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diff --git a/src/input/clifford.input.pamphlet b/src/input/clifford.input.pamphlet new file mode 100644 index 00000000..c6f22ac9 --- /dev/null +++ b/src/input/clifford.input.pamphlet @@ -0,0 +1,107 @@ +\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} |