-
Notifications
You must be signed in to change notification settings - Fork 4
Commit
This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository.
- Loading branch information
Showing
19 changed files
with
2,922 additions
and
17 deletions.
There are no files selected for viewing
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Large diffs are not rendered by default.
Oops, something went wrong.
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,126 @@ | ||
| <!DOCTYPE html> | ||
| <html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang=""> | ||
| <head> | ||
| <meta charset="utf-8" /> | ||
| <meta name="generator" content="pandoc" /> | ||
| <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" /> | ||
| <title>FeynCalc manual (development version)</title> | ||
| <style> | ||
| code{white-space: pre-wrap;} | ||
| span.smallcaps{font-variant: small-caps;} | ||
| span.underline{text-decoration: underline;} | ||
| div.column{display: inline-block; vertical-align: top; width: 50%;} | ||
| div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;} | ||
| ul.task-list{list-style: none;} | ||
| pre > code.sourceCode { white-space: pre; position: relative; } | ||
| pre > code.sourceCode > span { display: inline-block; line-height: 1.25; } | ||
| pre > code.sourceCode > span:empty { height: 1.2em; } | ||
| .sourceCode { overflow: visible; } | ||
| code.sourceCode > span { color: inherit; text-decoration: inherit; } | ||
| div.sourceCode { margin: 1em 0; } | ||
| pre.sourceCode { margin: 0; } | ||
| @media screen { | ||
| div.sourceCode { overflow: auto; } | ||
| } | ||
| @media print { | ||
| pre > code.sourceCode { white-space: pre-wrap; } | ||
| pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; } | ||
| } | ||
| pre.numberSource code | ||
| { counter-reset: source-line 0; } | ||
| pre.numberSource code > span | ||
| { position: relative; left: -4em; counter-increment: source-line; } | ||
| pre.numberSource code > span > a:first-child::before | ||
| { content: counter(source-line); | ||
| position: relative; left: -1em; text-align: right; vertical-align: baseline; | ||
| border: none; display: inline-block; | ||
| -webkit-touch-callout: none; -webkit-user-select: none; | ||
| -khtml-user-select: none; -moz-user-select: none; | ||
| -ms-user-select: none; user-select: none; | ||
| padding: 0 4px; width: 4em; | ||
| color: #aaaaaa; | ||
| } | ||
| pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa; padding-left: 4px; } | ||
| div.sourceCode | ||
| { } | ||
| @media screen { | ||
| pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; } | ||
| } | ||
| code span.al { color: #ff0000; font-weight: bold; } /* Alert */ | ||
| code span.an { color: #60a0b0; font-weight: bold; font-style: italic; } /* Annotation */ | ||
| code span.at { color: #7d9029; } /* Attribute */ | ||
| code span.bn { color: #40a070; } /* BaseN */ | ||
| code span.bu { } /* BuiltIn */ | ||
| code span.cf { color: #007020; font-weight: bold; } /* ControlFlow */ | ||
| code span.ch { color: #4070a0; } /* Char */ | ||
| code span.cn { color: #880000; } /* Constant */ | ||
| code span.co { color: #60a0b0; font-style: italic; } /* Comment */ | ||
| code span.cv { color: #60a0b0; font-weight: bold; font-style: italic; } /* CommentVar */ | ||
| code span.do { color: #ba2121; font-style: italic; } /* Documentation */ | ||
| code span.dt { color: #902000; } /* DataType */ | ||
| code span.dv { color: #40a070; } /* DecVal */ | ||
| code span.er { color: #ff0000; font-weight: bold; } /* Error */ | ||
| code span.ex { } /* Extension */ | ||
| code span.fl { color: #40a070; } /* Float */ | ||
| code span.fu { color: #06287e; } /* Function */ | ||
| code span.im { } /* Import */ | ||
| code span.in { color: #60a0b0; font-weight: bold; font-style: italic; } /* Information */ | ||
| code span.kw { color: #007020; font-weight: bold; } /* Keyword */ | ||
| code span.op { color: #666666; } /* Operator */ | ||
| code span.ot { color: #007020; } /* Other */ | ||
| code span.pp { color: #bc7a00; } /* Preprocessor */ | ||
| code span.sc { color: #4070a0; } /* SpecialChar */ | ||
| code span.ss { color: #bb6688; } /* SpecialString */ | ||
| code span.st { color: #4070a0; } /* String */ | ||
| code span.va { color: #19177c; } /* Variable */ | ||
| code span.vs { color: #4070a0; } /* VerbatimString */ | ||
| code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warning */ | ||
| </style> | ||
| <link rel="stylesheet" href="css/feyncalc.css" /> | ||
| <script src="js/katex.min.js"></script> | ||
| <script>document.addEventListener("DOMContentLoaded", function () { | ||
| var mathElements = document.getElementsByClassName("math"); | ||
| var macros = []; | ||
| for (var i = 0; i < mathElements.length; i++) { | ||
| var texText = mathElements[i].firstChild; | ||
| if (mathElements[i].tagName == "SPAN") { | ||
| katex.render(texText.data, mathElements[i], { | ||
| displayMode: mathElements[i].classList.contains('display'), | ||
| throwOnError: false, | ||
| macros: macros, | ||
| fleqn: true | ||
| }); | ||
| }}}); | ||
| </script> | ||
| <link rel="stylesheet" href="js/katex.min.css" /> | ||
|
|
||
| </head> | ||
| <body> | ||
| <header id="title-block-header"> | ||
| <h1 class="title">FeynCalc manual (development version)</h1> | ||
| </header> | ||
| <div class="sourceCode" id="cb1"><pre class="sourceCode mathematica"><code class="sourceCode mathematica"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a> </span></code></pre></div> | ||
| <h2 id="contractions">Contractions</h2> | ||
| <h3 id="see-also">See also</h3> | ||
| <p><a href="Extra/FeynCalc.html">Overview</a>.</p> | ||
| <h3 id="simplifications">Simplifications</h3> | ||
| <p>Now that we have some basic understanding of FeynCalc objects, let us do something with them. Contractions of Lorentz indices are one of the most essential operations in symbolic QFT calculations. In FeynCalc the corresponding function is called <code>Contract</code></p> | ||
| <div class="sourceCode" id="cb2"><pre class="sourceCode mathematica"><code class="sourceCode mathematica"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>FV<span class="op">[</span><span class="fu">p</span><span class="op">,</span> <span class="sc">\</span><span class="op">[</span>Mu<span class="op">]]</span> MT<span class="op">[</span><span class="sc">\</span><span class="op">[</span>Mu<span class="op">],</span> <span class="sc">\</span><span class="op">[</span>Nu<span class="op">]]</span></span> | ||
| <span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>Contract<span class="op">[</span><span class="sc">%</span><span class="op">]</span></span></code></pre></div> | ||
| <p><span class="math display">\overline{p}^{\mu } \bar{g}^{\mu \nu }</span></p> | ||
| <p><span class="math display">\overline{p}^{\nu }</span></p> | ||
| <div class="sourceCode" id="cb3"><pre class="sourceCode mathematica"><code class="sourceCode mathematica"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a>FV<span class="op">[</span><span class="fu">p</span><span class="op">,</span> <span class="sc">\</span><span class="op">[</span>Alpha<span class="op">]]</span> FV<span class="op">[</span><span class="fu">q</span><span class="op">,</span> <span class="sc">\</span><span class="op">[</span>Alpha<span class="op">]]</span></span> | ||
| <span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a>Contract<span class="op">[</span><span class="sc">%</span><span class="op">]</span></span></code></pre></div> | ||
| <p><span class="math display">\overline{p}^{\alpha } \overline{q}^{\alpha }</span></p> | ||
| <p><span class="math display">\overline{p}\cdot \overline{q}</span></p> | ||
| <p>Notice that when we enter noncommutative objects, such as Dirac matrices, we use <code>Dot</code> (<code>.</code>) and not <code>Times</code> (<code>*</code>)</p> | ||
| <div class="sourceCode" id="cb4"><pre class="sourceCode mathematica"><code class="sourceCode mathematica"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a>FV<span class="op">[</span><span class="fu">p</span><span class="op">,</span> <span class="sc">\</span><span class="op">[</span>Alpha<span class="op">]]</span> MT<span class="op">[</span><span class="sc">\</span><span class="op">[</span><span class="fu">Beta</span><span class="op">],</span> <span class="sc">\</span><span class="op">[</span><span class="fu">Gamma</span><span class="op">]]</span> GA<span class="op">[</span><span class="sc">\</span><span class="op">[</span>Alpha<span class="op">]]</span> . GA<span class="op">[</span><span class="sc">\</span><span class="op">[</span><span class="fu">Beta</span><span class="op">]]</span> . GA<span class="op">[</span><span class="sc">\</span><span class="op">[</span><span class="fu">Gamma</span><span class="op">]]</span></span> | ||
| <span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a>Contract<span class="op">[</span><span class="sc">%</span><span class="op">]</span></span></code></pre></div> | ||
| <p><span class="math display">\overline{p}^{\alpha } \bar{\gamma }^{\alpha }.\bar{\gamma }^{\beta }.\bar{\gamma }^{\gamma } \bar{g}^{\beta \gamma }</span></p> | ||
| <p><span class="math display">\left(\bar{\gamma }\cdot \overline{p}\right).\bar{\gamma }^{\gamma }.\bar{\gamma }^{\gamma }</span></p> | ||
| <p>This is because <code>Times</code> is commutative, so writing something like</p> | ||
| <div class="sourceCode" id="cb5"><pre class="sourceCode mathematica"><code class="sourceCode mathematica"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>GA<span class="op">[</span><span class="sc">\</span><span class="op">[</span>Delta<span class="op">]]</span> GA<span class="op">[</span><span class="sc">\</span><span class="op">[</span><span class="fu">Beta</span><span class="op">]]</span> GA<span class="op">[</span><span class="sc">\</span><span class="op">[</span>Alpha<span class="op">]]</span></span></code></pre></div> | ||
| <p><span class="math display">\bar{\gamma }^{\alpha } \bar{\gamma }^{\beta } \bar{\gamma }^{\delta }</span></p> | ||
| <p>will give you completely wrong results. It is also a very common beginner’s mistake!</p> | ||
| </body> | ||
| </html> |
Oops, something went wrong.