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DEPARTMENT OF MATHEMATICS TECHNICAL REPORT ON THE DECOMPOSITION OF CLIFFORD ALGEBRAS OF ARBITRARY BILINEAR FORM Rafal Ablamowicz Bertfried Fauser November 1999 No. 1999-10 TENNESSEE TECHNOLOGICAL UNIVERSITY Cookeville, TN 38505 On the Decomposition of Cli fford Algebras of Arbitrary Bilinear Form∗ Bertfried Fauser Universit¨ at Konstanz Fakult¨ at f¨ur Physik, Fach M678 D-78457 Konstanz E-mail: [email protected] Rafaà l AbÃlamowicz Department of Mathematics Box 5054 Tennessee Technological University Cookeville, TN 38505, USA E-mail: [email protected] September, 27 1999 Abstract Clifford algebras are naturally associated with quadratic forms. These algebras are Z2-graded by construction. However, only a Zn-gradation induced by a choice of a basis, or even better, by a Chevalley vectorspace isomorphism C`(V)↔!Vand an ordering, guarantees a multi- vector decomposition into scalars, vectors, tensors, and so on, mandatory in physics. We show that the Chevalley isomorphism theorem cannot begeneralized to algebras if the Z n-grading or other structures are added, e.g., a linear form. We work with pairs consisting of a Cli fford algebra and a linear form or a Zn-grading which we now call Clifford algebras of multi- vectors orquantum Cli fford algebras . It turns out, that in this sense, all multi-vector Cli fford algebras of the same quadratic but di fferent bilinear forms are non-isomorphic. The usefulness of such algebras in quantumÞeld theory and superconductivity was shown elsewhere. Allowing for ar- bitrary bilinear forms however spoils their diagonalizability which has a considerable e ffect on the tensor decomposition of the Cli fford algebras governed by the periodicity theorems, including the Atiyah-Bott-Shapiro mod 8 periodicity. We consider real algebras C` p,qwhich can be decom- p o s e di nt h es y m m e t r i cc a s ei n t oat e n s o rp r o d u c t C`p−1,q−1⊗C`1,1.The g e n e r a lc a s eu s e di nq u a n t u m Þeld theory lacks this feature. Theories ∗Paper presented at the 5th International Conference on Cli fford Algebras and their Ap- plications in Mathematical Physics, Ixtapa, Mexico, June 27 - July 4, 1999. 1 with non-symmetric bilinear forms are however needed in the analysis of multi-particle states in interacting theories. A connection to q-deformed structures through nontrivial vacuum states in quantum theories is out- lined.MSCS: 15A66; 17B37; 81R25; 81R50 Keywords: Clifford algebras of multi-vectors, Cli fford map, quantum Clifford algebras, periodicity theorems, index theorems, spinors, spin- tensors, Chevalley map, quadratic forms, bilinear forms, deformed tensor products, multi-particle geometric algebra, multi-particle states, compos- iteness, inequivalent vacua 1 Why study Cli fford algebras of an arbitrary bilinear form? 1.1 Notation, basics and naming 1.1.1 Notation ToÞx our notation, we want to give some preliminary material. If nothing is said about the ring linear spaces or algebras are build over, we denote it by R and assume usually that it is unital, commutative and not of characteristic 2 . In some cases we specialize our base ring to the Þeld of real or complex numbers denoted as RandC. Aquadratic form is a map Q:V7→Rwith the following properties ( α∈R, V∈V) i) Q(αV)=α2Q(V), ii)2 g(x,y)=Q(x−y)−Q(x)−Q(y), (1) where g(x,y) is bilinear and necessarily symmetric. g(x,y) is called polar bilinear form ofQ.Transposition is de Þned as g(x,y)T=g(y,x).Quadratic forms over the reals can always be diagonalized by a choice of a basis. That is, in every equivalence class of a representation there is a diagonal representative. We consider a quadratic space H=(V,Q) as a pair of a linear space V —over the ring R— and a quadratic form Q.This is extended to a reßexive space H0=(V,B) viewed as a pair of a linear space Vand an arbitrary non- degenerate bilinear form B=g+A,where g=gTandA=−ATare the symmetric and antisymmetric parts respectively. gis connected to a certain Q. We denote the Þnite additive group of nelements under addition modulo nasZn.This should not be confused with the ring Znalso denoted the same way. Algebras or modules can be graded by an Abelian group. If the linear space W—not the same as V—, of an algebra can be divided into a direct sum W=W0+W1+...+Wn−1and if the algebra product maps these spaces in a compatible way one onto another, see examples, so that the index labelsbehave like an Abelian group, one refers to a grading [8]. 2 Example 1: W=W0+W1andW0W0⊆W0,W 0W1'W1W0⊆W1 andW1W1⊆W0.The indices are added modulo 2 and form a group Z2.If W=W0+W1+...+Wn−1one has e.g. WiWj⊆Wi+jmod nwhich is a Zn-grading. In the case of Zn-grading, elements of Wmare called m-vectors or homoge- nous multi-vectors. The elements of W0'Rare also called scalars and the elements of W1arevectors . When the Z2-grading is considered, one speaks about even and odd elements collected in W0andW1respectively. However, observe that the Cli fford product is notgraded in this way since with V'W1andR'W0one has V×V=R+W2which is not group- like. Only the even/odd grading, sometimes called parity grading, is preserved,C` +C`+⊆C`+,C ` +C`−'C`−C`+⊆C`−andC`−C`−⊆C`+.Hence, C` isZ2graded and C`'C`++C`−'W0+W1. Clifford algebras are displayed as follows: C`(B,V )i s a quantum Cli fford algebra ,C`(Q, V )i s a b a s i s - f r e e C l i fford algebra, C`(g,V)i s a C l i fford algebra with a choice of a basis, C`p,qandC`nare real and complex Cli fford algebras of symmetric bilinear forms with signature p, q or of complex dimension n respectively. 1.1.2 Basic constructions of Cli fford algebras Constructions of Cli fford algebras can be found at various places in literature. We give only notation and refer the Reader to these publications [6, 8, 10, 12,14, 34, 48, 60]. Functorial: The main advantage of the tensor algebra method is its formal strength. Existence and uniqueness theorems are most easily obtained in this language. Mathematicians derive almost all algebras from the tensor algebra—the real mother of algebras— by a process called factorization. If one singles out a two-sided ideal Iof the tensor algebra, one can calculate ’modulo’ this ideal. That is all elements in the ideal are collected to form a class called’zero’ [0] 'I.Every element is contained in an equivalence class due to this construction. Denote the tensor algebra as T(V)=R⊕V⊕...⊗ nV⊕... and let x,y,...∈VandL ,M,... ∈T(V).This algebra is by construction naturally Z∞-graded for any dimension of V. I nt h ec a s eo fC l i fford algebras, one selects an ideal of the form IC`={X|X=L⊗(x⊗x−Q(x)1)⊗M} (2) which implements essentially the ’square law’ of Cli fford algebras. Note, that elements of di fferent tensor grades —scalar and grade two— are identi Þed. Hence this ideal is not grade-preserving and the factor algebra —the Cli fford algebra— cannot be Z∞-graded ( Þniteness of C`(V)) ; and not even multi-vector or Zn- graded with n=d i m Vbecause all indices are now mod 2 .However, the ideal IC`isZ2-graded, that is, it preserves the evenness and the oddness of the 3 tensor elements. One de Þnes now the Cli fford algebra as: C`(Q, V ): =T(V) IC`. (3) It is clear from the construction that a Cli fford algebra is unital and asso- ciative, a heritage from the tensor algebra. Generators and relations: Physicists and most people working in Cli fford analysis prefer another construction of Cli fford algebras by generators and re- lations [17]. One chooses a set of generators ei,images of some arbitrary basis elements xiofVunder the usual Cli fford map γ:V7→C`(V)i n t h e C l i fford algebra C`(V),and asserts the validity, in the case of R=RorC,of the normalized, ’square law’: e2 i=±1. (4) Using the linearity, that is polarizing this equations by ei7→ei+ej,one obtains the usual set of relations which have to be used to ’canonify’ the algebraic expressions: eiej+ejei=2g(ei,ej)1=2gij1. (5) The de Þnition of the Cli fford algebra reads: C`(gij,V)'Alg(ei)m o d eiej=2gij1−ejei. (6) W h i l et h e— i m a g eo ft h e —n u m b e r so ft h eb a s e Þeld are called scalars ,t h e eiand their linear combinations are called vectors . The entire algebra is constructed by multiplying and linear-combining the generators eimodulo the relation (5). This ’modulo relation’ is in fact nothing else as a ’cancellation law’ which pro- vides one with a unique representative of the class of tensor elements. A basis of the linear space underlying the Cli fford algebra is given by reduced monomials in the generators, where a certain ordering has to be chosen in the index set, e.g. ascending indices or antisymmetry. A monomial build out of ngenerators and the linear span of such monomials is called a homogenous n-vector. Thereby a unique Zn-grading is introduced by the choice of a basis and an ordering. This method has the advantage of being plain in construction, easy to re- member, and powerful in computational means. 1.1.3 Naming A very important and delicate point in mathematics and physics is the appro- priate naming of objects and structures. Since we deal with a very well known structure, but want to highlight special novel features, we have to give distin- guishing names to di fferent albeit well known objects, which otherwise could not be properly addressed. This section shall establish such a coherent naming, at least for this article. 4 Clifford algebra is often denoted, following Cli fford himself and Hestenes, as ’Geometric Algebra’, GA or ’Cli fford Geometric Algebra’ CGA or ’Cli fford Grassmann Geometric Algebra’ CGGA [58]. Having the advantage of being descriptive this notation has, however, also a peculiar tendency to call uponconnotations and intuitions which might not in all cases be appropriate. Even at this stage, one has to distinguish ’Metric Geometric Algebra’ MGA and ’Projective Geometric Algebra’ PGA which relies on the identi Þcation of the homogenous multi-vector objects and geometrical entities [42]. In the former case, ’vectors’ are identi Þed with ’places’ —position vectors— of pseudo-Euclidean or unitary spaces while in the second case ’vectors’ are identi Þed with ’points’ of a projective space. Both variants, metric or projective, use unquestionably the arti Þcial multi- vector structure introduced by the mere notation of a basis and foreign to Clifford algebras to assert ’ontological’ statements such as: ’ xis a place in Euclidean space’ or ’ xis a point in a projective space’. Both of these interpretations have one thing in common, namely, they assert anobject character to the Cli fford elements themselves. We will coin for this case the term ’ Classical Cli fford Algebra ’. To our current experience, the Wick isomorphism developed below guaran- tees that such interpretation of Cli fford algebras is independent from the chosen Z n-grading. That is, we make the following conjecture: if the Cli fford ele- ments themselves are ’ontologically’ interpreted as ’place’ or ’point’ then all Zn-gradings are isomorphic through the Wick isomorphism. We turn to the second aspect. In [56] Oziewicz introduced the term ’Cli fford algebras of multi-vectors’ to highlight the fact that he considered di fferent Zn- gradings or, equivalently, di fferent multi-vector structures. However, Cli fford algebras have in nearly every case been used as multi-vector Cli fford algebras since mathematicians and physicists want to consider the n-vectors or multi- vectors for di fferent purposes. Following the introduction of Cli fford algebras of arbitrary bilinear forms, implicitly in [12] and explicitly in [1, 24, 25, 26, 27, 28, 29, 31, 32, 49, 55], situations have occurred for good physical reasons where di fferent Zn-gradings have led to di fferent physical outcomes. In those situations a theory of gradings is mandatory. An e wp o i n ti st h e operational approach to Clifford elements. If one considers aC l ifford number to be an operator, it has to act on another object, a ’state vector’. This ’ quantum point of view ’ moves also the ontological assertions into the states. Their interpretation however is di fficult. Moreover, one has to deal with representation theory which was not neces- sary in the ’classical’ Cli fford algebraic approach —in both senses of classical, i.e. also as opposed to quantum, here. Adopting Wigner’s de Þnition of a particle as an irreducible representation —of the Poincar´ e group— one has to seek irreducible representations of Cli fford algebras. It is a well known fact that these represen- tations are faithfully realized in spinor spaces. It is exactly at this place where itwill be shown in this article that one obtains di fferent Z n-gradings or di fferent multi-vector structures leading to di fferent results. In fact we are able to Þnd 5 irreducible spinor spaces of dimension 8 in C`2,2(B,V ),where 2 ,2d e n o t e s t h e signature of the symmetric part gofB,and not of dimension 4 as predicted by the ’classical’ Cli fford algebra theory. For the case of Cli fford algebras of multi-vectors we coin the term quantum Clifford algebra .1,2I ti sc l e a rt ou st h a tw er i s kc r e a t i n gac o n f u s i o nw i t h this term, which looks like a q-deformed version of an ordinary Cli fford algebra, while also in our case the common ’square law’ is fully valid! However, this link is not wrong! As we show elsewhere in these proceedings [4], one is able to Þnd Hecke algebras and q-symmetry within the structure of the quantum Cli fford algebra. It is also in accord with the attempt of G. Fiore, presented at this conference, to describe q-deformed algebras in terms of undeformed generators. This is just a reverse of our argument. However, the characteristic point in ourconsideration is that we dismiss the classical ontological interpretation in favor of an operational interpretation. Thereby it is necessary to study states which are now Z n-grade dependent. Our approach should be contrasted by the recent developments excellently described in [15, 51]. A di fferent treatment of Cli fford algebras in connection with Hecke algebras was given in [57]. As a last point, we emphasize that indecomposable spinor representations of unconventionally large dimensions are expected to be spinors of bound systems, see [27]. Hence, studying decomposability is the Þrst step towards an algebraic theory of compositeness including stability of bound states. 1.2 Why study C`(B,V )and not C`(Q, V )?—P h y s i c s Clifford algebras play without any doubt a predominant role in physics and mathematics. This fact was clearly addressed and put forward by D. Hestenes [38, 39, 40, 41]. Based on this solid ground, we give an analysis of Cli fford algebras of an arbitrary bilinear form which exhibit novel features especiallyregarding their representation theory. The most distinguishing fact between our approach and usual treatments of Cli fford algebras e.g., [6, 10, 14, 48, 60], is that we seriously consider how the Z n-grading is introduced in Cli fford algebras. This is most important since Cli fford algebras are onlyZ2-graded by their natural —functorial— construction. The introduction of a further Þner grading does therefore put new assumptions into the theory. One might therefore ask, if theses additional structures are important or even necessary in physics and mathematics. Indeed, after examining various cases we notice that every application of Clifford algebras which is computational —not only functorial— deals in fact with the so called Cli fford algebras of multi-vectors [56] or quantum Cli fford algebras . However, the additional Zn-grading, even if mathematically and physically nec- essary for applications, is usually introduced without any ado. Looking at liter- a t u r ew ec a nh o w e v e r Þnd lots of places where Zn-graded Cli fford algebras are 1This is close to Saller’s notion of a “quantum algebra” which denotes however a special choice of grading [6 1]. 2Classical Cli fford algebras emerge as a particular case of quantum Cli fford algebras. 6 not only appropriate but needed. This is in general evident in every quantum mechanical setup. If one analyzes functional hierarchy equations of quantum Þeld theory (QFT), one is able to translate these functionals with a help of Cli fford algebras. Such attempts have already been made by Caianiello [11]. He noticed that at least two types of orderings are needed in QFT, namely the time-ordering and normal- ordering. Since one has —at least— two possibilities to decompose Cli fford al- gebras into basis monomials, he introduces Cli fford and Grassmann bases. A basis of a Cli fford algebra is usually given by monomials with totally ordered index sets. If one has a Þnite number of ’vector’ elements ei,one can, by using the anti-commutation relations of the Cli fford algebra, introduce the following bases i) {1;e1,... , en;e1e2,...;ei1ei2ei3(i1<i2<i3),...} ii) {1;e1,... , en;e[1e2],...;e[i1ei2ei3],...}. (7) We used the [ ...] bracket to indicate antisymmetrization in the index set. An ordering of index sets is inevitable since the eiejandejeimonomials are not algebraically independent due to the anti-commutation relations eiej= −ejei+gij1.Caianiello identi Þes then the two above choices with time- and normal-ordering. However, already at this point it is questionable why one uses’lexicographical’ ordering ’ <’ and not e.g. the ’anti-lexicographical’ ordering ’>’ or an ordering which results from a permutation of the index set. A detailed study shows that fermionic QFT needs antisymmetric index sets and that there are in Þnitely many such choices [25, 31]. Using this fact we have been able to show that singularities, which arise usually due to the reorderingprocedures such as the normal-ordering, are no longer present in such algebras [32]. Studying the transition from operator dynamics to functional hierarchies, the so-called Schwinger-Dyson-Freese hierarchies, in [25, 31] it turned out thatthe multi-vector structure, or, equivalently a uniquely chosen Z n-grading, was anecessary input to QFT. Multi-particle systems provide a further place where a careful study of grad- ings will be of great importance. It is a well known fact that one has the Clebsch-Gordan decomposition of two spin-1 2particles as follows [33, 37]: 1 2⊗1 2=0⊕1. (8) However, since this is an identity, it can be used either from left to right to form bosonic spin 0 and spin 1 ’composites’ orfrom right to left! There is no way —besides the experience— to distinguish if such a system is composed, that is, dynamically stable or not, see [26]. From a mathematical point of view onecannot distinguish nfree particles from an n-particle bound system by means of algebraic considerations. This is seen clearly in the decomposition theorems for Cli fford algebras where larger Cli fford algebras are decomposed into smaller blocks of Cli fford tensor factors. This cannot be true for bound objects which lose their physical character when being decomposed. An electron and proton 7 system is quite di fferent from a hydrogen atom. In this work, we will see, that one can indeed Þnd such indecomposable states in quantum Cli fford algebras. This raises a question how to distinguish such situations. One knows from QFT that interacting systems have to be described in non-Fock states and thatthere are in Þnitely many such representations [35]. It is thus necessary to intro- duce the concept of inequivalent states inÞnite dimensional systems [27, 44, 45]. Such states are necessarily non-Fock states, since Fock states belong to systems of non-interacting particles. This is the so-called free case which is however very useful in perturbation theory. The present paper supports the situation foundin [27]. Closely related to these inequivalent states are condensation phenomena . As it was shown in [27], one can algebraically determine boundedness usingan appropriate Z n-grading. Furthermore, it was shown that the dynamics determines correct grading. In BCS theory of superconductivity the fact that bound states can or cannot be build was shown to imply a gap-equation [27] which governs the phase transition. A further point related to Zn-graded Cli fford algebras is q-quantization. This can be seen when studying physical systems as in [30] and when adopting a more mathematical point of view as in [24, 28]. In these proceedings a detailed example was worked out to show how q-symmetry and Hecke algebras can be described within quantum Cli fford algebras [4]. It is quite clear that this structure should play a major role in the discussion of the Yang-Baxter equation, the knot theory, the link invariants and in other related Þelds which are crucial for the physics of integrable systems in statistical physics. However, the most important implication from these various applications is that the q-symmetry and more general deformations are symmetries of com- posites . This was already addressed in [30] and more recently in [24]. Also the present work provides full support for this interpretation, as the talk of G. Fioreat this conference. Providing as much evidence as possible to this fact was a major motivation for the present work. 1.3 Why study C`(B,V )and not C`(Q, V )?— M a t h e m a t i c s There are also arguments of purely mathematical character which force us to consider quantum Cli fford algebras. If we look at the construction of Cli fford algebras by means of the tensor algebra, we notice that C`is a functor. To every quadratic space H=(V,Q), ap a i ro fal i n e a rs p a c e Vover a ring Rand a quadratic form Q,there is a uniquely connected Cli fford algebra C`(Q, V ).That is, one can introduce the algebra structure without any further input or choices, so to say for free. One may further note that if the characteristic of the ring Ris not 2 ,then there is a one-to-one correspondence between quadratic forms and classes of symmetric matrices [62]. In other words, every symmetric matrix is a representation of a quadratic form in a special basis. Over the reals (complex numbers) the classesof quadratic forms can be labeled by dimension nand signature s(dimension n only, no signature in C).Equivalently one can use the numbers p, qof positive 8 and negative eigenvalues of the quadratic form. This leads to a classi Þcation (naming) of real (and complex) Cli fford algebras. One writes C`(Q, V )'C`p,q (C`n) where dim V=n=p+qandQhas signature s=p−q.The remarkable fact is that the ’square law’ for vectors Q(v)≡v2=α1∈C`(Q;V)(α∈R orα∈C) is a diagonal map determining only the symmetric part of the map Q(V)7→R.Following Cli fford one should note that the product operation can be seen as acting on the second factor 2 ×xas a doubling of x;t h a t i s ,2 ×is a doubling operator or endomorphism acting on the space of the second factor. In this sense any ’Cli fford number’ induces an endomorphism on the graded space Wunderlying the algebra and it is questionable why one should use only diagonal maps and their symmetric polarizations. Furthermore, note that one has quadratic forms 'bilinear forms alternating forms. (9) The dualization V7→V∗'lin-Hom( V,R) is performed by an arbitrary (non- degenerate) bilinear form. Endomorphisms have in general the following form End(V)'V⊗V∗, (10) so why do we restrict ourselves to the symmetric case? If we consider a pair (V,B)o f a s p a c e Vand an arbitrary bilinear form B,can we construct func- torially an algebra like the Cli fford algebra for the pair H=(V,Q)? It can be easily checked that if one insists on the validity of the ’square law’ v2=α1,theanti-commutation relations of the resulting algebra are the same as for usual Cli fford algebras while the commutation relations —and thus the meaning of ordering and grade— is changed. Let B=g+A, AT=−A, gT=g. We denote B(x,y)=xBy,A(x,y)=xAyandg(x,y)=xgy(the latter also denoted by Hestenes and Sobczyk as x·y).3Then, the B-dependent Clifford product xy Bof two 1 -vectors xandyinC`(B,V ) can be decomposed indifferent ways into scalar and bi-vector parts as follows xy B=x gy+xú∧y Hestenes, common case, A=0 xy B=x By+x∧y Oziewicz, Lounesto, Abà lamowicz, Fauser ,(11) where xú∧y=x∧y+A(x,y)=x∧y+xAy.Of course, for any 1 -vector xand any element uinC`(B,V )w e h a v e : xu B=x Bu+x∧u=x gu+x Au+x∧u=x gu+xú∧u. (12) Notice that the element xú∧u=xAu+x∧uis not even a homogenous multi-vector inVV.We have thus established that the multi-vector structure 3The symbols B,Aand gdenote the left contraction in C`(B,V )w i t h r e s p e c t t o B, A andgrespectively. 9 is uniquely connected with the antisymmetric part Aof the bilinear form, see also [1, 29, 31]. This has an immediate consequence: in some cases one Þnds bi-vector ele- ments which satisfy minimal polynomial equations of the Hecke type [24, 28].This feature is treated extensively elsewhere in this Volume [4]. Some mathematical formalisms, not treated here, are closely connected to this structure. One is the structure theory of Cli fford algebras over arbitrary rings [36] where a classi Þcation is still lacking. Connected to these questions is the arithmetic theory of Arf invariants and the Brauer-Wall groups. Much more surprising is the fact that due to central extensions the ungraded bi-vector Lie algebras turn into Kac-Moody and Virasoro algebras [54] and, as it is also shown in [4], to some q-deformed algebras. Since Cli fford algebras naturally contain re ßections, automorphisms gener- ated by non-isotropic vectors, we expect to Þnd in Þnite dimensional Coxeter groups [17, 43], a ffine Weyl groups etc., connected to Z n-graded or quantum Clifford algebras. Involutions connected to special elements, norms and traces [36] are also affected by di fferent gradings. This has considerable e ffects. One important point is that the Cauchy-Riemann di fferential equations are altered which makes probably the concept of monogeneity [48] grade dependent. However, this isspeculative. 2 Chevalley’s approach to Cli fford algebras 2.1 Confusion with Chevalley’s approach Chevalley’s book “The algebraic theory of spinors” [12] seems to have been badly accepted by working mathematicians and physicists despite its frequent citation. Albert Crumeyrolle stated the following in [14], p. xi: In spite of its depth and rigor, Chevalley’s book proved too abstract for most physicists and the notions explained in it have not been applied much until recently, which is a pity. The more compact and readable book “The study of certain important algebras” [13] seems to be little known. However, one can Þnd in many physical writings e.g. Berezin [7] very analogous structures, without mentioning the much more complete work of Chevalley. When looking for the most general construction of Cli fford algebras over arbitrary rings including the case where the characteristic of Ris 2,Chevalley constructed the so-called Clifford map. This map is an injection of the linear space Vinto the algebra C`(V) which establishes the ’square law’. This con- struction emphasizes the operator character of Cli fford algebras and establishes a connection between the spaces underlying the Zn-graded Grassmann algebra and the thereon constructed Cli fford algebra. For our purpose it is important 10 that only Chevalley’s construction allows a non-symmetric bilinear form in con- structing Cli fford algebras. However, this fact is not explicit in Chevalley’s writings but it is clearly emphasized in [55]. Ironically, a careful analysis of Lounesto shows that even Crumeyrolle made a mistake in describing the Chevalley isomorphism connecting Grassmann and Clifford algebra spaces. In [50] Lounesto points out that Crumeyrolle rejects the Chevalley isomorphism for any characteristic. This seems to be implied by Crumeyrolle’s frequent questioning, also in previous Cli fford conferences of this series: “What is a bi-vector?” [53]. However, an isomorphism can beuniquely given if the characteristic of Ris not 2 ,see [49, 50]. On the other hand, Lounesto points out that Lawson and Michelsohn [47] postulate such an isomorphism which is wrong in the exceptional case of characteristic 2 .One should note in this context that their point of view is taken by almost all working mathematicians and physicists. At this point we submit, that we insist on Chevalley’s construction even in t h ec a s eo fc h a r a c t e r i s t i cn o t 2 .Lounesto claims that in this cases C`(B,V )i s isomorphic to C`(Q, V )w i t h Qthe quadratic form associated to B.In fact, this is true for the Cli fford algebraic structure and was proved in [1] up to the dimension 9 of V.However, this, the so-called Wick isomorphism between C`(B,V )a n d C`(Q, V ),has to be rejected when the Z n-grading is considered, or, in other words, the multi-vector structure. Hence, we reject Lounesto’s judgment that it is worth studying C`(B,V ) only in characteristic 2 for the reason of carefully treating the involved Zn-grading or multi-vector structure. This is one of the main points of our analysis. 2.2 Chevalley’s construction of C`(B,V ) A detailed and mathematical rigorous development of quantum Cli fford algebras C`(B,V ) can be found in [24, 31]. We will develop only the notation and point out some peculiar features insofar as they appear in the present study, see also [1, 29]. The main feature of the Chevalley approach is that Cli fford algebras are constructed as special —satisfying the ’square law’— endomorphism algebras on —the linear space of— a Grassmann algebra. In this way the Grassmann algebra,which is naturally Z n-graded, induces via the Chevalley isomorphism a grading or multi-vector structure in the Cli fford algebra. This grading is however not preserved by the Cli ffo r dp r o d u c tw h i c hr e n d e r st h eC l i fford algebra to be a deformation of the Grassmann algebra. To proceed along this line we construct the Grassmann algebra as a factor algebra of the tensor algebra. Let IG:={X|X=A⊗(x⊗x)⊗B} (13) with notation as in (2) and de Þne ^ V:=T(V) IG,π:T(V)7→^ V. (14) 11 The projected tensor product π(⊗)7→∧ is denoted as wedge or outer product. The induced grading is ^ V=R⊕V∧V⊕...⊕∧nV⊕... . (15) As the next step, we consider re ßexive duals of the linear space V.DeÞne V∗:= lin-Hom( V,R)( 1 6 ) where dim V∗=d i m V(reßexivity). Using the action of the dual elements on Vwe de Þne the (left) contraction Bas: ix(y)=x By=B(x,y). (17) Note, that ix∈V∗is the dualized element xand that here a certain duality map is employed. If this is the usual duality map iei(ej)=δijone denotes this as Euclidean dual isomorphism and writes the map as ?[61]. The notation xBya n dm u c hm o r e B(x,y) is very peculiar since we have B:V×V7→V, B :V×V7→V. (18) Hence, BandBare in lin-Hom( V×V,R)'V∗×V∗.In this notation a dual isomorphism is implicitly involved, since we consider really maps of the form <.|.>:V∗×V7→R (19) which might be called ad u a lp r o d u c t orap a i r i n g [8, 61]. Having de Þned the action of V∗onV,we lift this action to the entire Grassmann algebrasVVandVV∗.Forx,y∈V,andu, v, w ∈VVwe have: i) x By=B(x,y), ii) x B(u∧v)=(x Bu)∧v+ˆu∧(x Bv), iii)( u∧v) Bw=u B(v Bw), (20) where ˆ is the involutive map —grade involution— ˆ : V7→−Vlifted toVV. The Cli fford algebra C`(B,V ) is then constructed in the following way. De Þne an operator L± x:VV7→VVfor any x∈Vas: (L± x)2:=x B·±x∧· (21) and observe that this is a Cli fford map [12, 24, 31] (L± x)2=±Q(x)1, (22) where Q(x)=B(x,y).This is nothing else as again the ’square law’, and one proceeds as in the case of generators and relations. Chevalley has thus established that C`(B,V )⊂End(^ V). (23) This inclusion is strict. 12 3 Wick isomorphism and Zn-grading 3.1 Wick isomorphism In this section we will prove the following Theorem: C`(B,V )∼=C`(Q, V )( 2 4 ) asZ2-graded Cli fford algebras. This isomorphism, denoted below by φ,is the Wick isomorphism since it is the well know normal-ordering transformation of the quantum Þeld theory [21, 31, 64]. This was not noticed for a long time which is another ’missed op- portunity’ [22]. Proof: The proof proceeds in various steps, numbered by letters a, b, c, etc. After de Þning the outer exponential, we prove the following important formulas: i) e−F ∧∧eF ∧=1, ii) e−F ∧∧x∧eF ∧∧u=x∧u, iii) e−F ∧∧(x g(eF ∧∧u)) = x gu+(x gF)∧u, (25) and Þnally we show that the Wick isomorphism φis given as: C`(B,V )= φ−1(C`(g,V)) =e−F ∧∧C`(Q, V )∧eF ∧ (26) ∼=(C`(g,V),<.>A r)( 2 7 ) where <.>A rdenotes the A-dependent Zn-grading. That is, the isomorphism is given by the following transformation of vector variables which is then algebraically lifted to the entire algebra: x g·→x B·=x g·+(x gF)∧· x∧·→x∧· (28) a)According to Hestenes and Sobczyk [39] it is possible to express every anti- symmetric bilinear form in the following way A(x,y): =F g(x∧y)( 2 9 ) where Fis an appropriately chosen bi-vector. Fcan be decomposed in a non- unique way into homogenous parts Fi=ai∧bi,F =PFi.We de Þne the outer exponential of this bi-vector as ( ∧0F=1) eF ∧:=X1 n!∧nF=1+F+1 2F∧F+...+1 n!∧nF+... . (30) This series is Þnite when the dimension of VisÞnite since in that case there exists a term of the highest grade. 13 b)Substitute the series expansion (30) into (25-i) and note that after applying the Cauchy product formula for sums we have e−F ∧∧eF ∧=∞X r=0ÃrX l=0(−1)lµr l¶! 1 r!∧rF. (31) The alternating sum of the binomial coe fficient is zero except in the case r=0 w h e nw eo b t a i n 1, which proves formula (25-i). c)To prove (25-iii) one needs the commutativity of xgFiwith Fj.If the contraction is zero, it commutes trivially, if not, the contraction is a vector y. From y∧F=F∧yf o re v e r yb i - v e c t o r ,w eh a v et h a t xgFicommutes with anyFjand thus with F.This allows us to write x g(∧nF)=n(x gF)∧(∧(n−1)F). (32) Once more using ∧0F=1,the Leibniz’ rule and the fact that ˆF=F,we obtain x g(eF ∧∧u)=eF ∧∧(x gu+(x gF)∧u), (33) which proves (25-iii). d)Since any vector ycommutes under the wedge with any bi-vector F,the case (25-ii) reduces to b). e)The Wick isomorphism is now given as C`(B,V )=φ−1(C`(g,V)) =e−F ∧∧ C`(Q, V )∧eF ∧.The same transformation can be achieved by decomposing every Clifford ’operator’ into vectorial parts and then into contraction and wedge parts w.r.t. ( g,∧) and then performing the substitution laws given in (28) and a Þnal renaming of the contractions; see [31] for an application in quantum Þeld theory. Note, that since the wedges are notaltered and the new contractions are given by xB·≡dx(·): =xg·+(xgF)∧·,this transformation does mix grades, but it respects the parity. It is thus a Z2-graded isomorphism. QED. An equivalent proof was delivered in [63] without using (explicitly) Cli fford algebras but index doubling —see below. The Wick isomorphism was called there’nonperturbative normal-ordering’. 3.2 C`(B,V )↔C`(Q, V )—I s o m o r p h i cy e td i fferent? We have already discussed that many researchers reject the idea that C`(B,V ) is of any use because of the Wick isomorphism. However, as our proof hasshown this isomorphism is only Z 2-graded. Indeed it was not the mathematical opportunity, but a necessity in modeling quantum physical multi-particle sys- tems and quantum Þeld theory which forced us to investigate quantum Cli fford algebras [25, 27, 31, 32]. 14 Decomposing Bintog,A as in (12) and noting that in our case, of charac- teristic not 2 one has Q(x)=g(x,x),one concludes that C`(Q, V )i s e x a c t l y the equivalence class of C`(B,V )'C`(g+A, V)w i t h Avarying arbitrarily: C`(Q, V )=[C`(g+A,V)]. (34) In other words, one does not have a single Cli fford algebra C`(Q, V ) but an entire class of equivalent —under the Z2-graded Wick isomorphism— Cli fford algebras C`(B,V ).This can be written as C`(Q, V )'C`(g+A, V)m o d A (35) which induces a unique projection from the class of quantum Cli fford algebras onto the classical Cli fford algebra. Such a projection πcan be de Þned as: i) π:T(V)7→C`(B,V ) ii) <.>A r:=π(⊗rV). (36) This is once more a sort of ’cancellation law’. The important fact is that only those properties belong to C`(Q, V ) which do notdepend on the particular choice of a representant parameterized by A.Physically speaking, only those properties belong to C`(Q, V ) which are homogenous over the entire equivalence class. As we will show now, especially the multi-vector Zn-grading is notof this simple type. Recall that it is possible to decompose the Cli fford product in various ways as in (11) and (12). Hence we obtain a relation between the ∧- and the ú∧-grading as: xú∧y=A(x,y)+x∧y (37) which shows that a ú∧-bi-vector is an inhomogeneous ∧-multi-vector and vice versa. Since the antisymmetric part can be absorbed in the wedge product, using the Wick isomorphism, we can give the grading explicitly by writing <.>A r=<.>˙∧ r (38) with respect to the doted wedge ú∧within the undeformed algebra C`(Q, V ), see G. Fiore’s talk. This gives us a second characterization of C`(B,V ),namely C`(B,V )'(C`(Q, V ),<.>A r). (39) That is, C`(B,V ) can be seen as a pair of a classical Z2-graded Cli fford algebra C`(Q, V ) and a unique multi-vector structure given by the projectors <.>A r. As a main result we have that these algebras are notisomorphic under the Wick isomorphism C`(g+A1,V)6' WickC`(g+A2,V)iffA16=A2. (40) 15 4 Periodicity theorems Our theory will have an impact on all famous periodicity theorems of Cli fford algebras, especially on the Atiyah-Bott-Shapiro mod 8 index theorem [5]. But to be as concrete and explicit as possible, we restrict ourself to the case C`p,q' C`p−1,q−1⊗C`1,1.Periodicity theorems can be found, for example, in [6, 10, 46, 52, 60]. We need some further notation. Let Vp,q=(gp,q,V) b eaq u a d r a t i cs p a c e , where g=d i a g ( 1 ,... , 1,−1,... ,−1) with pplus signs and qminus signs, and letVbe a linear space of dimension p+q.A c c o r d i n gt ot h eW i t tt h e o r e m[ 6 5 ] one can split o ffa quadratic space of the hyperbolic type M1,1.This split is orthogonal with respect to g: Vp,q=Np−1,q−1⊥gM1,1. (41) If one applies the Cli fford map γ:Vp,q7→C`p,qand de Þnes its natural restric- tions γ0:Np−1,q−17→C`p−1,q−1,γ00:M1,17→C`1,1,one obtains the following Periodicity Theorem: C`p,q'C`p−1,q−1⊗C`1,1. (42) While in this special case the tensor product may be ungraded, in general the tensor product in such decompositions may be graded or not, see [10, 46, 52]. Using the obvious notation C`(Vp,q)=C`p,q(Q) and introducing the re- strictions of the Wick isomorphism φ−1|Nand φ−1|M,(here N=Np−1,q−1 andM=M1,1),we can calculate the decomposition of C`p,q(B).However, if there are terms in the bi-vector Fwhich connect spaces NandM,that is, if F=PFiand if there exists Fs=as∧bswith as∈N,bs∈M,this part of the construction belongs neither to the restriction φ−1|Nnor toφ−1|M. We have either no tensor decomposition oradeformed tensor product . E x p r e s s e di nf o r m u l a sw eg e t : C`p,q =φ−1(C`p,q(Q)) =φ−1[C`p−1,q−1(Q|N)⊗C`1,1(Q|M)] =C`p−1,q−1(B|N)(φ−1⊗)C`1,1(B|M) =C`p−1,q−1(B|N)⊗φ−1C`1,1(B|M). (43) Remark: The deformed tensor product ⊗φ−1is not braided by construction, s i n c ew eh a v en or e s t r i c t i o n so n φ−1.But one is able to Þnd e.g., Hecke elements, etc., necessary for a common q-deformation or, more generally, a braiding. As the main result of our investigation we have shown that quantum Clif- ford algebras do not come in general with periodicity theorems as e.g. the famous Atiyah-Bott-Shapiro mod 8 index theorem. This has enormous impact on quantum manifold theory and the topological structure of such spaces as wellas on their analytical properties. However, we have constructed a deformed —not necessarily braided— tensor product ⊗ φ−1which gives a decomposition at the 16 cost of losing (anti)-commutativity. To fully support this view and convince also those Readers who might consider our reasoning too abstract and only formal in nature, we proceed to provide some examples. 5 Examples In this section we consider three examples each of them pointing out a peculiar feature of quantum Cli fford algebras and Zn-gradings. Two of these examples have been found by using CLIFFORD, a Maple V Rel. 5 package for quantum Clifford algebras [2, 3]. While the second example is generic, the third one was taken from [27] and provides an example of a physical theory which bene Þts extraordinarily from using quantum Cli fford algebras. 5.1 Example 1 This example shows that even in classical Cli fford algebras one does not have a unique access to the objects of the graded space. Consider the well-known Dirac γmatrices which generate the Dirac-Cli fford algebra C`1,3and satisfy γiγj+γjγi=2ηij1with the Minkowski metric ηij=d i a g ( 1 ,−1,−1,−1).The linear span of the γ-matrices (generators) contains 1 -vectors x=Pxiγi. DeÞneγ5=γ0γ1γ2γ3and note that γ2 5=−1.If we de Þnenew generators αi:=γiγ5which are 3-vectors(!), it is easily checked that they nevertheless fulÞllαiαj+αjαi=2ηij1.They might be called vectors on an equal right. DeÞne the map γ5:C`1,37→C`1,3,x7→x0:=xγ5,lifted to C`1,3.We have thus de Þned two different Clifford maps γ:V7→C`1,3and γ0:V7→C`1,3 with γ0:=γ5◦γ.That is one can’t know for sure which elements are ’vectors’ even in this case. We emphasized earlier that we did not expect the interpretation and the mathematical aspects of classical Cli fford algebras to change in such a trans- formation. However, see [16] for a far more elaborate application of a similar situation where both gradings are used. 5.2 Example 2 In this example we examine the split case C`2,2'C`1,1⊗C`1,1and show the existence and irreducibility of an 8 -dimensional representation not known in the classical representation theory of Cli fford algebras. We start with C`1,1(B)w h e r e Bis given as B:=µ1a 0−1¶ . (44) Ifais zero, we have two choices for an idempotent element generating a spinor space: f− 11:=1 2(1+e1), f+ 11:=1 2(1+e1∧e2). (45) 17 A spinor basis can be found in both cases by left multiplying by e2which yields S±=<f± 11,e2f± 11>.The spinor spaces S±are 2 -dimensional and the Cli fford elements are represented as 2 ×2 matrices. If ais not zero, an analogous construction runs through. Now let us put together two such algebras, as shown in [52], generated by C`1,1=<e1,e2>andC`1,1=<e3,e4>.The bilinear form Bwhich reduces in both cases to the above setting and which contains connecting elements is B:= 1an 11n12 0−1n21n22 00 1 a 00 0 −1 . (46) We expect the nijparameters to govern the deformation of the tensor product in the decomposition theorem. Searching with CLIFFORD for idempotents in this general case yields the following fact. Let λbe a Þxed parameter. Among six choices for an idempotent f,we found f:=1 2(1+X1)=1 4(2 + λa)1+1 4p 4−λ2a2−4λ2e1+1 2λe1∧e2 where X1is one of six di fferent, non-trivial, and general elements XinC`(B,V ) satisfying X2=1.This is an indecomposable idempotent which therefore gener- ates an irreducible 8dimensional representation since the regular representation ofC`(B,V )i s o f d i m e n s i o n1 6 .This fact depends on the appearance of the non-zero nijparameters. It was proved by brute force that none of the remain- ingÞve non-trivial elements Xi,i=2,... , 6,and squaring to 1commuted with X1.Thus, the search showed that there is no second Cli fford element X26=X1 which would square to 1and which would commute with X1.S u c ha ne l e m e n t would be necessary to decompose fi n t oap r o d u c t f=Q i1 2(1+Xi)w h e r e XiXj=XjXiandX2 i=1.Since this type of reasoning can be used to classify Clifford algebras [19] we have found a way to classify quantum Cli fford algebras. This type of an indecomposable exotic representation will occur in the next example of a physical model and is thereby not academic. 5.3 Example 3 5.3.1 Index doubling For a simple treatment with a computer algebra, using CLIFFORD package, and for physical reasons not discussed here, see [25, 27, 31], we introduce an index doubling which provides us with a possibility to map the contraction andthe wedge onto a new Cli fford product in the larger algebra. The bene Þts of such a treatment are: the associativity of the mapped products, only one algebra product needed during calculations, etc. DeÞne the self-dual (re ßexive) space V=V⊕V ∗and introduce generators eiwhich span VandV∗ V=<e1,... , en>, V∗=<en+1,... , e2n>. (47) 18 In this transition we require that the elements eifrom Vgenerate a Grassmann sub-algebra and the en+1,... , e2n∈V∗are duals which act via the contraction onV.This gives the following conditions on the form B:V×V7→R: i) e2 i=ei∧ei∧·=0 ii) e2 n+i=en+1Ben+iB·=(en+i∧en+i) B·=0. (48) Thus, with respect to the basis <e1,... , en,en+1,... , e2n>,Bhas the fol- lowing matrix: B:=µ0g gT0¶ +A=g+A, (49) where, with an abuse of notation, the symmetric part of Bis again denoted byg.Note that we have introduced here a further freedom since Amay be non-trivial also in the V-VandV∗-V∗sectors. This fact has certain physical consequences which were discussed in [27]. The ei’s from Vcan be identi Þed with Schwinger sources of quantum Þeld theory [31, 25]. 5.3.2 The U(2)-model We simply report here the result from [27] and strongly encourage the reader to consult this work since we quote here only a part of that work which shows the indecomposability of quantum Cli fford algebra representations and the there- from following physical consequences. DeÞneC`(B,V )'C`2,2(B) by specifying V=<ei>=<a† 1,a† 2,a3,a4> and B:=1 2µ01 I 1I 0¶ +A, (50) where 1 I is the 2 ×2 unit matrix and Ais an arbitrary but Þxed 4 ×4 antisymmetric matrix with respect to the eioraibasis. Note furthermore that the aianda† ifulÞll the canonical anti-commutation relations, CAR, of a quantum system: {ai,a† j}+=δij.DeÞne furthermore Cli fford elements N,S i∈ R⊕V∧V,i∈{1,2,3}such that the following relations hold: [N,a i]−=−ai,[N,a† i]−=+a† i,N†=N, [Sk,ai]−=σijaj,h.c., k∈{1,2,3}, [Sk,N]−=0,[Sk,Sl]−=i²klmSm,S† K=Sk, (51) where †is the anti-involutive map (includes a product reversion) interchanging ai↔a† i.This is the U(2) algebra if A≡0. DeÞne a ‘vacuum’, for a discussion see [27], simply be de Þning the expecta- tion function —linear functional— as the projector onto the scalar part <.>A 0 which depends now explicitly on A.In a physicist’s notation <0|ˆH|0>' <H>A 0for any operator ˆHresp. Cli fford element H. 19 An algebraic analysis which coincides in the positive de Þnite case with C∗-algebraic results shows that this linear functional called ‘vacuum’ can be uniquely decomposed in certain extremal, that is indecomposable, states. De- noting these states as spinor like S1,S2and exotic Ewe obtain the following identity: <.>A 0=λ1<.>S1+λ2<.>S2+λ3<.>E,X λi=1. (52) Since the regular representation of C`2,2(B) is 16 dimensional and we Þnd dim S1=d i m S2=4,dimE= 8 this is a direct sum decomposition into irreducible representations. The ’classical’ case would have led to four representations of the spinor type each 4 dimensional. The indecomposable exotic representation obtained from <.>Eis therefore new and it is a direct outcome of the structure of the quantum Cli fford algebra, see previous example. This representation decomposes into two spinor like parts if Avanishes identically A≡0. 00.20.40.60.810 0.2 0.4 0.6 0.8 1 v-axisv-w-plane of vacua simplex of positive states w-axis Fock dual-Fock edge-stateLegend: Bogoliubov-Valatin vacua quasi-free vacua quasi-free neg. shift quasi-free pos. shift edge-Fock vacua dual-Fock-edge vacua bifurcation path Figure 1: C`(B)- d e f o r m a t i o n o f U(2) algebra In [27] we obtained a v-w-plane of vacua while implementing thePλi=1 condition and renaming of variables into v,w. There it was shown, see Fig- ure 1, that we Þnd free systems of Fock and dual-Fock type which constitutes the spinor representations S1,S2and that the line connecting them contains 20 Bogoliubov-transformed ground-states of BCS-superconductivity. Quasi free, that is correlation free, states are on the displayed parabola. In the exotic state one Þnds spin 1 and spin 0 components which are beyond Bogoliubov transfor- mations. Every choice of AÞxesexactly one particular state in the v-w-plane. Hence, we have solved the problem of Þnding an algebraic condition on which side of the Clebsch-Gordan identity1 2⊗1 2=0⊕1our algebraic system has to betreated. Our model, even if only marginally discussed, shows all features we want to see in the composite and multi-particle theory. Moreover, exotic representationswhich describe ’bound objects’ not capable of a decomposition are beyond the treatment in [20] which mimics in Cli fford algebraic terms the usual tensor method which generically bears this problem. In this context we refer to theinteresting work of Daviau [18] on de Broglie’s spin fusion theory [9] and to the joint works with Stumpf and Dehnen [23, 26] which are connected with algebraic composite theories. Acknowledgment The Þr s ta u t h o r( B F )a c k n o w l e d g e sat r a v e lg r a n to ft h eD F Ga n dac r i t i c a l reading of the manuscript by Th. Konrad. References [1] R. Abà lamowicz, P. 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