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Clifford Algebr 1 wheeler

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Informal expository notes by Nicholas Wheeler, written for a thesis project by a student. They begin with complex algebra as a 2x2 matrix algebra and rotations, then treat the order-2 Clifford algebra: multiplication table, conjugation, modulus, and infinitesimal and finite similarity transformations giving Lorentz-type boosts and rotations, with Mathematica-aided calculations. This is a copy of Wheeler's work in Phil's archive; the text shown covers only the opening sections.

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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 timeto time over the years, but not recently. I do so because aspects of the subjecthave become the focus of a thesis eﬀort by Nakul Shankar, whose awkwardsituation is that he is going to have to change horses in midstream: I willdirect the ﬁrst phase of his project, but will be obliged to pass the baton toTom 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 notationaland conceptual commonality, and to construction of a clear statement of myown motivating interests in this area. 1. Complex algebra, revisited. Familiarly, x2+y2does not factor on the reals, but the related object ( x2+y2)Idoes factor (x2+y2)I=(xI+yiii)·(xI−yiii) (1) provided Iandiiiare objects with the stipulated properties I·I=I I·iii=iii iii·I=iii iii·iii=−I  (2) Equations (2) are collectively equivalent to the statement that if zzz 1=x1I+y1iii andzzz2=x2I+y2iiithen zzz1·zzz2=(x1x2−y1y2)I+(x1y2+y1x2)iii (3) Evidently zzz1·zzz2=zzz2·zzz1:/braceleftbigg The algebra is commutative , and is found by calculation to be also associative .(4) 2 Transformational principles derived from Clifford algebras It is evident also that x1y2+y1x2=0i ﬀx1/y1=−x2/y2: we are motivated therefore to introduce the operation zzz=xI+yiii−−−−−−−−−−−− → conjugationzzz=xI−yiii (5) Then zzz·zzz=(x2+y2)I (6) and we ﬁnd that zzz–1=zzz x2+y2exists unless x2+y2=0 ;i.e., unlesszzz=000 (7) We agree to call |zzz|≡/radicalbig x2+y2/greaterorequalslant0 (8) the “modulus” of zzz. By calculation we discover that |zzz1·zzz2|=|zzz1|·|zzz2| (9) We can mechanize the condition that zzzbe “unimodular” ( |zzz|= 1) by writing zzz= cosθ·I+ sinθ·iii=eiiiθ(10) Transformations of the form zzz/mapsto−→ZZZ≡eiiiθ·zzz=(xcosθ−ysinθ)I+(xsinθ+ycosθ)iii (11) are manifestly modulus-preserving. Notated /parenleftbigg x y/parenrightbigg /mapsto−→/parenleftbigg X Y/parenrightbigg =/parenleftbigg cosθ−sinθ sinθcosθ/parenrightbigg/parenleftbigg x y/parenrightbigg (12) they have clearly the structure characteristic of rotations . Writing R(θ)≡/parenleftbigg cosθ−sinθ sinθcosθ/parenrightbigg ≡cosθ·I+ sinθ·J (13) we arrive at a 2 ×2matrix representation of the algebra now in hand. The matrices I≡/parenleftbigg 10 01/parenrightbigg and J≡/parenleftbigg 0−1 10/parenrightbigg (14) Basic elements of complex algebra 3 are readily seen to satisfy (compare (2)) I·I=I I·J=J J·I=J J·J=−I  (15) so we are led to the identiﬁcation zzz=xI+yiii←−−→Z=xI+yJ=/parenleftbigg x−y yx/parenrightbigg (16) In this representation conjugation ←−−→transposition (17) and |zzz| 2= det Z (18) Alternative matrix representations can be obtained by similarity transformation Z−→Z/prime≡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−trZ·λ+ detZ =λ2−trZ·λ+1 2/braceleftbig trZ2−(trZ)2/bracerightbig so by the Cayley-Hamilton theorem we have Z2−2xZ+(x2+y2)I=Owhence Z–1=2xI−Z x2+y2 But 2xI−Z=ZT,s ow eh a v e =ZT detZ which is the matrix representation of (7). The eigenvalues of Zarex±iyand the associated eigenvalues are/parenleftbigg ±i 1/parenrightbigg , which is to say: we have /parenleftbigg x−y yx/parenrightbigg/parenleftbigg ±i 1/parenrightbigg =(x±iy)/parenleftbigg ±i 1/parenrightbigg 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 wecan 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 ofx2+y2. Or, as we ﬁnd it now more convenient to notate the assignment, to extract the square root of x1x1+x2x2≡δijxixjwhere /bardblδij/bardbl≡/parenleftbigg 10 01/parenrightbigg To that end we posit the existence of objects I,eee1andeee2such that (δijxixj)I=(xieeei)2: allxi(20) Immediately eeeieeej+eeejeeei=(δij+δji)I =2δijIbyδij=δji (21) which when spelled out in speciﬁc detail read eee1eee1=eee2eee2=I (22.1) eee1eee2+eee2eee1=000 (22.2) It follows that products of the general form eeei1eeei2eeei3···eeein, which we suppose to have been assembled from peee1’s andq=n−peee2’s, can (by eee2eee1=−eee1eee2) always be brought to “dictionary order” ±eee1eee1···eee1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipuprighteee2eee2······eee2/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright pfactors qfactors 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 ifpeven,qeven eee1ifpodd,qeven eee2ifpeven,qodd eee1eee2ifpodd,qodd and that this list exhausts the possibilities. We confront therefore an algebra with elements of the form aaa=a0I+a1eee1+a2eee2+a12eee1eee2 (23) Ifbbbis deﬁned similarly then, by computation, aaa·bbb=(a0b0+a1b1+a2b2−a12b12)I +(a0b1+a1b0−a2b12+a12b2)eee1 +(a0b2+a1b12+a2b0−a12b1)eee2 +(a0b12+a1b2−a2b1+a12b0)eee1eee2 (24) Clifford algebra of order 2 5 from which it follows as a corollary that aaa·bbb−bbb·aaa=2 ( −a2b12+a12b2)eee1 + 2(+a1b12−a12b1)eee2 + 2(+a1b2−a2b1)eee1eee2 (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 butnon-commutative . If we let the “conjugate” of aaabe deﬁned/denoted aaa=a0I−a1eee1−a2eee2−a12eee1eee2 (26) then it follows from (24) that aaa·aaa=(a0a0−a1a1−a2a2+a12a12)I (27) and from (25) that =aaa·aaa Evidently a right/left inverse of aaaexists iﬀ the “modulus” of aaa |aaa|≡a0a0−a1a1−a2a2+a12a12(28) does not vanish, and is given then by aaa–1=aaa |aaa|(29) ByMathematica -assisted calculation we establish that |aaa·bbb|=|aaa|·|bbb| (30) Transformations of the form aaa/mapsto−→AAA=uuu–1aaauuu (31) are therefore modulus-preserving, and we can in such a context assume without loss of generality that uuuis unimodular: |uuu|= 1. Equation (31) serves to establish a linear relationship between the coeﬃcients of AAAand those of aaa:  A0 A1 A2 A3 =U a0 a1 a2 a3  (32) Notational remark : I have at this point found it convenient to write a3in place of a12,eee3in place ofeee1eee2,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 uuu), but it is simpler and more sharply informative to 6 Transformational principles derived from Clifford algebras proceed on the assumption that uuudiﬀers only inﬁnitesimally from I:1 uuu=I+www: terms of 2ndorder in wwwwill be neglected Thenuuu–1=I−wwwin leading order, which on comparison with uuu–1=uuumeans that we can without loss of generality assume that w0=0 : www=w1eee1+w2eee2+w3eee3 (33) We now have AAA=aaa+[aaawww−wwwaaa]+··· in leading order. By calculation [aaawww−wwwaaa]= 2 ( −w3a2+w2a3)eee1 + 2(+w3a1−w1a3)eee2 + 2(+w2a1−w1a2)eee3 (34) so in matrix representation we have U=I+Wwhere W≡2 0 000 00 −w3+w2 0+w30−w1 0+w2−w10  (35) Notice now that the modulus of aaacan be written |aaa|= a0 a1 a2 a3 T G a0 a1 a2 a3 with G≡ 10 00 0−100 00 −10 00 01  (36) and that modulus presevation entails UTGU=Gwhence (in leading order) WTG+GW=Owhich can be written WT=−GWG–1or again (GW)T=−(GW) (37) We verify that the matrices WandGdeﬁned above do in fact satisfy that “G-antisymmetry” condition. We write W=2w1J1+2w2J2+2w3J3 (38) and observe that the matrices J1≡ 00 0 0 00 0 000 0 −1 00 −10 ,J 2≡ 0000 000 + 100000+ 10 0 ,J 3≡ 00 00 00 −10 0 + 10000 00  1The 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 notcandidates to provide matrix representatives of the algebraic objects eee1,eee3,eee3)—closed under commutation: J1J2−J2J1=−J3 J2J3−J3J2=+J1 J3J1−J1J3=+J2  (39) Except for the goofy signs these commutation relations resemble those we associate with the generators of O(3), the 3-dimensional rotation group. Entrusting all computational work to Mathematica , we discover that λ4−λ2= 0 is the characteristic equation of both J1andJ2 λ4+λ2= 0 is the characteristic equation of J3 so /braceleftbig +1,−1,0,0/bracerightbig are the eigenvalues of both J1andJ2/braceleftbig +i,−i,0,0/bracerightbig are the eigenvalues of J3 We verify that each of the J-matrices satisﬁes its own characteristic equation (as the Hamilton-Jacobi theorem requires), and discover that in fact J1andJ2satisfy the reduced characteristic equation J3−J=O J3satisﬁes the reduced characteristic equation J3+J=O More to the point: the characteristic equation of Wreads2 λ4−4(w2 1+w2 2−w2 3)λ2=0 which yields eigenvalues /braceleftbig +2/radicalBig w2 1+w2 2−w2 3,−2/radicalBig w2 1+w2 2−w2 3,0,0/bracerightbig and the reduced Hamilton-Jacobi statement W3−4(w2 1+w2 2−w2 3)W=O (40) Turning now from the inﬁnitesimal to the ﬁnite aspects of the theory, let the inﬁnitesimal w-triplet be written  w1 w2 w3 =1 Nθ k1 k2 k3 with  k2 1+k2 1+k2 1=+ 1,else k2 1+k2 1+k2 1=0,else k2 1+k2 1+k2 1=−1 where our obligation to distinguish three cases arises from the indeﬁnitness of 2Here—and occasionally hereafter—I allow myself to write (for example) w2 1 where I should more properly write w1w1or (w1)2. 8 Transformational principles derived from Clifford algebras the metric matrix G. Iteration of (35) then gives3 UN=/bracketleftBig I+1 N2θ/braceleftbig k1J1+k2J2+k3J3/bracerightbig/bracketrightBigN ↓ U(θ;kkk) = exp/bracketleftBig 2θ/braceleftbig k1J1+k2J2+k3J3/bracerightbig/bracketrightBig asN↑∞ (41) ≡e2θK We have now in hand enough algebraic information to develop and interpret the action of the transformation matrix e2θK. The technique is pretty,4but its details need not concern us at the moment. It is suﬃcient to notice that theG-antisymmetry of Kforces U≡e 2θKto be G-orthogonal: G–1KTG=−K=⇒G–1UTG=U–1(42) And that Mathematica today stands ready to do (in, typically, 0.0166 seconds!) all the work. Commands of the form MatrixExp[ matrix]//MatrixForm yielded the following illuminating results: e2θJ1= 10 0 0 01 0 00 0 cosh 2 θ−sinh 2θ 00 −sinh 2θcosh 2θ  e 2θJ2= 1000 0 cosh 2 θ0 sinh 2 θ 00100 sinh 2 θ0 cosh 2 θ  e 2θJ3= 10 0 0 0 cos 2θ−sin 2θ0 0 sin 2θcos 2θ0 00 0 1  e2θ(J2+J3)=I+2 00 0 0 00 −θ1θ1 0θ1−θ2θ2 0θ1−θ2θ2  Notice in the connection with •the ﬁrst example that −12−02+02=−1 •the second example that −02−12+02=−1 •the third example that −02−02+12=+ 1 •the fourth example that −02−12+12= 0 and the series terminates. 3I am running out of letters and fonts. In the notation advanced at (41) what I formerly called Uwould now be denoted U(δθ;kkk). 4For my most recent discussion of this subject, and references to earlier treatments, see §4 in “Extrapolated interpolation theory” ( ). Clifford algebra of order 2 9 The ﬁrst example describes what is, in eﬀect, a Lorentzian boost along the negative 2-axis (the 3-axis being identiﬁed with the “time” axis); the seconddescribes a boost along the positive 1-axis; the third describes a rotation in the (1,2)-plane. The ﬁnal example describes a transformation that is degenerate: its action is certainly describable, but I will not linger to do so. The 2-factor in the exponent at (41) is a story in itself: it is most familiar as the source of the double-valuedness of the spinor representations of O(3) , but that is only one of its manifestations: it arises in allsuch contexts. It remains only to construct a matrix representation of our Cliﬀord algebra. Here—in the absence of a deductive procedure—I am obliged to proceed byimprovisation, by modiﬁcation of rabbits pulled from Pauli’s hat. The Paulimatrices are standardly deﬁned 5 σσ1≡/parenleftbigg 01 10/parenrightbigg ,σσ2≡/parenleftbigg 0−i i0/parenrightbigg ,σσ3≡/parenleftbigg 10 0−1/parenrightbigg (43) though some authors adopt similarity-equivalent alternatives to those matrices. The Pauli matrices are traceless, Hermitian, and satisfy the relations σσ2 1=σσ2 2=σσ2 2=I (44.1) σσ1σσ2=iσσ3=−σσ2σσ1 σσ2σσ3=iσσ1=−σσ3σσ2 σσ3σσ1=iσσ2=−σσ1σσ3  (44.2) We are inspired to introduce ce·1≡σσ1=/parenleftbigg 01 10/parenrightbigg ce·2≡σσ2=/parenleftbigg 0−i i0/parenrightbigg ce·3≡ce·1ce·2=/parenleftbigg i0 0−i/parenrightbigg  (45) which evidently/demonstrably satisfy ce· 2 1=ce·2 2=I,ce·2 3=−I (46.1) ce·1ce·2=ce·3=−ce·2ce·1 ce·2ce·3=−ce·1=−ce·3ce·2 ce·3ce·1=−ce·2=−ce·3ce·2  (46.2) 5See David Griﬃths, Introduction to Quantum Mechanics (), page 156. See also page 2 in Chapter 1 of my Advanced Quantum Topics (). 10 Transformational principles derived from Clifford algebras which are just what we need. If we introduce aι=a0I+a1ce·1+a2ce·2+a3ce·3 lb=b0I+b1ce·1+b2ce·2+b3ce·3 and with Mathematica ’s assistance compute aιlbwe obtain a result which is precise agreement with (24). In aaa=a0I+a1eee1+a2eee2+a3eee3←−−→aι=a0I+a1ce·1+a2ce·2+a3ce·3(47) we have, therefore, a complex 2 ×2 matrix representation of the Cliﬀord algebra that was called into being at (4). We note with interest that detaι=a0a0−a1a1−a2a2+a3a3= modulus |aaa| (47) At (14) we encountered a 2 ×2 real matrix representation of i. Acting now on a hunch, we make substitutions 1/mapsto→/parenleftbigg 10 01/parenrightbigg ,0/mapsto→/parenleftbigg 00 00/parenrightbigg ,i/mapsto→/parenleftbigg 0−1 10/parenrightbigg into the equations (45) that deﬁned the ce·-matrices and obtain E1≡ 0010 000110000100  E 2≡ 0001 00 −10 0−100 1000  E3≡ 0−100 1000000 −1 0010   (49) We are informed by Mathematica that these real matrices satisfy relations identical to the relations (46) satisﬁed by the complex ce·-matrices. In aaa=a 0I+a1eee1+a2eee2+a3eee3←−−→A=a0I+a1E1+a2E2+a3E3(50) we have, therefore, a real 4 ×4 matrix representation of the Cliﬀord algebra of order 2, in which connection we observe that detA=(a0a0−a1a1−a2a2+a3a3)2= (modulus |aaa|)2(51) Quaternions 11 It is important not to confuse the 4-dimensionality of recent discussion with the 4-dimensionality that laid claim to our attention at (32). Our recent workhas placed us in position to display a 4 ×4 real representation (alternatively a complex 2 ×2 complex representation) of (31), whereas the work of pages 6–9 was concerned with the representation of (32). Those two transformationprinciples are of quite diﬀerent design ...yet—and this is the point—manage to subject the numbers/braceleftbig a 0,a1,a2,a3/bracerightbig to the same adventure . It would be a story well worth the telling if it ended there. But it doesn’t. We have busied ourselves thus far with spelling out the speciﬁc meaning—orat least the meaning assumed within C 2, the Cliﬀord algebra of order 2—of the upper (black) portion of the following ﬁgure, but ﬁnd ourselves in position now aaa/mapsto→AAA=uuu–1aaauuu←−−−−→ a0 ... a3 /mapsto→ A0 ... A3 =U a0 ... a3  vectors,tensors/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ↓ aι/mapsto→A=uι –1aι uι /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ↓ s 1 ... sn /mapsto→ S1 ... Sn =uι s1 ... sn  spinors to descend to its lower ( red) left corner, where we ﬁnd a transformation law that—though U,uuuanduιall encode the same data—is distinct from the transformation law seen at upper right: while the numbers/braceleftbig a0,a1,a2,a3/bracerightbig go adventuring so, in their wake, do the numbers/braceleftbig s1,...,sn/bracerightbig , but in their own distinctive way. What began at (20) as an attempt to construct a formal squareroot of (a 2-dimensional instance of) the familiar inner product has resultedﬁnally in what might, in a manner of speaking, be called the square root of vector algebra itself! 3. Quaternions: a digression. From the Pauli matrices (43) construct the traceless antihermitian matrices lhj≡(1/i)σσj:j=1,2,3. Working from (44) we have lh2 1=lh2 2=lh2 3=−I (52.1) lh1lh2=lh3=−lh2lh1 lh2lh3=lh1=−lh3lh2 lh3lh1=lh2=−lh1lh3  (52.2) which have the attractive property that the i-factors present in (44) have now 12 Transformational principles derived from Clifford algebras disappeared. The i-factors were of no concern to Pauli, but their presence would be unwelcome if our objective were to construct a generalization of complexalgebra. To the latter end, posit the existence of abstract objects hhh 1,hhh2,hhh3 that satisfy the relations (52), and after notational adjustments hhh1becomes iii hhh2becomes jjj hhh3becomes kkk obtain iii2=jjj2=kkk2=−I (53.1) iiijjj=kkk=−jjjiii jjjkkk=iii=−kkkjjj kkkiii=jjj=−iiikkk  (53.2) These are equations that, after a long period of frustrated thought, occurred to William Rowan Hamilton in a ﬂash on Monday, October ,a sh e strolled with his wife across Brougham Bridge, in Dublin, on his way to ameeting of the Royal Irish Academy. To establish his priority he scratchedequations (53) onto the bridge rail. 6That very afternoon he announced to the Academy his intention to read—and on Monday, November 13thdid read—the ﬁrst of his many papers on what he by then called the “theory of quaternions.”Hamilton’s motivation, which had a very strong but obscurely idiosyncraticphilosophical component, is diﬃcult for modern readers to grasp. 7But his accomplishment is easy to grasp—easier for us, no doubt, than it was forHamilton: he had introduced into vocabulary of mathematics the concept of non-commutivity . He had planted one of the seeds (Hermann Grassmann, at about the same time, planted another) from which the theory oﬀ algebras ingeneral, and Cliﬀord algebras in particular, were soon to sprout. It follows from (53) that if aaa=a 0I+a1iii+a2jjj+a3kkk bbb=b0I+b1iii+b2jjj+b3kkk 6In a letter written in , shortly before his death, Hamilton claimed to have written iii2=jjj2=kkk2=iiijjjkkk=−I, from which equations (53.2) can be recovered as corollaries. It seems doubtful that Hamilton was so sophisticatedat such an early point in his work, but perhaps he was: the physical evidencehas long since vanished. 7See the discussion in Chapters 6 & 7 of T. L. Hankins, Sir William Rowan Hamilton () and Chapter 2 of M. J. Crowe, A History of Vector Analysis : The Evolution of the Idea of a Vectorial System (). Quaternions 13 are quaternions then their product8can be described aaabbb=(a0b0−a1b1−a2b2−a3b3)I+(a0b1+a1b0+a2b3−a3b2)iii +(a0b2+a2b0+a3b1−a1b3)jjj +(a0b3+a3b0+a1b2−a2b1)kkk(54) It follows that if we deﬁne aaa≡a0I−a1iii−a2jjj−a3kkk (55) then aaaaaa=(a0a0+a1a1+a2a2+a3a3)I (56) =aaaaaa and that aaa–1can be described aaa–1=aaa a0a0+a1a1+a2a2+a3a3(57) which (on the assumption that the a’s are real) exists except in the case aaa=000. We agree to call |aaa|≡/radicalbig a0a0+a1a1+a2a2+a3a3/greaterorequalslant0 (58) the “modulus” of the quaternion aaaand establish by calculation that |aaa1·aaa2|=|aaa1|·|aaa2| (59) We stand now in need of some sharpened terminology: let the coeﬃcient a0ofIin the development of the quaternion aaabe called the “spur” of aaa: sp(a0I+a1iii+a2jjj+a3kkk)≡a0(60) It follows from (54) that sp(ababab) = sp(bababa) (61) 8It was Hamilton who, in order to drive the evil i’s from the temple, had been the ﬁrst to propose that complex numbers be construed to be ordered pairs of real numbers, subject to the multiplication law (x1,y1)·(x2,y2)=(x1x2−y1y2,x1y2+x2y1) It was, I presume, the 3-dimensionality of physical space that inspired his interest in ordered triplets . He recalled late in life, in a letter to his eldest son, that “every morning ...on my coming down to breakfast, [you and your brother] used to ask me, “Well, Papa, can you multiply triplets”? Whereto I was always obliged to reply, with a sad shake of the head: “No, I can only addand subtract them.” (My source here has been Crowe’s page 29.) Hamilton cannothave anticipated that his triplets would have to be embedded within quartets,or that his banished iwould return with two even more spooky friends. 14 Transformational principles derived from Clifford algebras It is in view of the fact that, in matrix theory, tr(AB) = tr( BA) and because I want to preserve the “trace” for matrix-theoretic applications ...that I have pressed into quaternionic service its German equivalent. To that terminology I add now more: we agree to say of a quaternion aaathat it is “pure” if and only if its spur vanishes. In short: aaa,if “pure,” has the form a1iii+a2jjj+a3kkk and when aaaisnotpure we will call a1iii+a2jjj+a3kkkits “pure part” (just as we speak of the “imaginary part” of a complex number). Ifxxxandyyyare pure then, by (54), we have xxxyyy=−(x1y1+x2y2+x3y3)I+(x2y3−x3y2)iii +(x3y1−x1y3)jjj +(x1y2−x2y1)kkk (62) Look now to the quaternionic similarity transformation aaa/mapsto−→AAA=uuu–1aaauuu (63) where one can, without loss of generality, assume uuuto be unimodular. Such transformations are, by (59) modulus-preserving. And they are, by (61), alsospur-preserving: A 0=a0: alluuu (64) It follows that we might as well assume from the outset that aaais pure. This we do, and emphasize by notational adjustment: in place of (63) we write (x1iii+x2jjj+x3kkk)/mapsto−→(X1iii+X2jjj+X3kkk)=uuu–1(x1iii+x2jjj+x3kkk)uuu(65) FromX1X1+X2X2+X3X3=x1x1+x2x2+x3x3we conclude that such transformations admit of the alternative description  x1 x2 x3 /mapsto−→ X1 X2 X3 =R x1 x2 x3  (66) where Ri sa3 ×3 rotation matrix. I will not proceed farther down this road: it is a road too well traveled ...though it leads pretty things, valuable things. Retreating to the lh-matrices that at (52) marked our point of departure, we have already in hand a 2 ×2 complex matrix representation of Hamilton’s Quaternions 15 quaternion algebra: iii←→lh1=−iσσ1=/parenleftbigg 0−i −i0/parenrightbigg jjj←→lh2=−iσσ2=/parenleftbigg 0−1 10/parenrightbigg kkk←→lh3=−iσσ3=/parenleftbigg −i0 0i/parenrightbigg  (67) The representative of aaa=a 0I+a1iii+a2jjj+a3kkktherefore reads A=/parenleftbigg a0−ia3−a2−ia1 a2−ia1a0+ia3/parenrightbigg and we have detA=a0a0+a1a1+a2a2+a3a3=|aaa|2(68.1) trA=2a0= 2 sp(aaa) (68 .2) which render explicit the relationships between the quaternionic “modulus” and “spur” and their matrix-theoretic counterparts. “Representation theory”leads also to good things, but here again they are things too familiar to requireexplicit review on this occasion. It is, by the way, my impression that we touchhere upon an aspect of his theory that Hamilton—who worked when the theoryof matrices was still in its infancy—did himself notexplore. 9 To summarize: the relationship between Hamilton’s quaternion algebra Q and the simplest Cliﬀord algebra C2is intimate, but curiously skew. Hamilton introduces a triple of algebraic objects/braceleftbig iii,jjj,kkk/bracerightbig , to which he assigns co-equal status. The object x2 1+x2 2+x2 3emerges naturally but incidentally from his theory: it is not an object to which generative signiﬁcance is assigned. Cliﬀord does assign generative signiﬁcance to x2 1+x2 2. He is led to an algebraic construct in which eee1=iiiiandeee2=ijjjplay the role of generators and into which eee3≡eee1eee2=−kkkis introduced simply to achieve algebraic closure. In the fully-elaborated theory it is not x2 1+x2 2+x2 3but−x2 1−x2 2+x2 3that acquires the status of a natural object. Hamilton devoted the last twenty-two years of his life to the development and promotion of the theory of quaternions, to which he “was inclined toimbue with cosmic signiﬁcance.” 10British and American mathematicians and 9I do not have access to Hamilton’s Lectures on Quaternions ()o rt oh i s posthumous Elements of Quaternions (), which should be consulted in this regard. The theory of matrices originates in work published by Arthur Cayleyin. 10The phrase is Carl Boyer’s: see page 625 in his A History of Mathematics (). 16 Transformational principles derived from Clifford algebras mathematical physicists, during the closing decades of the 19thCentury, tended fairly generally to be quaternionists (or at least to pay lip service to the newreligion), though some became outspoken critics of the trend, and others werecontent to entertain various shades of bemused indiﬀerence. 11T o d a yi ti s universally recognized that the invention of quaternions (which is to say: of non-commutivity) was a seminal event, though the quaternion algebra itself hasbecome a relatively insigniﬁcant detail within a vast subject. Such importanceas it does enjoy is due more to the work of Pauli than of Hamilton. And yet,echos of the former cult status of quaternion algebra persist to this day: Googleresponds with more than 59,000 items to the key-word “quaternion,” and muchof that work appears on cursory inspection to be fairly self-indulgent, havinglittle to do with anything. The situation with regard to Cliﬀord’s invention could hardly be more diﬀerent. The birthplace of “Cliﬀord algebra” is diﬃcult to discover withinCliﬀord’s Mathematical Papers : the essential thought was presented as but one idea among the bewilderingly many, an incidental bi-product of his interest inthe work of Grassmann ...and it was certainly not an idea he chose to cultivate, to promote. That work fell to others, decades later. Today, Cliﬀord is a cultﬁgure, and his algebra an object of worship. A journal Advances in Applied Cliﬀord Algebras exists, International Cliﬀord Algebra Conferences are held, Google responds with more than45,000 items to the key-word“Cliﬀord algebra.”I have learned to keep this work at arm’s length not because it is frivolous(though some of it certainly is) but because it tends to be seductive, the rewardsdisproportionate to the investment The present project represents a departurefrom that personal policy. Hamilton’s vision (shared most vocally/inﬂuentially by Peter Guthrie Tait) of a fully “quaternionized physics” was ultimately subverted by a combiniationof circumstances, among them •the accumulated weight of the formalism •the discovery of simpler, more direct ways to manage multi-dimensional objects •the discovery that physics has need sometimes of algebraic structures more complicated than (or at least alternative to) quaternions Under the second head we might cite the invention (beginning in the ’s) of tensor analysis, and the work of Gibbs & Heaviside who, in the early ’s, independently invented the formalism known today as vector algebra & analysis . Gibbs, though familiar with Hamilton, claimed Grassmann as his principal inﬂuence, while Heaviside(whoprobably never heard of Grassmann) worked in direct reaction to Hamilton.It was the idea of each to squeeze the juice from quaternions and discard the 11I am thinking here especially of Maxwell ...whose passing mention (in his Treatise ) of quaternions did, however, lead both Gibbs and Heaviside to take up—only to abandon—the subject. Metric generalization 17 rind. The implications of that idea were clearly spelled out in J. W. Gibbs & E. B. Wilson’s Vector Analysis (),which was based on class notes developed by Gibbs during the ’s and’s,and was the inﬂuential ﬁrst textbook in the ﬁeld. Gibbs (like Heaviside) considered/braceleftbig iii,jjj,kkk/bracerightbig to refer to objects no more mysterious than orthogonal unit vectors in 3-space . “Pure quaternions” xxx=x1iii+x2jjj+x3kkk yyy=y1iii+y2jjj+y3kkk become by this interpretation simple 3-vectors. Drawing inspiration from (62), Gibbs deﬁned two distinct kinds of “product”: number-valued dot productxxx···yyy≡x1y1+x2y2+x3y3 vector-valued cross product xxx×yyy≡/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleiiijjjkkk x 1x2x3 y1y2y3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle  (69) And he abandoned all thought of “dividing a vector by a vector ,” perhaps because the meaning Hamilton would so proudly assign to xxxyyy–1=−1 yyy···yyy/braceleftBig (xxx···yyy)I+(xxx×yyy)/bracerightBig (70) would be a hybrid object,vectorially meaningless unless either xxx···yyy=0 o r xxx×yyy=000. It is remarkable,when you think about it,how successful are the applications of vector analysis to solid geometry and 3-dimensional physics,given that vector analysis provides no concept of vector division . 12 4. Second order Clifford algebra with general metric. We turn now to study of the implications of writing,in place of (20), (gijxixj)=(xieeei)2: allxi(71) where g•g≡/parenleftbigg g11g12 g21g22/parenrightbigg is understood to be real,symmetric and non-singular ( g≡detg•g/negationslash= 0) but we will not impose the requirement that g•gbe positive-deﬁnite ( g>0). It has been my experience (in other,more complicated,contexts) that metric generalization /parenleftbigg 10 01/parenrightbigg /mapsto−→/parenleftbigg g11g12 g21g22/parenrightbigg 12For extended discussion of the mathematical developments that historically radiated from Hamilton’s invention,and more detailed references,see “Theoriesof Maxwellian design” ( ). 18 Transformational principles derived from Clifford algebras tends to complicate life at the outset,but that the added eﬀort is worthwhile in the longrun,for it exposes important distinctions that otherwise remaininvisible. Immediately eee ieeej+eeejeeei=2gijI (72) The antisymmetry condition (22.2) is now lost,though we have in its place the antisymmetry of eeeij≡eeeieeej−gijI:eeeij=−eeeji (73) It becomes therefore natural to close the/braceleftbig eeei,eeej/bracerightbig -generated algebra with the introduction of fff≡1 2εijeeeij (74.1) =eee12 =eee1eee2−g12I (74.2) =−eee2eee1+g21I (74.3) From (72)–(74) we extract the primative products eee1eee1=g11I eee2eee1=g21I−fff fffeee1=g21eee1−g11eee2eee1eee2=g12I+fff eee2eee2=g22I fffeee2=g22eee1−g12eee2eee1fff=g11eee2−g12eee1 eee2fff=g21eee2−g22eee1 ffffff=−gI where in the ﬁnal equation g≡g11g22−g12g21=detg•g.Therefore and equivalently: if ccc≡sI+v1eee1+v2eee2+pfff CCC≡SI+V1eee1+V2eee2+Pfff are arbitrary Cliﬀord numbers then cccCCC=s/bracketleftbig SI+V1eee1+V2eee2+Pfff/bracketrightbig +v1/bracketleftbig Seee1+V1(g11I)+V2(g12I+fff)+P(g11eee2−g12eee1)/bracketrightbig +v2/bracketleftbig Seee2+V1(g21I−fff)+V2(g22I)+P(g21eee2−g22eee1)/bracketrightbig +p/bracketleftbig Sfff+V1(g21eee1−g11eee2)+V2(g22eee1−g12eee2)+P(−gI)/bracketrightbig =I/bracketleftbig sS+(v1V1+v2V2)−gpP/bracketrightbig +eee1/bracketleftbig sV1+v1S−(v1g12+v2g22)P+p(g21V1+g22V2)/bracketrightbig +eee2/bracketleftbig sV2+v2S+(v1g11+v2g21)P−p(g11V1+g12V2)/bracketrightbig +fff/bracketleftbig sP+v1V2−v2V1+pS/bracketrightbig =I/bracketleftbig sS+vnVn−gpP/bracketrightbig +eee1/bracketleftbig sV1+v1S−v2P+pV2/bracketrightbig +eee2/bracketleftbig sV2+v2S+v1P−pV1/bracketrightbig +fff/bracketleftbig sP+v1V2−v2V1+pS/bracketrightbig (75.1) Metric generalization 19 Here I have used gmnto lower indices,in the manner standard to tensor analysis: vm≡gm1v1+gm2v2≡gmnvnandVm≡gmnVn. IMPORTANT REMARK: I should emphasize that the contravariant Levi-Civita symbol εijencountered at (74.1) can in every coordinate system be described εij=/braceleftbigg sgn/parenleftbigi 1j 2/parenrightbig :i&jdistinct 0 : otherwise if and only if it is understood to transform as a density of weight W= +1. That same weight then attaches automatically to its covariant companion εij=gimgjnεmn Note,however,that while the numerical values of εijrange on/braceleftbig −1,0,+1/bracerightbig those ofεijrange on/braceleftbig −g,0,+g/bracerightbig Similarly,the deﬁning statement <epsilonbij=/braceleftbigg sgn/parenleftbigi 1j 2/parenrightbig :i&jdistinct 0 : otherwise holds in every coordinate system iﬀ <epsilonbijis understood to transform as a density of weight W=−1. One has εij=g<epsilonbij and achieves consistency with the observation that g≡detg•gtransforms as a scalar density of weight W=+ 2 I honor the notational conventions adopted in “Electrodynamical applications of the exterior calculus” ( ). See pages 7 & 9 for more detailed discussion of the points at issue. The preceding remarks place us in position to consider the transformational properties of the results in hand (which,so long as the metric was required to be Euclidean,we were in only a very weak position to do). The constructiong ijxixjtransforms by invariance provided we assume •thexitransform as components of a weightless contravariant vector; •thegijtransform as ...a weightless covariant tensor. We are then forced by (71) to assume that •theeeeitransform as ...a weightless covariant vector. Looking to Cliﬀord’s construction ccc≡sI+vmeeem+pfff the established invariance of vmeeemmakes it natural to assume that •sandItransform by invariance. And since,as remarked above, •ffftransforms as a scalar density of weight W=+ 1 we are forced to stipulate that •ptransforms as a scalar density of weight W=−1. 20 Transformational principles derived from Clifford algebras The point of my s,v,p-notation is now clear: those symbols are intended to suggest scalar, vector andpseudo-scalar ,respectively. Let the conjugate ofcccbe deﬁned/denoted ccc≡sI−vmeeem−pfff From (75)—which we are now in position to write cccCCC=/bracketleftbig sS+vnVn−gpP/bracketrightbig I +/bracketleftbig sVn+vnS+εmnvmP−pεmnVm/bracketrightbig eeen +/bracketleftbig sP+g–1εmnvmVn+pS/bracketrightbig fff (75.2) —it then follows that cccccc=/bracketleftbig s2−(v,v)+gp2/bracketrightbig I=cccccc (76) =N(ccc)·I where N(ccc)≡s2−(v,v)+gp2(77) deﬁnes the norm ofccc. From (76) we see that C2[g•g]—the Cliﬀord algebra of order 2 with arbitrary metric g•g—is not a division algebra (but becomes one when theviare made imaginary:13not just the zero element,but) all elements with (v,v)=s2+gp2: deﬁnes a quadratic surface in v-space are non-invertible. Let N(ccc−λI)=λ2−2sλ+[s2−(v,v)+gp2] =λ2−2tr(ccc)·λ+N(ccc) (78) deﬁne the “characteristic polynomial” of the Cliﬀord number ccc. A quick calculation serves to establish that ccc2−2tr(ccc)·ccc+N(ccc)I=000: all Cliﬀord numbers ccc (79) In short: Every Cliﬀord number satisﬁes its own characteristic equation .14The zeros of (78) lie at λ=s±/radicalbig (v,v)−gp2 (80) 13One is brought thus back to Qin the Euclidean case. 14My guess is that it was as the quaternionic instance of this statement— not as a proposition about matrices (that was Cayley’s contribution)—thatHamilton knew the “Cayley-Hamilton theorem.” Metric generalization 21 Certainly we expect to have N(cccCCC)=N(ccc)·N(CCC) (81) but the direct demonstration (I know of no cunningly indirect demonstration) is a bit tedious. We have N(cccCCC)−N(ccc)·N(CCC) =/braceleftBig/bracketleftbig s2S2+2sS(v,V)−2gsSpP +(v,V)2−2gpP(v,V)+g2p2P2/bracketrightbig −/bracketleftbig s2(V,V)+2sS(v,V)+2sP(vmεmnVn)−0 +S2(v,v)+0+2pS(vmεmnVn) +P2(εkmεknvmvn)−2pP(εkmεknvmVn) +p2(εkmεknVmVn)/bracketrightbig +g/bracketleftbig s2P2+2g−1sP(vmεmnVm)+2spSP +g−2(vmεmnVn)2+2g−1pS(vmεmnVn)+p2S2/bracketrightbig/bracerightBig −/braceleftBig s2S2−s2(V,V)+gs2P2 −S2(v,v)+(v,v)(V,V)−gP2(v,v) +gp2S2−gp2(V,V)+g2p2P2/bracerightBig which after much cancellation becomes =/bracketleftbig g–1(vmεmnVn)2+(v,V)2−(v,v)(V,V)/bracketrightbig +2gpP/bracketleftbig g–1(εkmεknvmVn)−(v,V)/bracketrightbig −gP2/bracketleftbig g–1(εkmεknvmvn)−(v,v)/bracketrightbig −gp2/bracketleftbig g–1(εkmεknVmVn)−(V,V)/bracketrightbig =/bracketleftbig g–1εmnεij+gmjgni−gmignj/bracketrightbig vmviVnVj +2gpP/bracketleftbig g–1εkmεkn−gmn/bracketrightbig vmVn −gP2/bracketleftbig g–1εkmεkn−gmn/bracketrightbig vmvn −gp2/bracketleftbig g–1εkmεkn−gmn/bracketrightbig VmVn (82) It becomes clear on a moment’s thought that—in 2-dimensional instance of a very general proposition (see equation (21) in the material cited on page 19)— εi1i2<epsilonbj1j2=g–1εi1i2εj1j2=/vextendsingle/vextendsingle/vextendsingle/vextendsingleδ i1j1δi1j2 δi2j1δi2j2/vextendsingle/vextendsingle/vextendsingle/vextendsingle=δ i1j1δi2j2−δi1j2δi2j1 and therefore that g–1εmnεij=gmignj−gmjgni g–1εkmεkn=δkkgmn−δkngmk=( 2−1)gmn=gmn 22 Transformational principles derived from Clifford algebras Returning with this information to (82) we ﬁnd that that all the [stuﬀ]-terms vanish,completing the proof of (81).15 Proceeding on the assumption that uuuis aninvertible Cliﬀord number,16 we look now again to transformations of the form ccc/mapsto−→CCC=uuu–1cccuuu (83) Such transformations are,by (81), norm-preserving . It is clear also that we can, without loss of generality,assume uuuto be unimodular: N(uuu) = 1. Equation (83) sets up a linear relationship between the elements of CCCand those of ccc, which we emphasize by writing  s v1 v2 p /mapsto−→ S V1 V2 P =U s v1 v2 p  (84) The norm of ccccan in this notation be written N(ccc)= s v1 v2 p T G s v1 v2 p with G≡ 10 00 0−g11−g120 0−g21−g220 00 0 g  (85) (notice that the real symmetric matrix Ggives back (36) in the Euclidean case) and the representation (84) of (83) will itself be norm-preserving if and only ifUisG-orthogonal: U TGU=G. This was seen already on page to require that the inﬁnitesimal generator of UbeG-antisymmetric. It is with those points fresh in our minds that we turn to the details. Write uuu=I+<epsilonbwww: neglect terms of order <epsilonb2(86) whereepsilon is an inﬁnitesmal parameter (that,since it wears no indices,will not be confused with the Levi-Civita tensor). In leading order uuu–1=I−<epsilonbwwwand (83) becomes ccc/mapsto−→CCC=ccc+<epsilonb[cccwww−wwwccc]+··· (87) Write (to establish our notational conventions) ccc=sI+vneeen+pfff www=σI+wneeen+wfff 15The argument presented above tells us nothing useful about why(81) is valid,and can be expected to become rapidly more diﬃcult to carry tocompletion as the order of the Cliﬀord algebra ascends. What we need—butwhat I presently lack—is a simple, illuminating, dimensionally generalizable proof of fundamental statement (81). 16Note that the set of invertible Cliﬀord numbers has group structure with respect to the operation of multiplication. I’m sure mathematicians must havea name for such things. Metric generalization 23 and from (75.2) obtain [cccwww−wwwccc]=2/bracketleftbig wεmnvm−wmεmnp/bracketrightbig eeen−2g–1/bracketleftbig wmεmnvn/bracketrightbig fff (88.1) =2g/bracketleftbig w<epsilonbmnvm−wm<epsilonbmnp/bracketrightbig gnkeeek−2/bracketleftbig wm<epsilonbmnvn/bracketrightbig fff(88.2) Notice thatσis silent: it is without loss of generality that we henceforth assume wwwto be spurless,writing www=wneeen+wfff (89.1) where,by the assumed unimodularity of www, (w)2=1+wmgmnwn(89.2) Notice also that—relatedly— sdoes not participate in the transformation (87); i.e.,that it transforms by invariance (which is to say: “like a scalar”). Notice ﬁnally that because •ghas weightW=2 •εmnandfffhave weight W=1 •vn,wn,gmnandeeenare weightless: W=0 •<epsilonbmn,pandwhave weight W=−1 •g–1has weightW=−2 each of the terms on the right side of (88) is—as we require—weightless. In representation of (86) we have U=I+2<epsilonbW (90.1) and write W=wnJn+wK (90.2) to emphasize the fact that Wdepends linearly on the coordinates of www. The detailed designs of J1,J2andKcan be read oﬀ from (88),which supplies J1= 000 0 000 −g·g21 000 −g·g22 00 −10  J2= 000 0 000 + g·g11 000 + g·g12 0+ 1 0 0  K= 00 0 0 0g·g21−g·g110 0g·g22−g·g120 00 0 0   (91) 24 Transformational principles derived from Clifford algebras Quick (but interesting) calculation conﬁrms that each of the matrices (91) is in factG-antisymmetyric,and they are seen to give back the matrices encountered on page 6 in the Euclidean case. Mathematica -assisted calculation gives J1J2−J2J1=−K J2K−KJ2=g(g11J1+g12J2) KJ1−J1K=g(g21J1+g22J2)  (92) which assume the simple form (39) in the Euclidean case. Also det(J1−λI)=λ4−g·g22λ2 det(J2−λI)=λ4−g·g11λ2 det(K−λI)=λ4+gλ2 which in the Euclidean case reproduce results reported on page 7. Finally we have det(sI+v1J1+v2J2+pK)=s2(s2−vmgmnvm+gp2) =s2·N(sI+v1eee1+v2eee2+pfff) (93) by simpliﬁcation17of the result reported by Mathematica . We are brought thus to the conclusion that ccc/mapsto−→CCC=e−θwwwccceθwww:www=wneeen+wfff and s v1 v2 p /mapsto−→ S V1 V2 P =e2θW s v1 v2 p :W=wnJn+wK say the same thing in two diﬀerent ways. Such transformations generally involve “vector/psuedoscalar intermixing,” and in that respect relate unnaturally to our point of departure,which wastheg •g-rotationally invariant expression xmgmnxn. Transformations “natural” to that expression result in the present formalism from setting v1=v2= 0 and w= 1,in which connection I note that Mathematica ’sMatrixExp command instantly produces e2θK= 10 00 0 stuﬀ stuﬀ 00 stuﬀ stuﬀ 000 01  where the unsimpliﬁed “stuﬀ” terms are enormously complicated. We have 17Use g/parenleftbigg g11g12 g21g22/parenrightbigg =/parenleftbigg g22−g12 −g21g11/parenrightbigg Metric generalization 25 touched here upon a subject of some intrinsic interest,so I linger to develop some of the details: We need only concern ourselves with the 2 ×2 “nucleus” of K;i.e.,with k≡g/parenleftbigg g21−g11 g22−g12/parenrightbigg =/parenleftbigg −g21−g22 g11g12/parenrightbigg which is readily seen to be g•g-antisymmetric: g•gk=/parenleftbigg 0−g g0/parenrightbigg is antisymmetric From det( k−λI)=λ2+git follows that k2+gI=O. It is natural,therefore, to introduce ˆk≡1√gksince it satisﬁes the simpler equation ˆk2+I=O.W e now have uι≡exp/braceleftbig 2θk/bracerightbig = exp/braceleftbig 2ϑˆk/bracerightbig withϑ≡√gθ = cos 2ϑ·I+ sin 2ϑ·ˆk = cos 2ϑ/parenleftbigg 10 01/parenrightbigg +sin 2ϑ√g/parenleftbigg −g21−g22 g11g12/parenrightbigg (94) and verify that uιTg•guι=g•g. In the Euclidean case g•g/mapsto−→/parenleftbigg 10 01/parenrightbigg :√g=1 we recover the rotation matrix uι=/parenleftbigg cos 2θ−sin 2θ sin 2θcos 2θ/parenrightbigg while specialization to the Minkowski metric g•g/mapsto−→/parenleftbigg 10 0−1/parenrightbigg :√g=i gives the Lorentz matrix uι=/parenleftbigg cos 2iθ i–1sin 2iθ i–1sin 2iθ cos 2iθ/parenrightbigg =/parenleftbigg cosh 2θsinh 2θ sinh 2θcosh 2θ/parenrightbigg If we were Dirac-like inhabitants of a 1-dimensional world (2-dimensional spacetime) we would have essential interest in the least-dimensional matrix representations of the fundamental anticommutation relations (72). I turn now 26 Transformational principles derived from Clifford algebras to description of a method for construcing such matrices.18We proceed from the observation that these real,symmetric,traceless matrices ce·/prime/prime 1=/parenleftbigg 10 0−1/parenrightbigg ,ce·/prime/prime 2=/parenleftbigg 01 10/parenrightbigg (95.1) —convenient variants of some ce·-matrices introduced at (45)—would serve our needs in the Euclidean case g•gEuclidean =/parenleftbigg 10 01/parenrightbigg for by calculation ce·/prime/prime 1ce·/prime/prime 1=g11·I=1·I ce·/prime/prime 2ce·/prime/prime 2=g22·I=1·I ce·/prime/prime 1ce·/prime/prime 2+ce·/prime/prime 2ce·/prime/prime 1=2g12·I=0·I Therefore the matrices ce·/prime 1≡√g1ce·/prime/prime 1andce·/prime 2≡√g2ce·/prime/prime 2 (95.2) serve our needs in the diagonal case g•gdiagonal =/parenleftbigg g10 0g2/parenrightbigg for trivially ce·/prime 1ce·/prime 1=g11·I=g1·I ce·/prime 2ce·/prime 2=g22·I=g2·I ce·/prime 1ce·/prime 2+ce·/prime 2ce·/prime 1=2g12·I=0·I Now construct ce·1≡ce·/prime 1cosα−ce·/prime 2sinα ce·2≡ce·/prime 1sinα+ce·/prime 2cosα/bracerightBigg (95.3) and from the requirements ce·1ce·1=g11·I ce·1ce·2+ce·2ce·1=2g12·I ce·2ce·2=g22·I obtain g11=g1cos2α+g2sin2α=1 2(g1+g2)+1 2(g1−g2) cos 2α g12=g21=(g1−g2) cosαsinα=1 2(g1−g2) sin 2α g22=g1sin2α+g2cos2α=1 2(g1+g2)−1 2(g1−g2) cos 2α  (96) 18My primary source will be some penciled notes I wrote in October, ,in response to points raised in David Griﬃths’ elementary particles course,whichI attended that term. Metric generalization 27 g12=g21 g22 2α g2 g11g1 Figure 1 :Diagrammatic interpretation of (96), known to engineers asMohr’s construction . The circle is centered at1 2(g1+g2)and has radius1 2(g1−g2). Readingg11,g22andg12=g21from the ﬁgure, one obtains precisely (96). Evidently the triples/braceleftbig g11,g12=g21,g22/bracerightbig and/braceleftbig g1,g2,α/bracerightbig provide alternative but equivalent descriptions of the 2 ×2 real symmetric matrix g•g. This fact has been known and used for well more than a century by engineers,andits elegant diagrammatic interpretation—see the ﬁgure—is known as “Mohr’sconstruction.” 19Mathematica conﬁrms,by the way,that matrices of the design /parenleftbigg g1cos2α+g2sin2α (g1−g2) cosαsinα (g1−g2) cosαsinαg 1sin2α+g2cos2α/parenrightbigg have eigenvalues/braceleftbig g1,g2/bracerightbig for all values of α. The conclusion of interest is that ce·1≡/parenleftbigg√g1cosα−√g2sinα −√g2sinα−√g1cosα/parenrightbigg ce·2≡/parenleftbigg√g1sinα√g2cosα√g2cosα−√g1sinα/parenrightbigg  (97) 19See “Non-standard applications of Mohr’s construction” ( ). Mohr was a professor of civil engineering ﬁrst in Stuttgart,later in Dresden,and was ledto his construction ( ) as a means of clarifying a problem having to do with the fracture of brittle materials. In some respects he had been anticipated byanother civil engineer named Culmann ( ). Both were studying a problem that had been pioneered by Coulomb. 28 Transformational principles derived from Clifford algebras Which,indeed,check out: working from (97) we ﬁnd ce·1ce·1=/parenleftbigg g1cos2α+g2sin2α 0 0 g1cos2α+g2sin2α/parenrightbigg =/parenleftbigg g110 0g11/parenrightbigg ce·1ce·2+ce·2ce·1=/parenleftbigg (g1−g2) sin 2α 0 0(g1−g2) sin 2α/parenrightbigg =2/parenleftbigg g120 0g12/parenrightbigg ce·2ce·2=/parenleftbigg g1sin2α+g2cos2α 0 0 g1sin2α+g2cos2α/parenrightbigg =/parenleftbigg g220 0g22/parenrightbigg Our 1-dimensional Dirac would set g1=+ 1 ,g2=−1,α= 0 and by (97) obtain IΓ1=/parenleftbigg 10 0−1/parenrightbigg IΓ2=/parenleftbigg 0i i0/parenrightbigg which again check out: IΓ1IΓ1=/parenleftbigg 10 01/parenrightbigg IΓ1IΓ2+IΓ2IΓ1=/parenleftbigg 00 00/parenrightbigg IΓ2IΓ2=/parenleftbigg −10 0−1/parenrightbigg In the notes cited previously18I work out in fair detail the theory of the resulting “Dirac equation” (IΓm∂m+iκI)/parenleftbigg ψ1 ψ2/parenrightbigg =/parenleftbigg 0 0/parenrightbigg (98) There are no major surprises. Our 1-dimensional physicists might,however,be surprised by their discovery that transformations that are G-orthogonal with respect to (see again (85)) the “hyperdimensional metric” Gdirac≡ 1000 0−100 0010000 −1  are latent in the design of their little theory. Here—though one aspect of the subject remains to be developed—I bring to an end this review of the C 2[g•g] generated by/braceleftbig ce·1,ce·2/bracerightbig . That this discussion, which began on page 17,progressed as smoothly as it did can,I think,beattributed mainly to the fact that at (74) we chose fff=eee 1eee2−g12I(rather than eee1eee2) to close the algebra. Third order theory 29 5. Third order Clifford algebra with general metric. Equations (71) and (72)— (gijxixj)=(xieeei)2whenceeeeieeej+eeejeeei=2gijI —remain in force,the diﬀerence being that all indices range now on/braceleftbig 1,2,3/bracerightbig . C3[g•g] is an algebra of order 23= 8,generated by/braceleftbig eee1,eee2,eee3/bracerightbig . The initial question is: How most usefully to describe the general element cccofC3[g•g]? How most naturally to achieve algebraic closure ? I propose to adopt a practice standard to the exterior calculus. Let eeei1i2...ip≡1 p!eeei1∧eeei2∧···∧eeeip ≡1 p!/braceleftBigg antisymmetrized/summationdisplay permutations/bracerightBigg (99) which carries with it the implication that in the n-dimensional case there will be/parenleftbign p/parenrightbig distinct terms of order p. Look back again to the Casen=2 We have one element Iof orderp= 0,two elements/braceleftbig eee1,eee2/bracerightbig of orderp= 1,and one element eee12=1 2!(eee1eee2−eee2eee1) =1 2!(2eee1eee2−2g12I)=fff of orderp= 2 (which,however,has two distinct names: eee12=−eee21). Proceeding similarly to the case of immediate interest,we in Casen=3have one element Iof orderp= 0,three elements/braceleftbig eee1,eee2,eee3/bracerightbig of orderp= 1,three elements eee23=1 2!(eee2eee3−eee3eee2) =2 2!(eee2eee3−g23I)=−eee32 eee13=1 2!(eee1eee3−eee3eee1) =2 2!(eee1eee3−g13I)=−eee31 eee12=1 2!(eee1eee2−eee2eee1) =2 2!(eee1eee2−g12I)=−eee21 of orderp= 2,and one element eee123=1 3!(eee1eee2eee3−eee1eee3eee2+eee2eee3eee1−eee2eee1eee3+eee3eee1eee2−eee3eee2eee1) =6 3!(eee1eee2eee3−g23eee1+g31eee2−g12eee3) of orderp= 3 (which has 3! diﬀerent names). It becomes natural in this light 30 Transformational principles derived from Clifford algebras to write ccc=cI+cieeei+1 2!cijeeeij+1 3!cijkeeeijk (100) where the coeﬃcients are weightless antisymmetric tensors20of ascending order and we have politely “averaged over all alternative names.” But adoption of such a policy would carry with it the implication that to describe the elements of C2[g•g] we should write ccc=cI+cieeei+1 2!cijeeeij whereas it has been our established practice to write ccc≡sI+vieeei+pfff.H o w to achieve consistency? Let fffi1i2...in−p≡g−1 2·1 p!εi1i2...in−pj1j2...jpeeej1j2...jp =g+1 2·1 p!<epsilonbi1i2...in−pj1j2...jpeeej1j2...jp (101) deﬁne the population/braceleftbig fffi1i2...in−p/bracerightbig of elements dualto the population/braceleftbig eeei1i2...ip/bracerightbig . Theg±1 2-factors have been introduced to insure that elements of a population and its dual transform with the same weight (which is to say: weightlessly).The two populations contain identically many elements:/parenleftbig n n−p/parenrightbig =/parenleftbign p/parenrightbig . But each element of/braceleftbig eeei1i2...ip/bracerightbig hasp! distinct names,while each element of the dual population/braceleftbig fffi1i2...in−p/bracerightbig has (n−p)! distinct names: that distinction is greatest atp=n,and it disappears at p=1 2n(which requires that nbe even). From21 g+1 21 (n−p)!<epsilonbk1k2...kpi1i2...in−pfffi1i2...in−p =g1 p!(n−p)!<epsilonbk1k2...kpi1i2...in−p<epsilonbi1i2...in−pj1j2...jpeeej1j2...jp =1 p!(n−p)!(−)p(n−p)g<epsilonbk1k2...kpi1i2...in−p<epsilonbj1j2...jpi1i2...in−peeej1j2...jp =(−)p(n−p)1 p!δk1k2...kpj1j2...jpeeej1j2...jp =(−)p(n−p)eeek1k2...kp (102) we see that “double dualization”returns the original population except,perhaps, for an overall sign—a minus sign that is present if and only if nis even andp is odd. Look in particular to the case that precipitated this discussion: the case n=p= 2. Drawing upon (101) and (102) we have fff=g−1 2·1 2εj1j2eeej1j2=g−1 2·1 2(eee1eee2−eee2eee1) =g+1 2·1 2<epsilonbj1j2eeej1j2 20Use of the term “tensor” will remain technically unwarrented until we have given explicit attention to the transformational aspects of the theory. 21Here I allow myself to make free use of notions (for example: that of the “generalized Kronecker delta”) and identities—workhorses of exterior algebra—that (as was mentioned already on page 19) are developed on pages 7–9 of“Electrodynamical applications of the exterior calculus” ( ). Third order theory 31 (note that the ﬁrst of those equations diﬀers from (74) only by the inclusion of the weight-preserving√g-factor) and eeek1k2=(−)2g+1 2·<epsilonbk1k2fff (note that,because eeek1k2wears what is in case n= 2 a full complement of indices,fffis deprived of any). Introducing this last bit of information into ccc=cI+cieeei+1 2!cijeeeij we obtain =cI+cieeei+/braceleftbig√g1 2!<epsilonbijcij/bracerightbig fff (103.1) which diﬀers only notationally from our former =sI+vieeei+pfff (103.2) except in this detail:√g-factors have served in (103.1) to render both/braceleftbig etc./bracerightbig andfffweightless,while the fffin (103.2) has weight W= +1 and its coeﬃcient phas weightW=−1. We confront therefore a POLICY DECISION: Should or should not√g-factors be included? Inclusion seems to simplify discussion of general algebraic issues,but in cases where g<0 serves to introduce i’s that for physical reasons may be unwelcome. My policy will be to retain the√ g’s, with the understanding that in speciﬁc applications we may wantto drop them. The practice of writing√ |g|that is sometimes used in general relativity seems to me to create more problems than itsolves. In the past—especially when working in C 4[g•g]—I have found it most convenient to adopt the “symmetrized hybrid” notations that proceed inC2[g•g]:ccc=sI+sieeei+pfff inC3[g•g]:ccc=sI+sieeei+pifffi+pfff inC4[g•g]:ccc=sI+sieeei+1 2!sijeeeij+pifffi+pfff inC5[g•g]:ccc=sI+sieeei+1 2!sijeeeij+1 2!pijfffij+pifffi+pfff inC6[g•g]:ccc=sI+sieeei+1 2!sijeeeij+1 3!sijkeeeijk+1 2!pijfffij+pifffi+pfff ... These have at least the merit that they total minimize the number of indices and mimic the symmetry of the binomial distribution. The scheme does,however,become ambiguous “at the middle” when nis even (should one write 1 3!sijkeeeijk or1 3!pijkfffijk?) and in some applications it presents also other disadvantages ...as will emerge. When cccis presented as described above I will say it has been 32 Transformational principles derived from Clifford algebras presented in “symmetrized form,” and will use “canonical form” to refer to the presentation ccc=cI+cieeei+1 2!cijeeeij+1 3!cijkeeeijk+1 4!cijkleeeijkl+··· Suppose that aaaandbbb—elements of Cn[g•g]—have been presented in canonical form,and that it is desired to obtain the canonical description of their productababab. What we then need (and cannot do without!) are formulæ of the type eee i1...ip·eeej1...jq=cI+ckeeek+1 2!ck1k2eeek1k2 +···+1 (p+q)!ck1...kp+qeeek1...kp+q (104) If our interest shifted to a Cliﬀord algebra of higher order then we would need those same formulæ plus some of their higher order companions ,whereas if we shifted our interest to a Cliﬀord algebra of lower order we would ﬁnd thatwe had already in hand all the material we need ...though some terms would automatically blink oﬀ because eee k1...kp+q=000if any index is repeated and if the indices range on a reduced set such repeats become unavoidable. An element of “universality” ( n-independence) attaches therefore to formulæ of type (104). Note that the “symmetrized” notation does notlend itself well to the problem in hand,for the onset of ffftermisn-dependent. I describe an approach to the construction of such formulæ. lowest level analysis We have eee1eee2=eee1eee2 −eee2eee1=eee1eee2−2g12I Add and multiply by1 2!to obtaineee12=eee1eee2−g12I,the general implication being that eeei·eeej=eeeij+gijI (105) which,by the way,follows directly from eeeieeei=1 2(eeeieeej−eeejeeei)+1 2(eeeieeej+eeejeeei) and works even when i=j. As a check on the accuracy of (105.1) we have 1 2!/summationdisplay signed permutationseeei·eeej=1 2!(eeeij−eeeji)+1 2!(gij−gji)I =eeeij:g-terms cancel by symmetry next higher level Our objective will be to develop eeei·eeejkandeeeij·eeek. To that end,we look to each of the terms that contribute to eeeijkand use the “ﬂip principle” eeemeeen=−eeeneeem+2gmnIto bring each eee·eee·eeeto “dictionary order.” This will supply the canonical development of eeeieeejeeek,which we will use to assemble the formulæ of interest. Turning to the details,we have eee1eee2eee3=eee1eee2eee3 −eee1eee3eee2=eee1eee2eee3−2g23eee1 eee2eee3eee1=eee1eee2eee3+2g13eee2−2g12eee3 −eee2eee1eee3=eee1eee2eee3−2g12eee3 eee3eee1eee2=eee1eee2eee3+2g13eee2−2g23eee1 −eee3eee2eee1=eee1eee2eee3−2g23eee1+2g13eee2−2g12eee3 Third order theory 33 Adding those results together and dividing by 6, we have eee123=eee1eee2eee3−g12eee3+g31eee2−g23eee1 or eee1eee2eee3=eee123+g12eee3−g31eee2+g23eee1 which in the general case reads eeeieeejeeek=eeeijk+gijeeek−gkieeej+gjkeeei (106) and gives eeei·eeejk=eeeijk−gikeeej+gijeeek (107.1) eeejk·eeei=eeejki+gikeeej−gijeeek (107.2) Quickcalculation conﬁrms that these formulæ remain valid even in the cases i=jandi=k. And as a further checkon the accuracy of (106) we ﬁnd (with assistance from Mathematica ) that 1 3!/summationdisplay signed permutationseeeieeejeeek=eeeijk+(g-terms that cancel) If (as in C2[g•g]) our indices ranged on/braceleftbig 1,2/bracerightbig then eeej·eeejkandeeejk·eeejwould be essentially the only cases of interest, and we would “by descent”have eee j·eeejk=−gjkeeej+gjjeeek eeejk·eeej=+gjkeeej−gjjeeek next higher level To obtain the canonical development of eeeieeejeeekeeel we use eeeijkl=1 4(eeeijkeeel−eeelijeeek+eeeklieeej−eeelijeeek) in combination with results already in hand. Looking to the details: hitting (106) with eeelon the right we get eeeijkeeel=eeeieeejeeekeeel−gijeeekeeel+gkieeejeeel−gjkeeeieeel whence eeeijkl=1 4/summationdisplay signed cyclic permutationseeeijkeeel =1 4/braceleftbig eeeieeejeeekeeel−eeejeeekeeeleeei+eeekeeeleeeieeej−eeeleeeieeejeeek/bracerightbig +1 4/summationdisplay signed cyclic permutations(−gijeeekeeel+gkieeejeeel−gjkeeeieeel) But eeeieeejeeekeeel=eeeieeejeeekeeel −eeejeeekeeeleeei=eeeieeejeeekeeel−2gileeejeeek+2gikeeejeeel−2gijeeekeeel eeekeeeleeeieeej=eeeieeejeeekeeel+2gileeekeeej−2gikeeeleeej+2gjleeeieeek−2gjkeeeieeel −eeeleeeieeejeeek=eeeieeejeeekeeel−2gileeejeeek+2gjleeeieeek−2gkleeeieeej 34 Transformational principles derived from Clifford algebras Enlisting the assistance of Mathematica to pull these results together, we ﬁnd eeeieeejeeekeeel=eeeijkl+(gijeeekl+gkleeeij)−(gikeeejl+gjleeeik)+( gileeejk+gjkeeeil) +(gijgkl−gikgjl+gilgjk)I (108) To conﬁrm the accuracy of that statement it is suﬃcient to establish that the adjacent transpositional properties22of the expression on the right duplicate those of the expression on the left. For example, we have eeejeeeieeekeeel=−eeeieeejeeekeeel+2gijeeekeeel =−eeeieeejeeekeeel+2gij(eeekl+gklI) which is readily seen to be mimiced by the expression on the rightside of (107): the point to notice is that right side of (108) =[ ij-symmetric term] + [ ij-antisymmetric term] with [ij-symmetric term] = gij(eeekl+gklI) As an additional checkon the accuracy of (108) we have23 /summationdisplay signed permutations/braceleftBigg (gijeeekl+gkleeeij)−(gikeeejl+gjleeeik)+( gileeejk+gjkeeeil) +(gijgkl−gikgjl+gilgjk)I/bracerightBigg =0 Equation (108) puts us in position canonical representations of the products eeei·eeejkl,eeeij·eeekland eeejkl·eeei. We might now, with patient labor, use (108) to construct—in a moment, almost eﬀortlessly, willconstruct—these product formulæ: eeei·eeejkl=eeeijkl+gijeeekl+gikeeelj+gileeejk (109.1) eeejkl·eeei=eeejkli+gijeeekl+gikeeelj+gileeejk (109.2) eeeij·eeekl=eeeijkl−(gikeeejl+gjleeeik)+( gjkeeeil+gileeejk) −(gikgjl−gjkgil)I (109.3) 22I assume my reader to be familiar with the fact that every permutation can be expressed as the product of (a characteristically even/odd number of)transpositions of adjacent symbols : see J. S. Lomont, Applications of Finite Groups (), page 260 or W. Burnside’s classic Theory of Groups of Finite Order (),§11. 23Compare the “cancellations of g-terms” that were encountered on pages 32 & 33. I used resources discovered within Mathematica ’s “Combinatorica” package to carry out the calculation, but by an improvised procedure so clumsythat the worktookme nearly an hour, and that would place the case of nexthigher order (entails 4! →5!) well beyond the limits of my patience. The time has come to acquire some computational technique! Third order theory 35 DIGRESSION: Some Mathematica Technique. The properties —except for the transformation properties (weight)—that weassociate with the Levi-Civita symbol /epsilon1 i1i2...inare reproduced inMathematica by the command Signature[ {i ,j ,...,k }], the action of which is illustrated below: Signature[ {1,2,3}]=+ 1 Signature[ {1,1,3}]=0 Signature[ {2,1,3}]=−1 The command is powerful enough to read subscripts, thus Signature[ {α1,α2,α3}]=+ 1 Signature[ {α1,α1,α3}]=0 Signature[ {α2,α1,α3}]=−1 This permits us to use subscripts to orchestrate sums of the sort in which the Levi-Civita symbol is a frequent participant: 2/summationdisplay i=12/summationdisplay j=1Signature[ {i, j}]F[αi,αj]=F[α1,α2]−F[α2,α1] 2/summationdisplay i=12/summationdisplay j=1Signature[ {i, j}]ei,j=e1,2−e2,1 We must, however, be prepared to workaround the fact that Mathematica ’s natural instinct is to assume commutivity: 2/summationdisplay i=12/summationdisplay j=1Signature[ {i, j}]eiej=0 /negationslash=e1e2−e2e1 Here I use the technique described above to reconstruct the deﬁnition of determinant: det/parenleftbigg g1,1g1,2 g2,1g2,2/parenrightbigg −2/summationdisplay i=12/summationdisplay j=1Signature[ {i, j}]g1,ig2,j=0 The commas are, by the way, critical, for in their absence Mathematica would multiply the subscripts, as demonstrated below: 2/summationdisplay i=12/summationdisplay j=1Signature[ {i, j}]gi,j=g1,2−g2,1 2/summationdisplay i=12/summationdisplay j=1Signature[ {i, j}]gij=g1·2−g2·1=0 36 Transformational principles derived from Clifford algebras Finally—in anticipation of things to come—I use the technique to reproduce the derivation of (107.1) from (106). Let the latteridentity eee ieeejeeek=eeeijk+gijeeek−gkieeej+gjkeeei be notated eeei·eeej1eeej2=F(i, j1,j2) where F(i, j, k)≡eeei,j,k+1 2(gi,j+gj,i)eeek −1 2(gk,i+gi,k)eeej+1 2(gj,k+gk,j)eeei has been spelled out in such a way as (in eﬀect) to inform Mathematica that gij=gji. Thus prepared, we write eeem·eeen1n2=1 2!2/summationdisplay i=12/summationdisplay j=1Signature[/braceleftbig i, j/bracerightbig ]F(m, n i,nj) and instantly recover(107). Note also that we are in position now to reestablish—this time without labor—the g-independence of 3/summationdisplay i=13/summationdisplay j=13/summationdisplay k=1Signature[/braceleftbig i, j, k/bracerightbig ]F(ni,nj,nk) To prepare for application of those techniques to the derivation of the product formulæ (109) we deﬁne F(i, j, k, l ) by making substitutions gmn/mapsto− →gm,n+gn,m 2,eeeij/mapsto− →ei,j,eeeijkl/mapsto− →ei,j,k,l into the expression that appears on the right side of (108). Then, to obtain (109.1), we evaluate 1 3!3/summationdisplay j=13/summationdisplay k=13/summationdisplay l=1Signature[/braceleftbig j, k, l/bracerightbig ]F(i, j, k, l ) and make notational adjustments 1 /mapsto→j,2/mapsto→k,3/mapsto→l. To obtain (109.2) we evaluate 1 3!3/summationdisplay j=13/summationdisplay k=13/summationdisplay l=1Signature[/braceleftbig j, k, l/bracerightbig ]F(j, k, l, i ) and proceed similarly. To obtain (109.3) we evaluate 1 2!2!2/summationdisplay i=12/summationdisplay j=14/summationdisplay k=34/summationdisplay l=3Signature[/braceleftbig i, j/bracerightbig ] Signature[/braceleftbig k, l/bracerightbig ]F(i, j, k, l ) and make notational adjustments 1 /mapsto→1, 2/mapsto→j,3/mapsto→k,4/mapsto→l. Third order theory 37 But if general multiplication (inversion, similarity transformation, etc.) within C3[g•g] is our objective then our workis not yet done: we must 1) develop the canonical representation of eeeieeejeeekeeeleeemand use that information (and the preceding techniques) to construct descriptions of •eeei·eeejklm(not actually needed until we come to C4[g•g]) •eeeij·eeeklm •eeeklm·eeeij •eeejklm·eeei(not actually needed until we come to C4[g•g]) 2) develop the canonical representation of eeeieeejeeekeeeleeemeeenand use that information ...to construct descriptions of •eeei·eeejklmn (not actually needed until we come to C5[g•g]) •eeeij·eeeklmn(not actually needed until we come to C4[g•g]) •eeeijk·eeelmn •eeeklmn·eeeij(not actually needed until we come to C4[g•g]) •eeejklmn ·eeei(not actually needed until we come to C5[g•g]) Note that, because of the recursive design of the theory (calculations in any speciﬁed order make essential use of lower order results), those problems mustbe approached in the order stated. next higher level We proceed in imitation of the pattern established on page 33. Noting that the cyclic permutations of/braceleftbig i, j, k, l, m/bracerightbig are all even, we have eeeijklm=1 5(eeeijkleeem+eeemijkeeel+eeelmijeeek+eeeklmieeej+eeejklmeeei) eeeijkleeem=eeeieeejeeekeeeleeem−(gijeeekl+gkleeeij)eeem +(gikeeejl+gjleeeik)eeem −(gileeejk+gjkeeeil)eeem −(gijgkl−gikgjl+gilgjk)eeemby (108) giving eeeijklm=1 5/braceleftbig/braceleftbig eeeieeejeeekeeeleeem+eeemeeeieeejeeekeeel+eeeleeemeeeieeejeeek+eeekeeeleeemeeeieeej+eeejeeekeeeleeemeeei/bracerightbig/bracerightbig +1 5/summationdisplay cyclic permutations/braceleftbig −(gijeeekl+gkleeeij)eeem +(gikeeejl+gjleeeik)eeem −(gileeejk+gjkeeeil)eeem −(gijgkl−gikgjl+gilgjk)eeem/bracerightbig A computation as tedious as it is elementary (it makes use only of the basic identity eeeieeej=−eeejeeei+2gijI) supplies 38 Transformational principles derived from Clifford algebras /braceleftbig/braceleftbig etc./bracerightbig/bracerightbig =5eeeieeejeeekeeeleeem+2(gimpppjkl−gjmpppikl+gkmpppijl−glmpppijk) +2 (gimpppljk−gjmppplik+gkmppplij−gilpppjkm+gjlpppikm−gklpppijm) +2 (gimpppklj−gilpppkmj+gikppplmj−gjmpppikl+gjlpppikm−gjkpppilm) +2 (gimpppjkl−gilpppjkm+gikpppjlm−gijpppklm) with pppijk≡eeeieeejeeek =eeeijk+gijeeek−gkieeej+gjkeeei We are in position now to consign all remaining computational tedium to Mathematica . To that end, we enter these deﬁnitions24 G[i,j]:=gi,j+gj,i 2 p[i,j,k]:=ei,j,k+G[i, j]ek−G[k, i]ej+G[j, k]ei q[i,j,k]:=ei,j,k+G[k, j]ei−G[k, i]ej A[i,j,k,l,m]:=5ei,j,k,l,m +2 (G[i, m,]p[j, k, l]−···)... +2 (··· − G[i, j]p[k, l, m ]) B[i,j,k,l,m]:=−(G[i, j]q[k, l, m ]+G[k, l]q[i, j, m ]) +(G[i, k]q[j, l, m ]+G[j, l]q[i, k, m ]) −(G[i, l]q[j, k, m ]+G[j, k]q[i, l, m]) +(G[i, j]G[k, l]−G[i, k]G[j, l]+G[i, l]G[i, k])em and askfor the evaluation of 1 5A[i, j, k, l, m ]+1 5/braceleftbig B[i, j, k, l, m ]+B[m, i, j, k, l ]/bracerightbig +B[l, m, i, j, k ]+B[k, l, m, i, j ]+B[j, k, l, m, i ] Mathematica promptly disgorges a ﬂood of output: our non-trivial assignment is to sort though it, make patterned sense of it. Thus am I brought at lengthto the canonical decomposition of ppp ijklm ≡eeeieeejeeekeeeleeemthat is presented as equation (110) on the next page. I will not comment explicitly on the signdistribution, except to remarkthat it appears on its face to be semi-intelligible. 25 24The construction of p[i,j,k] reﬂects the description (106) of pppijk≡eeeieeejeeek, q[i,j,k] reﬂects the description (17.2) of eeeijeeek. The deﬁnitions A[i,j,k,l,m] andB[i,j,k,l,m] are motivated by the design of the ﬁnal equation on the preceding page. 25We have reached a point at which typographic accuracy has become a major consideration, and where by-hand simpliﬁcation—even though actually doneon-screen—has become hazardous. Third order theory 39 pppijklm ≡eeeieeejeeekeeeleeem=eeeijklm+gijeeeklm −gikeeejlm +gileeejkm −gimeeejkl +gjkeeeilm −gjleeeikm +gjmeeeikl +gkleeeijm −gkmeeeijl +glmeeeijk +(gjkglm−gjlgkm+gjmgkl)eeei −(gklgmi−gkmgli+gkiglm)eeej +(glmgij−gligmj+gljgmi)eeek −(gmigjk−gmjgik+gmkgij)eeel +(gijgkl−gikgjl+gilgjk)eeem (110) On this basis we carefully enter into our Mathematica notebookthe deﬁnition F[i,j,k,l,m]:=ei,j,k,l,m +G[i, j]ek,l,m −G[i, k]ej,l,m ... +(G[i, j]G[k, l]−···+G[i, l]G[j, k])em As checks on the accuracy of (110) we observe, for example, that pppijkmm =pppijk·gmm while F(i, j, k, m, m )=/braceleftbig eeeijk+gijeeek−gkieeej+gjkeeei/bracerightbig ·gmm =pppijk·gmmby (106) —the interesting point here being that high-order formulæ can be used to generate/reproduce lower-order formulæ (the catch being that the latter areneeded to derive the former!). We also ﬁnd that all the g-terms disappear from 1 5!5/summationdisplay i,j,k,l,m =1Signature[ i, j, k, l, m ]F(i, j, k, l, m ) —leaving us with what is, in fact, precisely the deﬁnition ofeeeijklm. Further 40 Transformational principles derived from Clifford algebras checks on the accuracy of (110) and of the transcription of F(i, j, k, l, m )i n t o our notebookare provided by F(i, j, k, m, m )=F(i, j, m, m, k )=F(i, m, m, j, k )=F(m, m, i, j, k ) and F(i, j, j, k, k )=F(j, j, i, k, k )=F(j, j, k, k, i ) Satisﬁed that all is correct,26we ask Mathematica to construct 1 1!4!5/summationdisplay j,k,l,m =2Signature[ j, k, l, m]F(1,j ,k ,l ,m ) 1 2!3!2/summationdisplay i,j=15/summationdisplay k,l,m=3Signature[ i, j]Signature[ k, l, m]F(i, j, k, l, m ) 1 3!2!5/summationdisplay k,l,m=32/summationdisplay i,j=1Signature[ k, l, m]Signature[ i, j]F(k, l, m, i, j ) 1 4!1!5/summationdisplay j,k,l,m =2Signature[ j, k, l, m]F(j, k, l, m, 1) and, by notational adjustment of its output, obtain eeei·eeejklm=eeeijklm+(gijeeeklm−gikeeejlm+gileeejkm−gimeeejkl) (111 .1) eeeij·eeeklm=eeeijklm −(gikeeejlm−gileeejkm+gimeeejkl) +(gjkeeeilm−gjleeeikm+gjmeeeikl) −(gilgjm−gimgjl)eeek −(gimgjk−gikgjm)eeel −(gikgjl−gilgjk)eeem (111.2) eeeklm·eeeij=eeeijklm+(gikeeejlm−gileeejkm+gimeeejkl) −(gjkeeeilm−gjleeeikm+gjmeeeikl) −(gilgjm−gimgjl)eeek −(gimgjk−gikgjm)eeel −(gikgjl−gilgjk)eeem (111.3) eeejklm·eeei=eeeijklm −(gijeeeklm−gikeeejlm+gileeejkm−gimeeejkl) (111 .4) To test—if only weakly—the accuracy of the preceding formulæ we might look in the Euclidean case to such products as eee1·eee2345andeee2·eee2345. 26It is of critical importance that everything be precisely correct, for errors at any given order propagate to all higher orders. Third order theory 41 next higher level As was remarked already on page 37, we must develop the canonical representation of pppijklmn ≡eeeieeejeeekeeeleeemeeen(whence, in particular, of eeeijk·eeelmn) before we will be in position to workout the theory of C3[g•g]. We proceed from eeeijklmn =1 6(eeeijklm eeen−eeenijkleeem+eeemnijk eeel−eeelmnij eeek+eeeklmni eeej−eeejklmn eeei) eeeijklm eeen=eeeieeejeeekeeeleeemeeen−gijeeeklmeeen +gikeeejlmeeen −gileeejkmeeen +gimeeejkleeen −gjkeeeilmeeen +gjleeeikmeeen −gjmeeeikleeen −gkleeeijmeeen +gkmeeeijleeen −glmeeeijkeeen −(gjkglm−gjlgkm+gjmgkl)eeeieeen +(gklgmi−gkmgli+gkiglm)eeejeeen −(glmgij−gligmj+gljgmi)eeekeeen +(gmigjk−gmjgik+gmkgij)eeeleeen −(gijgkl−gikgjl+gilgjk)eeemeeen ≡pppijklmn −Bijklmn Introducing the abbreviation aijklmn ≡gijpppklmn pppklmn≡eeekeeeleeemeeendeveloped at (108) we ﬁnd by careful pencil-&-paper workthat pppijklmn −pppnijklm +pppmnijkl −ppplmnijk +pppklmnij −pppjklmni =6pppijklmn +2 (−ainjklm +ajniklm −aknijlm +alnijkm −amnijkl ) +2 ( ainmjkl −ajnmikl +aknmijl −alnmijk +aimjkln −ajmikln +akmijln −almijkn ) +2 (−ainlmjk +ajnlmik −aknlmij +aimljkn −ajmlikn +akmlijn −ailjkmn +ajlikmn −aklijmn ) +2 ( ainklmj −ajnklmi +aimkljn −ajmklin +ailkjmn −ajlkimn +aikjlmn −ajkilmn ) +2 (−ainjklm +aimjkln −ailjkmn +aikjlmn −aijklmn ) ≡6pppijklmn −Aijklmn 42 Transformational principles derived from Clifford algebras Assembly of those results gives eeeieeejeeekeeeleeemeeen=eeeijklmn +1 6Aijklmn +1 6/braceleftbig Bijklmn −Bnijklm +Bmnijkl −Blmnijk +Bklmnij −Bjklmni/bracerightbig (112) This equation describes pppijklmn ≡eeeieeejeeekeeeleeemeeenas a linear combination of eeeijklmn ,gijeeeklm·eeen,gijgkleeem·eeenand gijpppklmn-type terms. Using (109.2), (105) and (108) to describe the canonical representations of eeem·eeen,eeeklm·eeen andpppklmn, we carefully feed the right side of (112) into Mathematica (tookme the better part of an hour) and in a few seconds obtain an enormously (!) longstring of g ijeeeklmn,gijgkleeemnand gijgklgmnIterms (plus a solitary eeeijklmn ). Carefully exploiting the symmetry of gijand the total antisymmetry of eeeij, eeeijklto consolidate those terms, I at length (which is to say: after a long afternoon’s work) obtained eeeieeejeeekeeeleeemeeen=eeeijklmn +gijeeeklmn+eeeijgklmn+gijgklmnI −gikeeejlmn−eeeikgjlmn−gikgjlmnI +gileeejkmn+eeeilgjkmn+gilgjkmnI −gimeeejkln−eeeimgjkln−gimgjklnI +gineeejklm+eeeingjklm+gingjklmI +gjkeeeilmn+eeejkgilmn −gjleeeikmn−eeejlgikmn +gjmeeeikln+eeejmgikln −gjneeeiklm−eeejngiklm +gkleeeijmn+eeeklgijmn −gkmeeeijln−eeekmgijln +gkneeeijlm+eeekngijlm +glmeeeijkn+eeelmgijkn −glneeeijkm−eeelngijkm +gmneeeijkl+eeemngijkl ≡F(i, j, k, l, m, n ) (113) with gijmn≡gijgmn−gimgjn+gingjm (114) We note that the signs are precisely those that would result if subscripts were introduced into /epsilon1......and then brought to standard ijklmn order. Informing Third order theory 43 Mathematica of the deﬁnition of F(i,j,k,l,m,n ), we ﬁrst construct27 1 6!6/summationdisplay i,j,k,l,m,n =1Signature[ {i,j,k,l,m,n }]F(i,j,k,l,m,n ) and obtain a string of the 6! = 720 signed permutations of e1,2,3,4,5,6from which all reference to gijhas vanished—leaving us (compare page 39) with what is, in fact, precisely the deﬁnition ofeeeijklmn . This I take to be strong evidence that (113) is correct, and has been accurately transcribed into our Mathematica notebook. Further evidence is provided—here as on pages 39 & 40—by veriﬁedstatements of the types F(i,j,k,l,m,m )=F(m,m,i,j,k,l ),etc. F(i,j,m,m,n,n )=(eee ij+gijI)gmmgnn We are in position now to construct canonical developments of all ﬁve of the sixth-order products listed on page 37. I will concern myself, however, onlywith the product eee ijk·eeelmnthat is directly relevant to the theory of C3[g•g]. From the reported value of 1 3!3!3/summationdisplay i,j,k=16/summationdisplay l,m,n=4Signature[ {i,j,k}]Signature[ {i,j,k}]F(i,j,k,l,m,n ) we extract eeeijk·eeelmn=eeeijklmn +gileeejkmn−eeeil(gjmgkn−gjngkm) −gimeeejkln+eeeim(gjlgkn−gjngkl) +gineeejklm−eeein(gjlgkm−gjmgkl) −gjleeeikmn+eeejl(gimgkn−gingkm) +gjmeeeikln−eeejm(gilgkn−gingkl) −gjneeeiklm+eeejn(gilgkm−gimgkl) +gkleeeijmn−eeekl(gimgjn−gingjm) −gkmeeeijln+eeekm(gilgjn−gingjl) +gkneeeijlm−eeekn(gilgjm−gimgjl)+gikjlmn I(115.1) wheregijklmn =1 8(eight permuted copies of six terms) and can, we notice, be described gijklmn =−det gilgimgin gjlgjmgjn gklgkmgkn  (116) 27No small assignnment, this: it took Mathematica 5 , running at 1.6 GHz on my PowerMac G5, 145.34 seconds to accomplish the feat, and required 18.3MB of memory. 44 Transformational principles derived from Clifford algebras Similar calculations supply eeeij·eeeklmn eeeklmn·eeeij/bracerightbigg =eeeijklmn ∓gikeeejlmn±gileeejkmn∓gimeeejkln±gineeejklm ±gjkeeeilmn∓gjleeeikmn±gjmeeeikln∓gjneeeiklm −(gimgjn−gingjm)eeekl +(gilgjn−gingjl)eeekm −(gilgjm−gimgjl)eeekn −(gikgjn−gingjk)eeelm +(gikgjm−gimgjk)eeeln −(gikgjl−gilgjk)eeemn (115.2) We are inspired by the point remarked at (116) to observe that if we proceed from  gilgimgin gjlgjmgjn gklgkmgkn  to the associated matrix of cofactors (or “signed minors”)  GilGimGin GjlGjmGjn GklGkmGkn ≡ +/vextendsingle/vextendsingle/vextendsingle/vextendsingleg jmgjn gkmgkn/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/vextendsingle/vextendsingle/vextendsingle/vextendsingleg jlgjn gklgkn/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingleg jlgjm gklgkm/vextendsingle/vextendsingle/vextendsingle/vextendsingle −/vextendsingle/vextendsingle/vextendsingle/vextendsingleg imgin gkmgkn/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingleg ilgin gklgkn/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/vextendsingle/vextendsingle/vextendsingle/vextendsingleg ilgim gklgkm/vextendsingle/vextendsingle/vextendsingle/vextendsingle +/vextendsingle/vextendsingle/vextendsingle/vextendsingleg imgin gjmgjn/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/vextendsingle/vextendsingle/vextendsingle/vextendsingleg ilgin gjlgjn/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingleg ilgim gjlgjm/vextendsingle/vextendsingle/vextendsingle/vextendsingle  then (115.1) assumes the more orderly form eee ijk·eeelmn=eeeijklmn +gileeejkmn−eeeilGil(117.1) −gimeeejkln−eeeimGim +gineeejklm−eeeinGin −gjleeeikmn−eeejlGjl +gjmeeeikln−eeejmGjm −gjneeeiklm−eeejnGjn +gkleeeijmn−eeeklGkl −gkmeeeijln−eeekmGkm +gkneeeijlm−eeeknGkn−det gilgimgin gjlgjmgjn gklgkmgkn I and that similar notational simpliﬁcations can be brought to (115.2). Look in this light back to the description (109.3) of eeeij·eeekl, which if we proceed Third order theory 45 /parenleftbigg gikgil gjkgjl/parenrightbigg /mapsto− →matrix of cofactors/parenleftbigg GikGil GjkGjl/parenrightbigg =/parenleftbigg +gjl−gjk −gil+gik/parenrightbigg can be cast into the form eeeij·eeekl=eeeijkl−eeeikGik(117.2) −eeeilGil −eeejkGjk −eeejlGjl−det/parenleftbigg gikgil gjkgjl/parenrightbigg I As a check on the accuracy of (115) we verify that (non-obviously) the expression on the right possesses both the/braceleftbig i,j,k/bracerightbig -antisymmetry and the/braceleftbig l,m,n/bracerightbig -antisymmetry that are manifest on the left. InC3[g•g] all indices range on/braceleftbig 1,2,3/bracerightbig . It follows in that instance that eeeijk·eeelmn=±eee123·eee123if it does not vanish And from (115) it follows by quick calculation that eee123·eee123=−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleg 11g12g13 g21g22g23 g31g32g33/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleI (118.1) Turning for purposes of comparison to C 2[g•g], where all indices range on/braceleftbig 1,2/bracerightbig and eeeij·eeekl=±eee12·eee12if it does not vanish Working from (109.3/117.2) we have eee12·eee12=eee1212−(g11eee22+g22eee11)+(g21eee12+g12eee21)−(g11g22−g12g21)I =−/vextendsingle/vextendsingle/vextendsingle/vextendsingleg11g12 g21g22/vextendsingle/vextendsingle/vextendsingle/vextendsingleI (118.2) Equations (118) generalize statements that in the Euclidean case (where both determinants become unity) are obvious: eee1eee2eee3·eee1eee2eee3=(−)3eee1eee1eee2eee2eee3eee3=−I eee1eee2·eee1eee2=(−)1eee1eee1eee2eee2=−I Our problem now is to make digestable sense of the information we have worked so hard to obtain. To that end I have entered (105), (107), (109) and(111) into Mathematica as functional deﬁnitions, writing oneone[i ,j]:=ei,j+gi,j onetwo[i ,j,k]:=ei,j,k−gi,kej+gi,jek twoone[j ,k,i]:=ei,j,k+gi,kej−gi,jek 46 Transformational principles derived from Clifford algebras etc. Working ﬁrst within C2[g•g], we write ccc≡sI+v1eee1+v2eee2+peee12 CCC≡SI+V1eee1+V2eee2+Peee12 and, making free use of such little facts as eee2,1=−eee1,2andeee1,1,2=000, obtain a result that can be expressed ccc·CCC=I/bracketleftbig sS+(g11v1V1+g12v1V2+g21v2V1+g22v2V2)−(g11g22−g12g21)pP/bracketrightbig +eee1/bracketleftbig sV1+v1S−(v1g12+v2g22)P+p(V1g12+V2g22)/bracketrightbig +eee2/bracketleftbig sV2+v2S+(v1g11+v2g21)P−p(V1g11+V2g21)/bracketrightbig +eee12/bracketleftbig sP+v1V2−v2V1+pS/bracketrightbig Here we have recovered precisely the multiplication formula that was presented as (75.1) on page 18 (and in notationally compacted form as (75.2) on page20)...which is gratifying ...and, as will soon emerge, useful in a surprising connection. Turning now at last to C 3[g•g], we discover that everything hinges upon how we elect to display the Cliﬀord numbers in question. Suppose, for example, we were to yield to the natural temptation to write ccc≡sI+v1eee1+v2eee2+v3eee3+a1eee23+a2eee31+a3eee12+peee123 CCC≡SI+V1eee1+V2eee2+V3eee3+A1eee23+A2eee31+A3eee12+Peee123 We would confront then a fairly formidable computational problem: the 42 terms that in C3[g•g] entered into the development of ccc·CCChave become now 82 terms, and some if those are fairly complicated.28There is, I claim, a better way, but to describe it I must back up a bit: Look by way of orientation to the case Cn[g•gEuclidean ], wherein eeeijbecomes eeeieeej=/braceleftbigg −eeejeeei:i/negationslash=j I :i=j Look more particularly to a property of the element fff≡eee1,2,...,n=eee1eee2···eeen. Clearly eeei1eeei2···eeeipfff=[ (−)n−1]pfffeeei1eeei2···eeeip:i1<i2<···<ip&p/lessorequalslantn =/braceleftbigg(−)pfffeeei1eeei2···eeeip:neven fffeeei1eeei2···eeeip:nodd —the implication being that if nis odd then fffcommutes with everything :fff has joined Ias an element of the “center” of Codd[g•gEuclidean ], the general element 28In what is for me the case C12[g•g] of ultimate interest those would have expanded to a total of [212]2=1 6,777,216verycomplicated terms! Third order theory 47 of which can be written xI+yfff. Easily, fff2=−I, so the center of such a Cliﬀord algebra provides an abstract copy of the ﬁeld of complex numbers. Those properties of Codd[g•gEuclidean ] can be obtained as specialized instances of some properties of Codd[g•g]—properties that I presently prepared to discuss only as they become manifest within C3[g•g]. Let fff≡eee123 and understand that in non-Euclidean cases fffmustnotbe confused with eee1eee2eee3. We are informed by Mathematica (who we supplied with all relevant information on the preceding page) that fffeeei−eeeifff=2eee123i =000in all cases: i∈/braceleftbig 1,2,3/bracerightbig (119.1) fffeeeij−eeeijfff=2 (gi1eeej23−gj1eeei23)+2 (gi2eeej31−gj2eeei31)+2 (gi3eeej12−gj3eeei12) =000in all cases: i,j∈/braceleftbig 1,2,3/bracerightbig (119.2) and that fff2=−gIwithg≡/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleg 11g12g13 g21g22g23 g31g32g33/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(120) Now a trick. Take the basic elements of C 3[g•g]t ob e /braceleftbig I,eee1,eee2,eee12,fff,fffeee1,fffeee2,fffeee12/bracerightbig (121) —noting that while in the Euclidean case C3[g•gEuclidean ] fff≡eee123 fffeee1=eee123eee1 fffeee2=eee123eee2 fffeee12=eee123eee12becomes eee1eee2eee3 becomes eee2eee3 becomes −eee1eee3=eee3eee1 becomes −eee3  (122) three of the last four members of the proposed basic element set are in the general case somewhat goofy ( notthe sort of thing we would have plucked from thin air): Mathematica supplies fffeee 1=g11eee23+g12eee31+g13eee12 fffeee2=g21eee23+g22eee31+g23eee12 fffeee12=−(g12g23−g13g22)eee1 −(g13g21−g11g23)eee2 −(g11g22−g12g21)eee3  (123) from which, it will be noted, we can recover the Euclidean statements (122) as specialized consequences. 48 Transformational principles derived from Clifford algebras The point of the trick—of the seemingly unnatural element-selection (120) —is that it permits us portray C3[g•g] as a kind of “complex extension” of C2[g•g] C3[g•g]=C2[g•g]+fffC2[g•g] (124) and to extract the rule for multiplying within C3[g•g] from the much simpler rule (75.2) that describes multiplication within C2[g•g].29To see how that works, and where it leads, let aaaandbbb,AAAandBBBbe arbitrary elements of C2[g•g], and from them form ccc≡aaa+fffbbbandCCC≡AAA+fffBBB: elements of C3[g•g] (125) Then, drawing extensively upon (119) and (120), ccc·CCC=(aaaAAA+fff2bbbBBB)+fff(aaaBBB+bbbAAA) =(aaaAAA−gbbbBBB)+fff(aaaBBB+bbbAAA) To assign speciﬁc meaning to the expression on the right we have only to work out four products within C2[g•g]. Some valuable conclusions are fairly immediate. Suppose, for example, we were to setCCC≡aaa−fffbbb (126.1) We would then have ccc·CCC=(aaaaaa+gbbbbbb)−fff(aaabbb−bbbaaa) (126 .2) which, however, puts us within sight of a formula for ccc–1if and only if aaaand bbbcommute ( aaabbb−bbbaaa=000), which is a severe restriction. To escape the force of this diﬃculty we note that the following elements of C3[g•g] PPP+≡1 2(I+ig−1 2fff) and PPP−≡1 2(I−ig−1 2fff) (127) 29It should be noticed that the g•gon the right side of (124) is 2 ×2, while the g•gon the left is 3 ×3. The metric components  g13 g23 g31g32g33  are, however, not actually missing on the right: they are, according to (123), sequestered in the deﬁnitions offffeee1,fffeee2andfffeee12. Third order theory 49 mimic the properties of a complete set of orthogonal projection operators :30 PPP++PPP−=I (128.1) PPP+·PPP−=PPP−·PPP+=000 (128.2) PPP2 +=PPP+andPPP2 −=PPP− (128.3) Moreover, each commutes with every element of C3[g•g]: PPP±ccc=cccPPP±: allcccinC3[g•g] (128 .4) It follows that every such ccccan be presented as a sum of orthogonal components : ccc=ccc++ccc−:ccc+·ccc−=000withccc±≡PPP±ccc (129) Fundamental to the inversion problem within C3[g•g] is the observation that so, in particular, can Ibe thus decomposed: I=I++I−:I+·I−=000with I±≡PPP±I (130) We are in position now to write ccc·CCC=(ccc++ccc−)·(CCC++CCC−)=ccc+·CCC++ccc−·CCC− (131) The inversion of cccwould be accomplished if we could arrange to have ccc+·CCC+=I+ andccc−·CCC−=I−. Drawing nowupon (this slight adjustment ccc≡aaa+g−1 2fffbbb (132) of) the decomposition introduced on page 47 we ﬁnd that31 ccc+=(aaa−ibbb)I+ ccc−=(aaa+ibbb)I−/bracerightBigg (133) 30In one writes PPP≡α(I+βfff) and requires PPP2=PPPone ﬁnds that α=1 2and β=±ig−1 2are forced. P. K. Ra˘ sevski˘ ı—in “The theory of spinors,” American Mathematical Society Translations, Series 2, Volume 6 (1957)—has providedan elaborate account of the theory of Cliﬀord algebras in the Euclidean case.In the following discussion I enlarge upon material to be found in his §5. 31Use PPP+·g−1 2fff=−iPPP+ PPP−·g−1 2fff=+iPPP−/bracerightBigg (137) which follow quickly from the projectivity statements PPP+·1 2(I+ig−1 2fff)=PPP+andPPP−·1 2(I−ig−1 2fff)=PPP− It should be noted also that PPP+andI+are diﬀerent names for the same thing (dittoPPP−andI−). 50 Transformational principles derived from Clifford algebras But (aaa−ibbb) and (aaa+ibbb) live in C2[g•g], where the inversion problem has already been solved. We are in position, therefore, to construct ccc–1≡(aaa−ibbb)–1I++(aaa+ibbb)–1I− (134) and to observe that ccc·ccc–1=I++I−=I. I ﬁnd it easy to resist any temptation to pursue the general-metric aspect of this discussion to its ﬁner details.32The lesson, in general terms, is that C3[g•g]=C/prime 2[g•g]⊕C/prime/prime 2[g•g] with C/prime 2[g•g]⊥C/prime/prime 2[g•g] (135) If matrices of the form/parenleftbigg •• ••/parenrightbigg serve to represent elements of C2[g•g] then we expect to have •• 00 •• 00 00 •• 00 ••  (136) in representation of C3[g•g]. And we expect the transformation theory latent in C3[g•g] to be relatively uninteresting—to consist simply of duplex copies of the theory latent already in C2[g•g]. In the preceding discussion I surpressed ﬁne details in order to expose most clearly the essential drift of the idea, but did so at cost: the discussion tookplace at such an abstract level that it is diﬃcult to gain a vivid sense of whatit was we actually accomplished. To remedy this defect I propose to revert nowto the Euclidean metric, where everything is especially simple. Let aaa=a 0I+a1eee1+a2eee2+a3eee1eee2 bbb=b0I+b1eee1+b2eee2+b3eee1eee2/bracerightBigg (138) and with the aid of fff=eee1eee2eee3 (139) construct ccc=aaa+fffbbb =(a0I+a1eee1+a2eee2+a3eee1eee2)+(b0eee1eee2eee3+b1eee2eee3+b2eee3eee1−b3eee3) UsePPP±≡1 2(I±ieee1eee2eee3)≡I±to construct ccc+=PPP±ccc =1 2/braceleftbig (a0−ib0)I+(a1−ib1)eee1+(a2−ib2)eee2+(a3−ib3)eee1eee2/bracerightbig +i1 2/braceleftbig (a0−ib0)eee1eee2eee3+(a1−ib1)eee2eee3+(a2−ib2)eee3eee1+(a3−ib3)eee3/bracerightbig =(aaa−ibbb)I+ (140.1) ccc−=(aaa+ibbb)I− (140.2) 32Were we to do so we would, in particular, want to make clear where the “border elements29of the 3 ×3 metric matrix” have come ﬁnally to rest. Third order theory 51 in terms of which we have ccc=ccc++ccc− =(aaa−ibbb)I++(aaa+ibbb)I− =aaa(I++I−)+bbb(−iI++iI−) =aaa(I++I−)+bbbfff(I++I−) =aaa+fffbbb The claim is that ccc–1can be described ccc–1=(aaa−ibbb)–1I++(aaa+ibbb)–1I− (141) where—as was established already on page 5— (aaa+ibbb)–1=(a0+ib0)I−(a1+ib1)eee1−(a2+ib2)eee2−(a3+ib3)eee1eee2 (a0+ib0)2−(a1+ib1)2−(a2+ib2)2+(a3+ib3)2 (aaa−ibbb)–1= result of obvious adjustment: i→−i I propose now to develop a matrix representation of the algebra described on the preceding page, but to gain the advantage of expository eﬃciency mustﬁrst digress to summarize the essential properties of the so-called Kronecker product .33 The “Kronecker product” (sometimes called the “direct product”) of •anm×nmatrix Aonto •ap×qmatrix B is themp×nqmatrix deﬁned34 A⊗B≡/bardblaijB/bardbl (142) Manipulation of expressions involving Kronecker products is accomplished by appeal to general statements such as the following: k(A⊗B)=(kA)⊗B=A⊗(kB) (143 .1) (A+B)⊗C=A⊗C+B⊗C A⊗(B+C)=A⊗B+A⊗C/bracerightBigg (143.2) A⊗(B⊗C)=(A⊗B)⊗C≡A⊗B⊗C (143.3) 33The following material was lifted directly from Chapter 1 page 24 of my advanced quantum topics (). 34The alternative deﬁnition A⊗B≡/bardblAbij/bardblgives rise to a “mirror image” of the standard theory. Good discussions can be found in E. P. Wigner, Group Theory and its Application to the Quantum Theory of Atomic Spectra (), Chapter 2; P. Lancaster, Theory of Matrices (),§8.2; Richard Bellman, Introduction to Matrix Analysis (2ndedition), Chapter 12, §§5–13. 52 Transformational principles derived from Clifford algebras (A⊗B)T=AT⊗BT(143.4) tr(A⊗B)=t r A·trB (143.5) —all of which are valid except when meaningless.35Less obviously (but often very usefully) (A⊗B)(C⊗D)=AC⊗BDif/braceleftBigAandCarem×m BandDaren×n(143.6) from which one can extract36 A⊗B=(A⊗In)(Im⊗B) (143 .7) det(A⊗B) = (det A)n(detB)m(143.8) (A⊗B)–1=A–1⊗B–1(143.9) Here I have used Imto designate the m×midentity matrix; when the dimension is obvious from the context I will, in the future, allow myself to omit thesubscript. The identities (143) are proven in each case by direct computation,and their great power will soon become evident. Mathematica can be enlisted to perform computations in this area (and can, in particular, be used to demon-strate the accuracy of (143)), but the procedure is a little fussy. If AandBare presented as lists of lists then the command Outer[Times, A, B]//MatrixForm permits one to inspect the design of A⊗B: EXAMPLE : Construct /parenleftbigg ab cd/parenrightbigg and/parenleftbigg pq rs/parenrightbigg and let the outputs be called AandB: A={{a,b},{c,d}}andB={{p,q},{r,s}} TheOuter command then produces  /parenleftbigg ap aq ar as/parenrightbigg/parenleftbigg bp bq br bs/parenrightbigg /parenleftbigg cp cq cr cs/parenrightbigg/parenleftbigg dp dq dr ds/parenrightbigg  35Recall that one cannot add matrices unless they are co-dimensional, and does not speak of the trace of a matrix unless it is square. 36See Lancaster32for the detailed arguments. Third order theory 53 Which is informative. But the interior braces—which are not easy to remove by hand—cause that object to behave improperly whensubjected to such basic matrix commands as Det[ ] ,Inverse[ ] , Transpose[ ] . I have devised a command that is free from that limitation—that yields output thatissusceptible to routine matrix manipulation—but it is complicated: 37 A⊗B:= Flatten[ Table[Flatten[Table[Part[Outer[Times,A,B],i,j,k], {j, Dimensions[A][ [2]]}]],{i, Dimensions[A][ [1]]}, {k, Dimensions[B][ [1]]}],1] It would be interestingto learn of a briefer command that serves equally well to create A⊗B= ap aq bp bq ar as br bs cp cq dp dq cr cs dr ds  To construct my matrix representation I pull from my intuitive hat the hunch that the projection numbers PPP+andPPP−might most naturally/usefully be represented by the complete pair of orthogonal projection matrices P+≡ 1000 010000000000 =/parenleftbigg 10 00/parenrightbigg ⊗I 2 P−≡ 0000 000000100001 =/parenleftbigg 00 01/parenrightbigg ⊗I 2 which, while they do not commute with every 4×4 matrix, do commute with every matrix of the form (136). From the matrix representation P+=1 2(I+iF) and P−=1 2(I−iF) of (127) we are brought to the forced conclusion that F=−i(P+−P−)= −i000 0−i00 00 + i0 000 + i  37To create [[,]]and ⊗at the Mathematica keyboard type ESC[[ESC,ESC]]ESC and ESCc*ESC, respectively. 54 Transformational principles derived from Clifford algebras We will borrow our representations of eee1andeee2from (45) on page 11: E1=II⊗ce·1= 0100 100000010010 : II≡I2 E2=II⊗ce·2= 0−i00 +i000 000 −i 00 + i0  —in which connection we note that F, as developed above, can be described F=ce·2ce·1⊗II Fromeee3=(eee1eee2)–1fff=eee2eee1fffit then follows that necessarily E3=E2E1F=(II⊗ce·2)(II⊗ce·1)(ce·2ce·1⊗II) =ce·1ce·2⊗ce·1ce·2= −1000 0+ 1 0 00 0 +1 0000 −1  It would at this point be very easy to demonstrate that the multiplicative properties of/braceleftbig I,E 1,E2,E3,E2E3,E3E1,E1E2,E1E2E3/bracerightbig precisely mimic those of/braceleftbig I,eee1,eee2,eee3,eee2eee3,eee3eee1,eee1eee2,eee1eee2eee3/bracerightbig ...but I won’t. Now take A≡a0I+a1E1+a2E2+a3E1E2 B≡b0I+b1E1+b2E2+b3E1E2 and from them construct C=A+FB =a0I+a1E1+a2E2+b3E3+b1E2E3+b2E3E1+a3E1E2+b0E1E2E3 Mathematica reports that Cis of the form (136), and more speciﬁcally that C=/parenleftbigg C+O OC−/parenrightbigg with C+=/parenleftbigg (a0+ia3)−i(b0+ib3)(a1−ia2)−i(b1−ib2) (a1+ia2)−i(b1+ib2)(a0−ia3)−i(b0−ib3)/parenrightbigg C−=/parenleftbigg (a0+ia3)+i(b0+ib3)(a1−ia2)+i(b1−ib2) (a1+ia2)+i(b1+ib2)(a0−ia3)+i(b0−ib3)/parenrightbigg Third order theory 55 Clearly C/prime·C/prime/prime=/parenleftbigg C/prime +·C/prime/prime + O OC/prime −·C/prime/prime −/parenrightbigg The problem of multiplyingtwo 4 ×4 matrices has been reduced to two instances of the problem of multiplyingtwo 2 ×2 matrices. It is clear also that C–1=/parenleftbigg C–1 +O OC–1 −/parenrightbigg and will exist if and only if detC= det( C+) det(C−)/negationslash=0 These sweet results are made even sweeter by the observation that (see again equations (45) on page 9) the 2 ×2 matrices C±can be developed C+=(a0−ib0)II+(a1−ib1)ce·1+(a2−ib2)ce·2+(a3−ib3)ce·1ce·2 C−=(a0+ib0)II+(a1+ib1)ce·1+(a2+ib2)ce·2+(a3+ib3)ce·1ce·2 so we have ( Mathematica concurs) C–1 +=(a0−ib0)II−(a1−ib1)ce·1−(a2−ib2)ce·2−(a3−ib3)ce·1ce·2 detC+ C–1 −=(a0+ib0)II−(a1+ib1)ce·1−(a2+ib2)ce·2−(a3+ib3)ce·1ce·2 detC− with detC±=(a0∓ib0)2−(a1∓ib1)2−(a2∓ib2)2+(a3∓ib3)2 It follows that if ccc=a0I+a1eee1+a2eee2+b3eee3+b1eee2eee3+b2eee3eee1+a3eee1eee2+b0eee1eee2eee3 is real (in the sense that the a’s andb’s are real) then det Cis real, and given by detC=|(a0∓ib0)2−(a1∓ib1)2−(a2∓ib2)2+(a3∓ib3)2|2 Notice also that ↓ =(a0a0−a1a1−a2a2+a3a3)2in the special case bbb=000 All reference to eee3has disappeared: we have recovered not the “modulus” previously encountered in the theory of C2[g•gEuclidean ],38but its square. The real regular representation of C3[g•gEuclidean ] is 8-dimensional. The complex representation described above—extracted from the representationtheory of C 2[g•gEuclidean ]—is 4-dimensional and it seems clear (though I have 38See again equations (28) & (47) on pages 5 & 10. 56 Transformational principles derived from Clifford algebras not proven) that its dimension is least-possible . I exists in many variants. An alternative representation would result if, for example, we sent E1/mapsto→E/prime 1=E2 E2/mapsto→E/prime 2=E3 E3/mapsto→E/prime 3=E1 which is, in eﬀect, to proceed from a diﬀerent C2sub-algebra of C3: to assign to eee1the special role formerly assigned to eee3. Or we could assume that P±project onto some other/any orthogonal pair of planes in 4-space. Or we could subjectthe representation in hand to an arbitrary similarity transformation E 1/mapsto→E/prime 1=S–1E1S E2/mapsto→E/prime 2=S–1E2S E3/mapsto→E/prime 3=S–1E3S We anticipate—though it could conceivably turn out to be otherwise—that allleast-dimensional (or “irreducible”) representations of C3[g•gEuclidean ] are interrelated in this manner. And that the representation theory of C3[g•g]i s simply (or not so simply!) a fussed-up variant of the Euclidean theory. 6. Fourth order Clifford algebra with general metric. C4[g•g] is generated by objects/braceleftbig eee1,eee2,eee3,eee4/bracerightbig that satisfy relations of a sort eeeieeej+eeejeeei=2gijI (144) that are characteristic of Cliﬀord algebras in general, and that were ﬁrst encountered in these pages at (72). Physicists, or course, have relativisticinterest in the 4-metric g •gLorentz ≡/bardblgµν/bardbl≡ 1000 0−100 00 −10 000 −1  (145) It was Dirac who ﬁrst noticed the relevance of (144)—in that special instance— to relativistic quantum mechanics: he wrote39 γγγµγγγν+γγγνγγγµ=2gµνI (146) gµνtaken to be Lorentzian 39Without reference to Cliﬀord, whose name appears, so far as I am aware, nowhere in any of Dirac’s published work. Fourth order theory 57 ...pulled a matrix representation IΓ0≡ 1000 010000 −10 000 −1 IΓ 1≡ 000 −1 00 −10 01001000  IΓ 2≡ 000 i 00 −i0 0−i00 i000 IΓ3≡ 00 −10 000110000−100  out of thin air, 40and was led directly to the celebrated Dirac equation (), which can (in this or any other irreducible representation) be considered todescribe the motion of a 4-component wavefunction. The algebraic structure latent in (146) is known amongphysicists as the “Dirac alg ebra,” about whicha great deal has been written. InI was motivated to consider what becomes of the Dirac algebra when the metric is allowed to become arbitrary: I was, in short, motivated (byconsiderations that I today ﬁnd not very urgent!) to study C 4[g•g]. I will allow myself to borrow freely from that ancient material, which resides in what I willcall my geneva notebook . In view of the heavy demands which we found it necessary to make upon Mathematica in our eﬀort to develop the theory of C 3[g•g] I ﬁnd it remarkable that—workingonly with pen and (larg e sheets of) paper—I was able to make any progress at all toward a theory of C4[g•g]. But I was in fact able to carry that theory through to a kind of completion. I will beinterested in remindingmyself how that feat was accomplished. The way the theory plays out depends critically upon what one takes to comprise the “basis set,” in terms of which the elements cccofC 4are to be developed as linear combinations. In the Euclidean case—or, more generally, ifthe metric is diagonal g •g= g1000 0g200 00 g30 000 g4  40Not quite: the Pauli matrices (43) were already in circulation by , and in terms of them we have the highly structured statements IΓ0=/parenleftbigg IO O−I/parenrightbigg IΓ1=/parenleftbigg O−σσ1 σσ1O/parenrightbigg ,IΓ2=/parenleftbigg O−σσ2 σσ2O/parenrightbigg ,IΓ3=/parenleftbigg O−σσ3 σσ3O/parenrightbigg My conventions here conform to those adopted in Appendix C of David Griﬃths’ Introduction to Elementary Particles (), and are fairly standard. 58 Transformational principles derived from Clifford algebras —it might make eﬃcient good sense to work with (say) I eee1,eee2,eee2,eee4 eee1eee2,eee1eee3,eee1eee4,eee2eee3,eee2eee4,eee3eee4 eee2eee3eee4,eee3eee4eee1,eee4eee1eee2,eee1eee2eee3 eee1eee2eee3eee4 But if the metric has non-zero oﬀ-diagonal elements then permuting the elements of such products does not simply introduce occasional minus signs:permutation brings additive shifts into play, as in eee 2eee1=−eee1eee2+g12I.I tw a s to blunt the force of this circumstance that at (99) we adopted the strategy ofantisymmetrized averaging : •In place of eee1eee2adopt (compare (73) on page 18) eee12≡eee1eee2−eee2eee1 2!=eee1eee2−g12I •In place of eee1eee2eee3adopt eee123≡eee1eee2eee3−eee1eee3eee2+eee2eee3eee1−eee2eee1eee3+eee3eee1eee2−eee3eee2eee1 3! •In place of eee1eee2eee3eee4adopt eee1234=1 4!εijkleeeieeejeeekeeel We are by this strategy led to a set of basis elements that organizes itself into “binomial piles:” we have /parenleftbig4 0/parenrightbig = 1 term of type I /parenleftbig4 1/parenrightbig = 4 terms of type eeei/parenleftbig4 2/parenrightbig = 6 terms of type eeeij/parenleftbig4 3/parenrightbig = 4 terms of type eeeijk/parenleftbig4 1/parenrightbig = 1 term of type eeeijkl, call it fff=eee1234 In Dirac algebra (relativistic quantum applications) the eeei’s are usually denoted γγγ0,γγγ1,γγγ2,γγγ3. We established at (108) that eeeijkl=eeeieeejeeekeeel−(gijeeekl+gkleeeij)+(gikeeejl+gjleeeik)−(gileeejk+gjkeeeil) −(gijgkl−gikgjl+gilgjk)I wherei,j,k,lare necessarily distinct (a permutation of 1,2,3,4 in C4). It follows in particular that eee1234=eee1eee2eee3eee4−(g12eee34+g34eee12)+(g13eee24+g24eee13)−(g14eee23+g23eee14) −(g12g34−g13g24+g14g23)I ↓ =eee1eee2eee3eee4if and only if the metric is diagonal Fourth order theory 59 In Dirac algebra—where the metric isdiagonal— γγγ0γγγ1γγγ2γγγ3is usually called γγγ5 and is represented by IΓ0IΓ1IΓ2IΓ3≡IΓ5= 00i0 000 i i000 0i00  It is sometimes useful to notice that eee1234=1 4!εijkleeeieeejkl=1 4!εjklieeejkleeei=1 4!εijkleeeijeeekl The point, if not made obvious by a moment’s thought, can be established by Mathematica -assisted computation: construct 4/summationdisplay i=14/summationdisplay j=14/summationdisplay k=14/summationdisplay l=1Signature[ {i,j,k,l }]⋆ where⋆refers serially to the expressions that appear on the right sides of equations (109), page 34. Which brings me to the tricky case eeeijk. These elements are of four types: eee234,eee341,eee412andeee123. We would eliminate indicial clutter if we agreed to label each by the “missing index,” as was suggested already on page 30. Tothat end we might write fff 1≡−1 3!ε1ijkeeeijk fff2≡−1 3!ε2ijkeeeijk fff3≡−1 3!ε3ijkeeeijk fff4≡−1 3!ε4ijkeeeijk  (147.1) —the curious minus signs will be motivated in a moment—and from those objects construct fff 1≡g1mfffm fff2≡g2mfffm fff3≡g3mfffm fff4≡g4mfffm  (147.2) In the diagonal case we would then have fff 1=−g11eee234=−g11eee2eee3eee4 fff2=+g22eee341=+g22eee3eee4eee1 fff3=−g33eee412=−g33eee4eee1eee2 fff4=+g44eee123=+g44eee1eee2eee3 60 Transformational principles derived from Clifford algebras In the geneva notebook I chose, on the other hand, to introduce elements λλλi by the rule λλλi≡fffeeei (148) In diagonal cases this is readily seen41to amount to a mere change of notation λλλi=fffi (149) To see what happens in non-diagonal cases we look to (111.4) on page 40, which supplies λλλi=eee1234eeei=−(gi1eee234−gi2eee134+gi3eee124−gi4eee123) whence λλλi=gimλλλm=−(δi 1eee234−δi 2eee134+δi 3eee124−δi 4eee123)/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright |—precisely the object constructed on the right side of (147.1) The implication is that (149) holds universally —for allmetrics. We have accomplished two things: we have secured contact with the conventions adoptedin the geneva notebook , and we have added to the variety of ways in which the basic elements of C 4[g•g] can be described. TheGGGeneral element of C4[g•g] will be denoted GGG=SI+Vieeei+1 2Tijeeeij+Ajfffj+Pfff where (as was anticipated already on page 20, and for reasons not yet explained) •Sis intended to suggest “Scaler” •Vis intended to suggest “Vector” •Tis intended to suggest “antisymmetric Tensor” •Ais intended to suggest “Axial vector” (or “pseudo-vector”) •Pis intended to suggest “Pseudo-scaler” Let a second element HHHbe constructed similarly HHH=sI+vieeei+1 2tijeeeij+ajfffj+pfff 41A typical calculation runs λλλ1=eee1eee2eee3eee4eee1=(−)3eee1eee1eee2eee3eee4=−g11eee2eee3eee4=fff1 The point of the “curious minus signs” has now become clear: they serve to establish precise agreement with the conventions adopted in the geneva notebook . Fourth order theory 61 Our ability to develop the algebraic theory42ofC4[g•g] hinges on our ability to describe the product GGG·HHH. We have already in hand much of the requisite information ...for recall:43 •we obtained at (105) the canonical development of eeei·eeej •from the canonical development (106) of pppijkwe extracted descriptions of eeei·eeejkandeeejk·eeei •from the canonical development (108) of pppijklwe extracted descriptions of eeei·eeejkl,eeeij·eeeklandeeejkl·eeei •from the canonical development (110) of pppijklmwe extracted descriptions ofeeei·eeejklm,eeeij·eeeklm,eeeklm·eeeijandeeejklm·eeei •from the canonical development (113) of pppijklmn we extracted descriptions ofeeeij·eeeklmn,eeeijk·eeelmnandeeeklmn·eeeij This information places us in position to describe products GGG·HHHin cases where P=p= 0. What we still lack are descriptions of fffi·fff,fff·fffiandfff·fff. To construct those we might use our established procedures to ◦produce the canonical development of pppijklmnr and extract descriptions of eeeijk·eeelmnrandeeelmnr·eeeijk ◦produce the canonical development of pppijklmnrs and extract a description ofeeeijkl·eeemnrs. Such an approach would, however, be enormously laborious (so fast does n! grow), even with the assistance of Mathematica . I will refrain from attempting to pursue this route until I have learned how to get Mathematica to doallthe work. How, then, to proceed? My plan is ( i) to review how far we can go toward the description of GGG·HHHon the basis of what we already know, then ( ii) to bring into play the quite diﬀerent methods employed in the geneva notebook and (iii) check for consistency in the region where the two methods overlap. For computational purposes we abandon the fff-notation, writing GGG=SI+Vieeei+1 2Tijeeeij−1 3!Amεmijkeeeijk+P1 4!εijkleeeijkl HHH=sI+vieeei+1 2tijeeeij−1 3!amεmijkeeeijk+p1 4!εijkleeeijkl but will revert to fff-notation when statingour ﬁnal results. The products SI·HHH andGGG·sIare trivial, and contribute to GGG·HHHthe followingpopulation of terms: SsI+(Svi+sVi)eeei+1 2(Stij+sTij)eeeij+(Saj+sAj)fffj+(Sp+sP)fff From (105) we obtain Vivjeeeieeej=(Vmvm)I+Vivjeeeij 42As distinguished from (say) the irreducible representation theory, which is in many respects a separate problem. 43In the followingremarks I revert to the notation pppijk...n ≡eeeieeejeeek···eeen 62 Transformational principles derived from Clifford algebras where (as will henceforth be our casual practice) we have used gijto manipulate indices, writing Vm≡gmnVn. Resolving Vivjinto its symmetric/antisymmetric parts Vivj=1 2(Vivj+Vjvi)+1 2(Vivj−Vjvi) we note that—because eeeijis itself antisymmetric—only the antisymmetric part ofVivjsurvives the summation process: we therefore have Vivjeeeieeej=(Vmvm)I+1 2(Vivj−Vjvi)eeeij (150.1) Drawingnext upon (107) we obtain 1 2(Vitjkeeeieeejk+viTjkeeejkeeei)=(Vmtmi−vmTmi)eeei+1 2(Vktmn+vkTmn)eeekmn But from fffj=−1 3!εjpqreeepqr(see again (147.1) on page 59) it follows44that εjkmnfffj=−1 3!εjkmnεjpqreeepqr=−geeekmn so we have 1 2(Vitjkeeeieeejk+viTjkeeejkeeei) =(Vmtmi−vmTmi)eeei−1 2gεjlmn(Vltmn+vlTmn)fffj(150.2) These implications of (109) and (111) are relatively straightforward:45 −1 3!(Viameeeieeejkl+viAmeeejkleeei)εmjkl =1 2(Vman−vmAn)εmnijeeeij−1 3!(Viam−viAm)εmjkleeeijkl =1 2(Vman−vmAn)εmnijeeeij−(Vmam−vmAm)fff (150.3) 1 4!(Vipeeeieeejklm+viPeeejklmeeei)εjklm=1 3!(Vjp−vjP)εjklmeeeklm =−(Vjp−vjP)fffj (150.4) In consequence again of (109) we ﬁnd 1 4Tijtkleeeijeeekl=1 4Tijtkleeeijkl−Ti mtjmeeeij−1 2TmntmnI Buteeeijkl=1 gεijklfffso (dismissingthe irrelevant symmetric part of Timtjm)w e have =−1 2TmntmnI−1 2(Ti mtjm−Tj mtim)eeeij +1 4gεklmnTkltmnfff (150.5) 44I draw here upon properties of the Levi-Civita symbols and of the closely related “generalized Kronecker symbols” that are developed on pages 8 & 9 of“Electrodynamical applications of the exterior calculus” ( ). 45The only tricky point: because only four values are available to the indices, the expressions εmjkleeeijklvanish unless m=i(and each of the non-vanishing expressions comes in 3! ﬂavors). Fourth order theory 63 Revisiting(111) we ﬁnd −1 3!2(Tijaneeeijeeeklm+tijAneeeklmeeeij)εnklm =1 3!2(Tijan−tijAn)gikεnklmeeejlm+ 5 similar terms −1 3!2εnklm(Tijan−tijAn)gilgjmeeek+ 5 similar terms We write −1 gεpjlmfffpin place of eeejlm, draw upon the identity44 1 gεnklmεpjlm=2δnk pj≡2(δn pδk j−δn jδk p) and—workingvery carefully on a larg e piece of paper—notice that all “trace terms” (terms proportional to the Tmm) cancel: we are led at last to a result that can be written =−1 2εilmn(Altmn+alTmn)eeei+(Amtmj−amTmj)fffj (150.6) Drawingﬁnally upon (115.1) we sharpen our pencils, take another larg e piece of paper and—after much consolidation—obtain 1 3!3!Apaqεpijkεqlmneeeijkeeelmn=1 36Apaq/braceleftBig 9εipjkεiqmneeejkmn −18εpimnεqj mneeeij−6εpijkεq ijkI/bracerightBig But44 εipjkεiqmneeejkmn=g/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleg pqgpmgpn gjqgjmgjn gkqgkmgkn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleeee jkmn=000 for the same reason that gijeeeij••vanishes (antisymmetry kills symmetry). Drawingsimilarly upon εpimnεqj mn=2 !g/vextendsingle/vextendsingle/vextendsingle/vextendsingleg pqgpj giqgij/vextendsingle/vextendsingle/vextendsingle/vextendsingle and ε pijkεq ijk=3 !ggpq we are led straightforwardly to 1 3!3!Apaqεpijkεqlmneeeijkeeelmn=−gAnanI−1 2g(Aiaj−Ajai)eeeij (150.7) From equations (150) it now follows that if GGG=SI+Vieeei+1 2Tijeeeij+Ajfffj+Pfff HHH=sI+vieeei+1 2tijeeeij+ajfffj+pfff/bracerightBigg (151) 64 Transformational principles derived from Clifford algebras then GGG·HHH=I/braceleftBig Ss+Vmvm−1 2Tmntmn−gAnan/bracerightBig +eeei/braceleftBig (Svi+sVi)+(Vmtmi−vmTmi) −1 2εilmn(Altmn+alTmn)/bracerightBig +1 2eeeij/braceleftBig (Stij+sTij)+(Vivj−Vjvi) +1 2εijmn(Vman−Vnam) −(Ti mtjm−Tj mtim) −1 2εijmn(Amvn−Anvm) −g(Aiaj−Ajai) −1 2εijmn(Ptmn+pTmn)/bracerightBig +fffj/braceleftBig (Saj+sAj)−1 2gεjlmn(Vltmn+vlTmn) +(Amtmj−amTmj)+(Pvj−pVj)/bracerightBig +fff/braceleftBig Sp−Vmam+1 4gεklmnTkltmn+Anvn+Ps/bracerightBig +Ajpfffjfff+Pajffffffj+Ppffffff which is—so far as it goes (we are not yet in position to evaluate the red terms )— in precise agreement with the result reported on page 115 of the geneva notebook .46 The method used to obtain the preceding(still frag mentary) result may have some claim to conceptual elegance (it is, in any event, conceptuallystraightforward), but is —at every turn—computationally quite burdensome.The method employed in the geneva notebook is, on the other hand, quite inelegant, but is computationally so relatively eﬃcient that—working only withpen and (large sheets of) paper I was able in to carry the GGG·HHHproblem all the way to completion. Here—workingwith our present set of notationalconventions—I undertake to construct a sketch that “Geneva method,” whichI will use to obtain descriptions of the missingred terms . The basic plan of attack is familiar already from page 26, and might be symbolized  g 1000 0g200 00 g30 000 g4 /mapsto− → g11g12g13g14 g21g22g23g24 g31g32g33g34 g41g42g43g44  46See also page 11 of “Aspects of the theory of Cliﬀord algebras,” which are the notes for a seminar presented Marchand can be found in collected seminars –. Fourth order theory 65 We assume initially that the metric is diagonal , which is to say: that the generators satisfy eee/prime ieee/prime j+eee/prime jeee/prime i=2g/prime ijI:g/prime ij≡/braceleftbigggi:i=j 0: i/negationslash=j(152) Takingthe elements to be a familiar notational reﬁnement (pag e 59) of those listed on page 58 I eee/prime 1,eee/prime 2,eee/prime 3,eee/prime 4 eee/prime 12,eee/prime 13,eee/prime 14,eee/prime 23,eee/prime 24,eee/prime 34:eee/prime ij≡eee/prime ieee/prime j=/braceleftbigggiI:i=j −eee/prime jeee/prime i:i/negationslash=j fff/prime≡eee/prime 1eee/prime 2eee/prime 3eee/prime 4 fff/prime i≡fff/primeeee/prime i=  −g1eee/prime 2eee/prime 3eee/prime 4:i=1 +g2eee/prime 1eee/prime 3eee/prime 4:i=2 −g3eee/prime 1eee/prime 2eee/prime 4:i=3 +g4eee/prime 1eee/prime 2eee/prime 3:i=4 we write GGG=S/primeI+V/prime1eee/prime 1+V/prime2eee/prime 2+V/prime3eee/prime 3+V/prime4eee/prime 4 +T/prime12eee/prime 12+T/prime13eee/prime 13+T/prime14eee/prime 14+T/prime23eee/prime 23+T/prime24eee/prime 24+T/prime34eee/prime 34 +A/prime1fff/prime 1+A/prime2fff/prime 2+A/prime3fff/prime 3+A/prime4fff/prime 4 +P/primefff/prime HHH= same with lower case coeﬃcients and work out by hand all the 162= 256 terms that enter into the construction ofGGG·HHH.47We are led to a result that can be written 47This is fairly easy: there are only 52= 25 categories of terms, and 10 of those are trivial (contain Ias a factor). Patterns present within each category help one spot careless errors. Some typical calculations: T/prime12v/prime1eee/prime 12eee/prime 1=−T/prime12v/prime1eee/prime 1eee/prime 1eee/prime 2=−g1T/prime12v/prime1eee/prime 2=−v/prime 1T/prime12eee/prime 2 T/prime12v/prime3eee/prime 12eee/prime 3=T/prime12v/prime3eee/prime 1eee/prime 2eee/prime 3=( 1/g4)T/prime12v/prime3fff/prime 4 =T/prime12v/prime3fff/prime4 A/prime1v/prime1fff/prime 1eee/prime 1=−g1A/prime1v/prime1eee/prime 2eee/prime 3eee/prime 4eee/prime 1=+g1A/prime1v/prime1fff/prime A/prime1v/prime2fff/prime 1eee/prime 2=−g1A/prime1v/prime2eee/prime 2eee/prime 3eee/prime 4eee/prime 2=−g1g2A/prime1v/prime2eee/prime 34=−(g/g3g4)A/prime1v/prime2eee/prime 34 =−gA/prime1v/prime2eee/prime34 Hereg=g1g2g3g4is the determinant of the diagonal metric, and g–1 iare the diagonal elements of its inverse. Dangling giandg–1 ifactors have been absorbed into index lowering/raising procedures of the sort standard to tensor algebra. 66 Transformational principles derived from Clifford algebras GGG·HHH=I/braceleftBig Ss+Vmvm−1 2Tmntmn−gAnan+gPp/bracerightBig +eeei/braceleftBig (Svi+sVi)+(Vmtmi−vmTmi) −1 2εilmn(Altmn+alTmn)+g(Pai−pAi)/bracerightBig +1 2eeeij/braceleftBig (Stij+sTij)+(Vivj−Vjvi) +1 2εijmn(Vman−Vnam) −(Ti mtjm−Tj mtim) −1 2εijmn(Amvn−Anvm) −g(Aiaj−Ajai) −1 2εijmn(Ptmn+pTmn)/bracerightBig +fffj/braceleftBig (Saj+sAj)−1 2gεjlmn(Vltmn+vlTmn) +(Amtmj−amTmj)+(Pvj−pVj)/bracerightBig +fff/braceleftBig Sp−Vmam+1 4gεklmnTkltmn+Anvn+Ps/bracerightBig (153) where all coeﬃcients and Cliﬀord elements are understood to wear primes . Index raising/lowering manipulations have used to put all gi-factors discretely out of sight: they survive only in the combination g≡g1g2g3g4. Familiarly, every real symmetric g•gcan be rotated to diagonal form, and the numbers that then stand on the diagonal are the eigenvalues of g•g. We will use that technique—but in reverse—to relax the diagonality assumption thatwas basic to the derivation of (153). To that end, introduce new/alternativegenerators eee ithat are real linear combinations of the old ones (and vice versa ): eee/prime i=Rp ieeep (154) Then (152) becomes Rp iRq j(eeepeeeq+eeeqeeep)=2g/prime ijI LetS≡/bardblSim/bardbldenote the inverse of R≡/bardblRpi/bardbl:RpiSim=δpm. Then eeemeeen+eeeneeem=2gmnIwithgmn≡Si mSj ng/prime ij /arrowbothv g•g=STg•g/primeS The matrices g•gandg•g/primewill be spectrally identical if and only if ST=S–1, which is to say: if and only if S—whence also R, its inverse—is a rotation matrix,48 48This means that the vectors RRRiassembled from the respective columns ofR—the components of which appear as coeﬃcients in the ithinstance of (154)—are orthonormal . For the purposes at hand they can be anysuch set of vectors. Fourth order theory 67 ...which we will henceforth assume to be the case. From (154) it follows that V/primeieee/prime i=V/primeiRp ieeep=Vpeeep provided V/primeitransforms (as we will assume) as a contravariant vector: Vp=Rp iV/primei Removal of the primes from eee/prime 1eee/prime 2,eee/prime 1eee/prime 3,...involves a subtlty traceable to the circumstance that when we resolve eee/prime ieee/prime jinto its symmetric/antisymmetric parts eee/prime ieee/prime j=1 2(eee/prime ieee/prime j+eee/prime jeee/prime i)+1 2(eee/prime ieee/prime j−eee/prime jeee/prime i) the symmetric part1 2(eee/prime ieee/prime j+eee/prime jeee/prime i)=g/prime ijIvanishes (i/negationslash=j) owingto the assumed diagonality of g/prime ij. Its transform1 2(eeeieeej+eeejeeei)=gijI—already discussed— is fundamental, but is always proportional to the transformationally invariantobject I. Reference to “the transform of eee /prime ij=eee/prime ieee/prime j(i/negationslash=j)” must be understood as a reference to the antisymmetric part ofeee/prime ij: eee/prime ij≡1 2(eee/prime ieee/prime j−eee/prime jeee/prime i)=Rp iRq j1 2(eeepeeeq−eeeqeeep) 1 2(eeepeeeq−eeeqeeep)=eeepeeeq−gpqI≡eeepq We then have T/primeijeee/prime ij=Tpqeeepq ifT/primeijis understood to transform as an antisymmetric contravariant tensor of second rank: Tpq=Rp iRq jT/primeij To remove the primes from the totally antisymmetric object eee/prime 1eee/prime 2eee/prime 3eee/prime 4we writefff/prime=1 4!εijkleee/prime ieee/prime jeee/prime keee/prime las a way of automatingthe requirement that all indices be distinct (thus excludingproducts of—say—the type eee/prime 1eee/prime 1eee/prime 3eee/prime 4) and by transformation to unprimed generators obtain fff/prime=1 4!εijklRp iRq jRr kRs leeepeeeqeeereees =R·1 4!εpqrseeepeeeqeeereeeswithR≡detR=±1 =R·fff We then have P/primefff/prime=Pfff provided P/primeis understood to transform as a scalar density of weight W=−1: P=R·P/primewhich is to say: P/prime=R−1·P 68 Transformational principles derived from Clifford algebras Lookingﬁnally to fff/prime j≡fff/primeeee/prime j, it follows immediately from results now in hand that fff/prime j=R·Rq jfffqwithfffq≡fffeeeq from which we obtain A/primejfff/prime j=Aqfffq provided A/primejis understood to transform as a contravariant vector density of weightW=−1: Aq=R·Rq jA/primejwhich is to say: A/primej=R−1·Sj qAq The precedingarg ument establishes the structural invariance of (153) under all transformations that preserve the prescribed eigenvalues of g•g. But (153) holds whatever those eigenvalues might be . We conclude that the product formula is of unrestricted generality . Notingthat ◦gtransforms as an object of weight W=+ 2 ◦εijkl,fffandfffjtransform as objects of weight W=+ 1 ◦S,s,Vi,vi,Tij,tij,eeeiandeeeijtransform as objects of weight W=0 ◦Aj,aj,Pandptransform as objects of weight W=−1 ◦g–1transforms as an object of weight W=−2 we observe that •the coeﬃcient of Iin (153) presents all weightless scalars —bilinear in the coeﬃcients of GGGandHHH—that can be assembled from the above material •the coeﬃcient of eeeiin (153) presents all weightless contravariant vectors that can be assembled ... •the coeﬃcient of eeeijin (153) presents all weightless antisymmetric contravariant second rank tensors that can be assembled ... •the coeﬃcient of fffjin (153) presents all contravariant vectors of negative unit weight that can be assembled ... •the coeﬃcient of fffin (153) presents all scalars of negative unit weight that can be assembled ... Had we possessed this information in advance it would not, however, have permitted us to simply write down (153), for it speaks not at all to signs and numerics. I turn now to discussion of some of the implications of (153). Inversion problem & norm in the fourth order theory 69 7. Implications of the product formula. Given GGG=SI+Vieeei+1 2Tijeeeij+Ajfffj+Pfff we introduce “conjugates” of two ﬂavors: GGGT=SI−Vieeei−1 2Tijeeeij+Ajfffj+Pfff (155.1) GGGt=SI+Vieeei+1 2Tijeeeij−Ajfffj−Pfff (155.2) It then follows as a corollary of (153) that GGGGGGT=I/braceleftBig SS−VmVm+1 2TmnTmn−gAnAn+gPP/bracerightBig +fffj/braceleftBig 2SAj+1 gεjlmnVlTmn−2AmTmj−2PVj/bracerightBig +fff/braceleftBig 2SP−2VmAm−1 4gεklmnTklTmn/bracerightBig (156) from which—remarkably—all eeeiandeeeijterms have vanished: we are left with an expression of what I will call the “pseudo-simple” form49↓ GGG≡SI+Ajfffj+Pfff(157) which entails—again as a quick corollary of (153)—that GGGGGGt=GGGtGGGis a simple multiple of I. Speciﬁcally GGGGGGt=N(GGG)I (158) where50 N(GGG)≡S2+gAjAj−gP2(159) deﬁnes what I will call the “norm” of GGG. Evidently GGG–1=1 N(GGG)GGGT(GGGGGGT)t: exists if and only if N(GGG)/negationslash= 0 (160) It was in prospect of this important result that the operationsTandtwere introduced, and I digress now to review their basic properties: It is immediately evident thatTandtare both linear operations, that they commute (GGGT)t=(GGGt)T(161.1) and that (GGGT)T=GGG:(GGGt)t=GGG (161.2) It is a (not so immediately evident) implication of (153) that (GGGHHH)T=HHHTGGGT(161.3) In this respectTmimics a familiar property of the transposition operation, and we are brought by this remark to the realization that “pseudo-simplicity” and 49It is perhaps worth noting that if GGGandHHHare pseudo-simple their product is, in general, notpseudo-simple. 50Compare (77) on page 20. 70 Transformational principles derived from Clifford algebras and “symmetry”—in the sense “invariant under the action ofT”—are equivalent notions: the Cliﬀord numbers1 2(GGG+GGGT),GGGGGGTandGGGTGGGare all pseudo-simple, all sent into themselves byT. It becomes interesting in this light to observe that it is generally the case—even when GGGandHHHare both pseudo-simple—that (according to (153)) (GGGHHH)t= neither HHHtGGGtnorGGGtHHHt I describe now some algebraic problems that, while they lie near the heart of the theory, we seem to be not yet in position to attack. Equation (160)—which can be formulated GGGGGG T(GGGGGGT)t=N(GGG)I —describes the rightinverse of GGG. Application ofTgives (GGGGGGT)tGGGGGGT=N(GGG)I (according to which GGGGGGTand (GGGGGGT)tcommute ) which, since valid for all GGG, must remain valid when GGGis replaced by GGGT: (GGGTGGG)tGGGTGGG=N(GGGT)I Evidently right inverse of GGG=1 N(GGG)GGGT(GGGGGGT)t left inverse of GGG=1 N(GGGT)(GGGTGGG)tGGGT  (162) We know on generalgrounds51that the expressions on the right side of (162) must be equal, but are not presently in position to argue that they are“obviously” so. If by brute force appeal to (153) we could show that GGG T(GGGGGGT)t=(GGGTGGG)tGGGT (which in the Euclidean case I have, with the assistance of Mathematica , actually done) one would have N(GGG)=N(GGGT) (163) which would appear to be even harder (sixteen times harder) to establish by brute force calculation. Arguing similarly from ( GGGHHH)–1=HHH–1GGG–1we expect to have 1 N(GGGHHH)HHHTGGGT(GGGHHHHHHTGGGT)t=1 N(GGG)N(HHH)HHHT(HHHHHHT)tGGGT(GGGGGGT)t 51IfAX=XB= 1 then multiplication by Bon the right supplies A=B. Inversion problem & norm in the fourth order theory 71 which—if we could (whether by frontal attack or by indirection) establish HHHTGGGT(GGGHHHHHHTGGGT)t=HHHT(HHHHHHT)tGGGT(GGGGGGT)t —would entail that N(GGG) possesses the “determinantalproperty” N(GGGHHH)=N(GGG)N(HHH) (164) It is clear from its deﬁnition (159) that N(GGG) is a multinomial of 4thorder in the coeﬃcients of GGG. I undertake here to develop its explicit structure. Looking to (156/157) we see that S,AjandPcan be described S=S2+σ Aj=αjS+βj P=πS+ρ  (165) where I have isolated the S-dependence for reasons that will soon emerge. In this notation N(GGG)=(S2+σ)2+g(αjS+βj)(αjS+βj)−g(πS+ρ)2 =S4+( 2σ+gαjαj−gπ2)S2+2g(αjβj−πρ)S+(σ2+gβjβj−gρ2) ≡S4+ (noS3-term) + N2S2+N3S1+N4S0(166) This result puts us in position to develop N(GGG−λI) in powers of λ: N(GGG−λI)=λ4−4Sλ3+( 6S2+N2)λ2 −(4S3+2N2S+N3)λ +(S4+N2S2+N3S+N4) ≡λ4+N3λ3+N2λ2+N1λ1+N0λ0(167) Tentatively assuming N(GGGHHH)=N(GGG)N(HHH) to have been established , we have the similarity-transform invariance of the norm N(UUU–1GGGUUU)=N(GGG) (168) which implies the similarity-transform invariance of/braceleftbig N0,N1,N2,N3/bracerightbig . And this—by Mathematica -assisted inversion of the equations that describe the N’s in terms of the N’s, a process that supplies S=−1 4N3 N2=N2−3 8N2 3 N3=−N1+1 2N2N3−1 8N3 3 N4=N0−1 4N1N3+1 16N2N2 3−3 256N4 3 —implies (as it is also implied by) the invariance of/braceleftbig S,N0,N1,N2/bracerightbig . And by 72 Transformational principles derived from Clifford algebras formal extension of the Cayley-Hamilton theorem we expect GGGitself to be a solution of its own characteristic equation: GGG4+N3GGG3+N2GGG2+N1GGG1+N0GGG0=000 (169) Direct computationalvariﬁcation of this statement—imitative of what within C2[g•g] was accomplished at (79) on page 20—would appear, however, to lie far beyond the bounds of feasibility. At (165) we found it convenient to introduce these abbreviations: σ≡−VmVm+1 2TmnTmn−gAnAn+gPP αj≡2Aj βj≡1 gεjlmnVlTmn−2AmTmj−2PVj π≡2P ρ≡−2VmAm−1 4gεklmnTklTmn Returning with that information to (166) we obtain N2=−2VmVm−TmnTnm+2gAnAn−2gP2 N3=Pεklmn/bracketleftbig Tkl+2 (AkVl−AlVk)/bracketrightbig Tmn N4=1 4(TmnTnm)2−1 16g(εklmnTklTmn)2 +(TmnTnm)(VkVk+gAkAk) +1 gεklmnVlTmnεkrstVrTst −4gAmTmkTknAn +(VkVk)2+g2(AkAk)2 −4g(VkAk)2+2g(VjVj)(AkAk) +8gP(AmTmnVn)−gP2(TmnTnm) +2gP2(VkVk−gAkAk)+g2P4  (170) after simpliﬁcations. 52My notation has been designed to underscore a fact now evident: Nkis homogeneous of degree kin/braceleftbig V,T,A,P/bracerightbig The terms that enter into the description of N4can be grouped in a great variety of “natural” ways: which is most useful was found in the geneva notebook to depend upon the context. It became obvious at (166) that N(GGG)=N4in cases where S=0 52See the geneva notebook , page 209. Inversion problem & norm in the fourth order theory 73 Notice that the expressions on the right side of (170) are invariant under/braceleftbig S,V,T,A,P/bracerightbig →/braceleftbig S,−V,−T,A,P/bracerightbig . We can on this basis consider (163) to be an established fact:53N(GGG)=N(GGGT). We are, however, no closer than before to proof of the conjectured identity (164), upon which our recent remarks are critically dependent. Proof would beimmediate if we could set up a matrix representation ofC4[g•g] GGG←→G within which54 N(GGG) = det G A procedure that might in principle work proceeds from the fact that C4[g•g]i s associative :i fw ew e r e( i)t o writeGGG=15/summationdisplay p=0Gpeeepin place of GGG=SI+Vieeei+1 2Tijeeeij+Ajfffj+Pfff then ( ii) to work out the values of the 163= 4096 (real-valued) structure constants cprqthat enter into the statements eeepeeeq=15/summationdisplay r=0cpr qeeer and (iii) used them to assemble 16 ×16 realmatrices Ep≡/bardblcprq/bardblwe would—as an expression of ( eeepeeeq)eees=eeep(eeeqeees)—arrive at the “regular representation” EpEq=15/summationdisplay r=0cpr qEr The demonstration that if G=15/summationdisplay p=0GpEprepresents GGG=15/summationdisplay p=0Gpeeep 53It is, on the other hand, notgenerally the case that N(GGG)=N(GGGt):N0and N2are invariant under/braceleftbig S,V,T,A,P/bracerightbig →/braceleftbig S,V,T, −A,−P/bracerightbig butN1reverses its sign. If, however, GGGis pseudo-simple ( i.e.,i fV=T= 0) then N1= 0. The short of it: we have N(GGG)=N(GGGt) if and only if GGGis pseudo-simple. 54A relation of the weaker form [N(GGG)]characteristic power= detG would serve just as well. 74 Transformational principles derived from Clifford algebras then detG=  [N(GGG)]1,else [N(GGG)]2,else [N(GGG)]4 would appear, however, to be enormously tedious (and for that very reason not deeply instructive). Because N(GGG) is of order 4 in the coeﬃents of GGGit becomes naturalto l ook for 4×4 matrices Ep, for then det Gwould also be of order 4. Under favorable circumstances that I will, for the moment, not attempt to characterize it mayhappen that detGis invariably real, even though Gis complex. It becomes then feasible that det G=N(GGG). I illustrate how this works out in the case C 4[g•gEuclidean ]. Let the generators/braceleftbig eee1,eee2,eee3,eee4/bracerightbig be represented by “Euclideanized” variants of the Dirac matrices encountered on page 57: E1≡ 1000 010000 −10 000 −1  E 2≡ 000 −i 00 −i0 0i00 i000  E3≡ 000 −1 00100100 −1000  E 4≡ 00 −i0 000 i i000 0−i00  Those matrices happen to be traceless hermitian , and (as it turns out) so is the implied representative of fff. The representatives of eeeijandfffjare found, however, to be traceless antihermitian : the representative GofGGGis therefore curiously non-descript! Nevertheless ...Mathematica responds to the command Det[G]//ComplexExpand with many lines of manifestly real output. Turning our attention now to the representation GGT(GGT)t—which we intend to compare with N(GGG)I—given G=SI+ViEi+1 2TijEij+AjFj+PF it is easy enough to construct GT=SI−ViEi−1 2TijEij+AjFj+PF and to compute G.GT, but the output is a formless mess. How to demonstrate that it has the form SI+AjFj+PF? And how to construct SI−AjFj−PF? The ﬁrst question is resolved by appeal to the fact that 1 4tr(EpEq)=±δpq:/braceleftbiggupper sign for I,EiandF lower sign for EijandFj(171) Similarity transformations in the fourth order theory 75 For it is a computationally demonstrable property of the “formless mess” that 1 4tr(EiG.GT)=1 4tr(EijG.GT)=0 soG.GTdoes indeed have the form SI+AjFj+PF. Noting that SI−AjFj−PF=2SI+(SI+AjFj−PF) we construct (G.GT)t=1 2tr(G.GT)I−G.GT (which is manageable on the computer, even though all the terms involved are gigantic). This done, we are informed that indeed 1 4tr/braceleftbig (G.GT).(G.GT)t/bracerightbig = detG (172) So GGGGGGT(GGGGGGT)t=N(GGG)I (173.1) has acquired the representation G.GT.(G.GT)t= (det G)I (173.2) which establishes the point at issue. By the “rescale and rotate” procedure  1000 010000100001 /mapsto− → g 1000 0g200 00g30 000 g4 /mapsto− → g11g12g13g14 g21g22g23g24 g31g32g33g34 g41g42g43g44  developed on pages 64–68 we expect to be able to abandon the Euclidean metric presumption that entered this discussion with the construction of our Ei matrices, but I am presently disinclined to pursue those details. 8. Similarity transformations in the fourth order theory. We look now55to the norm-preserving transformations GGG/mapsto− →GGG/prime=UUU–1GGGUUU (174) Proceding initially on the assumption that UUUdiﬀers only inﬁnitesimally from the identity UUU=III+epsilonνHHH we have GGG/prime=GGG+epsilonν[GGGHHH−HHHGGG]+··· (175) 55See again pages 22 et seq and page 56. 76 Transformational principles derived from Clifford algebras The product rule (153) supplies GGGHHH−HHHGGG=2eeei/braceleftBig (Vmtmi−vmTmi)+g(Pai−pAi)/bracerightBig +eeeij/braceleftBig (Vivj−Vjvi)−(Ti mtjm−Tj mtim)−g(Aiaj−Ajai)/bracerightBig +2fffj/braceleftBig (Amtmj−amTmj)+(Pvj−pVj)/bracerightBig +2fff/braceleftBig Amvm−Vmam/bracerightBig =2/braceleftBig −ti mVm−vmTmi−gpAi+gaiP/bracerightBig eeei +/braceleftBig −(viVj−vjVi)−(ti mTmj−tj mTmi)+g(aiAj−ajAi)/bracerightBig eeeij +2/braceleftBig −pVj−amTmj−tj mAm+vjP/bracerightBig fffj +2/braceleftBig −amVm+vmAm/bracerightBig fff (176) S, the coeﬃcient of Iin the development of GGG, is similarity invariant: S/prime=S. We concentrate therefore on the relation of the 15 numbers/braceleftbig V/prime,T/prime,A/prime,P/prime/bracerightbig to their unprimed counterparts, and for this purpose me may as well—will— assumeGGGto be “pure”: S= 0. Nor (as we have just seen) does its scalar part sIcontribute to the action of HHH, so we assume also that HHHis pure:s= 0. Let theGGG-coeﬃcients be strung out as a column vector: vectorG≡ V 1 V2 V3 V4 T12 T13 T12 T14 T23 T24 T34 A1 A2 A3 A4 P (177) It follows now from (176) that in this notation the inﬁnitesimal similarity transformation (175) can be described vectorG/mapsto− →vectorG /prime=(III+2epsilonνHHH+···)vectorG Similarity transformations in the fourth order theory 77 where IIIis the 15 ×15 unit matrix and HHHis a 15 ×15 matrix the detailed structure of which can be read oﬀ from (176). By iteration we ﬁnd (compare page 24)that GGG/mapsto− →GGG /prime=e−θHHHGGGeθHHH(178.1) can be described vectorG/mapsto− →vectorG/prime=e2θHvectorG (178.2) The quadratic similarity-invariant N2(see again (170)) acquires in this notation a fairly natural description. Writing −1 2N2=gmnVmVn−1 2gmpgnqTmnTpq−g·gmnAmAn+g·P2 we have =vectorGTMMMvectorG (179) where MMMis a 15 ×15 symmetric matrix assembled from elements of the 4 ×4 metric matrix /bardblgmn/bardbl. In an obvious sense, MMMinjects “induced metric structure” into 15-space. From the similarity invariance of vectorGTMMMvectorGwe infer that UUU≡e2θH is “MMM-orthogonal” MMM–1UUUTMMM=UUU–1 and therefore that its logarithm HHHis “MMM-antisymmetric” MMM–1HHHTMMM=−HHH:HHHTMMM, therefore, is literally antisymmetric What to do about—what lesson is to be drawn from—the fact that the cubic and quartic invariants N3andN4ﬁnd no natural dwelling place within such a scheme ? These are the questions with which the work recorded in the geneva notebook is largely concerned, and it is upon that work that I now draw. I will begin by describing the basic idea, then labor to develop the detailsin the instance that concerns us. LetM≡/bardblm jk/bardblbe a non-singular symmetric N×Nmatrix, and let its inverse be denoted W≡/bardblwij/bardbl:wijmjk=δik. Let A≡/bardblajk/bardbland, upon agreement that WandMwill be used to raise/lower indices, write MA≡/bardblajk/bardbl: ajk=mjpapk. With the tensor rule Xij/mapsto− →X/prime ij=UpiUqjXpqin mind we study transformations of the form MA−λM/mapsto− →(MA−λM)/prime=UT(MA−λM)U (180) withU≡/bardblUjk/bardbl. We now impose upon Uthe restrictive assumption that M/prime≡UTMU=M:Uis “M-orthogonal” We then have UT(MA−λM)U=M(U–1AU−λI): the transformation (180), after multiplication on the left by W, has assumed the form A−λI/mapsto− →A/prime−λI=U–1(A−λI)U (181) 78 Transformational principles derived from Clifford algebras of asimilarity transformation . Immediately det(A/prime−λI) = det( A−λI) The implication is that the coeﬃcients Qnthat enter into the construction of the characteristic polynomial56 p(λ)≡det(A−λI)=N/summationdisplay n=01 n!Qn(−λ)N−n(182) areU-invariant functions of the elements of A. I describe now two distinct methods for constructing theQn: It is a fact—as little known as it is pretty—that Qncan be described Q0=1 Qn=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleT 1T2T3T4... T n 1T1T2T3... T n−1 02T1T2... T n−2 003 T1... T n−3 ............... 0000 ... T 1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle:n=1,2,3,... (183.1) where T n≡trAn(183.2) We are led from this result to the recursion relation Qn=n/summationdisplay m=1(−)m+1(n−1) ! (n−m)!TmQn−m from which it follows (with a little assistance from Mathematica ) that Q0=1 Q1=T1 Q2=T2 1−T2 Q3=T3 1−3T1T2+2T3 (184) Q4=T4 1−6T2 1T2+3T2 2+8T1T3−6T4 Q5=T5 1−10T3 1T2+1 5T1T2 2+2 0T2 1T3−20T2T3−30T1T4+2 4T5 Q6=T6 1−15T4 1T2+4 5T2 1T2 2−15T3 2+4 0T3 1T3−120T1T2T3 +4 0T2 3−90T2 1T4+9 0T2T4+ 144T1T5−120T6 ... Worthy of specialnote is the universality property that attaches to the preceding formulæ: they read the same whatever the dimension N, but numerically Qn>N= 0 as a consequence of the Cayley-Hamilton theorem (185) 56I borrow my Q-notation from “A mathematicalnote: al ogorithm for the eﬃcient evaluation of the trace of the inverse of a matrix” ( ) so that I can most smoothly borrow certain results from that same source. Similarity transformations in the fourth order theory 79 Of even greater relevance to my intended application is the fact that if Ais “M-antisymmetric” Akj=−Ajk⇔(MA)T=−MA ⇔WATM=−A then by an easy argument tr An=(−)n·trAnwhich supplies Todd= 0 (186) and (184) simpliﬁes very greatly: Q0=1 Q1=0 Q2=−T2 Q3= 0 (187) Q4= +3(T2 2−2T4) Q5=0 Q6=−15(T3 2−6T2T4+8T6) ... EXAMPLE: Let us, in the case N=4 ,(i) identify Mwith the Lorentz metric g•g= 1000 0−100 00 −10 000 −1  (ii) recognize that the g•g-orthogonaltransformation matrices Uhave become Lorentz matrices (descriptive of Lorentz transformations, and often notated /\\\) and (iii) identify Awith the electromagnetic ﬁeld tensor57 Abecomes F≡/bardblFµ ν/bardbl= 0E1E2E3 E1 0B3−B2 E2−B3 0B1 E3B2−B1 0  Mathematica ’s Tr[MatrixPower[F,2]]//Expand and Tr[MatrixPower[F,2]]2-2Tr[MatrixPower[F,4]]//Expand 57Seeprinciples of classical electrodynamics (/), page 108. 80 Transformational principles derived from Clifford algebras commands instantly supply Q2=−2/braceleftbig EEE···EEE−BBB···BBB/bracerightbig Q4=−24/braceleftbig EEE···BBB/bracerightbig2 which are familiar as Lorentz invariant properties of the electromagnetic ﬁeld tensor .58It would appear on this evidence that we can expect invariants to be objects of physicalimportance .59From (182) we expect, in the case N=4 ,t o have1 4!Q4= detA, and indeed: we ﬁnd by computation that60 detF=/braceleftbig EEE···BBB/bracerightbig2 Of course, it is clear on tensor-theoretic grounds that the traces of allpowers of the ﬁeld tensor (all contractions Fµα1Fα1α2Fα2α3···Fαpµ) are invariant. But •Todd≡trFodd=0 •T6,T8,T10,...are redundant with T2andT4in consequence of the Cayley- Hamilton theorem, which in the electromagnetic instance reads F4−/braceleftbig EEE···EEE−BBB···BBB/bracerightbig F2−/braceleftbig EEE···BBB/bracerightbig2I=O and can be veriﬁed by (instantaneous) calculation. I describe now an alternative appropach to expansion of p(λ)≡det(A−λI) which oﬀers computational advantages in my intended application. Laplacewould have us develop det Aby expansion along some arbitrarily selected row or column. An inherently more symmetrical procedure was devised by Cayley. 61It involves “expansion along the principal diagonal,” and is most simply explainedby example: /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea 11a12a13a14 a21a22a23a24 a31a32a33a34 a41a42a43a44/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 12a13a14 a210a23a24 a31a320a34 a41a42a430/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+a 11/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 23a24 a320a34 a42a430/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle +a 22/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 13a14 a310a34 a41a430/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+a 33/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 12a14 a210a24 a41a420/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+a 44/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 12a13 a210a23 a31a320/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle +a 11a22/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 34 a430/vextendsingle/vextendsingle/vextendsingle/vextendsingle+a 11a33/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 24 a420/vextendsingle/vextendsingle/vextendsingle/vextendsingle+a 11a44/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 23 a320/vextendsingle/vextendsingle/vextendsingle/vextendsingle +a 22a33/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 14 a410/vextendsingle/vextendsingle/vextendsingle/vextendsingle+a 22a44/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 14 a410/vextendsingle/vextendsingle/vextendsingle/vextendsingle+a 33a44/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 12 a210/vextendsingle/vextendsingle/vextendsingle/vextendsingle +a 11a22a33|0|+a11a22|0|a44+a11|0|a33a44+|0|a22a33a44+a11a22a33a44 58See page 184 in the notes just cited. 59In view of the importance of the role assigned by plane waves in electro- magnetic theory it is interesting that both invariants vanish for those specialized solutions of Maxwell’s equations. 60That det Fis aperfect square is no accident: it is a particular instance of a generalcircumstance to which we wil lsoon attach major importance. 61See§125 in Thomas Muir, A Treatise on the Theory of Determinants (), which was reprinted by Dover in . Similarity transformations in the fourth order theory 81 The ﬁrst determinant on the right has been “invertebrated.” The second is the invertebrate (Sylvester’s terminology) from which the 1strow & column have been struck. At the sixth term on the right a11a22multiplies the invertebrate from which the 1stand 2ndrows & columns have been struck. So it goes: on the right we ﬁnd invertebrates multiplied by diagonal elements taken in all possible combinations . Look now to det( A−λI) in a case in which Ais antisymmetric. Writing µ≡−λsimply to avoid some distracting minus signs, we have /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleµa 12a13a14 a21µa 23a24 a31a32µa 34 a41a42a43µ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 12a13a14 a210a23a24 a31a320a34 a41a42a430/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+µ/braceleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 23a24 a320a34 a42a430/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle +/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 13a14 a310a34 a41a430/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 12a14 a210a24 a41a420/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 12a13 a210a23 a31a320/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracerightBigg +µ 2/braceleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 34 a430/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 24 a420/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 23 a320/vextendsingle/vextendsingle/vextendsingle/vextendsingle +/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 14 a410/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 14 a410/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle0a 12 a210/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracerightBigg +µ 4 But easily, if AT=−Athen det A= 0 in all odd-dimensional cases, while (less obviously) in all even-dimensional cases det Ais a perfect square—the square of the so-called “Pfaﬃan” of A: detA=/braceleftbigg0 if antisymmetric Ais odd-dimensional (PfA)2if antisymmetric Ais even-dimensional The terms linear in µtherefore drop away: we are left with det(A−λI)=(a12a34−a13a24+a14a23)2 −(a2 12+a2 13+a2 14+a2 23+a2 24+a2 34)λ2+λ4 We verify computationally that indeed (a12a34−a13a24+a14a23)2=1 4!Q4=3(T2 2−2T4) 4! −(a2 12+a2 13+a2 14+a2 23+a2 24+a2 34)=1 2!Q2=−T2 2! The point is that those and similar expressions are much easier to compute (by hand, if not in the opinion of Mathematica ) by the Cayley-Pfaﬀ method than by assembling traces of powers. So much for methodological preliminaries, for “the basic idea.” Turning now to the speciﬁc questions posed on page 77, we have learned that we can ﬁnd 82 Transformational principles derived from Clifford algebras natural dwelling places for multiple invariants if we associate the pure elements ofC4[g•g] with tensors oﬀ higher order than the vectors contemplated at (177). It seems most naturalto consider “tensors of higher order” to mean “tensorsof second rank,” which are representable by square matrices . Look ﬁrst to the numerology: •The pure elements of C 4[g•g] are 24−1 = 15 component objects. •Similarity transformations within C4[g•g] possess 3 invariants. We are mindfulthat •the space of traceless 4 ×4 hermitian matrices is 42−1 = 15 dimensional and, additionally, that •such objects do support a population of 3 real unitary invariants: they can be taken to be62 Q2=−T2 Q3=2T3 Q4=3T2 2−6T4 But down that road lies a metric generalization of the standard theory of Dirac matrices which, valuable though it is, is not my destination. We will instead proceed from the observation that •the space of real6 ×6 antisymmetric matrices i s1+2+3+4+5=1 5 dimensional, and •supports a population of 3 real rotational invariants, which at (187) were calledQ2,Q4andQ6. The immediate question: How to deploy the coordinates/braceleftbig V,T,A,P/bracerightbig ofGGG∈C4[g•g] among the slots provided by such matrix? In the geneva notebook Iw a s guided by the fact that I was concerned there with the relationship betweenC 4[g•gEuclidean ] and the “E-numbers” EEEµν=−EEEνµ:µ,ν∈/braceleftbig 0,1,2,3,4,5/bracerightbig 62Read from (184) with T1set equalto zero. It is instructive to write H= a11a12+ib12a13+ib13a14+ib14 a12−ib12a22a23+ib23a24+ib24 a13−ib13a23−ib23a33 a34+ib34 a14−ib14a24−ib24a34−ib34−(a11+a22+a33)  and then to ask Mathematica to execute the commands CharacteristicPolynomial[H, λ] ComplexExpand[ %] Simplify[ %] One obtains det( H−λI)=λ4+0λ3+1 2!Q2λ2−1 3!Q3λ1+1 4!Q4λ0and ﬁnds that all theQ’s are indeed real: the i’s have done no damage. Similarity transformations in the fourth order theory 83 to which A. S. Eddington ( –) assigned centralimportance in the work of his ﬁnalyears.63But even in the absence of any such “Eddingtonian bias” it seems entirely natural to construct  0 P 0T12T13T14 0T23T24 0T34 0 0  where the only remaining question is where to insert V’s, where A’s, and this is matter that can be settled by experimentation. Thus was I led to construct /bardblG ab/bardbl≡ 0A 1A2A3A4P 0T12T13T14V1 0T23T24V2 0T34V3 0V4 0 (188.1) and /bardblγ ab/bardbl≡ g 0000 0 0g 11g12g13g14 0 0g21g22g23g24 0 0g31g32g33g34 0 0g41g42g43g44 0 0 0000 −1 (188.2) from which it follows that G≡/bardblG a b/bardbl≡/bardblGakγkb/bardbl= 0A 1A2A3A4−P −gA10T12T13T14−V1 −gA2T210T23T24−V2 −gA3T31T320T34−V3 −gA4T41T42T430 −V4 −gP −V1−V2−V3−V4 0  63SeeRelativity Theory of Protons & Electrons ()—especially Chapter 2: “The sixteenfold frame”—and Fundamental Theory , which was published () posthumously by E. T. Whittaker (who gave the book its unfortunate title). See also N. B. Slater, The Development & Meaning of Eddington’s ‘Fundamental Theory’ (), which provides an account of the substance of Eddington’s unpublished manuscripts, and of what light they may cast upon theevolution of his thought. Slater, by the way, was a friend of Eddington’s and—during the period ( /) when he was working on his book—my frequent dining companion at Cornell, where I was a ﬁrst-year graduate student and hea visiting scholar. 84 Transformational principles derived from Clifford algebras A few moments with pen and paper are suﬃcient to establish that (quoting now from (187), and writing Tn≡trGnto emphasize that all traces are understood to refer speciﬁcally to the 6 ×6 matrix G) Q2=−T2 =−/braceleftbig VmVm+1 2Tm nTn m−gAnAn+gP2/bracerightbig =1 2N2: see again (170) It was the prospect of such a result that guided me in the deﬁnitions (188). Discussion of the anticipated relationships that link Q4=3 ( T2 2−2T4) Q6=−15(T3 2−6T2T4+8T6) = 6! det G to the invariants N3andN4of (170) is computationally more burdensome. It is accomplished in the geneva notebook by pen-and-paper work based upon the methods described on pages 78–81. Mathematica stands ready to assist, but by unfortunate quirk reads superscripts as powers. To work around thisdiﬃculty I proceed step-wise: phase one : Assume the metric g ijto be Euclidean, so that sub/superscript distinctions are irrelevant. Deﬁne T[m,n]:=Tm,n−Tn,m 2 so as to inform Mathematica thatTmnis antisymmetric, enter G= 0A 1A2A3A4−P −A1 0T[1,2]T[1,3]T[1,4]−V1 −A2T[2,1] 0 T[2,3]T[2,4]−V2 −A3T[3,1]T[3,2] 0 T[3,4]−V3 −A4T[4,1]T[4,2]T[4,3] 0 −V4 −P −V1 −V2 −V3 −V4 0  command Det[G]//Expand and get a very long expression which, however, the command Simplify[ %]brings to the form detG=− 1 16/parenleftbig relatively brief 3rd-order expression/parenrightbig2 Now command 4/summationdisplay k=14/summationdisplay l=14/summationdisplay m=14/summationdisplay n=1Signature[ {k,l,m,n }] /parenleftBig PT[k,l]+2 (AkVl−AlVk)/parenrightBig T[m,n]//Simplify and get −2(same “relatively brief 3rd-order expression”), and conclude that 1 6!Q6= detG=−1 16/parenleftbig −1 2N3/parenrightbig2 whence Q6=−45 4N2 3 Similarity transformations in the fourth order theory 85 The invariant N4was seen at (170) to be quartic, and to contain a P4term. TheQ4-inspired command 3/parenleftBig Tr[MatrixPower[Gam, 2]]2-2Tr[MatrixPower[Gam, 4]]/parenrightBig //Expand gives, on the other hand, an expression that, though quartic, presents no P-powers higher than P2. We are led therefore to construct N4−1 4N2 2: quartic, no P4term according to (170) and by command Solve[N4−1 4N2 2+xQ4== 0,x]//Simplify obtainx=1 6. Pulling these results together, we have (in the Euclidean case) Q2=1 2N2 Q4=−6N4+3 2N2 2 Q6=−45 4N2 3  (189) phase two is addressed to the demonstration that (189) hold even after the Euclidean assumption is abandoned. Let G, its subscripts notwithstanding, be understood to to mean /bardblGab/bardbl. Enter g[m,n]:=gm,n+gn,m 2 into the keyboard construction of /bardblγab/bardbl. Multiply those matrices as indicated on paged 83, to inform Mathematica what we have in mind when we write G. UseGto construct the trace representations (187) of Q2,Q4andQ6= 6! det G. Write N2=−24/summationdisplay a=1g[m,a]/parenleftbig VaVm−gAaAm/parenrightbig −4/summationdisplay a=14/summationdisplay b=1g[m,a]g[n,b]Ta,bTm,n−2gP2 to describe N2, and provide similarly detailed descriptions of N2andN4.A ll then proceeds as before ...but slowly, even when Mathematica 5 runs at several GHz, for the calculations are immense. What we have established is that •N2,N3andN4are invariant under similarity transformations of the sort (see again 174) and (178.1) encountered within C4(g•g): GGG/mapsto−→GGG/prime=e−θHHHGGGeθHHH •a distinct but equivalent set of objects Q2,Q4andQ6arises when one looks to the response of 6-dimensional antisymmetric tensors Gpq 86 Transformational principles derived from Clifford algebras toγ•γ-orthogonaltransformations. The argument—a reﬁnement of that encountered already on page 77—runs as follows: let Gpq/mapsto−→G/primepq=Up aGabUq b be notated IΓ/mapsto−→IΓ/prime=UIΓUT and assume Uto beγ•γ-orthogonal: γ•γ=UTγ•γU. ThenIΓ/prime=UIΓγ•γU–1γ•γ–1 becomes G/prime=UGU–1 withG≡IΓγ•γ=/bardblGpaγaq/bardbl≡/bardblGpq/bardbl. Clearly G/primeandGshare the same characteristic polynomial, which by the assumed antisymmetry of IΓ (γ•γ-antisymmetry of G) has the form det(G−λI)=λ6+1 2!Q2λ4+1 4!Q4λ2+1 6!Q6λ0 Which brings us to the core of the matter: if U=e−βHisγ•γ-orthogonalthen (as argued already on page 77) His necessarily γ•γ-antisymmetric. Assuming β to be inﬁnitesimal, we have G/prime=G+β[GH−HG]+··· Theγ•γ-antisymmetry of GandHimplies that of [ GH−HG]. So—taking Gto have the design indicated on page 83, and Hto be64the lower case version of that matrix—to we can write GH−HG= 0 A 1 A2 A3 A4−P −gA10 T12T13T14−V1 −gA2T210 T23T24−V2 −gA3T31T320 T34−V3 −gA4T41T42T430−V4 −gP−V1−V2−V3−V40 (190) where by straightforward calculation V i=/braceleftBig −ti mVm−vmTmi−gpAi+gaiP/bracerightBig Tij=1 2/braceleftBig −(viVj−vjVi)−(ti mTmj−tj mTmi)+g(aiAj−ajAi)/bracerightBig Aj=/braceleftBig −pVj−amTmj−tj mAm+vjP/bracerightBig P=/braceleftBig −amVm+vmAm/bracerightBig  (191) Comparison with (176) on page 76 establishes that if GGG↔G(in the sense “share the same/braceleftbig V,T,A,P/bracerightbig coeﬃcients) and if also HHH↔H, then GGGHHH−HHHGGG←→ 2(GH−HG) (192) 64See again (151) on page 63. Numerological source of interest in the twelfth-order theory 87 ...from which follows the important conclusion that GGG/mapsto−→GGG/prime=e−1 2θHHHGGGe1 2θHHH(193.1) and G/mapsto−→G/prime=e−θHGeθH(193.2) achieve the same action :/braceleftbig V,T,A,P/bracerightbig /mapsto−→/braceleftbig V/prime,T/prime,A/prime,P/prime/bracerightbig . And (193.2)— becausee−θHisγ•γ-orthogonal—can be phrased Gpq/mapsto−→G/primepq=/parenleftbig e−θH/parenrightbigp a/parenleftbig e−θH/parenrightbigq bGab(193.3) which is to say: similarity transformations within C4(g•g) are equivalent to the γ•γ-orthogonaltransformations of antisymmetric tensors in 6-space. It becomes naturalin view of (193.3) to l ook to the transformation of contravariant6-vectors ξ p/mapsto−→ξ/primep=/parenleftbig e−θH/parenrightbigp aξa(194) whereγ•γ-orthogonality entails ξ/primepγpqξ/primeq=ξpγpqξq(195) It will be appreciated that we have in (193.2) a 6 ×6 matrix representation not of C4(g•g) itself, but only of the associated commutator sub-algebra : GGGHHH=1 2(GGGHHH+HHHGGG)+1 2(GGGHHH−HHHGGG)/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright |—admits of 6-dimensional representation When we to write G=SI+ViEi+1 2TijEij+AjFj+PF we ﬁnd, for example, that EiEj+EjEi/negationslash=2gijI 9. Numerological source of interest in the twelfth-order theory. To identify a pure element of C2—a structure that can be considered to be rooted in the 2-dimensionalgroup O(2)—one must assign value to 22−1 = 3 coeﬃcients. The physically important relationship between •similarity transformations within C2on the one hand, and •the group O(3) of rotations in 3-space on the other owes something to the fact that 1 + 2 = 3 is the number also of the matrix elements that must be speciﬁed to identify an antisymmetric 3 ×3 matrix (logarithm of a 3 ×3 rotation matrix). The 2-dimensionality of the irreducible complex matrix representations of C2brings complex 2-vectors (simple spinors) into pl ay as naturalcompanions of real3-vectors.65Similarly ... 65And 2-spinors of higher rank into play as companions of real 3-tensors of higher rank. While we stress here the generative relation of O(2) to C2, it should be borne in mind that O(2) has a “downwardly natural” relationship also to the complex numbers (rotations on the complex plane). 88 Transformational principles derived from Clifford algebras To identify a pure element of C4—a structure that can be considered to be rooted in the 2-dimensionalgroup O(4)—one must assign value to 24−1=1 5 coeﬃcients. The relationship—developed above—between •similarity transformations within C4on the one hand, and •the group O(6) of rotations in 6-space on the other owes something to the fact that 1+2 +3+4+5=15isthen um beralsoofthe matrix elements that must be speciﬁed to identify an antisymmetric 6 ×6 matrix (logarithm of a 6 ×6 rotation matrix). The 4-dimensionality of the irreducible complex matrix representations of C4brings complex 4-spinors into play as naturalcompanions of real4-vectors/tensors. Physicalimportance attachesfamiliarly to the scalar/vector/tensor/pseudovector/pseudoscalar latent in thetransform theory of C 4(Dirac algebra). Less familiar is the demonstrated fact that those objects are latent also in the theory of O(6), and no work (so far as I am aware) has been assigned by physicists to the associated 6-vectors,66or to 6-tensors of higher order. It was with these points in mind that, in , I was led to ask: Are there yet other instances in which a Mersenne number 2p−1 is triangular? Are there higher instances of 2p−1=n−1/summationdisplay k=1k=1 2n(n−1) =/parenleftbign 2/parenrightbig (196.1) We have already in hand the cases 21−1=2−1/summationdisplay k=1k=1 22−1=3−1/summationdisplay k=1k=3 24−1=6−1/summationdisplay k=1k=1 5 Laborious work with a Frieden calculator exposed also the case 212−1=91−1/summationdisplay k=1k= 4095 Though further searching provided no additionalexampl es, I recorded at the time my guess that “the number of triangular Mersennes is probably inﬁnite.”I consulted my then-colleagues in the Reed College Mathematics Department,and was informed by Burrows Hunt that “there are very few theorems refer tothe intersection of sparse sequences.” So there I left it ... 66These are not to be confused with the 6-vectors that ion electrodynamics are sometimes associated with the antisymmetric 4 ×4 ﬁeld tensor. Numerological source of interest in the twelfth-order theory 89 ...untilJune , when it came to my attention that Brian Tuckerman, of IBM, had (in ) discovered the 24thMersenne prime.67I wrote to him to discover what he might tell me about triangular Mersenne numbers. Six daysafter my letter was posted he wrote back to remark ( i) that the substitution n= 1 2(m+ 1) casts 2p−1=1 2n(n−1) into the form 2p+3−7=m2=( 2n−1)2(196.2) (ii) that—except for the case p= 1—the pin (196) must certainly be even, and (iii) that he had searched up to p=1 05and found no solution beyond myp= 12. Tuckerman guessed that my problem must have been studied, and referred me to D. H. Lehmer (celebrated number theorist at Berkeley)for references. I wrote immediately to Lehmer, who (again within six days)reported that the list (p,n)=/braceleftbig (0,0),(1,2),(2,3),(4,6),(12,91)/bracerightbig is exhaustive !, as had been shown by D. J. Lewis in , and that proof can be found also on pages 205–6 in L. J. Mordell’s Diophantine Equations (). Upon consulting Lewis68I learned that my 2p+3−7=m2problem is the simplest instance of a class of problems that can be shown “by means of a p-adicargument” to possess ﬁnitely many solutions. Reference is made to earlierpapers by T. Nagell ( ,and) and by Th. Skolem, S. Chowla & D.J. Lewis. 69The latter begins with these words: “Ramanujan70observed that the equation 2n+2−7=x2has...integralsol utions for n=1,2,3,5,13; and he conjectured that these are the only solutions. The authors cite earlier work, butclaim to be the ﬁrst to establish the validity of Ramanujan’s conjecture. Thatclaim inspired an indignant T. Nagell to publish “The Diophantine equationx 2+7=2n,” Arkiv f¨ ur Matematik 4, 185 (year not recorded), in which he draws attention to the fact that proof of Ramanujan’s conjecture appears asProblem 165 on page 272 of Nagell’s Introduction to Number Theory (): Nagell then presents an English translation of his own “quite elementary” proofof. Upon consulting Collected Papers 70we ﬁnd that pages 322–334 record questions & solutions submitted by Ramanujan to the Journalof the Indian 67It is 219937−1 and runs to 6002 decimaldigits. At present the l argest known Mersenne prime—discovered only a few days ago (November ) and thought to be the 40th—is 220996011−1, which runs to 6,320,430 digits. The distributed calculations that have identiﬁed the last few Mersenne primes havemade critical use of an algorithm devised by Richard Crandall. 68“Two classes of Diophantine equations,” Paciﬁc Journal of Mathematics 11, 1063 (1961). 69“The Diophantine equation 2n+2−7=x2and related problems,” Proc. Amer. Math. Soc. 10, 663 (1959). 70G. H. Hardy et al (editors), Collected Papers of Srinivasa Ramanujan (), page 327, Problem 464. 90 Transformational principles derived from Clifford algebras MathematicalSociety. question 464 reads “2n−7 is a perfect square for the values 3, 4, 5, 7, 15 of n. Find other values.” This I don’t read as a conjecture that there are noother values ...but perhaps it can be argued that if there were other values Ramanujan would have had no interest in the problem. So we have basically three and only three cases C2←→O(3) C4←→O(6) C12←→O(91) in which the numerology works out. The ﬁrst two can, in fact, be developed in detail, and are of established physical importance. The question thereforearises: Can the “last case” be developed in similar detail, and has it a role to play in the description of the real world ? Possibly relevant is the observation that 91 is itself triangular: 1+2+ ···+1 3=9 1 = number of elements in a 14 ×14 antisymmetric matrix This suggests that we might adopt antisymmetrized double indexing to describe the elements x ij=−xji(i,j=1,2,...,14) of a 91-vector, and in that same (Eddingtonian) spirit write Aij,kl=  −Akl,ij −Aji,kl −Aij,lk to describe the 4095 elements of a 91 ×91 antisymmetric matrix. But how to make that convention mesh with the convention eeepeeeq+eeeqeeep=2gpqI:p,q=1,2,...,12 naturalto the devel opment of C12?