Maxima by Example, Ch. 12: Dirac Algebra and Quantum Electrodynamics, Version 3
Ted Woollett
Loading the Maxima code file dirac3.mac loads the rest of the Dirac package code files: simplifying-new.lisp, dgcon3.mac, dgtrace3.mac,
dgeval3.mac, and dgmatrix3.mac. The text file dgfunctions3.txt has an alphabetical list of the package functions, as well as the name of the Maxima code file in which the function is defined.
Our intent is to present the detailed cross-section examples as wxMaxima worksheets, which include detailed comments and explanations of the Dirac3 tools, rather than as batch files.
Separate sections of Ch. 12 are devoted to a review of high energy physics notation (which agrees with Peskin
and Schroeder), trace and contraction theorems, covariant polarization 4-vectors for physical (external) photons, and to the use of simplifying-new.lisp.
The text file mandelstam.txt contains copy and paste code for defining the contents of the Maxima list invarR. The list invarR is initially defined to be an empty list by dgeval3.mac at launch of Dirac3. The user can then populate the list invarR using the function set_invarR(...), and the file
mandelstam.txt contains examples of approriate set_invarR(...) invocations, depending on whether some or all of the particle masses are equal. The resulting list invarR then becomes a "replacement list" for replacing the dot product of a pair of particle 4-momenta, represented by D(pa,pb),
by an expression depending on the mass symbols and the Mandelstam variables s, t, and u. The example worksheets show this process in
detail.
Of course one does not need to introduce Mandelstam variables to work out amplitudes and cross sections, and we provide many examples of how to get the right answers using the tools in the Dirac3 package, using both symbolic and explicit matrix methods.
- --mbe12dirac3.pdf : Ch. 12, Dirac Algebra and
Quantum Electrodynamics, June 9, 2019, Maxima 5.36.1, 83 pages,
- --mbe12dirac3.tex : Latex code file for Ch. 12, Dirac Algebra and
Quantum Electrodynamics, June 9, 2019, Maxima 5.36.1
- --simplifying-new.lisp : Recursive simplification code, 4-15-11
- --dirac3.mac : The Dirac3 package driver, Jan. 10, 2019
- --dgcon3.mac : Symbolic contraction code, Sept. 11, 2018
- --dgtrace3.mac : Symbolic trace code, Sept. 11, 2018
- --dgeval3.mac : Frame dependent evaluation code, Dec. 21, 2018
- --dgmatrix3.mac : Explicit Dirac spinors and matrices, July 29, 2019
- --dgfunctions3.txt : Alphabetical list of the Dirac3 package functions, Jan. 10, 2019
- --mandelstam.txt : Copy and Paste Examples for invoking the function set_invarR(...), Sept. 18, 2018
- --ee-mumu-HE.wxm : unevaluated wxMaxima worksheet for the high energy limit of e(-) e(+) --> mu(-) mu(+), Oct. 18, 2018
- --ee-mumu-HE.wxmx : evaluated wxMaxima worksheet for the high energy limit of e(-) e(+) --> mu(-) mu(+), Oct. 18, 2018
- --ee-mumu-AE.wxm : unevaluated wxMaxima worksheet for the arbitrary energy case of e(-) e(+) --> mu(-) mu(+), Oct. 19, 2018
- --ee-mumu-AE.wxmx : evaluated wxMaxima worksheet for the arbitrary energy case of e(-) e(+) --> mu(-) mu(+), Oct. 19, 2018
- --e-mu-scatt.wxm : unevaluated wxMaxima worksheet for the elastic scattering process
e(-) mu(-) --> e(-) mu(-), Oct. 17, 2018
- --e-mu-scatt.wxmx : evaluated wxMaxima worksheet for the elastic scattering process
e(-) mu(-) --> e(-) mu(-), Oct. 17, 2018
- --em-em-scatt-HE.wxm : unevaluated wxMaxima worksheet for the elastic scattering process
e(-) e(-) --> e(-) e(-), Moller scattering, for high energy case, Oct. 24, 2018
- --em-em-scatt-HE.wxmx : evaluated wxMaxima worksheet for the elastic scattering process
e(-) e(-) --> e(-) e(-), Moller scattering, for the high energy case, Oct. 24, 2018
- --em-em-scatt-AE.wxm : unevaluated wxMaxima worksheet for the elastic scattering process
e(-) e(-) --> e(-) e(-), Moller scattering, for arbitrary energy case, Nov. 2, 2018
- --em-em-scatt-AE.wxmx : evaluated wxMaxima worksheet for the elastic scattering process
e(-) e(-) --> e(-) e(-), Moller scattering, for the arbitrary energy case, Nov. 2, 2018
- --em-ep-scatt-HE.wxm : unevaluated wxMaxima worksheet for the elastic scattering process
e(-) e(+) --> e(-) e(+), Bhabha scattering, for high energy case, Oct. 30, 2018
- --em-ep-scatt-HE.wxmx : evaluated wxMaxima worksheet for the elastic scattering process
e(-) e(+) --> e(-) e(+), Bhabha scattering, for the high energy case, Oct. 30, 2018
- --em-ep-scatt-AE.wxm : unevaluated wxMaxima worksheet for the elastic scattering process
e(-) e(+) --> e(-) e(+), Bhabha scattering, for arbitrary energy case, Nov. 2, 2018
- --em-ep-scatt-AE.wxmx : evaluated wxMaxima worksheet for the elastic scattering process
e(-) e(+) --> e(-) e(+), Bhabha scattering, for the arbitrary energy case, Nov. 2, 2018
- --photon1.wxm : unevaluated wxMaxima worksheet for proving two theorems satisfied by the real linear
polarization vectors of a physical photon; uses Maxima lists to define 3-vectors, and uses Dirac3 package function vprod (A,B) to return the vector cross product of the 3-vectors A and B, Nov. 6, 2018
- --photon1.wxmx : evaluated wxMaxima worksheet proving two theorems satisfied by the real linear polarization vectors of a physical photon; uses Maxima lists to define 3-vectors, and uses Dirac3 package function vprod (A,B) to return the vector cross product of the 3-vectors A and B, Nov. 6, 2018
- --photon2.wxm : unevaluated wxMaxima worksheet which derives explicit circular polarization
4-vectors for external photons, Nov. 23, 2018
- --photon2.wxmx : evaluated wxMaxima worksheet which derives explicit circular polarization
4-vectors for external photons, Nov. 23, 2018
- --ee-gaga-HE.wxm : unevaluated wxMaxima worksheet for pair annihilation process
e(-) + e(+) --> gamma + gamma at high energy. Both the unpolarized cross section case and the specific helicity
amplitudes are derived, the latter using the explicit circular polarization 4-vectors derived in photon2.wxm. Dec. 13, 2018
- --ee-gaga-HE.wxmx : evaluated wxMaxima worksheet for pair annihilation process
e(-) + e(+) --> gamma + gamma at high energy. Both the unpolarized cross section case and the specific helicity
amplitudes are derived, the latter using the explicit circular polarization 4-vectors derived in photon2.wxm. Dec. 13, 2018
- --ee-gaga-AE.wxm : unevaluated wxMaxima worksheet for pair annihilation process
e(-) + e(+) --> gamma + gamma for arbitrary energy. Both the unpolarized cross section case and the specific helicity
amplitudes are derived, the latter using the explicit circular polarization 4-vectors derived in photon2.wxm. Dec. 12, 2018
- --ee-gaga-AE.wxmx : evaluated wxMaxima worksheet for pair annihilation process
e(-) + e(+) --> gamma + gamma for arbitrary energy. Both the unpolarized cross section case and the specific helicity
amplitudes are derived, the latter using the explicit circular polarization 4-vectors derived in photon2.wxm. Dec. 12, 2018
- --compton1.wxm : unevaluated wxMaxima worksheet for Compton scattering of a
electron in the rest frame of the initial electron. Dec. 6, 2018
- --compton1.wxmx : evaluated wxMaxima worksheet for the Compton scattering of a
electron in the rest frame of the initial electron. Dec. 6, 2018
- --compton-CMS-HE.wxm : unevaluated wxMaxima worksheet for the high energy Compton scattering of a electron in the center of momentum frame, includes calculation of helicity amplitudes. Dec. 12, 2018
- --compton-CMS-HE.wxmx : evaluated wxMaxima worksheet for the high energy Compton scattering of a electron in the center of momentum frame, includes calculation of helicity amplitudes. Dec. 12, 2018
- --piKaon.wxm : unevaluated wxMaxima worksheet for the QED elastic scattering of a
positive pion by a positive Kaon in the center of momentum frame, Dec. 19, 2018
- --piKaon.wxmx : evaluated wxMaxima worksheet for the QED elastic scattering of a
positive pion by a positive Kaon in the center of momentum frame, Dec. 19, 2018
- --pi-pi-scatt.wxm : unevaluated wxMaxima worksheet for the QED elastic scattering of a
positive pion by a positive pion in the center of momentum frame, Dec. 22, 2018
- --pi-pi-scatt.wxmx : evaluated wxMaxima worksheet for the QED elastic scattering of a
positive pion by a positive pion in the center of momentum frame, Dec. 22, 2018
- --ePi-HE.wxm : unevaluated wxMaxima worksheet for the QED elastic scattering of a
electron by a positive pion in the center of momentum frame, in the high energy limit, Feb. 12, 2019
- --ePi-HE.wxmx : evaluated wxMaxima worksheet for the QED elastic scattering of a
electron by a positive pion in the center of momentum frame, in the high energy limit, Feb. 12, 2019
- --compton0.wxm : unevaluated wxMaxima worksheet for Compton scattering of a
pion in the rest frame of the initial pion. Jan. 10, 2019
- --compton0.wxmx : evaluated wxMaxima worksheet for the Compton scattering of a
pion in the rest frame of the initial pion. Jan. 10, 2019
- --massive-spin1-pol.wxm : unevaluated wxMaxima worksheet for derivation of
polarization 4-vectors for a massive spin 1 particle, June 3, 2019
- --massive-spin1-pol.wxmx : evaluated wxMaxima worksheet for derivation of
polarization 4-vectors for a massive spin 1 particle, June 3, 2019
- --W-decay.wxm : unevaluated wxMaxima worksheet for decay rate of the W-boson, July 26, 2019
- --W-decay.wxmx : evaluated wxMaxima worksheet for decay rate of the W-boson, July 26, 2019
The Dirac package files are free software: you can redistribute them and/or modify
them under the terms of the GNU GENERAL PUBLIC LICENSE, Version 2, June 1991,
as published by the Free Software Foundation. For more information see
the license information at the top of the file.
Quark Gluon Hard Scattering Processes
A simple tree level QCD hard scattering calculation. The batch file includes references and a copy of the batch file run. The version 2 Dirac package (dirac2.mac) needs to be loaded, and in addition the file qcd1.mac needs to be loaded as well. The latter file contains explicit matrix definitions of the eight SU(3) group
3 x 3 color matrices T[a], a = 1,2,..,8, and also calculates the antisymmetric gauge group structure contants f[a,b,c], and the symmetric structure constants d[a,b,c] for use in calculations.
- --qcd1.mac Color matrices and structure constants
- --qq-qq1.mac Quark-quark scattering via gluon exchange, April 30, 2011, Maxima 5.24.0
A Clebsch-Gordan Coefficient Package for Maxima
Contains (not very efficient) Maxima code for Wigner vector coupling coefficients (Clebsch-Gordan coefficients)
cj (j1,j2,j3,m1,m2,m3) and the simply related Wigner 3j coefficients
threej (j1,j2,j3,m1,m2,m3), both for explicit integer and half-integer arguments only.
--clebsch.mac : Oct. 22, 2010, Maxima 5.21.1
The text file clebsch.mac is free software: you can redistribute it and/or modify
it under the terms of the GNU GENERAL PUBLIC LICENSE, Version 2, June 1991,
as published by the Free Software Foundation. For more information see
the license information at the top of the file.
A Simple Maxima Lagrangian Mechanics Package
--lag1.mac : Oct. 26, 2010, Maxima 5.22.1. This is a batch file which contains some simple
examples of deriving the second order equations of motion for a system described by a Lagrangian function of
generalized coordinates and velocities and for which no constraint equations are needed.
The results from the batch file run are in a comment at the end of lag1.mac.
--lag2.mac : Oct. 26, 2010, Maxima 5.22.1. This Maxima code file contains code
which is automatically loaded in by lag1.mac.
The text files lag1.mac and lag2.mac are free software: you can redistribute them and/or modify
them under the terms of the GNU GENERAL PUBLIC LICENSE, Version 2, June 1991,
as published by the Free Software Foundation. For more information see
the license information at the top of the file.
Symbolic Pauli Matrix Commutator Expansions
--pauli1.mac : Nov. 29, 2010, Maxima 5.22.1. This code file loads commutators.mac and uses
tellsimpafter methods to complete the symbolic expansion of Pauli matrices. Also included is code for
comparing explicit matrix calculations of commutators with the symbolic results. Note that the code
uses the convention that sig[k], for k = 1,2,3, (lower case first letter) represents the symbolic
Pauli matrix, and Sig[k], for k = 1,2,3, (upper case first letter) represents the explicit Pauli matrix.
At the end of this file are examples of use.
--pauli2.mac : Nov. 29, 2010, Maxima 5.22.1. This code file loads commutators.mac and
uses defrule methods to complete the symbolic expansion of Pauli matrices.
At the end of this file are examples of use.
--commutators.mac : Nov. 29, 2010, Maxima 5.22.1.
This Maxima code file, written by Maxima coordinator Robert Dodier, first creates a
new property 'bilinear', which can then be used in declare(comm,bilinear) declarations.
The file then uses defrule methods to create rules for the extraction of scalars,
numbers, and factors of %i to be removed from 'comm' arguments and rules for the
basic product rule expansion of a commutator of two operators or matrices.
At the end of this file are examples of use.
The text files commutators.mac, pauli1.mac, and pauli2.mac are free software:
you can redistribute them and/or modify them under the terms of
the GNU GENERAL PUBLIC LICENSE as published by the Free Software Foundation.