The Erwin Schrodinger International Pasteurgasse ESI Institute for Mathematical Physics A Wien Austria Pure Spinors in Odd Dimensions Wo jciech Kop czynski Vienna Preprint ESI Octob er Supp orted by Federal Ministry of Science and Research Austria Available via WWWESIACAT Pure spinors in o dd dimensions Wo jciech Kop czynski Instytut Fizyki Teoretycznej Uniwersytet Warszawski ul Hoza PL Warszawa Poland email kopczfuwedupl Abstract The pap er contains description of pure simple spinors in a complex vector space this particularly simple approach uses extensively Fo ck bases of the under lying vector space The pap er describ es pure spinors in a complexied vector space using a complexied version of Fo ck bases b etter known in relativity theory as null bases and intro ducing an imp ortant notion of real index Both these pap ers concern spinors in an evendimensional vector space The present pap er aims at descrption of pure spinors in o dddimensional vector spaces it is complementary to and however in one es sential place the sucient conditions for spinor to b e pure it go es b eyond using the PauliKonk typ e identity found in General spinors in o dd dimensions Let W b e a complex vector space of dimension m m and the central dot denotes the scalar pro duct in W Let ClW End S b e one of two irreducible representations of the Cliord algebra ClW in the spinor space S of complex dimension m In contrast to I have to make a distinction in notation b etween elements of ClW and corresp onding element of End S since is nonfaithful Eg the letter u denotes a vector and also its image in the Cliord algebra but not a corresp onding endomorphism of the spinor space the latter is denoted by u The fundamental multiplication prop erty in the Cliord algebra will b e written in the form u v v u u v where u v W ClW If e e is an orthonormal basis of W 1 2m+1 e e e e then we can intro duce an almost nul l basis n n p p e by the formulas 1 m 1 m e ie p e ie m e e n 21 2 21 2 2m+1 The elements of this basis satisfy the relations 2 n n n n p p p p e n p p n n e e n p e e p Null vectors of b oth kinds span a maximal total ly nul l MTN subspaces of W N spanfn n g P spanfp p g and we have the direct sum decomp osition 1 m 1 m W N P spanfeg Conversely given an MTN subspace N one can nd an MTN subspace P such that N P fg and a unit vector e orthogonal to N and P determined up to a sign such that the decomp osition holds One can also nd bases n n in N and 1 m p p in P such that the orthogonality conditions hold An arbitrary vector in 1 m W admits a unique decomp osition u n p z e where n N p P z C Writing m m X X n x n p y p =1 =1 we have 2 2 u u u x y x y z 1 1 m m The orthogonal group OV also SOV acts transitively on the set of all MTN subspaces of V this set has a natural structure of a complex manifold of dimension mm Let us x the orientation of an orthonormal basis e The pro duct of all e is prop ortional to the identity endomorphism of S The two inequivalent representations of ClW in S are distinguished by the sign b elow m i r mid 1 2m+1 S This formula can b e written in terms of the almost null basis as m n p n p e id 1 1 m m S The transp osing mapping B S S S is the dual of the spinor space S is dened by T m 1 B B It satises T m(m+1)2 B k With any two spinors one bilinearly asso ciate a k vector B W dened by k B v hB v i B v k where v is an arbitrary k vector The conventions for the scalar pro duct of k vectors are the same as in The formulae and lead to the statement that the symmetry prop erties of the blinear forms B are determined by k (mk )(mk +1)2 B B k k Therefore B for m k mo d k Pure spinors Following I shall intro duce two invariantly determined subspaces M and V asso ciated with any spinor M fu W u g V fB S g 1 M is always totally null TN ie M M Consider V It consists of all u W such that B u S 1 but B u hB ui 1 implies u since B is an isomorphism therefore V M Since it is quite obvious that M V we have V M It follows also that M V V These remarks apply also in the evendimensional case For economy of notation I shall avoid writing V in the sequel The spinor is pure if the subspace M is maximal totally null MTN In the evendimensional cale it simply means M M in o dd dimensions the situation seems to b e slightly more complicated dimM M Do there exist pure spinors in dimensions Since the representation of the even Clifford algebra Cl W is faithful there should exist S such that either + n n or n n e dep ending on the parity of m is nonzero 1 m 1 m Then M N and is pure The spinor similarly as in the evendimensional case can b e thought as a vacuum m state annihilated by the op erators n n The collection of spinors 1 m p p 1 m p p p p 1 2 m1 m p p 1 m is linearly indep endent in virtue of the anticommutation relations satised by n and p Therefore this is a basis of S All the spinors o ccuring in are pure eg the subspace M p p spanfn n p p g k +1 m 1 k k +1 m is TN and of dimension m The direction of a pure spinor is determined by corresp onding MTN subspace Indeed let S b e such that M M and let p p p 0 1 1 1m 1 m Multiplication by succesive pro ducts of n gives for any multiindex I I Therefore analogously like in the evendimensional case there holds Prop osition There is a natural onetoone corresp ondence b etween the set of all MTN subspaces of W and the set of directions of pure spinors What can b e said ab out the spinor e which is not present in The anticommu tation relations for e and n lead immediately to the conclusion that M e M 2 It follows from Prop osition that e and since e we have e Prop osition Let and b e linearly indep endent pure spinors There is a basis of S of the form such that is one of the basis vectors other than Proof mutatis mutandis can b e transferred from Prop osition The group Spinm acts transitively on the set of directions of all simple spinors Proof Following Prop osition if and are two pure spinors having dierent directions then there holds a relation of the form p p k +1 m p n p n k +1 k +1 m m Since all factors ab ove contain reections the group Pinm acts transitively At the same time holds p n p n e k +1 k +1 m m One of the pro ducts ab ove must represent an element of the group Spinm QED Prop osition A necessary and sucient condition for a spinor to b e simple is that the multivectors B for k m m m Moreover k the only nonvanishing multivectors of this form are B n n m 1 m and B B e m+1 m Proof To prove necessity consider the action of the endomorphism n n 1 m n n on elements of a basis of S constructed out of the pure spinor It is 1 m clear the result will b e nonvanishing only for the last element of the basis n n p p 1 m 1 m Therefore im n n spanf g This means that there exists a spinor such 1 m that n n B 1 m But according to B B which gives m m B n n B n n 1 m 1 m and this formula together with yields B B B B Since the spinors and ab ove are arbitrary we get thus B n n 1 m The tensor pro duct B can always b e expressed in terms of the basis 1 k where m and k m of End S 1 k m X m B B k k =0 Applying to we get the pro of of necessity and moreover the result To prove notice that if u is a vector and v a p vector then p v u u c v v u For any mvector v we have then m+1 m+1 B v n B n c v B n c v m+1 m1 b ecause the rst factor ab ove is zero Similarly B v e B v e B v B v m+1 m From that follows immediately the formula The suciency pro of follows closely that of Urbantke and is based on his form of the identity of the PauliKonk typ e m1 X m B B B B B B d 1 1 k k k k =0 where and are arbitrary spinors and co ecients d whose exact form can b e derived k from do not vanish for m k mo d We notice that the nullity conditions B lead consistently with the identity k to m B B B B 1 1 Consider the map P B and the subspace f S B g of S 1 Since B is nondegenerate the subspace has co dimension Thus dim S dim im P dim ker P dim M dim ker P and dim dim im P j ? dim ker P j ? But if follows from that im P j ? M and also that ker P j ? ker P Sub tracting from we get dim M dim M which means that M is MTN QED Prop osition If and are pure spinors then the dimension of the intersection M M is the least integer k such that B The multivector B is k k prop ortional to the exterior pro duct of the vectors of a basis of this intersection Proof We can nd an almost null basis such that M spanfn n g 1 m M spanfn n p p g 1 k k +1 m p p k +1 m and k dim M M Since m1 p n p n n n n m m m 1 m 1 m1 the recursive pro cedure gives mk k (2m+1k )2 p p n n n n k +1 m 1 m 1 k k vector