Modulo 2 periodicity of complex Clifford algebras and electromagnetic field

Transcription

1 Modulo 2 periodicity of complex Clifford algebras and electromagnetic field Vadim V. Varlamov Applied Mathematics, Siberian State Academy of Mining & Metallurgy, Novokuznetsk, Russia Abstract Electromagnetic field is considered in the framework of a Clifford algebra C 2 over a field of complex numbers. It is shown here that a modulo 2 periodicity of complex Clifford algebras may be connected with electromagnetic field. root@varlamov.kemerovo.su

2 1 INTRODUCTION It is well-known that algebra introduced by Clifford [1] be simultaneously a synthesis and generalization of Hamilton theory of quaternions and Grassmann algebras. In the other words, Clifford algebra is generalization of quaternion algebra onto a case of manydimensional spaces. Thanks to this fact Lipschitz [2] was determined the tight connection between Clifford algebras and rotation groups of manydimensional spaces. The Clifford algebras becomes the object of fixed attention of physicists after introduction in theory of electron the noncommutative algebra of Dirac γ-matricies. We shall denote further Clifford algebra as l K n,wherekis algebraic field of characteristic 0 (K R, K Ω, K C); Ω is a field of double numbers (so-called singular field [3]). In essence, the problem of periodicity of algebras l K n tobeginonthefirstclifford paper (see ref.1), where the each algebra l K n is considered as a product of the some number of quaternion algebras. After construction of spinor representation of Clifford algebras [4] the structure of periodicity was finally elucidated by Atiyah, Bott, and Shapiro [5]. Namely, the Clifford algebras over a field of real numbers K R are possess a modulo 8 periodicity. By force of identity Ω R R the algebras over a field of double numbers are possess the same periodicity. In the case of a field K C we have a modulo 2 periodicity. The application of these periodicities in physics to begin on the paper of Coquereaux [6], where the modulo 8 periodicity of real algebras was used to particle physics. In present paper the modulo 2 periodicity of complex algebras is connected with electromagnetic field. This connection is possible by force of realization of the basic notions of electromagnetic field (such that vectors E and H, Maxwell equations) in the terms of an algebra C 2, which by Clifford terminology is called the algebra of hyperbolic biquaternions. 2 CLIORD ALGEBRAS OVER A IELD K A Clifford algebra R n 1 over a field of real numbers R be an algebra with 2 n 1 basic elements: e 0 (unit of algebra), e 1, e 2,...,e n 1 and products e i1i 2...i k e i1 e i2 e ik. Multiplication in R n 1 defined by a following rule: e 2 i e 0, e i e j e j e i (1) urtner on, l R n 1 be an algebra which we obtain from R n 1 by substitution (1) onto e 2 i σ(i l)e 0, e i e j e j e i, (2) where σ(n) { 1, if n 0 1, if n>0 The algebra R n 1 be a particular case of l R n 1 when l 0. 1

3 A general element A of l R n 1 represented by a following formal polynomial: n 1 A a 0 e 0 + a i e i + i1 a ij e ij n 1 n 1 i1 j1 n 1 i 11 n 1 i k 1 n a 12...n 1 e 12...n 1 a i1i 2...i k e i1i 2...i k. k0 a i1...i k e i1...i k + It is obvious that Clifford algebra l R n 1 is associative. A center of l R n 1 consist of unit e 0 and volume element e 12...n 1. The volume element is belong to the center only if n 1 is odd. Since e 12...n 1 e i σ(i l)( 1) n i 1 e 12...i 1i+1...n 1, e i e 12...n 1 σ(i l)( 1) i 1 e 12...i 1i+1...n 1, then e 12...n 1 is belong to the center if n i 1 i 1 (mod2).whenn 1iseven,the element e 12...n 1 commutes with elements e i1i 2...i 2k+1, and anticommutes with elements e i1i 2...i 2k. The transition from l R n 1 to l R n or l+1 R n may be represented as the transition from the real numbers in l R n 1 to complex coordinates a + bω, whereωis additional basis element e 12...n. By force of (1)-(2) in the case of transition from l R n 1 to l R n we have: and in case l R n 1 l+1 R n : e n n(n 1) ( 1)l+ 2 e 2 n(n 1) l n ( 1) 2. Therefore, in the first case 1, if n 4m+2orn4m 1,lis even and ω 2 if n 4m or n 4m +1,lis odd; +1, if n 4m or n 4m +1,lis even and if n 4m+2orn4m 1,lis odd; and in the second case 1, if n 4m+2orn4m 1,lis odd and ω 2 if n 4m or n 4m +1,lis even; +1, if n 4m or n 4m +1,lis odd and if n 4m+2orn4m 1,lis even; where m 0,1,...;m 1,2,... Hence it follows that for l is even we have: l R 4m 1 l C 4m 2, l+1 R 4m +1 l C 4m, (3) 2

4 and for l is odd l R 4m +1 l Ω 4m, l R 4m +1 l C 4m, l+1 R 4m 1 l Ω 4m 2, (4) l+1 R 4m 1 l C 4m 2, (5) l R 4m 1 l Ω 4m 2, l+1 R 4m +1 l Ω 4m, (6) where l C n and l Ω n are Clifford algebras over the field of complex numbers C and field of double numbers Ω, respectively. When n is even, the identities of type (3)-(6) excepted, since in this case volume element ω e 12...n is not belong to a center of extended algebra. In particular case, when l 0 we have a transition R n 1 R n and e n ( 1) n(n 1) 2 Therefore, { ω 2 1, if n 4m+2orn4m 1, +1, if n 4m or n 4m +1. In this case, for n is odd we have the following identities: R 4m 1 C 4m 2, R 4m +1 Ω 4m. (7) 3 THE ALGEBRAS R 3, C 2 AND ELECTROMAG- NETIC IELD Consider now an algebra R 3. The squares of units of this algebra are equal to e 2 i 1(i 0,1,2,3). The general element of R 3 has a following form: A a 0 e a i e i + i1 3 i1 j1 3 a ij e ij + a 123 e 123. urther on, for a general element of R 3 there exists a decomposition A 3 A 2 +ωa 2, where ω e 123 R 3 and A 2 is a general element of R 2 : a 0 e 0 + a 1 e 1 + a 2 e 2 + a 12 e 12. In fact A 3 A 2 + ωa 2 a 0 e 0 + a 1 e 1 + a 2 e 2 + a 12 e 12 + ω(b 0 e 0 + b 1 e 1 + b 2 e 2 + b 12 e 12 ) (a 0 +ωb 0 )e 0 +(a 1 +ωb 1 )e 1 +(a 2 +ωb 2 )e 2 +(a 12 + ωb 12 )e 12 (8) a 0 e 0 +a 1 e 1 +a 2 e 2 b 12 e 3 + a 12 e 12 b 2 e 13 + b 1 e 23 + b 0 e 123. Since in this case ω e 123 is belong to the center of R 3,andω 2 1, then in (8) the expressions a k + ωb k may be replaced by a k + ib k,whereiis imaginary unit. This way, we have the identity R 3 C 2, which be a particular case of (7). The algebra C 2 is called the algebra of complex quaternions (or hyperbolic biquaternions). 3

5 Analogously, for the algebra 3 R 3 with ω 2 +1 we have in accordance with (6) the identity 3 R 3 2 Ω 2,where 2 Ω 2 is the algebra of elliptic biquaternions. urther on, there exists a realization of electromagnetic field in the terms of R 3. Let A 0 0 e e e e 3, (9) A 1 A 0 e 0 + A 1 e 1 + A 2 e 2 + A 3 e 3, where A 0 and A 1 are elements of R 3. The coefficients of these elements be partial derivatives and components of vector-potential, respectively. Make up now the exterior product of elements (9): A 0 A 1 ( 0 e e e e 3 )(A 0 e 0 + A 1 e 1 + A 2 e 2 + A 3 e 3 ) ( 0 A A A A }{{ 3 )e } 0 +( 0 A A 0 )e }{{} 0 e 1 + E 0 E 1 ( 0 A A }{{} 0 )e 0 e 2 +( 0 A A 0 )e }{{} 0 e 3 +( 2 A 3 3 A 2 )e }{{} 2 e 3 + (10) E 2 E 3 H 1 +( 3 A 1 1 A }{{} 3 )e 3 e 1 +( 1 A 2 2 A 1 )e }{{} 1 e 2. H 2 H 3 The scalar part E 0 0, since the first bracket in (10) be a Lorentz condition 0 A 0 +diva 0. It is easily seen that the other bracket be components of electric and magnetic fields: E i ( i A A i ),H i (curla) i. Since ω e 123 is belong to the center of R 3,then ωe 1 e 1 ωe 2 e 3, ωe 2 e 2 ω e 3 e 1, ωe 3 e 3 ω e 1 e 2. In accordance with these correlations may be written (10) as A 0 A 1 (E 1 +ωh 1 )e 1 +(E 2 +ωh 2 )e 2 +(E 3 +ωh 3 )e 3 (11) It is obvious that the expression (11) is coincide with the vector part of complex quaternion (hyperbolic biquaternion) when e 1 ie 1, e 2 ie 2, e 3 ie 1 ie 2. urther on, make up the exterior product, where is the first element from (9) and is an expression of type (10): divee 0 ((curlh) 1 0 E 1 )e 1 ((curlh) 2 0 E 2 )e 2 ((curlh) 3 0 E 3 )e 3 + ((curle) H 1 )e 2 e 3 + ((curle) H 2 )e 3 e 1 + (12) +((curle) H 3 )e 1 e 2 +divhe 1 e 2 e 3. It is easily seen that the first coefficient of the product be a left part of equation dive ϱ. The following three coefficients are make up a left part of equation curlh 0 E j, the other coefficients are make up the equations curle + 0 H 0 and divh 0, respectively (for more details see [7]). 4

6 4 MODULO 2 PERIODICITY O COMPLEX CLIORD ALGEBRAS In dependence from the sign of the square of volume element ω all totality of Clifford algebras is divided into two classes. Namely, if ω 2 1Clifford algebra l K n over a field K (K R, K Ω, K C) is called positive, and negative in contrary case (ω 2 1). urther on, the each algebra l K n is associated to a vector space V over a field K. This space is endowed with a nondegenerate quadratic form Q. It is obvious that the dimensionality dim V of vector space is equal to a number of units of the algebra l K n. In accordance with this most general definition of Clifford algebra we shall denote this algebra as l K n (V,Q). Theorem 1 (Karoubi [8]) 1) If l1 K n1 (V 1,Q 1 )is positive, and if dim V 1 is even, then l 1+l 2 K n1+n 2 (V 1 V 2,Q 1 Q 2 ) l1 K n1 (V 1,Q 1 ) l2 K n2 (V 2,Q 2 ). 2) If l1 K n1 is negative, and if dim V 1 is even, then l 1+l 2 K n1+n 2 (V 1 V 2,Q 1 Q 2 ) l1 K n1 (V 1,Q 1 ) l2 K n2 (V 2, Q 2 ). Theorem 2 The algebra l+m K n+2m is isomorphic to M 2 m( l K n ). Indeed, it is well-known that the algebra l K n with even dimensionality (n 2ν) is isomorphic to a matrix algebra M 2 ν (K). urther on, since 1 K 2 ( 1 R 2, 1 Ω 2, 1 C 2 )is positive, then in accordance with theorem 1 we obtain l R n 1 R 2 l+1 K n+2 l K n 1 K 2 l Ω n 1 Ω 2 l C n 1 l R n M 2 (R) l Ω n M 2 (Ω) C l M 2 ( l R n ) M 2 ( l Ω n ) 2 C n M 2 (C) M 2 ( l C n ) Therefore, l+m K n+2m l K n M 2 (K)... M 2 (K) l K n M 2 m(k) M 2 m( l K n ). (13) Theorem 3 The algebra C n+2 is isomorphic to M 2 (C n ). It is obvious that in the case of a field K C we have an isomorphism l C n Cn. urther on, in accordance with theorem 1 we obtain C n+2 Cn C 2 (the algebra C 2 is positive if ω ie 12, e 2 1 e2 2 1orifωe 12, e 2 1 1,e2 2 1). Since C 2 M 2 (C), hence it follows that C n+2 Cn C 2 Cn M 2 (C) M 2 (C n ). (14) rom adduced above theorems immediately follows that for any Clifford algebra with even dimensionality (n 2ν)over a field Cthere exists a decomposition : C 2ν C2 C 2... C }{{ 2. (15) } ν times 5

7 In case of odd dimensionality (n 2ν+ 1) the algebra C 2ν+1 is decomposed to a direct sum of two subalgebras which are isomorphic to C 2ν. This decomposition is realized by means of mutually orthogonal idempotents where ω e ν+1 and ω 2 1;or ε 1 2 (1 + ω), ε 1 (1 ω), 2 ε 1 2 (1 + iω), ε 1 (1 iω), 2 where ω e ν+1 and ω 2 1. It is easily seen that ε + ε 1,εε 0,ε 2 ε, ε 2 ε. Since in this case volume element ω e ν+1 is belong to a center of C 2ν+1, then the idempotents ε, ε are commutes with all elements of the algebra C 2ν+1. Hence it follows that [9] C 2ν+1 C2ν C 2ν, where the each algebra C 2ν consist of the elements of type εa or ε A, here A C 2ν+1. Therefore, in accordance with (15) we have C 2ν+1 C2 C 2... C 2 C 2 C 2... C 2. (16) urther on, by force of C 2 M2 (C) we have for a base of spinor representation of C 2 the following matrices which are correspond to the units e 1 and e 2 : [ ] [ ] i E 1, E (17) i 0 Let e 3 e 1 e 2, then for a vector part of biquaternion (13) we obtain by means of (17) the following matrix: [ ] i 3 1 i 2, (18) 1 + i 2 i 3 here i E i + ih i, since in this case ω i. This way, the algebra C 2 with general element A 0 e e e e 3 is isomorphic to a matrix algebra of type (18), where 0 0 A 0 + diva 0 and i (i 1,2,3) are components of a complex electromagnetic field. We shall denote this matrix algebra as M 2 (C). Hence by force of modulo 2 periodicity of complex Clifford algebras from (15) and (16) immediately follows that C 2ν M 2 ν (C) M 2 (C) M 2 (C)... M 2 (C), (19) C 2ν+1 C2ν C 2ν M 2 ν (C) M 2 (C) ν (20) M 2 (C) M 2 (C)... M 2 (C) M 2 (C) M 2 (C)... M 2 (C). 6

8 orexample,considernowthealgebrac 4. In the spinor representation this algebra is isomorphic to a matrix algebra M 4 (C), the base of which consist of well-known γ- matrices. In the base of Weyl for these matrices we have [ ] 0 σ γ m m σ m, (21) 0 where m 0,1,2,3and [ σ ],σ 1 [ ] [,σ 2 0 i i 0 ] [,σ σ 0 σ 0, σ 1,2,3 σ 1,2,3. It is easily seen that σ-matrices are make up the base of spinor representation of C 2 M2 (C) and are coincide with matrices E i if σ 3 1 i E 1E 2. or the algebra C 4 in accordance with (19) we have C 4 C2 C 2 M 2 (C) M 2 (C). urther on, by general definition, the algebra C 4 is associated to a complex vector space C 4. or the vector a i e i C 4 in the base (21) we have a matrix A i γ i : i i i i The all matrix algebra M 4 (C) in the base (17) we obtain by means of theorem 3. Indeed, in the case when dimensionality of l K n is even the volume element ω e 12...n is not belong to a center of algebra l K n. However, when i 2m we have e m2m+k e i ( 1) 2m+1 i σ(i l)e 12...i 1i+1...2m2m+k, e i e m2m+k ( 1) i 1 σ(i l)e 12...i 1i+1...2m2m+k and, therefore, the condition of commutativity of elements e m2m+k and e i is 2m +1 i i 1 (mod 2). Thus, the elements e m2m+1 and e m2m+2 are commute with all elements e i whose indexes are not exceed 2m. Therefore, the transition from algebra K 2m to algebras K 2m+2, 1 K 2m+2 or 2 K 2m+2 and from algebra l K 2m to l K 2m+2, l+1 K 2m+2 or l+2 K 2m+2 may be represented as transition from the real (complex or double) coordinates of elements of algebras K 2m and l K 2m to the coordinates of type a + bφ + cψ + dφψ, where φ and ψ are additional basis elements e m2m+1 and e m2m+2. The elements e i1i 2...i k φ are contain index 2m +1 and not contain index 2m+2, and the elements e i1i 2...i k ψ are contain index 2m+2 and not contain index 2m + 1. Respectively, the elements e i1i 2...i k φψ are contain both indexes ], 7

9 2m +1 and 2m+ 2. Hence it immediately follows that a general element of l K 2m+2 or l+1 K 2m+2, l+2 K 2m+2, K 2m+2 may be represented as l K 0 2me 0 + l K 1 2me m2m+1 + l K 2 2me m2m+2 + l K 3 2me 2m+12m+2, (22) where l K i 2m (i 0,1,2,3) are represent the algebras with general element A 2m k0 ai1i2...i k e i1i 2...i k. When the elements φ e m2m+1 and ψ e m2m+2 are satisfy to condition φ 2 ψ 2 1 we see that the basis {e 0,φ,ψ,φψ}is isomorphic to a basis of quaternion algebra. Therefore, in this case from (22) we have ] l l K 2m+2 M2 ( l K K 2m )[ 0 2m i l K 3 2m l K 1 2m+i l K 2 2m. l K 1 2m +il K 2 2m l K 0 2m +il K 3 2m Analoguosly, when the elements φ and ψ are satisfy to conditions φ 2 ψ 2 1or φ 2 1,ψ 2 1 we have the transitions from the real (complex or double) coordinates in l K 2m+2 to the anti-quaternionic and pseudo-quaternionic coordinates, respectively (about anti-quaternions and pseudo-quaternions see [10, p. 434]). In our case by force of isomorphism l C n Cn we see that any algebra l C 4 C 4 (l 0,1,2,3,4) may be represented by the following quaternion: C 0 2e 0 + C 1 2φ + C 2 2ψ + C 3 2φψ, where φ e 123, ψ e 124 and C i 2 the algebras of hyperbolic biquaternions with general element (in our case) A 0 e e e 2 + {[ 3 e 12. In ] spinor [ representation ] [ ] for[ the quaternion ]} basis {e 0,φ,ψ,φψ} we have i i 0,,,. Hence it immediately follows that i 0 0 i C 4 C2 C 2 C2 M 2 (C) M 2 (C 2 ) inally, using (18) we obtain that M 4 (C) has a form: [ C 0 2 ic 3 2 C 1 2+iC 2 2 C 1 2 +ic2 2 C 0 2 +ic3 2 (1+i) 3 (1 i) 1 (1+i) 2 (1+i) 3 (1 i) 1+(1+i) 2 (1 i) 1+(1+i) 2 (1+i) 3 (1 i) 1 (1+i) 2 (1+i) 3 (1 i) 3 (1+i) 1+(1 i) 2 (1 i) 3 (1+i) 1+(1 i) 2 (1+i) 1 (1 i) 2 (1 i) 3 (1+i) 1 (1 i) 2 (1+i) 3 ]. (23) Analoguosly, for the algebra C 6 M 8 (C) we have the anti-quaternion C 0 4 e 0 + C 1 4φ + C 2 4ψ + C 3 4φψ,whereφe 12345, ψ e and φ 2 ψ 2 1. Theantiquaternion basis {e 0, φ, ψ, φψ } in spinor representation defined by the following 8

10 {[ 1 0 matricies ] [ 0 1, 1 0 ] [ 0 i, i 0 ] [ i 0, 0 i ]}. Using (23) we have i 3 1 i 2 i 3 1+i i i 1 1+i 2 i 3 1 i 2 i 3 2 i i i i 1 i 3 1 i 2 i 3 1 i 2 1+i i i 2 i 3 1+i 2 i 3 i 3 1 i 2 i 3 1+i i i 1 1+i 2 i 3 1 i 2 i 3 2 i i i i 1 i 3 1 i 2 i 3 1 i 2 1+i i i 2 i 3 1+i 2 i 3. or the algebras with odd dimensionality from (20) we obtain C 5 M 4 (C) M 4 (C), C 7 M 8 (C) M 8 (C) andsoon. References [1] W. K. Clifford, Applications of Grassmann s extensive algebra,am. J. Math. 1, 350 (1878). [2] R. Lipschitz, Untersuchungen über die Summen von Quadraten (Bonn, 1886). [3] N. Salingaros, Algebras with three anticommuting elements. II. Two algebras over a singular field, J.Math.Phys.22, 2096 (1981). [4] R. Brauer and H. Weyl, Spinors in n dimensions, Am. J. Math. 57, 425 (1935). [5] M.. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3, (suppl.1), 3 (1964). [6] R. Coquereaux, Modulo 8 periodicity of real Clifford algebras and particle physics, Phys. Lett. B 115, 389 (1982). [7] V.V.Varlamov,Physical fields and Clifford algebras, hep-th/ [8] M. Karoubi, K-Theory (Springer-Verlag, Berlin, 1979). [9] P. K. Rashevskii, The theory of spinors, Am. Math. Soc. Transl. (Ser.2) 6, 1 (1957). [10] B. A. Rozenfeld, Neevklidovi geometrii (Moscow, 1955). 9