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.

  1. --mbe12dirac3.pdf : Ch. 12, Dirac Algebra and Quantum Electrodynamics, June 9, 2019, Maxima 5.36.1, 83 pages,
  2. --mbe12dirac3.tex : Latex code file for Ch. 12, Dirac Algebra and Quantum Electrodynamics, June 9, 2019, Maxima 5.36.1
  3. --simplifying-new.lisp : Recursive simplification code, 4-15-11
  4. --dirac3.mac : The Dirac3 package driver, Jan. 10, 2019
  5. --dgcon3.mac : Symbolic contraction code, Sept. 11, 2018
  6. --dgtrace3.mac : Symbolic trace code, Sept. 11, 2018
  7. --dgeval3.mac : Frame dependent evaluation code, Dec. 21, 2018
  8. --dgmatrix3.mac : Explicit Dirac spinors and matrices, July 29, 2019
  9. --dgfunctions3.txt : Alphabetical list of the Dirac3 package functions, Jan. 10, 2019
  10. --mandelstam.txt : Copy and Paste Examples for invoking the function set_invarR(...), Sept. 18, 2018
  11. --ee-mumu-HE.wxm : unevaluated wxMaxima worksheet for the high energy limit of e(-) e(+) --> mu(-) mu(+), Oct. 18, 2018
  12. --ee-mumu-HE.wxmx : evaluated wxMaxima worksheet for the high energy limit of e(-) e(+) --> mu(-) mu(+), Oct. 18, 2018
  13. --ee-mumu-AE.wxm : unevaluated wxMaxima worksheet for the arbitrary energy case of e(-) e(+) --> mu(-) mu(+), Oct. 19, 2018
  14. --ee-mumu-AE.wxmx : evaluated wxMaxima worksheet for the arbitrary energy case of e(-) e(+) --> mu(-) mu(+), Oct. 19, 2018
  15. --e-mu-scatt.wxm : unevaluated wxMaxima worksheet for the elastic scattering process e(-) mu(-) --> e(-) mu(-), Oct. 17, 2018
  16. --e-mu-scatt.wxmx : evaluated wxMaxima worksheet for the elastic scattering process e(-) mu(-) --> e(-) mu(-), Oct. 17, 2018
  17. --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
  18. --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
  19. --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
  20. --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
  21. --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
  22. --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
  23. --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
  24. --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
  25. --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
  26. --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
  27. --photon2.wxm : unevaluated wxMaxima worksheet which derives explicit circular polarization 4-vectors for external photons, Nov. 23, 2018
  28. --photon2.wxmx : evaluated wxMaxima worksheet which derives explicit circular polarization 4-vectors for external photons, Nov. 23, 2018
  29. --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
  30. --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
  31. --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
  32. --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
  33. --compton1.wxm : unevaluated wxMaxima worksheet for Compton scattering of a electron in the rest frame of the initial electron. Dec. 6, 2018
  34. --compton1.wxmx : evaluated wxMaxima worksheet for the Compton scattering of a electron in the rest frame of the initial electron. Dec. 6, 2018
  35. --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
  36. --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
  37. --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
  38. --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
  39. --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
  40. --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
  41. --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
  42. --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
  43. --compton0.wxm : unevaluated wxMaxima worksheet for Compton scattering of a pion in the rest frame of the initial pion. Jan. 10, 2019
  44. --compton0.wxmx : evaluated wxMaxima worksheet for the Compton scattering of a pion in the rest frame of the initial pion. Jan. 10, 2019
  45. --massive-spin1-pol.wxm : unevaluated wxMaxima worksheet for derivation of polarization 4-vectors for a massive spin 1 particle, June 3, 2019
  46. --massive-spin1-pol.wxmx : evaluated wxMaxima worksheet for derivation of polarization 4-vectors for a massive spin 1 particle, June 3, 2019
  47. --W-decay.wxm : unevaluated wxMaxima worksheet for decay rate of the W-boson, July 26, 2019
  48. --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.
  1. --qcd1.mac Color matrices and structure constants
  2. --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.