Transformational principles latent in the theory of CLIFFORD ALGEBRAS Nicholas Wheeler, Reed College Physics Department October 2003 Introduction. My purpose in this informal material will be to refresh/consolidate my own thought concerning a subject that has engaged my attention from time to time over the years, but not recently. I do so because aspects of the subject have become the focus of a thesis effort by Nakul Shankar, whose awkward situation is that he is going to have to change horses in midstream: I will direct the first phase of his project, but will be obliged to pass the baton to Tom Wieting & Darrell Schroeter at mid-year. A large part of my effort here, therefore, will be directed to the the establishment of some degree of notational and conceptual commonality, and to construction of a clear statement of my own motivating interests in this area. 1. Complex algebra, revisited. Familiarly, x2 + y2 does not factor on the reals, but the related object (x2 + y2)I does factor (x2 + y2)I =(xI + yii) · (xI − yii) (1) provided I and i are objects with the stipulated properties · I I = I I · i = i (2) i · I = i i · i = −I Equations (2) are collectively equivalent to the statement that if z1 = x1I + y1i and z2 = x2I + y2i then z1· z2 =(x1x2 − y1y2)I +(x1y2 + y1x2)i (3) Evidently The algebra is commutative, and is found z · z = z · z : (4) 1 2 2 1 by calculation to be also associative. 2 Transformational principles derived from Clifford algebras It is evident also that x1y2 + y1x2 =0iffx1/y1 = −x2/y2: we are motivated therefore to introduce the operation z = xI + yii −−−−−−−−−−−−→ z = xI − yii (5) conjugation Then z ·z =(x2 + y2)I (6) and we find that z z –1 = z exists unless x2 + y2 =0;i.e., unless z =00 (7) x2 + y2 We agree to call |z|≡ x2 + y2 0 (8) the “modulus” of z. By calculation we discover that |z1· z2| = |z1|·|z2| (9) We can mechanize the condition that z be “unimodular” (|z| = 1) by writing z = cos θ · I + sin θ · i = eiθ (10) Transformations of the form z −→ Z ≡ eiθ · z =(x cos θ − y sin θ)I +(x sin θ + y cos θ)i (11) are manifestly modulus-preserving. Notated x X cos θ − sin θ x −→ = (12) y Y sin θ cos θ y they have clearly the structure characteristic of rotations. Writing cos θ − sin θ R(θ) ≡ ≡ cos θ · I + sin θ · J (13) sin θ cos θ we arrive at a 2 × 2 matrix representation of the algebra now in hand. The matrices 10 0 −1 I ≡ and J ≡ (14) 01 10 Basic elements of complex algebra 3 are readily seen to satisfy (compare (2)) I · I I = I · J = J (15) J · I = J J · J = −I so we are led to the identification x −y z = xI + yii ←−−→ Z = x I + y J = (16) yx In this representation conjugation ←−−→ transposition (17) and |z|2 = det Z (18) Alternative matrix representations can be obtained by similarity transformation Z −→ Z ≡ S –1ZS (19) Such transformations preserve (15) and (18), and preserve also the spectral features of Z, which are instructive, and to which I now turn: the characteristic polynomial = λ2 − 2xλ +(x2 + y2) = λ2 − tr Z · λ + det Z 2 − Z · 1 Z2 − Z 2 = λ tr λ + 2 tr (tr ) so by the Cayley-Hamilton theorem we have Z2 −2x Z+(x2 +y2)I = O whence I − Z Z –1 = 2x x2 + y2 But 2x I − Z = Z T,sowehave Z T = det Z which is the matrix representation of (7). The eigenvalues of Z are x ± iy and ±i the associated eigenvalues are , which is to say: we have 1 x −y ±i ±i =(x ± iy) yx 1 1 I hope my reader will forgive me for belaboring the familiar: my effort has been to establish a pattern, the first rough outline of a template to which we can adhere when we turn to less familiar subject matter. 4 Transformational principles derived from Clifford algebras 2. Clifford algebra of order 2. This subject arises when we ask not—as at (1)—to factor but to extract the formal square root of x2 + y2. Or, as we find it now more convenient to notate the assignment, to extract the square root of 10 x1x1 + x2x2 ≡ δ xixj where δ ≡ ij ij 01 To that end we posit the existence of objects I, e1 and e2 such that i j i 2 i (δijx x )I =(x ei) : all x (20) Immediately ei ej + ej ei =(δij + δji)I =2δijI by δij = δji (21) which when spelled out in specific detail read e1 e1 = e2 e2 = I (22.1) e1 e2 + e2 e1 =00 (22.2) ··· It follows that products of the general form ei1 ei2 ei3 ein , which we suppose to have been assembled from pee1’s and q = n−pee2’s, can (by e2 e1 = − e1e2) always be brought to “dictionary order” ± ··· ······ e 1e1 e1 e 2e2 e2 p factors q factors where (±)=(−)number of transpositions required to achieve dictionary order. Drawing now upon (22.1) we find that the expression presented just above can be written I if p even, q even e if p odd, q even = ± 1 e2 if p even, q odd e1e2 if p odd, q odd and that this list exhausts the possibilities. We confront therefore an algebra with elements of the form 0 1 2 12 a = a I + a e1 + a e2 + a e1e2 (23) If b is defined similarly then, by computation, a·b =(a0b0 + a1b1 + a2b2 − a12b12) I 0 1 1 0 2 12 12 2 +(a b + a b − a b + a b ) e1 0 2 1 12 2 0 12 1 +(a b + a b + a b − a b ) e2 0 12 1 2 2 1 12 0 +(a b + a b − a b + a b ) e1e2 (24) Clifford algebra of order 2 5 from which it follows as a corollary that 2 12 12 2 a·b − b·a =2(−a b + a b ) e1 1 12 12 1 + 2(+a b − a b ) e2 1 2 2 1 + 2(+a b − a b ) e1e2 (25) Equation (24) serves in effect as a “multiplication table,” while (25) makes vivid the fact—evident already in (22)—that we have now in hand an algebra that is (as calculation would confirm) associative but non-commutative. If we let the “conjugate” of a be defined/denoted 0 1 2 12 a = a I − a e1 − a e2 − a e1e2 (26) then it follows from (24) that a ·a =(a0a0 − a1a1 − a2a2 + a12a12) I (27) and from (25) that = a ·a Evidently a right/left inverse of a exists iff the “modulus” of a |a |≡a0a0 − a1a1 − a2a2 + a12a12 (28) does not vanish, and is given then by a a –1 = a (29) |a| By Mathematica-assisted calculation we establish that |a ·b| = |a |·|b| (30) Transformations of the form a −→ A = u –1 a u (31) are therefore modulus-preserving, and we can in such a context assume without loss of generality that u is unimodular: |u| = 1. Equation (31) serves to establish a linear relationship between the coefficients of A and those of a: A0 a0 A1 a1 = U (32) A2 a2 A3 a3 Notational remark: I have at this point found 3 12 it convenient to write a in place of a , e3 in place of e1e2, etc. One could—quickly enough, with the assistance of Mathematica—work out explicit descriptions of the elements of U (they are assembled quadratically from the elements of u), but it is simpler and more sharply informative to 6 Transformational principles derived from Clifford algebras proceed on the assumption that u differs only infinitesimally from I:1 u = I + w : terms of 2nd order in w will be neglected Then u –1 = I − w in leading order, which on comparison with u –1 = u means that we can without loss of generality assume that w0 =0: 1 2 3 w = w e1 + w e2 + w e3 (33) We now have A = a +[a w − w a]+··· in leading order. By calculation 3 2 2 3 [a w − w a]= 2(−w a + w a )e1 3 1 1 3 + 2(+w a − w a )e2 2 1 1 2 + 2(+w a − w a )e3 (34) so in matrix representation we have 0000 00−w3 +w2 U = I + W where W ≡ 2 (35) 0+w3 0 −w1 0+w2 −w1 0 Notice now that the modulus of a can be written a0 T a0 10 00 a1 a1 0 −100 |a | = G with G ≡ (36) a2 a2 00−10 a3 a3 00 01 and that modulus presevation entails U TGU = G whence (in leading order) W TG + GW = O which can be written W T = −GWG–1 or again (GW)T = −(GW) (37) We verify that the matrices W and G defined above do in fact satisfy that “G-antisymmetry” condition. We write 1 2 3 W =2w J1 +2w J2 +2w J3 (38) and observe that the matrices 00 0 0 0000 00 00 00 0 0 000+1 00−10 J ≡ , J ≡ , J ≡ 1 00 0 −1 2 0000 3 0+100 00−10 0+100 00 00 1 The plan is to construct finite similarity transformations by iteration of such infinitesimal transformations. Clifford algebra of order 2 7 thus defined are—though not closed multiplicatively (therefore not candidates to provide matrix representatives of the algebraic objects e1, e3, e3)—closed under commutation: J1J2 − J2J1 = −J3 J2J3 − J3J2 =+J1 (39) J3J1 − J1J3 =+J2 Except for the goofy signs these commutation relations resemble those we associate with the generators of O(3), the 3-dimensional rotation group.