\documentclass{article} \usepackage{axiom} \begin{document} \title{\$SPAD/src/input clifford.input} \author{The Axiom Team} \maketitle \begin{abstract} \end{abstract} \eject \tableofcontents \section{License} <<license>>= --Copyright The Numerical Algorithms Group Limited 1991. @ <<*>>= <<license>> -- CliffordAlgebra(n, K, Q) defines a vector space of dimension 2**n -- over K, given a quadratic form Q on K**n. -- -- If e[i] 1<=i<=n is a basis for K**n then -- 1, e[i] 1<=i<=n, e[i1]*e[i2] 1<=i1<i2<=n,...,e[1]*e[2]*..*e[n] -- is a basis for the Clifford Algebra. -- -- The algebra is defined by the relations -- e[i]*e[j] = -e[j]*e[i] i ^= j, -- e[i]*e[i] = Q(e[i]) -- -- Examples of Clifford Algebras are: -- gaussians, quaternions, exterior algebras and spin algebras. -- Choose rational functions as the ground field. )clear all K := FRAC POLY INT --% The complex numbers as a Clifford Algebra )clear p qf qf: QFORM(1, K) := quadraticForm(matrix([[-1]])$(SQMATRIX(1,K))) C := CLIF(1, K, qf) i := e(1)$C x := a + b * i y := c + d * i x * y recip % x*% %*y --% The quaternions as a Clifford Algebra )clear p qf qf:QFORM(2, K) :=quadraticForm matrix([[-1, 0], [0, -1]])$(SQMATRIX(2,K)) H := CLIF(2, K, qf) i := e(1)$H j := e(2)$H k := i * j x := a + b * i + c * j + d * k y := e + f * i + g * j + h * k x + y x * y y * x --% The exterior algebra on a 3 space. )clear p qf qf: QFORM(3, K) := quadraticForm(0::SQMATRIX(3,K)) Ext := CLIF(3,K,qf) i := e(1)$Ext j := e(2)$Ext k := e(3)$Ext x := x1*i + x2*j + x3*k y := y1*i + y2*j + y3*k x + y x * y + y * x -- In n space, a grade p form has a dual n-p form. -- In particular, in 3 space the dual of a grade 2 element identifies -- e1*e2->e3, e2*e3->e1, e3*e1->e2. dual2 a == coefficient(a,[2,3])$Ext * i + _ coefficient(a,[3,1])$Ext * j + _ coefficient(a,[1,2])$Ext * k -- The vector cross product is then given by dual2(x*y) --% The Dirac Algebra used in Quantum Field Theory. )clear p qf K := FRAC INT g: SQMATRIX(4, K) := [[1,0,0,0],[0,-1,0,0],[0,0,-1,0],[0,0,0,-1]] qf: QFORM(4, K) := quadraticForm g D := CLIF(4,K,qf) -- The usual notation is gamma sup i. gam := [e(i)$D for i in 1..4] -- There are various contraction identities of the form -- g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t) = -- 2*(gam(s)gam(m)gam(n)gam(r) + gam(r)*gam(n)*gam(m)*gam(s)) -- where the sum over l and t is implied. -- Verify this identity for m=1,n=2,r=3,s=4 m := 1; n:= 2; r := 3; s := 4; lhs := reduce(+,[reduce(+,[g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t) for l in 1..4]) for t in 1..4]) rhs := 2*(gam s * gam m*gam n*gam r + gam r*gam n*gam m*gam s) @ \eject \begin{thebibliography}{99} \bibitem{1} nothing \end{thebibliography} \end{document}