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Appendix a a Matrix Representation of a Clifford Algebra

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Appendix a a Matrix Representation of a Clifford Algebra Appendix A A Matrix Representation of a Clifford Algebra To construct a matrix representation for a Clifford algebra corresponding to a Euclidean or pseudo-Euclidean space, it is not sufficient to construct a set fe1; e2;:::;eng such that ej ek C ekej D 2njkI, where (A.1) nkk D hek; eki D 1 for k D 1;2;:::;p, nkk D hek; eki D1 for k D p C 1; p C 2;:::;pC q D n,and ˝ ˛ njk D ej ; ek D 0 for j ¤ k. We must also consider all possible products of the form M1M2:::Mn where n Mk D ek or I. These products form a set of 2 matrices that should be linearly independent. For our purposes, this set should be linearly independent. One’s first thought is that to check that these products are linearly independent would be a formidable task. However due to a theorem proven by Ian Porteus, there is a simple test to check for this required linear independence (Porteus 1981, pp. 243–245). Theorem 291. Asetfe1; e2;:::;eng consisting of n matrices representing or- thonormal 1-vectors generates a vector space (and therefore an algebra) of n dimension 2 unless the product e1e2 :::en D e12:::n is a scalar multiple of I. Proof. With one exception, a product of one or more distinct orthonormal 1-vectors will anticommute with at least one of the 1-vectors. To see this, we note that a product of an even number of the 1-vectors will anticommute with any 1-vector appearing in the product. We also see that the product of an odd number of Dirac matrices will anticommute with any Dirac matrix that does not appear in the product. The one product that commutes with all the 1-vectors in the set is the product e12:::n where n is odd. The proof now proceeds by self-contradiction. Suppose the products are not linearly independent. In that case, there exists a set of coefficients Aj1j2:::jk (not all zero) such that J. Snygg, A New Approach to Differential Geometry using Clifford’s Geometric Algebra, 431 DOI 10.1007/978-0-8176-8283-5, © Springer Science+Business Media, LLC 2012 432 A A Matrix Representation of a Clifford Algebra Xn X j1j2:::jk A ej1j2:::jk D 0.(A.2) kD0 j1<j2<<jk If the coefficient of I in (A.2) is not zero, divide (A.2) by that coefficient and obtain the equation X j1j2:::jk I C B ej1j2:::jk D 0, where (A.3) j1<j2<<jk the sum does not include the identity matrix. If the coefficient of I in (A.2)is zero, pick out a term in the equation with a nonzero coefficient (say em1m2:::mk )and m1m2:::mk 1 1 multiply by (.1=A /.em1m2:::mk / )where.em1m2:::mk / D˙em1m2:::mk .In this fashion, one can always obtain an equation with the form of (A.3) from (A.2). If the Bj1j2:::jk ’s are all zero, we already have the desired contradiction. If the sum in (A.3) contains a product (say em1m2:::mk ) that anti-commutes with em, 1 then one can multiply (A.3) on the left by em and on the right by .em/ and obtain X j1j2:::jk 1 I C B emej1j2:::jk .em/ D 0.(A.4) j1<j2<<jk We note that 1 emem1m2:::mk .em/ Dem1m2:::mk . Therefore, we can add (A.3)and(A.4) and thereby obtain a new equation that except for a factor of 2 is identical to (A.3) except at least one less term (em1m2:::mk ) will now appear in the sum. If n is even, this process can be continued until the sum reduces to zero and thus obtain the contradiction I D 0.Ifn is odd, the process can be continued until we have I C ˛e12:::n D 0.(A.5) Thus, we see that our desired contradiction occurs unless e12:::n is a scalar multiple of I and so the theorem is proved. ut Before continuing, I should make a few remarks. Suppose a set of 1-vectors n fe1; e2;:::;eng satisfies (A.1). If this set also generates an algebra of dimension 2 , I have defined the resulting algebra to be a Clifford algebra. However, it is possible to have a set of 1-vectors fe1; e2;:::;eng that satisfies (A.1) but generates an algebra with dimension less than 2n.Whatistobesaidofsuchanalgebra?Fromapurely algebraic point of view, such an algebra is also a Clifford algebra. Indeed, it can be shown that such an algebra is isomorphic to a Clifford algebra associated with a Euclidean or pseudo-Euclidean space of dimension n 1. For the study of geometry (at least the geometry presented in this book), such a Clifford algebra would be unsuitable to study an n-dimensional space. If the Dirac matrices satisfy (A.1) and generate an algebra of dimension 2n, Ian Porteus A A Matrix Representation of a Clifford Algebra 433 describes the resulting algebra as an universal Clifford algebra. Universal Clifford algebras are the only kind of Clifford algebras used in this book. Other authors use more abstract definitions to avoid nonuniversal Clifford algebras. Some physicists have written papers describing five-dimensional theories using a nonuniversal Clifford algebra. Since these papers do not mention the fact that they are using a nonuniversal Clifford algebra, it suggests that they have inadvertently made a mistake. This is understandable since in 1958, Marcel Riesz thought he had proven that nonuniversal Clifford algebras do not exist (1993, pp. 10–12). Having proven Theorem 291, we are now in a much better position to construct a matrix representation for a Clifford algebra. Perhaps, the best way to construct explicit representations is by using the Kronecker product of matrices. Suppose A is an n n matrix and B is an m m matrix. In particular if A D aij , then the Kronecker product is defined by the partitioned matrix: 2 3 a BaB a B 6 11 12 1n 7 6 a BaB a B 7 6 21 22 2n 7 6 7 6 7 A ı B D 6 7 (A.6) 6 7 4 5 an1Ban1 B annB For example if Ä Ä a a b b A D 11 12 and B D 11 12 ,then a21 a22 b21 b22 2 3 a11b11 a11b12 a12b11 a12b12 6 7 6 a11b21 a11b22 a12b21 a12b22 7 A ı B D 4 5 . a21b11 a21b12 a22b11 a22b12 a21b21 a21b22 a22b21 a22b22 The properties of the Kronecker product are thoroughly discussed in a book entitled Kronecker Products and Matrix Calculus with Applications by Alexander Graham (1981). Using a blackboard or large sheet of paper, it is not too difficult to convince oneself that .A ı B/.C ı D/ D AC ı BD, where (A.7) .A ı B/.C ı D/ represents the ordinary matrix product of .A ı B/ with .C ı D/. Similarly, AC and BD are, respectively, the ordinary matrix products of A with C and B with D. Furthermore, it can be shown that A ı .B ı C/D .A ı B/ ı C .(A.8) 434 A A Matrix Representation of a Clifford Algebra From this result, it is not difficult to show that .A1 ı A2 ı :::Ap/.B1 ı B2 ı :::ı Bp / D .A1B1/ ı .A2 ı B2/ ı :::ı .An ı Bp/: To see that our proposed matrix representations satisfy (A.1), it will be useful to note that .A1 ıA2 ı:::ıAp/.B1 ıB2 ı:::ıBp/ D.B1 ıB2 ı:::ıBp/.A1 ıA2 ı:::ıAp/ if Aj Bj DBj Aj for an odd number of j ’s between 1 and p and Aj Bj D Bj Aj for the remainder of the j ’s between 1 and p: To carry out our construction, I will use the Pauli spin matrices: Ä Ä Ä 01 0 i 10 D , D ,and D .(A.9) 1 10 2 i0 3 0 1 2 2;0 For R2;0 (the Clifford algebra that corresponds to the Euclidean space E (or R ), we use e1 D 1 and e2 D 2.Note!e1e2 D 1 2 D i 3 ¤˙I.ForR4;0, we can let e1 D 1 ı 1, e2 D 1 ı 2, e3 D 1 ı 3,and e4 D 2 ı I.(e1e2e3e4 D‹). This process can be continued by induction. Suppose it is possible to construct a m m matrix representation of size 2 2 for R2m;0. Suppose also that we designate the matrix representation of the Dirac matrices for this space by ek.2m/ for k D 1;2;:::;2m. We can then obtain a matrix representation for the Dirac matrices of R2mC2;0 as follows: ek.2m C 2/ D 1 ı ek.2m/ for k D 1;2;:::;2m. (A.10) m e2mC1.2m C 2/ D i 1 ı e12:::2m.2m/, and (A.11) e2mC2.2m C 2/ D 2 ı I . (A.12) m (The factor i that appears in (A.11) has been chosen to guarantee that .e2mC1.2mC 2//2 D I.) It is not difficult to show that the system of matrices just constructed is indeed an orthonormal system of 1-vectors, where 2 .ek/ D I and ej ek C ekej D 0 for j ¤ k. A A Matrix Representation of a Clifford Algebra 435 From Theorem 291, this is all that is required to generate the universal Clifford algebra R2mC2;0 since 2m C 2 is an even integer.
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