |||
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==Finite-dimensional Feynman Diagrams== | ==Finite-dimensional Feynman Diagrams== | ||
| − | + | * {{수학노트|url=윅_정리}} | |
* [http://www.math.sunysb.edu/%7Etony/whatsnew/column/feynman-1101/feynman1.html Finite-dimensional Feynman Diagrams] | * [http://www.math.sunysb.edu/%7Etony/whatsnew/column/feynman-1101/feynman1.html Finite-dimensional Feynman Diagrams] | ||
* [http://www.quantumdiaries.org/2010/02/14/lets-draw-feynman-diagams/ Let’s draw Feynman diagrams!] | * [http://www.quantumdiaries.org/2010/02/14/lets-draw-feynman-diagams/ Let’s draw Feynman diagrams!] | ||
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| − | + | ==m-점 함수(m-point functions)== | |
| + | For any choice of m (not necessarily different) indices <math>i_1 ,\dots , i_m</math> between 1 and d, define the m-point function as follows: | ||
| − | + | :<math> | |
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\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}. | \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}. | ||
| − | + | </math> | |
| − | 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 | + | 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 <math>Z_{\bf b}</math> |
For example, note that | For example, note that | ||
| − | + | :<math> | |
\begin{aligned} | \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})\\ | \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})\\ | ||
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{} &= \int_{{\bf R}^d} d{\bf v} ~~ \exp({\scriptstyle\frac{ 1}{ 2}}{\bf v}^tA~{\bf v} + {\bf b}^t{\bf v}) v^i | {} &= \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} | \end{aligned} | ||
| − | + | </math> | |
| − | So the 1-point function | + | So the 1-point function <math>v^i</math> is given by |
| − | + | :<math> | |
| − | \langle v^i \rangle = \frac{1}{Z_0} \frac{\partial Z_{\bf b}}{\partial b^i}\vert _{{\bf b} =0} = \frac{\partial}{\partial b^i} \exp({\scriptstyle\frac{1}{2}}{\bf b}^tA^{-1}{\bf b})\vert _{{\bf b} =0} | + | \langle v^i \rangle = \frac{1}{Z_0} \frac{\partial Z_{\bf b}}{\partial b^i}\vert _{{\bf b} =0} = \frac{\partial}{\partial b^i} \exp({\scriptstyle\frac{1}{2}}{\bf b}^tA^{-1}{\bf b})_{\vert _{{\bf b} =0}} |
| − | + | </math> | |
| − | Similarly the m-point function | + | Similarly the m-point function <math>\langle v^{i_1}\dots v^{i_m}\rangle</math> is given by |
| − | + | :<math> | |
\begin{aligned} | \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}}\\ | \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}} | {}=& \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} | \end{aligned} | ||
| − | + | </math> | |
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== Wick's Theorem == | == Wick's Theorem == | ||
| − | Calculating high-order derivatives of a function like | + | Calculating high-order derivatives of a function like <math>\exp(\frac{1}{2}{\bf b}^tA^{-1}{\bf b})</math> can be very messy. A useful theorem reduces the calculation to combinatorics. |
| − | + | ;Wick's theorem | |
| + | :<math>\displaystyle \frac{\partial}{\partial b^{i_1}}\cdots \frac{\partial}{\partial b^{i_m}}\exp(\frac{1}{2}{\bf b}^tA^{-1}{\bf b})_{\vert _{{\bf b} =0}}= | ||
| + | A^{-1}_{\textstyle i_{p_1},i_{p_2}} \cdots A^{-1}_{\textstyle i_{p_{m-1}},i_{p_m}},</math> | ||
| − | + | where the sum is taken over all pairings <math>(i_{p_1},i_{p_2}), \dots, (i_{p_{m-1}},i_{p_m})</math> of <math>i_1,\cdots, i_m</math> | |
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| − | where the sum is taken over all pairings | ||
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. | 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. | ||
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Let us calculate a couple of examples. | Let us calculate a couple of examples. | ||
| − | To begin, it is useful to write | + | To begin, it is useful to write <math>\exp(\frac{1}{2}{\bf b}^tA^{-1}{\bf b})</math> with <math>{\bf b}^tA^{-1}{\bf b}=\sum A_{i,j}^{-1}b^ib^j</math> |
| − | (the sums running from 1 to < | + | (the sums running from 1 to <math>d</math>) using the series expansion |
| + | :<math> | ||
| + | \exp x=1+\frac{x}{1!}+\frac{x^2}{2!}+\cdots | ||
| + | </math> | ||
| + | The typical term will be <math>(1/n!)(1/2^n)(\sum A^{-1}_{i,j}b^ib^j)^n</math> . This term is a homogeneous polynomial in the <em style="">b<sup style="">i</sup></em> of degree 2<em style="">n</em> | ||
| − | Differentiating <em style="">k</em> times a homogeneous polynomial of degree 2<em style="">n</em> and evaluating at zero will give zero unless <em style="">k</em> = 2<em style="">n</em>. So the job is to analyze the result of 2<em style="">n</em> differentiations on | + | Differentiating <em style="">k</em> times a homogeneous polynomial of degree 2<em style="">n</em> and evaluating at zero will give zero unless <em style="">k</em> = 2<em style="">n</em>. So the job is to analyze the result of 2<em style="">n</em> differentiations on <math>(1/n!)(1/2^n)(\sum A^{-1}_{i,j}b^ib^j)^n</math>. |
The differentiation carried out most frequently in these calculations is | The differentiation carried out most frequently in these calculations is | ||
| − | + | :<math>\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,</math> | |
| − | where we use the symmetry of the matrix | + | where we use the symmetry of the matrix <math>A^{-1}</math>, a direct consequence of the symmetry of <math>A</math>. |
| − | In what follows | + | In what follows <math>\frac{\partial}{\partial b^{i}}</math> will be abbreviated as <math>\partial_i</math> . |
====n=1==== | ====n=1==== | ||
| − | + | :<math> | |
| − | + | \partial_2 \partial_1(\frac{1}{2}\sum A^{-1}_{i,j}b^ib^j) = A^{-1}_{1,2} | |
| + | </math> | ||
| + | using the symmetry of the matrix <math>A^{-1}</math>. The same calculation shows that | ||
| + | :<math> | ||
\partial_1 \partial_1(\frac{1}{2}\sum A^{-1}_{i,j}b^ib^j) = A^{-1}_{1,1} | \partial_1 \partial_1(\frac{1}{2}\sum A^{-1}_{i,j}b^ib^j) = A^{-1}_{1,1} | ||
| − | + | </math> | |
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Note that (1,2) and (2,1) count as the same pairing. | Note that (1,2) and (2,1) count as the same pairing. | ||
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====n=2==== | ====n=2==== | ||
| − | + | :<math> | |
\begin{aligned} | \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 | \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 | ||
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{}=& A^{-1}_{2,3}A^{-1}_{1,4} + A^{-1}_{2,4}A^{-1}_{1,3} + A^{-1}_{3,4}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} | \end{aligned} | ||
| − | + | </math> | |
Similarly: | Similarly: | ||
| − | + | :<math> | |
\begin{aligned} | \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_3\partial_1\partial_1~~{\rm gives~~}2A^{-1}_{1,4}A^{-1}_{1,3} + A^{-1}_{3,4}A^{-1}_{1,1}\\ | ||
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\partial_1\partial_1\partial_1\partial_1~~{\rm gives~~} 3 A^{-1}_{1,1}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} | \end{aligned} | ||
| − | + | </math> | |
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== The first appearance of graphs == | == The first appearance of graphs == | ||
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In the last section we calculated some 2 and 4-point functions: | In the last section we calculated some 2 and 4-point functions: | ||
| − | + | :<math>\langle v^1,v^2 \rangle=A^{-1}_{1,2}</math> | |
| − | + | :<math>\langle v^1,v^1 \rangle=A^{-1}_{1,1}</math> | |
| − | + | :<math>\langle v^1,v^2,v^3,v^4 \rangle=A^{-1}_{2,3}A^{-1}_{1,4}+A^{-1}_{2,4}A^{-1}_{1,3}+A^{-1}_{3,4}A^{-1}_{1,2}</math> | |
| − | + | :<math>\langle v^1,v^1,v^3,v^4 \rangle=2A^{-1}_{1,4}A^{-1}_{1,3}+A^{-1}_{3,4}A^{-1}_{1,1}</math> | |
| − | + | :<math>\langle v^1,v^1,v^1,v^4 \rangle=3A^{-1}_{1,4}A^{-1}_{1,1}</math> | |
| − | + | :<math>\langle v^1,v^1,v^4,v^4 \rangle=2A^{-1}_{1,4}A^{-1}_{1,4}+A^{-1}_{4,4}A^{-1}_{1,1}</math> | |
| − | + | :<math>\langle v^1,v^1,v^1,v^1 \rangle=3A^{-1}_{1,1}A^{-1}_{1,1}</math> | |
| − | It is convenient to represent each of products appearing on the right as a <em style="">graph</em>, where the vertices represent the indices of the coordinates | + | It is convenient to represent each of products appearing on the right as a <em style="">graph</em>, where the vertices represent the indices of the coordinates <math>v_i</math> appearing in the m-point function, and each :<math>A^{-1}_{i,j}</math> becomes an edge from vertex i to vertex j. Here are the graphs corresponding to the terms in the 4-point functions above. |
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== Calculations with a potential function, ``Feynman Rules'' == | == Calculations with a potential function, ``Feynman Rules'' == | ||
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The integrals of interest in Physics have the form | The integrals of interest in Physics have the form | ||
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If <em style="">U</em> is a polynomial in the coordinate functions <em style="">v</em><sup style="">1</sup>, ...<em style="">v<sup style="">d</sup></em>, then each term in the sum of integrals is a sum of <em style="">m</em>-point functions, and can be evaluated by our method, which can be written symbolically as: | If <em style="">U</em> is a polynomial in the coordinate functions <em style="">v</em><sup style="">1</sup>, ...<em style="">v<sup style="">d</sup></em>, then each term in the sum of integrals is a sum of <em style="">m</em>-point functions, and can be evaluated by our method, which can be written symbolically as: | ||
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<em style="">Example:</em> This example is formally like the `` theory.'' We take and analyze | <em style="">Example:</em> This example is formally like the `` theory.'' We take and analyze | ||
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These terms will involve 6 derivatives; their sum is: | These terms will involve 6 derivatives; their sum is: | ||
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By Wick's Theorem we can rewrite this sum as | By Wick's Theorem we can rewrite this sum as | ||
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These pairings can also be represented by graphs, very much in the same way that we used for <em style="">m</em>-point functions: there will be one trivalent vertex for each <em style="">u</em> factor, and one edge for each <em style="">A</em><sup style="">-1</sup>. In this case there will be exactly two distinct graphs, according as the number of (unprimed, primed) index pairs is 1 or 3. | These pairings can also be represented by graphs, very much in the same way that we used for <em style="">m</em>-point functions: there will be one trivalent vertex for each <em style="">u</em> factor, and one edge for each <em style="">A</em><sup style="">-1</sup>. 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 ``theta''are 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. | 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. | ||
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| − | + | All eight of these labelings correspond to the same product: <em style="">u</em><sub style="">123</sub> <em style="">u</em><sub style="">456</sub> <em style="">A</em><sup style="">-1</sup><sub style="">13</sub> <em style="">A</em><sup style="">-1</sup><sub style="">25</sub> <em style="">A</em><sup style="">-1</sup><sub style="">46</sub>. | |
| − | + | All six of these labelings, and their six left-right mirror images, correspond to the same product: <em style="">u</em><sub style="">123</sub> <em style="">u</em><sub style="">456</sub> <em style="">A</em><sup style="">-1</sup><sub style="">14</sub> <em style="">A</em><sup style="">-1</sup><sub style="">25</sub> <em style="">A</em><sup style="">-1</sup><sub style="">36</sub>. | |
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The ``dumbbell'' graph has an <em style="">automorphism</em> (symmetry) group of order eight, whereas the ``theta'' graph has an automorphism group of order twelve. | The ``dumbbell'' graph has an <em style="">automorphism</em> (symmetry) group of order eight, whereas the ``theta'' graph has an automorphism group of order twelve. | ||
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Keeping this in mind, we may rewrite the coefficient of as: | Keeping this in mind, we may rewrite the coefficient of as: | ||
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In general, the ``Feynman rules'' for computing the coefficient of in the expansion of <em style="">Z<sub style="">U</sub></em> are stated in exactly this way, except that the sum is over trivalent graphs with 2<em style="">n</em> vertices (and 3<em style="">n</em> edges). | In general, the ``Feynman rules'' for computing the coefficient of in the expansion of <em style="">Z<sub style="">U</sub></em> are stated in exactly this way, except that the sum is over trivalent graphs with 2<em style="">n</em> vertices (and 3<em style="">n</em> edges). | ||
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== 7. Correlation functions == | == 7. Correlation functions == | ||
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The way path integrals are used in quantum field theory is, very roughly speaking, that the probability amplitude of a process going from point <em style="">v</em><sub style="">1</sub> to point <em style="">v</em><sub style="">2</sub> is an integral over all possible ways of getting from <em style="">v</em><sub style="">1</sub> to <em style="">v</em><sub style="">2</sub>. In our finite-dimensional model, each of these ``ways'' is represented by a point '''v''' in '''R'''<em style=""><sup style="">n</sup></em> and the probability measure assigned to that way is . The integral is what we called before a 2-point function | The way path integrals are used in quantum field theory is, very roughly speaking, that the probability amplitude of a process going from point <em style="">v</em><sub style="">1</sub> to point <em style="">v</em><sub style="">2</sub> is an integral over all possible ways of getting from <em style="">v</em><sub style="">1</sub> to <em style="">v</em><sub style="">2</sub>. In our finite-dimensional model, each of these ``ways'' is represented by a point '''v''' in '''R'''<em style=""><sup style="">n</sup></em> and the probability measure assigned to that way is . The integral is what we called before a 2-point function | ||
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In terms of Wick's Theorem and our graph interpretation of pairings, this becomes: | In terms of Wick's Theorem and our graph interpretation of pairings, this becomes: | ||
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where now the sum is over all graphs <em style="">G</em> with two single-valent vertices (the ends) labeled 1 and 2, and <em style="">n</em> 3-valent vertices. | where now the sum is over all graphs <em style="">G</em> with two single-valent vertices (the ends) labeled 1 and 2, and <em style="">n</em> 3-valent vertices. | ||
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| − | + | This graph occurs in the calculation of the coefficient of in <<em style="">v</em><sup style="">1</sup>,<em style="">v</em><sup style="">2</sup>>. | |
The <em style="">k</em>-point correlation functions are similarly defined and calculated. Here is where we begin to see the usual ``Feynman diagrams.'' | The <em style="">k</em>-point correlation functions are similarly defined and calculated. Here is where we begin to see the usual ``Feynman diagrams.'' | ||
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This graph occurs in the calculation of the coefficient of in <<em style="">v</em><sup style="">1</sup>,<em style="">v</em><sup style="">2</sup>,<em style="">v</em><sup style="">3</sup>,<em style="">v</em><sup style="">4</sup>>. | This graph occurs in the calculation of the coefficient of in <<em style="">v</em><sup style="">1</sup>,<em style="">v</em><sup style="">2</sup>,<em style="">v</em><sup style="">3</sup>,<em style="">v</em><sup style="">4</sup>>. | ||
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* https://docs.google.com/file/d/0B8XXo8Tve1cxSU5CQm1IVmZoQlk/edit | * https://docs.google.com/file/d/0B8XXo8Tve1cxSU5CQm1IVmZoQlk/edit | ||
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==books== | ==books== | ||
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==expositions== | ==expositions== | ||
| + | * Henn, Johannes M. “Lectures on Differential Equations for Feynman Integrals.” arXiv:1412.2296 [hep-Ph, Physics:hep-Th, Physics:math-Ph], December 6, 2014. http://arxiv.org/abs/1412.2296. | ||
* [http://www.science20.com/crawler_superland/richard_feynmans_functional_integral_1_action-89975 Feynman's functional integrals] | * [http://www.science20.com/crawler_superland/richard_feynmans_functional_integral_1_action-89975 Feynman's functional integrals] | ||
| − | + | * [http://www.lepp.cornell.edu/~pt267/undergradparticles.html Particle Physics for Undergrads] | |
| − | + | * Fiorenza, Domenico, and Riccardo Murri. 2001. “Feynman Diagrams via Graphical Calculus.” arXiv:math/0106001 (May 31). http://arxiv.org/abs/math/0106001. | |
| + | * [http://www.phy.pmf.unizg.hr/~kkumer/articles/feynman_for_beginners.pdf Feynman diagrams for beginners] | ||
| + | * Coecke, Bob. “Kindergarten Quantum Mechanics.” arXiv:quant-ph/0510032, October 4, 2005. http://arxiv.org/abs/quant-ph/0510032. | ||
==articles== | ==articles== | ||
| − | * | + | * Timothy Nguyen, The Perturbative Approach to Path Integrals: A Succinct Mathematical Treatment, arXiv:1505.04809 [math-ph], May 18 2015, http://arxiv.org/abs/1505.04809 |
| − | + | * http://arxiv.org/abs/1601.03646 | |
| − | + | * http://arxiv.org/abs/1112.4360 | |
| + | * Kotikov, A. V. “The Property of Maximal Transcendentality: Calculation of Feynman Integrals.” arXiv:1601.00486 [hep-Ph, Physics:hep-Th], January 4, 2016. http://arxiv.org/abs/1601.00486. | ||
| + | * http://arxiv.org/abs/1512.06407 | ||
| + | * Purkart, Julian. “The Signed Permutation Group on Feynman Graphs.” arXiv:1512.02408 [hep-Th, Physics:math-Ph], December 8, 2015. http://arxiv.org/abs/1512.02408. | ||
| + | * Borinsky, Michael, and Dirk Kreimer. “Feynman Diagrams and Their Algebraic Lattices.” arXiv:1512.05237 [hep-Th, Physics:math-Ph], December 16, 2015. http://arxiv.org/abs/1512.05237. | ||
| + | * Fiorentin, Michele Re. “FaRe: A Mathematica Package for Tensor Reduction of Feynman Integrals.” arXiv:1507.03527 [hep-Ph, Physics:hep-Th], July 13, 2015. http://arxiv.org/abs/1507.03527. | ||
[[분류:physics]] | [[분류:physics]] | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
| + | [[분류:QFT]] | ||
2020년 11월 16일 (월) 10:01 기준 최신판
Finite-dimensional Feynman Diagrams
m-점 함수(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 \[ \langle v^i \rangle = \frac{1}{Z_0} \frac{\partial Z_{\bf b}}{\partial b^i}\vert _{{\bf b} =0} = \frac{\partial}{\partial b^i} \exp({\scriptstyle\frac{1}{2}}{\bf b}^tA^{-1}{\bf b})_{\vert _{{\bf b} =0}} \]
Similarly the m-point function \(\langle v^{i_1}\dots v^{i_m}\rangle\) is given by
\[ \begin{aligned} \langle v^{i_1}, \dots, v^{i_m}\rangle =& \frac{1}{Z_0} (\frac{\partial}{\partial b^{i_1}}\cdots \frac{\partial}{\partial b^{i_m}}Z_{\bf b})_{\textstyle \vert _{{\bf b} =0}}\\ {}=& \frac{\partial}{\partial b^{i_1}}\cdots \frac{\partial}{\partial b^{i_m}} \exp(\frac{1}{2}{\bf b}^tA^{-1}{\bf b})_{\textstyle \vert _{{\bf b} =0}} \end{aligned} \]
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
\[\displaystyle \frac{\partial}{\partial b^{i_1}}\cdots \frac{\partial}{\partial b^{i_m}}\exp(\frac{1}{2}{\bf b}^tA^{-1}{\bf b})_{\vert _{{\bf b} =0}}= A^{-1}_{\textstyle i_{p_1},i_{p_2}} \cdots A^{-1}_{\textstyle i_{p_{m-1}},i_{p_m}},\]
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.
examples
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 \[ \exp x=1+\frac{x}{1!}+\frac{x^2}{2!}+\cdots \] 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
\[ \partial_2 \partial_1(\frac{1}{2}\sum A^{-1}_{i,j}b^ib^j) = A^{-1}_{1,2} \] 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: \[ \begin{aligned} \partial_4\partial_3\partial_1\partial_1~~{\rm gives~~}2A^{-1}_{1,4}A^{-1}_{1,3} + A^{-1}_{3,4}A^{-1}_{1,1}\\ \partial_4\partial_1\partial_1\partial_1~~{\rm gives~~} 3A^{-1}_{1,4}A^{-1}_{1,1} \\ \partial_4\partial_4\partial_1\partial_1~~{\rm gives~~} 2 A^{-1}_{1,4}A^{-1}_{1,4} + A^{-1}_{4,4}A^{-1}_{1,1}\\ \partial_1\partial_1\partial_1\partial_1~~{\rm gives~~} 3 A^{-1}_{1,1}A^{-1}_{1,1} \\ \end{aligned} \]
The first appearance of graphs
In the last section we calculated some 2 and 4-point functions: \[\langle v^1,v^2 \rangle=A^{-1}_{1,2}\] \[\langle v^1,v^1 \rangle=A^{-1}_{1,1}\] \[\langle v^1,v^2,v^3,v^4 \rangle=A^{-1}_{2,3}A^{-1}_{1,4}+A^{-1}_{2,4}A^{-1}_{1,3}+A^{-1}_{3,4}A^{-1}_{1,2}\] \[\langle v^1,v^1,v^3,v^4 \rangle=2A^{-1}_{1,4}A^{-1}_{1,3}+A^{-1}_{3,4}A^{-1}_{1,1}\] \[\langle v^1,v^1,v^1,v^4 \rangle=3A^{-1}_{1,4}A^{-1}_{1,1}\] \[\langle v^1,v^1,v^4,v^4 \rangle=2A^{-1}_{1,4}A^{-1}_{1,4}+A^{-1}_{4,4}A^{-1}_{1,1}\] \[\langle v^1,v^1,v^1,v^1 \rangle=3A^{-1}_{1,1}A^{-1}_{1,1}\]
It is convenient to represent each of products appearing on the right as a graph, where the vertices represent the indices of the coordinates \(v_i\) appearing in the m-point function, and each \[A^{-1}_{i,j}\] becomes an edge from vertex i to vertex j. Here are the graphs corresponding to the terms in the 4-point functions above.
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>.
computational resource
books
expositions
- Henn, Johannes M. “Lectures on Differential Equations for Feynman Integrals.” arXiv:1412.2296 [hep-Ph, Physics:hep-Th, Physics:math-Ph], December 6, 2014. http://arxiv.org/abs/1412.2296.
- Feynman's functional integrals
- Particle Physics for Undergrads
- Fiorenza, Domenico, and Riccardo Murri. 2001. “Feynman Diagrams via Graphical Calculus.” arXiv:math/0106001 (May 31). http://arxiv.org/abs/math/0106001.
- Feynman diagrams for beginners
- Coecke, Bob. “Kindergarten Quantum Mechanics.” arXiv:quant-ph/0510032, October 4, 2005. http://arxiv.org/abs/quant-ph/0510032.
articles
- Timothy Nguyen, The Perturbative Approach to Path Integrals: A Succinct Mathematical Treatment, arXiv:1505.04809 [math-ph], May 18 2015, http://arxiv.org/abs/1505.04809
- http://arxiv.org/abs/1601.03646
- http://arxiv.org/abs/1112.4360
- Kotikov, A. V. “The Property of Maximal Transcendentality: Calculation of Feynman Integrals.” arXiv:1601.00486 [hep-Ph, Physics:hep-Th], January 4, 2016. http://arxiv.org/abs/1601.00486.
- http://arxiv.org/abs/1512.06407
- Purkart, Julian. “The Signed Permutation Group on Feynman Graphs.” arXiv:1512.02408 [hep-Th, Physics:math-Ph], December 8, 2015. http://arxiv.org/abs/1512.02408.
- Borinsky, Michael, and Dirk Kreimer. “Feynman Diagrams and Their Algebraic Lattices.” arXiv:1512.05237 [hep-Th, Physics:math-Ph], December 16, 2015. http://arxiv.org/abs/1512.05237.
- Fiorentin, Michele Re. “FaRe: A Mathematica Package for Tensor Reduction of Feynman Integrals.” arXiv:1507.03527 [hep-Ph, Physics:hep-Th], July 13, 2015. http://arxiv.org/abs/1507.03527.