Understanding the second quantization of fermions in Clifford and in Grassmann space -- New way of second quantization of fermions, Part I
UUnderstanding the second quantization of fermions in Clifford and inGrassmann spaceNew way of second quantization of fermionsPart I
N.S. Mankoˇc Borˇstnik and H.B.F. Nielsen University of Ljubljana, Slovenia Niels Bohr Institute, Denmark
Both algebras, Clifford and Grassmann, offer ”basis vectors” for describing the internaldegrees of freedom of fermions [5, 6, 12]. The oddness of the ”basis vectors”, transfered to thecreation operators, which are tensor products of the finite number of ”basis vectors” and theinfinite number of momentum basis, and to their Hermitian conjugated partners annihila-tion operators, offers the second quantization of fermions without postulating the conditionsproposed by Dirac [1–3], enabling the explanation of the Dirac’s postulates. But while theClifford fermions manifest the half integer spins — in agreement with the observed prop-erties of quarks and leptons and antiquarks and antileptons — the ”Grassmann fermions”manifest the integer spins. In Part I properties of the creation and their Hermitian conju-gated annihilation operators of integer spins ”Grassmann fermions” are presented, solvingthe proposed equations of motion, applied on the vacuum state and on the Hilbert spaceof infinite number of ”Slater determinants” with all the possibilities of empty and occupied”fermion states”. In Part II the conditions are discussed under which the Clifford algebrasoffer the appearance of the second quantized fermions and the family quantum numbers. Inboth parts, Part I and Part II, the relation between the Dirac way and our way of the secondquantization is presented.
I. INTRODUCTION
In a long series of works we, mainly one of us N.S.M.B. ([5, 6, 12, 15–20] and the referencestherein), have found phenomenological success with the model named by N.S.M.B the spin-charge-family theory, with fermions, the internal degrees of freedom of which is describable with theClifford algebra of all linear combinations of products of γ a ’s in d = (13 + 1) (may be with dinfinity), interacting with only gravity. The spins of fermions from higher dimensions, d > (3 + 1),manifest in d = (3 + 1) as charges of the standard model , gravity in higher dimensions manifest asthe standard model gauge vector fields as well as the scalar Higgs and Yukawa couplings.There are two anticommuting kinds of algebras, the Grassmann algebra and the Clifford algebra a r X i v : . [ phy s i c s . g e n - ph ] O c t (of two independent subalgebras), expressible with each other. The Grassmann algebra, withelements θ a , and their Hermitian conjugated partners ∂∂θ a [12], can be used to describe the internalspace of fermions with the integer spins and charges in the adjoint representations, the two Cliffordalgebras, we call their elements γ a and ˜ γ a , can each of them be used to describe half integerspins and charges in fundamental representations. The Grassmann algebra is equivalent to the twoClifford algebras and opposite.The two papers explain how do the oddness of the internal space of fermions manifests in thesingle particle wave functions, relating the oddness of the wave functions to the correspondingcreation and annihilation operators of the second quantized fermions, in the Grassmann case andin the Clifford case, explaining therefore the postulates of Dirac for the second quantized fermions.We also show that the requirement that the Clifford odd algebra represents the observed quarks andleptons and antiquarks and antileptons reduces the Clifford algebra for the factor of two, reducingat the same time the Grassmann algebra, disabling the possibility for the integer spin fermions.In this paper it is demonstrated how do the Grassmann algebra — in Part I — and the twokinds of the Clifford algebras — in Part II — if used to describe the internal degrees of freedomof fermions, take care of the second quantization of fermions without postulating anticommutationrelations [1–3]. Either the odd Grassmann algebra or the odd Clifford algebra offer namely theappearance of the creation operators, defined on the tensor products of the ”basis vectors” of theinternal space and of the momentum space basis. These creation operators, together with theirHermitian conjugated partners anihilation operators, inherit oddness from the ”basis vectors”determined by the odd Grassmann or the odd Clifford algebras, fulfilling correspondingly, theanticommutation relations postulated by Dirac for the second quantized fermions, if they apply onthe corresponding vacuum state, Eq. (7) (defined by the sum of products of all the annihilationtimes the corresponding Hermitian conjugated creation operators). Oddness of the ”basis vectors”,describing the internal space of fermions, guarantees the oddness of all the objects entering thetensor product.In d -dimensional Grassmann space of anticommuting coordinates θ a ’s, i = (0 , , , , , · · · , d ),there are 2 d ”basis vectors”, which are superposition of products of θ a . One can arrange them intothe odd and the even irreducible representations with respect to the Lorentz group. There are aswell derivatives with respect to θ a ’s, ∂∂θ a ’s, taken in Ref. [12] as, up to a sign, Hermitian conjugatedto θ a ’s, ( θ a † = η aa ∂∂θ a , η ab = diag { , − , − , · · · , − } ), which form again 2 d ”basis vectors”. Againhalf of them odd and half of them even (the odd Hermitian conjugated to odd products of θ a ’s, theeven Hermitian conjugated to the even products of θ a ’s). Grassmann space offers correspondingly2 · d degrees of freedom.There are two kinds of the Clifford ”basis vectors”, which are expressible with θ a and ∂∂θ a : γ a = ( θ a + ∂∂θ a ), ˜ γ a = i ( θ a − ∂∂θ a ) [6, 13, 14]. They are, up to η aa , Hermitian operators. Each ofthese two kinds of the Clifford algebra objects has 2 d operators. ”Basis vectors” of Clifford algebrahave together again 2 · d degrees of freedom.There is the odd algebra in all three cases, θ a ’s, γ a ’s, ˜ γ a ’s, which if used to generate the creationand annihilation operators for fermions, and correspondingly the single fermion states, leads tothe Hilbert space of second quantized fermions obeying the anticommutation relations of Dirac [1]without postulating these relations: the anticommutation properties follow from the oddness ofthe ”basis vectors” in any of these algebras.Let us present steps which lead to the second quantized fermions: i. The internal space of a fermion is described by either Clifford or Grassmann algebra of an oddClifford character (superposition of an odd number of Clifford ”coordinates” (operators) γ a ’s orof an odd number of Clifford ”coordinates” (operators) ˜ γ a ’s) or of an odd Grassmann character(superposition of an odd number of Grassmann ”coordinates” (operators) θ a ’s). ii. The eigenvectors of all the (chosen) Cartan subalgebra members of the corresponding Lorentzalgebra are used to define the ”basis vectors” in the odd part of internal space of fermions. (TheCartan subalgebra is in all three cases chosen in the way to be in agreement with the ordinarychoice.) The algebraic application of this ”basis vectors” on the corresponding vacuum state(either Clifford | ψ oc > , defined in Eq. (18) of Part II, or Grassmann | φ og > , Eq. (7), which is in theGrassmann case just the identity) generates the ”basis states”, describing the internal degrees offreedom of fermions. The members of the ”basis vectors” manifest together with their Hermitianconjugated partners properties of creation and annihilation operators which anticommute, Eq. (11)in Part I and Eq. (18) in Part II, when applying on the corresponding vacuum state, due to thealgebraic properties of the odd products of the algebra elements. iii. The plane wave solutions of the corresponding Weyl equations (either Clifford, Eq. (23) orGrassmann, Eq. (21)) for free massless fermions are the tensor products of the superposition of themembers of the ”basis vectors” and of the momentum basis. The coefficients of the superpositioncorrespondingly depend on a chosen momentum (cid:126)p , with | p | = | (cid:126)p | , for any of continuous manymoments (cid:126)p . iv. The creation operators defined on the tensor products, ∗ T , of superposition of finite number of”basis vectors” defining the final internal space and of the infinite (continuous) momentum space,Eq. (24) in the Clifford case and Eq. (22) in the Grassmann case, have infinite basis. v. Applied on the vacuum state these creation operators form anticommuting single fermionstates of an odd Clifford/Grassmann character. vi.
The second quantized Hilbert space H consists of ”Slater determinants” with no single particlestate occupied (with no creation operators applying on the vacuum state), with one single particlestate occupied (with one creation operator applying on the vacuum state), with two single particlestates occupied (with two creation operator applying on the vacuum state), and so on. ”Slaterdeterminants” can as well be represented as the tensor product multiplication of all possible singleparticle states of any number. vii. The creation operators together with their Hermitian conjugated partners annihilationoperators fulfill, due to the oddness of the ”basis vectors”, while the momentum part commutes,the anticommutation relations, postulated by Dirac for second quantized fermion fields, not onlywhen they apply on the vacuum state, but also when they apply on the Hilbert space H , Eq. (39)in the Clifford case and Eq. (34) in the Grassmann case. In the Clifford case this happensonly after ”freezing out” half of the Clifford space, as it is shown in Part II, Sect. 2.2, whatbrings besides the correct anticommutation relations also the ”family” quantum number to eachirreducible representation of the Lorentz group of the remaining internal space.The oddness of the creation operators forming the single fermion states of an odd character,transfers to the application of these creation operators on the Hilbert space of the second quantizedfermions in the Clifford and in the Grassmann case. viii. Correspondingly the creation and annihilation operators with the internal space describedby either odd Clifford or odd Grassmann algebra, since fulfilling the anticommutation relationsrequired for the second quantized fermions without postulates, explain the Dirac’s postulates forthe second quantized fermions.In the subsection I A of this section we discuss in a generalized way our assumption, that theoddness of the ”basis vectors” in the internal space transfer to the corresponding creation andannihilation operators determining the second quantized single fermion states and correspondinglythe Hilbert space of the second quantized fermions.We present in Sect. II properties of the Grassmann odd (as well as, for our study of anticommut-ing ”Grassmann fermions” not important, the Grassmann even) algebra and of the chosen ”basisvectors” for even ( d = (2(2 n + 1) , n ), n is an integer) dimensional space-time, d = ( d −
1) + 1, andillustrate anticommuting ”basis vectors” on the case of d = (5 + 1), Subsect. II A, chapter A.b. .We define the action for the integer spin ”Grassmann fermions” in Subsect. II B. Solutions of thecorresponding equations of motion, which are the tensor products of finite number of ”basis vectors”and of infinite number of basis in momentum space, define the creation operators depending oninternal quantum numbers and on (cid:126)p in d -dimensional space-time. We illustrate the correspondingsuperposition of ”basis vectors”, solving the equation of motion in d = (5 + 1) in chapter B.a. .We present in Sect. III the Hilbert space H of the tensor multiplication of one fermion creationoperators of all possible single particle states of an odd character and of any number, representing”Slater determinants” with no ”Grassmann fermion” state occupied with ”Grassmann fermions”,with one ”Grassmann fermion” state occupied, with two ”Grassmann fermion” states occupied,up to the ”Slater determinant” with all possible ”Grassmann fermion” states of each of infinitenumber of momentum (cid:126)p occupied. The Hilbert space H is the tensor product (cid:81) ∞ ⊗ N of finitenumber of H (cid:126)p of a particular momentum (cid:126)p , for (continues) infinite possibilities for (cid:126)p .On H the creation and annihilation operators manifest the anticommutation relations of secondquantized ”fermions” without any postulates. These second quantized ”fermion” fields, manifestingin the Grassmann case an integer spin, offer in d -dimensional space, d > (3 + 1), the descriptionof the corresponding charges in adjoint representations. We follow in this paper to some extentRef. [12].In Subsect. III C relation between the by Dirac postulated creation and annihilation operatorsand the creation and annihilation operators presented in this Part I — for integer spins ”Grassmannfermions” — are discussed.In Sect. IV we comment on what we have learned from the second quantized ”Grassmannfermion” fields with integer spin when internal degrees of freedom are described with Grassmannalgebra and compare these recognitions with the recognitions, which the Clifford algebra is offering,discussions on which appear in Part II.In Part II we present in equivalent sections properties of the two kinds of the Clifford algebrasand discuss conditions under which odd products of odd elements (operators), γ a and ˜ γ a ’s of thetwo Clifford algebras, demonstrate the anticommutation relations required for the second quantizedfermion fields on the Hilbert space H = (cid:81) ∞ ⊗ N H (cid:126)p , this time with the half integer spin, offering in d -dimensional space, d > (3 + 1), the description of charges, as well as the appearance of familiesof fermions [12], both needed to describe the properties of the observed quarks and leptons andantiquarks and antileptons, appearing in families.In Part II we discuss relations between the Dirac way of second quantization with postulatesand our way using Clifford algebra.This paper is a part of the project named the spin-charge-family theory of one of the authors(N.S.M.B.), so far offering the explanation for all the assumptions of the standard model , with theappearance of the scalar fields included.The Clifford algebra offers in even d -dimensional spaces, d ≥ (13 + 1) indeed, the descrip-tion of the internal degrees of freedom for the second quantized fermions with the half integerspins, explaining all the assumptions of the standard model : The appearance of charges of theobserved quarks and leptons and their families, as well as the appearance of the correspondinggauge fields, the scalar fields, explaining the Higgs scalar and the Yukawa couplings, and in ad-dition the appearance of the dark matter, of the matter/antimatter asymmetry, offering severalpredictions [5, 6, 15–21]. A. Our main assumption and definitions
In this subsection we clarify how does the main assumption of Part I and Part II, the decision todescribe the internal space of fermions with the ”basis vectors” expressed with the superposition ofodd products of the anticommuting members of the algebra , either the Clifford one or the Grassmannone, acting algebraically, ∗ A , on the internal vacuum state | ψ o > , relate to the creation andannihilation anticommuting operators of the second quantized fermion fields.To appreciate the need for this kind of assumption, let us first have in mind that algebra withthe product ∗ A is only present in our work, usually not in other works, and thus has no well knownphysical meaning. It is at first a product by which you can multiply two internal wave functions B i and B j with each other, C k = B i ∗ A B j ,B i ∗ A B j = ∓ B j ∗ A B i , the sign ∓ depends on whether B i and B j are products of odd or even number of algebra elements:The sign is − if both are (superposition of) odd products of algebra elements, in all other casesthe sign is +.Let R d − define the external spatial or momentum space. Then the tensor product ∗ T extendsthe internal wave functions into the wave functions C (cid:126)p, i defined in both spaces C (cid:126)p, i = | (cid:126)p > ∗ T | B i > , where again B i represent the superposition of products of elements of the anticommuting algebras,in our case either θ a or γ a or ˜ γ a , used in this paper.We can make a choice of the orthogonal and normalized basis so that < C (cid:126)p,i | C (cid:126)p (cid:48) ,j > = δ ( (cid:126)p − (cid:126)p (cid:48) ) δ ij . Let us point out that either B i or C (cid:126)p, i apply algebraically on the vacuum state, B i ∗ A | ψ o > and C (cid:126)p, i ∗ A | ψ o > .Usually a product of single particle wave functions is not taken to have any physical meaningin as far as most physicists simply do not work with such products at all.To give to the algebraic product, ∗ A , and to the tensor product, ∗ T , defined on the spaceof single particle wave functions, the physical meaning, we postulate the connection between theanticommuting/commuting properties of the ”basis vectors”, expressed with the odd/even productsof the anticommuting algebra elements and the corresponding creation operators, creating secondquantized single fermion/boson statesˆ b † C (cid:126)p,i ∗ A | ψ o > = | ψ (cid:126)p,i > , ˆ b † C (cid:126)p,i ∗ T | ψ (cid:126)p (cid:48) ,j > = 0 , if (cid:126)p = (cid:126)p (cid:48) and i = j , in all other cases it followsˆ b † C (cid:126)p,i ∗ T ˆ b † C (cid:126)p (cid:48) ,j ∗ A | ψ o > = ∓ ˆ b † C (cid:126)p (cid:48) ,j ∗ T ˆ b † C (cid:126)p,i ∗ A | ψ o > , with the sign ± depending on whether ˆ b † C (cid:126)p,i have both an odd character, the sign is − , or not, thenthe sign is +.To each creation operator ˆ b † C (cid:126)p,i its Hermitian conjugated partner represents the annihilationoperator ˆ b C (cid:126)p,i ˆ b C (cid:126)p,i = (ˆ b † C (cid:126)p,i ) † , with the propertyˆ b C (cid:126)p,i ∗ A | ψ o > = 0 , defining the vacuum state as | ψ o > : = (cid:88) i ( B i ) † ∗ A B i | I > where summation i runs over all different products of annihilation operator × its Hermitian conju-gated creation operator, no matter for what (cid:126)p , and | I > represents the identity, ( B i ) † representsthe Hermitian conjugated wave function to B i .Let the tensor multiplication ∗ T denotes also the multiplication of any number of single particlestates, and correspondingly of any number of creation operators.What further means that to each single particle wave function we define the creation operatorˆ b † C (cid:126)p,i , applying in a tensor product from the left hand side on the second quantized Hilbert space— consisting of all possible products of any number of the single particle wave functions — addingto the Hilbert space the single particle wave function created by this particular creation operator.In the case of the second quantized fermions, if this particular wave function with the quantumnumbers and (cid:126)p of ˆ b † C (cid:126)p,i is already among the single fermion wave functions of a particular productof fermion wave functions, the action of the creation operator gives zero, otherwise the numberof the fermion wave functions increases for one. In the boson case the number of boson secondquantized wave functions increases always for one.If we apply with the annihilation operator ˆ b C (cid:126)p,i on the second quantized Hilbert space, then theapplication gives a nonzero contribution only if the particular products of the single particle wavefunctions do include the wave function with the quantum number i and (cid:126)p .In a Slater determinant formalism the single particle wave functions define the empty or occupiedplaces of any of infinite numbers of Slater determinants. The creation operator ˆ b † C (cid:126)p,i applies on aparticular Slater determinant from the left hand side. Jumping over occupied states to the placewith its i and (cid:126)p . If this state is occupied, the application gives in the fermion case zero, in theboson case the number of particles increase for one. The particular Slater determinant changessign in the fermion case if ˆ b † C (cid:126)p,i jumps over odd numbers of occupied states. In the boson case thesign of the Slater determinant does not change.When annihilation operator ˆ b C (cid:126)p,i applies on particular Slater determinant, it is jumping overoccupied states to its own place. giving zero, if this space is empty and decreasing the number ofoccupied states of this space is occupied. The Slater determinant changes sign in the fermion case,if the number of occupied states before its own space is odd. In the boson case the sign does notchange.Let us stress that choosing antisymmetry or symmetry is a choice which we make when treatingfermions or bosons, respectively, namely the choice of using oddness or evenness of basis vectors,that is the choice of using odd products or even products of algebra anticummuting elements.To describe the second quantized fermion states we make a choice of the basis vectors, whichare the superposition of the odd numbers of algebra elements, of both Clifford and Grassmannalgebras.The creation operators and their Hermitian conjugation partners annihilation operators there-fore in the fermion case anticommute. The single fermion states, which are the application of thecreation operators on the vacuum state | ψ o > , manifest correspondingly as well the oddness. Thevacuum state, defined as the sum over all different products of annihilation × the correspondingcreation operators, have an even character.Let us end up with the recognition:One usually means antisymmetry when talking about Slater-determinants because otherwise onewould not get determinants.In the present paper [5, 6, 13, 15] the choice of the symmetrizing versus antisymmetrizing relatesindeed the commutation versus anticommutation with respect to the a priori completely differentproduct ∗ A , of anticommuting members of the Clifford or Grassmann algebra. The oddness orevenness of these products transfer to quantities to which these algebras extend. II. PROPERTIES OF GRASSMANN ALGEBRA IN EVEN DIMENSIONAL SPACES
In Grassmann d -dimensional space there are d anticommuting operators θ a , { θ a , θ b } + = 0, a = (0 , , , , , .., d ), and d anticommuting derivatives with respect to θ a , ∂∂θ a , { ∂∂θ a , ∂∂θ b } + = 0,offering together 2 · d operators, the half of which are superposition of products of θ a and anotherhalf corresponding superposition of ∂∂θ a . { θ a , θ b } + = 0 , { ∂∂θ a , ∂∂θ b } + = 0 , { θ a , ∂∂θ b } + = δ ab , ( a, b ) = (0 , , , , , · · · , d ) . (1)Defining [12] ( θ a ) † = η aa ∂∂θ a , it follows( ∂∂θ a ) † = η aa θ a . (2)The identity is the self adjoint member. The signature η ab = diag { , − , − , · · · , − } is assumed.It appears useful to arrange 2 d products of θ a into irreducible representations with respect tothe Lorentz group with the generators [6] S ab = i ( θ a ∂∂θ b − θ b ∂∂θ a ) , ( S ab ) † = η aa η bb S ab . (3)2 d − members of the representations have an odd Grassmann character (those which are superpo-sition of odd products of θ a ’s). All the members of any particular odd irreducible representationfollow from any starting member by the application of S ab ’s.0If we exclude the self adjoint identity there is (2 d − −
1) members of an even Grassmanncharacter, they are even products of θ a ’s. All the members of any particular even representationfollow from any starting member by the application of S ab ’s.The Hermitian conjugated 2 d − odd partners of odd representations of θ a ’s and (2 d − − θ a ’s are reachable from odd and even representations,respectively, by the application of Eq. (2).It appears useful as well to make the choice of the Cartan subalgebra of the commuting operatorsof the Lorentz algebra as follows S , S , S , · · · , S d − d , (4)and choose the members of the irreducible representations of the Lorentz group to be the eigen-vectors of all the members of the Cartan subalgebra of Eq. (4) S ab √ θ a + η aa ik θ b ) = k √ θ a + η aa ik θ b ) , S ab √ ik θ a θ b ) = 0 , or S ab √ ik θ a θ b = 0 , (5)with k = η aa η bb . The eigenvector √ ( θ ∓ θ ) of S has the eigenvalue k = ± i , the eigenvaluesof all the other eigenvectors of the rest of the Cartan subalgebra members, Eq. (4), are k = ± √ ( θ a + η aa ik θ b ) and of evenobjects ik θ a θ b , with eigenvalues k = ± i and 0, respectively.Let us check how does S ac = i ( θ a ∂∂θ c − θ c ∂∂θ a ) transform the product of two ”nilpotents” √ ( θ a + η aa ik θ b ) and √ ( θ c + η cc ik (cid:48) θ d ). Taking into account Eq. (3) one finds that S ac √ ( θ a + η aa ik θ b ) √ ( θ c + η cc ik (cid:48) θ d ) = − η aa η cc k ( θ a θ b + kk (cid:48) θ c θ d ). S ac transforms the product of two Grassmann odd eigenvectorsof the Cartan subalgebra into the superposition of two Grassmann even eigenvectors.”Basis vectors” have an odd or an even Grassmann character, if their products contain an oddor an even number of ”nilpotents”, √ ( θ a + η aa ik θ b ), respectively. ”Basis vectors” are normalized,up to a phase, in accordance with Eq. (A1) of A.The Hermitian conjugated representations of (either an odd or an even) products of θ a ’s can beobtained by taking into account Eq. (2) for each ”nilpotent”1 √ θ a + η aa ik θ b ) † = η aa √ ∂∂θ a + η aa − ik ∂∂θ b ) , ( ik θ a θ b ) † = ik ∂∂θ a ∂∂θ b . (6)1Making a choice of the identity for the vacuum state, | φ og > = | > , (7)we see that algebraic products — we shall use a dot , · , or without a dot for an algebraic productof eigenstates of the Cartan subalgebra forming ”basis vectors” and ∗ A for the algebraic productof ”basis vectors” — of different θ a ’s, if applied on such a vacuum state, give always nonzerocontributions, ( θ ∓ θ ) · ( θ ± iθ ) · · · ( θ d − ∓ θ d ) | > (cid:54) = zero , (this is true also, if we substitute any of nilpotents √ ( θ a + η aa ik θ b ) or all of them with the cor-responding even operators ( ik θ a θ b ); in the case of odd Grassmann irreducible representations atleast one nilpotent must remain). The Hermitian conjugated partners, Eq. (6), applied on | > ,give always zero ( ∂∂θ ∓ ∂∂θ ) · ( ∂∂θ ± i ∂∂θ ) · · · ( ∂∂θ d − ± i ∂∂θ d ) | > = 0 . Let us notice the properties of the odd products θ a ’s and of their Hermitian conjugatedpartners: i. Superposition of products of different θ a ’s, applied on the vacuum state | > , give nonzerocontribution. To create on the vacuum state the ”fermion” states we make a choice of the ”basisvectors” of the odd number of θ a ’s, arranging them to be the eigenvectors of all the Cartansubalgebra elements, Eq. (4). ii. The Hermitian conjugated partners of the “basis vectors”, they are products of derivatives ∂∂θ a ’s, give, when applied on the vacuum state | > , Eq. (7), zero. Each annihilation operatorannihilates the corresponding creation operator. iii. The algebraic product, ∗ A , of a “basis vector” by itself gives zero, the algebraic anticommutatorof any two ”basis vectors” of an odd Grassmann character (superposition of an odd products of θ a ’s) gives zero (”basis vectors” of the two decuplets in Table I and the ”basis vector” of Eq. (13) ( θ ∓ θ ), for example, demonstrate this property). iv. The algebraic application of any annihilation operator on the corresponding Hermitianconjugated ”basis vector” gives identity, on all the rest of ”basis vectors” gives zero. Correspond-ingly the algebraic anticommutators of the creation operators and their Hermitian conjugatedpartners, applied on the vacuum state, give identity, all the rest anticommutators of creation and2annihilation operators applied on the vacuum state, give zero. v. Correspondingly the “basis vectors” and their Hermitian conjugated partners, applied onthe vacuum state | > , Eq. (7), fulfill the properties of creation and annihilation operator,respectively, for the second quantized ”fermions” on the level of one ”fermion” state. A. Grassmann ”basis vectors”
We construct 2 d − Grassmann odd ”basis vectors” and 2 d − − | > ) Grassmann even ”basis vectors” as superpositionof odd and even products of θ a ’s, respectively. Their Hermitian conjugated 2 d − odd and 2 d − − ∂∂θ a ’s, respectively [22]. A . a . Grassmann anticommuting ”basis vectors” with integer spins
Let us choose in d = 2(2 n + 1)-dimensional space-time, n is a positive integer, the startingGrassmann odd ”basis vector” ˆ b θ † , which is the eigenvector of the Cartan subalgebra of Eqs. (4, 5)with the egenvalues (+ i, +1 , +1 , · · · , +1), respectively, and has the Hermitian conjugated partnerequal to (ˆ b θ † ) † = ˆ b θ , ˆ b θ † : = ( 1 √ d ( θ − θ )( θ + iθ )( θ + iθ ) · · · ( θ d − + iθ d ) , ˆ b θ : = ( 1 √ d ( ∂∂θ d − − i ∂∂θ d ) · · · ( ∂∂θ − ∂∂θ ) . (8)In the case of d = 4 n , n is a positive integer, the corresponding starting Grassmann odd ”basisvector” can be chosen asˆ b θ † : = ( 1 √ d − ( θ − θ )( θ + iθ )( θ + iθ ) · · ·· · · ( θ d − + iθ d − ) θ d − θ d . (9)All the rest of ”basis vectors”, belonging to the same irreducible representation of the Lorentzgroup, follow by the application of S ab ’s.We denote the members i of this starting irreducible representation k by ˆ b θk † i and their Hermitianconjugated partners by ˆ b θki , with k = 1.3”Basis vectors”, belonging to different irreducible representations k = 2, will be denoted by ˆ b θ † j and their Hermitian conjugated partners by ˆ b θ j = (ˆ b θk † j ) † . S ac ’s, which do not belong to the Cartan subalgebra, transform step by step the two by two”nilpotents”, no matter how many ”nilpotents” are between the chosen two, up to a constant, asfollows: S ac √ ( θ a + η aa ik θ b ) · · · √ ( θ c + η cc ik (cid:48) θ d ) ∝ − η aa η cc k ( θ a θ b + kk (cid:48) θ c θ d ) · · · ,leaving at each step at least one ”nilpotent” unchanged, so that the whole irreducible representationremains odd.The superposition of S bd and i S bc transforms − η aa η cc k ( θ a θ b + kk (cid:48) θ c θ d ) into √ ( θ a − η aa ik θ b ) √ ( θ c − η cc ik (cid:48) θ d ), and not into √ ( θ a + η aa ik θ b ) √ ( θ c − η cc ik (cid:48) θ d ) or into √ ( θ a − η aa ik θ b ) √ ( θ c + η cc ik (cid:48) θ d ).Therefore we can start another odd representation with the ”basis vector” ˆ b θ † as followsˆ b θ † : = ( 1 √ d ( θ + θ )( θ + iθ )( θ + iθ ) · · · ( θ d − + iθ d ) , (ˆ b θ † ) † = ˆ b θ : = ( 1 √ d ( ∂∂θ d − − i ∂∂θ d ) · · · ( ∂∂θ − ∂∂θ ) . (10)The application of S ac ’s determines the whole second irreducible representation ˆ b θ † j .One finds that each of these two irreducible representations has
12 d ! d2 ! d2 ! members, Ref. [12].Taking into account Eq. (1), it follows that odd products of θ a ’s anticommute and so do theodd products of ∂∂θ a ’s. Statement 1:
The oddness of the products of θ a ’s guarantees the anticommuting propertiesof all objects which include odd number of θ a ’s.One further sees that ∂∂θ a θ b = η ab , while ∂∂θ a | > = 0, and θ a | > = θ a | > . and { ˆ b θki , ˆ b θl † j } ∗ A + =We can therefore conclude { ˆ b θki , ˆ b θl † j } ∗ A + | > = δ ij δ kl | > , { ˆ b θki , ˆ b θlj } ∗ A + | > = 0 · | > , { ˆ b θk † i , ˆ b θl † j } ∗ A + | > = 0 · | > , ˆ b θkj ∗ A | > = 0 · | > , (11)where { ˆ b θki , ˆ b θl † j } ∗ A + = ˆ b θki ∗ A ˆ b θl † j + ˆ b θlj ∗ A ˆ b θk † i is meant.These anticommutation relations of the ”basis vectors” of the odd Grassmann character, mani-fest on the level of the Grassmann algebra the anticommutation relations required by Dirac [1] forsecond quantized fermions.4The ”Grassmann fermion basis states” can be obtained by the application of creation operatorsˆ b θk † i on the vacuum state | > | φ ko i > = ˆ b θk † i | > . (12)We use them to determine the internal space of ”Grassmann fermions” in the tensor product ∗ T of these ”basis states” and of the momentum space, when looking for the anticommuting singleparticle ”Grassmann states”, which have, according to Eq. (5), an integer spin, and not half integerspin as it is the case for the so far observed fermions. A . b . Illustration of anticommuting ”basis vectors” in d = (5 + 1) -dimensional space Let us illustrate properties of Grassmann odd representations for d = (5 + 1)-dimensional space.Table I represents two decuplets, which are ”egenvectors” of the Cartan subalgbra ( S , S , S ), Eq. (4),of the Lorentz algebra S ab . The two decuplets represent two Grassmann odd irreducible representations of SO (5 , SO (3 , S and S alone, while the eigenvector of S has, as a ”spectator”, the eigenvalue either +1 (the firsttriplet in both decuplets) or − S equal to zero ((7 th , th , th , th ) lines in each of the two decuplets(Table II in Ref. [6])).Paying attention on the eigenvectors of S alone one recognizes as well even and odd representations of SO (1 , θ θ and θ ± θ , respectively.The Hermitian conjugated ”basis vectors” follow by using Eq. (6) and is for the first ”basis vector”of Table I equal to ( − ) ( √ ) ( ∂∂θ − i ∂∂θ ) ( ∂∂θ − i ∂∂θ ) ( ∂∂θ + ∂∂θ ). One correspondingly finds that when( √ ) ( ∂∂θ − i ∂∂θ ) ( ∂∂θ − i ∂∂θ ) ( ∂∂θ + ∂∂θ ) applies on ( √ ) ( θ − θ )( θ + iθ )( θ + iθ ) the result is identity.Application of ( √ ) ( ∂∂θ − i ∂∂θ ) ( ∂∂θ − i ∂∂θ ) ( ∂∂θ + ∂∂θ ) on all the rest of ”basis vectors” of the decuplet I as well as on all the ”basis vectors” of the decuplet II gives zero. ”Basis vectors” are orthonormalizedwith respect to Eq. (A1). Let us notice that ∂∂θ a on a ”state” which is just an identity, | > , gives zero, ∂∂θ a | > = 0, while θ a | > , or any superposition of products of θ a ’s, applied on | > , gives the ”vector”back.One easily sees that application of products of superposition of θ a ’s on | > gives nonzero contribution,while application of products of superposition of ∂∂θ a ’s on | > gives zero.The two by S ab decoupled Grassmann decuplets of Table I are the largest two irreducible representationsof odd products of θ a ’s. There are 12 additional Grassmann odd ”vectors”, arranged into irreducible S , S , S , for SO (5 , S ab ’s and are decoupled fromanother decuplet. The two operators of handedness, Γ (( d − for d = (6 , (5+1) for the whole decuplet, Γ (3+1) for the ”triplets” and ”fourplets”. I i decuplet of eigenvectors S S S Γ (5+1) Γ (3+1) √ ) ( θ − θ )( θ + iθ )( θ + iθ ) i √ ) ( θ θ + iθ θ )( θ + iθ ) 0 0 1 1 13 ( √ ) ( θ + θ )( θ − iθ )( θ + iθ ) − i − √ ) ( θ − θ )( θ − iθ )( θ − iθ ) i − − −
15 ( √ ) ( θ θ − iθ θ )( θ − iθ ) 0 0 − −
16 ( √ ) ( θ + θ )( θ + iθ )( θ − iθ ) − i − −
17 ( √ ) ( θ − θ )( θ θ + θ θ ) i √ ) ( θ + θ )( θ θ − θ θ ) − i √ ) ( θ θ + iθ θ )( θ + iθ ) 0 1 0 1 010 ( √ ) ( θ θ − iθ θ )( θ − iθ ) 0 − II i decuplet of eigenvectors S S S γ (5+1) γ (3+1) √ ) ( θ + θ )( θ + iθ )( θ + iθ ) − i − −
12 ( √ ) ( θ θ − iθ θ )( θ + iθ ) 0 0 1 − −
13 ( √ ) ( θ − θ )( θ − iθ )( θ + iθ ) i − − −
14 ( √ ) ( θ + θ )( θ − iθ )( θ − iθ ) − i − − − √ ) ( θ θ + iθ θ )( θ − iθ ) 0 0 − − √ ) ( θ − θ )( θ + iθ )( θ − iθ ) i − − √ ) ( θ + θ )( θ θ + θ θ ) − i − √ ) ( θ − θ )( θ θ − θ θ ) i − √ ) ( θ θ − iθ θ )( θ + iθ ) 0 1 0 − √ ) ( θ θ + iθ θ )( θ − iθ ) 0 − − representations of six singlets and six sixplets( 12 ( θ ∓ θ ) ,
12 ( θ ± iθ ) ,
12 ( θ ± iθ ) ,
12 ( θ ∓ θ ) θ θ θ θ ,
12 ( θ ± iθ ) θ θ θ θ ,
12 ( θ ± iθ ) θ θ θ θ ) . (13)The algebraic application of products of superposition of ∂∂θ a ’s on the corresponding Hermitian conju-gated partners, which are products of superposition of θ a ’s, leads to the identity for either even or oddGrassmann character [23].Besides 32 Grassmann odd eigenvectors of the Grassmann Cartan subalgebra, Eq. (4), there are (32 − θ a ’s. The even self adjoint operator identity (which is indeed the normalized product ofall the annihilation times ∗ A creation operators) is used to represent the vacuum state.It is not difficult to see that Grassmann ”basis vectors” of an odd Grassmann character anticommuteamong themselves and so do odd products of superposition of ∂∂θ a ’s, while equivalent even products commute. The Grassmann odd algebra (as well as the two odd Clifford algebras) offers, due to the oddness of theinternal space giving oddness as well to the elements of the tensor products of the internal space and of themomentum space, the description of the anticommuting second quantized fermion fields, as postulated byDirac. But the Grassmann ”fermions” carry the integer spins, while the observed fermions — quarks andleptons — carry half integer spin. A . c . Grassmann commuting ”basis vectors” with integer spins
Grassmann even ”basis vectors” manifest the commutation relations, and not the anticommutation onesas it is the case for the Grassmann odd ”basis vectors”. Let us use in the Grassmann even case, that isthe case of superposition of an even number of θ a ’s in d = 2(2 n + 1), the notation ˆ a θk † j , again chosen to beeigenvectors of the Cartan subalgebra, Eq. (4), and let us start with one representativeˆ a θ † j : = ( 1 √ d − ( θ − θ )( θ + iθ )( θ + iθ ) · · · ( θ d − + iθ d − ) θ d − θ d . (14)The rest of ”basis vectors”, belonging to the same Lorentz irreducible representation, follow by the applica-tion of S ab . The Hermitian conjugated partner of ˆ a θ † is ˆ a θ = (ˆ a θ † ) † ˆ a θ : = ( 1 √ d − ∂∂θ d ∂∂θ d − ( ∂∂θ d − − i ∂∂θ d − ) · · · ( ∂∂θ − ∂∂θ ) . (15)If ˆ a θk † j represents a Grassmann even creation operator, with index k denoting different irreducible rep-resentations and index j denoting a particular member of the k th irreducible representation, while ˆ a θkj represents its Hermitian conjugated partner, one obtains by taking into account Sect. II, the relations { ˆ a θki , ˆ a θk (cid:48) † j } ∗ A − | > = δ ij δ kk (cid:48) | > , { ˆ a θki , ˆ a θk ‘ j } ∗ A − | > = 0 · | > , { ˆ a θk † i , ˆ a θk (cid:48) † j } ∗ A − | > = 0 · | > , ˆ a θki ∗ A | > = 0 · | > , ˆ a θk † i ∗ A | > = | φ ke i > . (16)Equivalently to the case of Grassmann odd ”basis vectors” also here { ˆ a θki , ˆ a θl † j } ∗ A − = ˆ a θki ∗ A ˆ a θl † j − ˆ a θlj ∗ A ˆ a θk † i is meant. B. Action for free massless ”Grassmann fermions” with integer spin [12]
In the Grassmann case the ”basis vectors” of an odd Grassmann character, chosen to be theeigenvectors of the Cartan subalgebra of the Lorentz algebra in Grassmann space, Eq. (4), manifest7the anticommutation relations of Eq. (11) on the algebraic level.To compare the properties of creation and annihilation operators for ”integer spin fermions”,for which the internal degrees of freedom are described by the odd Grassmann algebra, withthe creation and annihilation operators postulated by Dirac for the second quantized fermionsdepending on the quantum numbers of the internal space of fermions and on the momentum space,we need to define the tensor product ∗ T of the odd ”Grassmann basis states”, described by thesuperposition of odd products of θ a ’s (with the finite degrees of freedom) and of the momentum(or coordinate) space (with the infinite degrees of freedom), taking as the basis the tensor productof both spaces. Statement 2:
For deriving the anticommutation relations for the ”Grassmann fermions”, tobe compared to anticommutation relations of the second quantized fermions, we need to define thetensor product of the Grassmann odd ”basis vectors” and the momentum space basis ( p a ,θ a ) = | p a > ∗ T | θ a > . (17)We need even more, we need to find the Lorentz invariant action for, let say, free massless”Grassmann fermions” to define such a ”basis”, that would manifest the relation | p | = | (cid:126)p | . Wefollow here the suggestion of one of us (N.S.M.B.) from Ref. [12]. A G = (cid:90) d d x d d θ ω { φ † γ G θ a p a φ } + h.c. ,ω = d (cid:89) k =0 ( ∂∂θ k + θ k ) , (18)with γ aG = (1 − θ a ∂∂θ a ), ( γ aG ) † = γ aG , for each a = (0 , , , , , · · · , d ). We use the integral over θ a coordinates with the weight function ω from Eq. (A1, A2). Requiring the Lorentz invariance weadd after φ † the operator γ G , which takes care of the Lorentz invariance. Namely S ab † (1 − θ ∂∂θ ) = (1 − θ ∂∂θ ) S ab , S † (1 − θ ∂∂θ ) = (1 − θ ∂∂θ ) S − , S = e − i ω ab ( L ab + S ab ) , (19)while θ a , ∂∂θ a and p a transform as Lorentz vectors.The Lagrange density is up to the surface term equal to [24] L G = 12 φ † γ G ( θ a − ∂∂θ a ) (ˆ p a φ )= 14 { φ † γ G ( θ a − ∂∂θ a ) ˆ p a φ − (ˆ p a φ † ) γ G ( θ a − ∂∂θ a ) φ } , (20)8leading to the equations of motion [25]12 γ G ( θ a − ∂∂θ a ) ˆ p a | φ > = 0 , (21)as well as the the ”Klein-Gordon” equation,( θ a − ∂∂θ a ) ˆ p a ( θ b − ∂∂θ b ) ˆ p b | φ > = 0 = ˆ p a ˆ p a | φ > . The eigenstates φ of equations of motion for free massless ”Grassmann fermions”, Eq. (21),can be found as the tensor product, Eq.(17) of the superposition of 2 d − Grassmann odd ”basisvectors” ˆ b θk † i and the momentum space, represented by plane waves, applied on the vacuum state | > . Let us remind that the ”basis vectors” are the ”eigenstates” of the Cartan subalgebra,Eq. (4), fulfilling (on the algebraic level) the anticommutation relations of Eq. (11). And sincethe oddness of the Grassmann odd ”basis vectors” guarantees the oddness of the tensor productsof the internal part of ”Grassmann fermions” and of the plane waves, we expect the equivalentanticommutation relations also for the eigenstates of the Eq. (21), which define the single particleanticommuting states of ”Grassmann fermions”.The coefficients, determining the superposition, depend on momentum p a , a = (0 , , , , ,. . . , d ), | p | = | (cid:126)p | , of the plane wave solution e − ip a x a .Let us therefore define the new creation operators and the corresponding single particle ”Grass-mann fermion” states as the tensor product of two spaces, the Grassmann odd ”basis vectors” andthe momentum space basis ˆ b θk s † ( (cid:126)p ) def = (cid:88) i c ksi ( (cid:126)p ) ˆ b θk † i , | p | = | (cid:126)p | , ˆ b θk s † tot ( (cid:126)p ) def = ˆ b θk s † ( (cid:126)p ) · e − ip a x a , | p | = | (cid:126)p | ,< x | φ kstot ( (cid:126)p ) > = ˆ b θks † tot ( (cid:126)p ) | > , | p | = | (cid:126)p | , (22)with s representing different solutions of the equations of motion and k different irreducible repre-sentations of the Lorentz group, (cid:126)p denotes the chosen vector ( p , (cid:126)p ) in momentum space.One has further | φ ks ( x , (cid:126)x ) > = (cid:90) + ∞−∞ d d − p ( √ π ) d − ˆ b θks † ( (cid:126)p ) | | p | = | (cid:126)p | | > (23)The orthogonalized states | φ ks ( (cid:126)p ) > fulfill the relation < φ ks ( (cid:126)p ) | φ k (cid:48) s (cid:48) ( (cid:126)p (cid:48) ) > = δ kk (cid:48) δ ss (cid:48) δ pp (cid:48) , | p | = | (cid:126)p | ,< φ k (cid:48) s (cid:48) ( x , (cid:126)x (cid:48) ) | φ ks ( x , (cid:126)x ) > = δ kk (cid:48) δ ss (cid:48) δ (cid:126)x (cid:48) ,(cid:126)x , (24)9where we assumed the discretization of momenta (cid:126)p and coordinates (cid:126)x .In even dimensional spaces ( d = 2(2 n + 1) and 4 n ) there are 2 d − Grassmann odd superpositionof ”basis vectors”, which belong to different irreducible representations, among them twice
12 d ! d2 ! d2 ! of the kind presented in Eqs. (8, 9) and discussed in the chapter A.b. of the subsect. II A andin Table I for a particular case d = (5 + 1). The illustration for the superposition ˆ b θk s † ( (cid:126)p ) andˆ b θk s † tot ( (cid:126)p ) is presented, again for d = (5 + 1), in chapter B.a. .We introduced in Eq. (22) the creation operators ˆ b θk s † tot ( (cid:126)p ) as the tensor product of the ”ba-sis vectors” of Grassmann algebra elements and the momentum basis. The Grassmann algebraelements transfer their oddness to the tensor products of these two basis. Correspondingly mustˆ b θk s † tot ( (cid:126)p ) together with their Hermitian conjugated annihilation operators (ˆ b θk s † tot ( (cid:126)p )) † = ˆ b θk stot ( (cid:126)p )fulfill the the anticommutation relations equivalent to the anticommutation relations of Eq. (11) { ˆ b θk stot ( (cid:126)p ) , ˆ b θk (cid:48) s (cid:48) † tot ( (cid:126)p (cid:48) ) } ∗ T + | > = δ kk (cid:48) δ ss (cid:48) δ ( (cid:126)p − (cid:126)p (cid:48) ) | > , { ˆ b θk stot ( (cid:126)p ) , ˆ b θk (cid:48) s (cid:48) tot ( (cid:126)p (cid:48) ) } ∗ T + | > = 0 · | > , { ˆ b θk s † tot ( (cid:126)p ) , ˆ b θk (cid:48) s (cid:48) † tot ( (cid:126)p (cid:48) ) } ∗ T + | > = 0 · | > , ˆ b θk stot ( (cid:126)p ) ∗ T | > = 0 · | > , | p | = | (cid:126)p | . (25) k labels different irreducible representations of Grassmann odd “basis vectors”, s labels different —orthogonal and normalized — solutions of equations of motion and (cid:126)p represent different momentafulfilling the relation ( p ) = ( (cid:126)p ) . Here we allow continuous momenta and take into account that < | ˆ b θk stot ( (cid:126)p ) ∗ T ˆ b θk (cid:48) s (cid:48) † tot ( (cid:126)p (cid:48) ) | > = δ kk (cid:48) δ ss (cid:48) δ ( (cid:126)p − (cid:126)p (cid:48) ) , (26)in the case of continuous values of (cid:126)p in even d -dimensional space.For each momentum (cid:126)p there are 2 d − members of the odd Grassmann character, belongingto different irreducible representations. The plane wave solutions, belonging to different (cid:126)p , areorthogonal, defining correspondingly ∞ many degrees of freedom for each of 2 d − ”fermion” states,defined by ˆ b θk s † tot ( (cid:126)p ), when applying on the vacuum state | >, Eq. (7).With the choice of the Grassmann odd ”basis vectors” in the internal space of ”Grassmannfermions” and by extending these ”basis states” to momentum space to be able to solve the equa-tions of motion, Eq. (21), we are able to define the creation operators ˆ b θk stot ( (cid:126)p ) of the odd Grassmanncharacter, which together with their Hermitian conjugated partners annihilation operators, fulfillthe anticommutation relations of Eq. (25), manifesting the properties of the second quantized0fermion fields. Anticommutation properties of creation and annihilation operators are due to theodd Grassmann character of the ”basis vectors”.To define the Hilbert space of all possible ”Slater determinants” of all possible occupied andempty fermion states and to discuss the application of ˆ b θk stot ( (cid:126)p ) and ˆ b k s † tot ( (cid:126)p ) on ”Slater determi-nants”, let us see what the anticommutation relations, presented in Eq. (25), tell. We realize fromEq. (25) the propertiesˆ b θk s † tot ( (cid:126)p ) ∗ T ˆ b θk (cid:48) s (cid:48) † tot ( (cid:126)p (cid:48) ) = − ˆ b θk (cid:48) s (cid:48) † tot ( (cid:126)p (cid:48) ) ∗ T ˆ b θk s † tot ( (cid:126)p ) , ˆ b θk stot ( (cid:126)p ) ∗ T ˆ b θk (cid:48) s (cid:48) tot ( (cid:126)p (cid:48) ) = − ˆ b θk (cid:48) s (cid:48) tot ( (cid:126)p (cid:48) ) ∗ T ˆ b θk stot ( (cid:126)p ) , ˆ b θk stot ( (cid:126)p ) ∗ T ˆ b θk (cid:48) s (cid:48) † tot ( (cid:126)p (cid:48) ) = − ˆ b θk (cid:48) s (cid:48) † tot ( (cid:126)p (cid:48) ) ∗ T ˆ b θk stot ( (cid:126)p ) , if at least one of ( k, s, (cid:126)p ) distinguishes from( k (cid:48) , s (cid:48) , (cid:126)p (cid:48) ) , ˆ b θk s † tot ( (cid:126)p ) ∗ T ˆ b θk s † tot ( (cid:126)p ) = 0 , ˆ b θk stot ( (cid:126)p ) ∗ T ˆ b θk stot ( (cid:126)p ) = 0 , ˆ b θk stot ( (cid:126)p ) ∗ T ˆ b θk s † tot ( (cid:126)p ) | > = | > , ˆ b θk stot ( (cid:126)p ) | > = 0 , | p | = | (cid:126)p | . (27)From the above relations we recognize how do the creation and annihilation operators apply on”Slater determinants” of empty and occupied states, the later determined by ˆ b θk s † tot ( (cid:126)p ): i. The creation operator ˆ b θk s † tot ( (cid:126)p ) jumps over the creation operator defining the occupied state,which distinguish from the jumping creation one in at least one of ( k, s, (cid:126)p ), changing sign of the”Slater determinant” every time, up to the last step when it comes to its own empty state, the onewith its quantum numbers ( k, s, (cid:126)p ), occupying this empty state, or if this state is already occupied,gives zero. ii. The annihilation operator changes sign of the ”Slater determinant” when ever jumping overthe occupied state carrying different internal quantum numbers ( k, s ) or (cid:126)p , unless it comes to theoccupied state with its own ( k, s, (cid:126)p ), emptying this state or, if this state is empty, gives zero.We show in Part II that the Clifford odd ”basis vectors” describe fermions with the half integerspin, offering as well the corresponding anticommutation relations, explaining Dirac’s postulatesfor second quantized fermions.We discuss in Sect. III the properties of the ”Slater determinants” of the occupied and empty”Grassmann fermion states”, created by ˆ b θk s † tot ( (cid:126)p ).In Subsect. B.a. we present one solution of the equations of motion for free massless ”Grassmann1fermions”. B . a . Plane wave solutions of equations of motion , Eq. (21), in d = (5 + 1) -dimensional space One of such plane wave massless solutions of the equations of motion in d = (5 + 1)-dimensional space formomentum p a = ( p , p , p , p , , p = | p | , is the superposition of ”basis vectors”, presented in Table Iamong the first three members of the first decuplet, k = I . This particular solution ˆ b θk s † tot ( (cid:126)p ) carries thespin S = 1 (”up”) and the “charge” S = 1 (both from the point of view of d = (3 + 1))ˆ b θ † tot ( (cid:126)p ): = β ( 1 √ { √ θ − θ )( θ + iθ ) − | p | − | p | ) p − ip ( θ θ + iθ θ ) − ( ( p + ip ) ( | p | + | p | ) ) 1 √ θ + θ )( θ − iθ ) }× ( θ + iθ ) · e − i ( | p | x − (cid:126)p · (cid:126)x ) , | p | = | (cid:126)p | ,β is the normalization factor. The notation ˆ b θ † tot ( (cid:126)p ) means that the creation operator represents the planewave solution of the equations of motion, Eq. (21), for a particular | p | = | (cid:126)p | .Applied on the vacuum state the creation operator defines the second quantized single particle state ofparticular momentum (cid:126)p . States, carrying different (cid:126)p , are orthogonal (due to the orthogonality of the planewaves of different momenta and due to the orthogonality of ˆ b θk (cid:48) s (cid:48) † tot ( (cid:126)p ) and ˆ b θk stot ( (cid:126)p ) with respect to k and s , Eqs. (24, 26, 25)).More solutions can be found in [12] and the references therein. III. HILBERT SPACE OF ANTICOMMUTING INTEGER SPIN “GRASSMANNFERMIONS”
The Grassmann odd creation operators ˆ b θ k s † tot ( (cid:126)p ), with | p | = | (cid:126)p | , are defined on the tensorproducts of 2 d − ”basis vectors”, defining the internal space of integer spin ”Grassmann fermions”,and on infinite basis states of momentum space for each component of (cid:126)p , chosen so that they solvefor particular ( (cid:126)p ) the equations of motion, Eq. (21). They fulfill together with their Hermitianconjugated annihilation operators ˆ b θ k stot ( (cid:126)p ) the anticommutation relations of Eq. (25).These creation operators form the Hilbert space of ”Slater determinants”, defining for each”Slater determinant” places with either empty or occupied ”Grassmann fermion” states. Statement 3 : Introducing the tensor product multiplication ∗ T of any number of single ”Grass-mann fermion” states of all possible internal quantum numbers and all possible momenta (that isof any number of ˆ b θ k s † tot ( (cid:126)p ) and with the identity included, applying on the vacuum state of any( k, s, (cid:126)p )), we generate the Hilbert space of the second quantized ”Grassmann fermion” fields.2It is straightforward to recognize that the above definition of the Hilbert space is equivalentto the space of ”Slater determinants” of all possible empty or occupied states of any momentumand any quantum numbers describing the internal space. The identity in this tensor productmultiplication, for example, represents the ”Slater determinant” of no single fermion state present.The 2 d − Grassmann odd creation operators of particular momentum (cid:126)p , if applied on the vacuumstate | > , Eq. (7), define 2 d − states. Since any state can be occupied or empty, the Hilbert space H (cid:126)p of a particular momentum (cid:126)p consists correspondingly of N H (cid:126)p = 2 d − . (28)”Slater determinants”, namely the one with no occupied state, those with one occupied state, thosewith two occupied states, up to the one with all 2 d − states occupied.The total Hilbert space H of anticommuting integer spin ”Grassmann fermions” consists ofinfinite many ”Slater determinants” of particular (cid:126)p , H (cid:126)p , due to infinite many degrees of freedomin the momentum space H = ∞ (cid:89) (cid:126)p ⊗ N H (cid:126)p , (29)with the infinite number of degrees of freedom N H = ∞ (cid:89) (cid:126)p d − . (30) A. ”Slater determinants” of anticommuting integer spin “Grassmann fermions” ofparticular momentum (cid:126)p
Let us write down explicitly these 2 d − contributions to the Hilbert space H (cid:126)p of particularmomentum (cid:126)p , using the notation that ks˜p represents the unoccupied state ˆ b θk s † tot ( (cid:126)p ) | > (of the s th solution of the equations of motion belonging to the k th irreducible representation), while ks˜p represents the corresponding occupied state.The number operator is according to Eq. (11) and Eq. (27) equal to N θk s(cid:126)p = ˆ b θ k s † tot ( (cid:126)p ) ∗ T ˆ b θ k stot ( (cid:126)p ) ,N θks(cid:126)p ∗ T ks(cid:126)p = 0 , N θks(cid:126)p ∗ T ks(cid:126)p = 1 . (31)Let us simplify the notation so that we count for k = 1 empty states as r˜p , and occupied statesas r˜p , with r = (1 , . . . , s max ), for k = 2 we count r = s max + 1 , . . . , s max + s max , up to r = 2 d − .3Correspondingly we can represent H (cid:126)p as follows | , , , . . . , d − ˜p > , | , , , . . . , d − ˜p >, | , , , . . . , d − ˜p > , | , , , . . . , d − ˜p >, ... | , , , . . . , d − ˜p > , | , , , . . . , d − ˜p >, ... | , , , . . . , d − ˜p > , (32)with a part with none of states occupied ( N r(cid:126)p = 0 for all r = 1 , . . . , d − ), with a part with onlyone of states occupied ( N r(cid:126)p = 1 for a particular r = 1 , . . . , d − while N r (cid:48) (cid:126)p = 0 for all the others r (cid:48) (cid:54) = r ), with a part with only two of states occupied ( N r(cid:126)p = 1 and N r (cid:48) (cid:126)p = 1, where r and r (cid:48) runfrom 1 , . . . , d − ), and so on. The last part has all the states occupied.Taking into account Eq. (27) is not difficult to see that the creation operator ˆ b θk s † tot ( (cid:126)p ) and theannihilation operators ˆ b θk stot ( (cid:126)p ), when applied on this Hilbert space H (cid:126)p , fulfill the anticommutationrelations for the second quantized “fermions”. { ˆ b θk stot ( (cid:126)p ) , ˆ b θk (cid:48) s (cid:48) † tot ( (cid:126)p ) } ∗ T + H (cid:126)p = δ kk (cid:48) δ ss (cid:48) H (cid:126)p , { ˆ b θk stot ( (cid:126)p ) , ˆ b θk (cid:48) s (cid:48) tot ( (cid:126)p ) } ∗ T + H (cid:126)p = 0 · H (cid:126)p , { ˆ b θk s † tot ( (cid:126)p ) , ˆ b θk (cid:48) s (cid:48) † tot ( (cid:126)p ) } ∗ T + H (cid:126)p = 0 · H (cid:126)p . (33)The proof for the above relations easily follows if taking into account that, when ever thecreation or annihilation operator jumps over an odd products of occupied states, the sign changes.Then one sees that the contribution of the application of ˆ b θk stot ( (cid:126)p ) ∗ T ˆ b θk (cid:48) s (cid:48) † tot ( (cid:126)p ) H (cid:126)p has the oppositesign than the contribution of ˆ b θk (cid:48) s (cid:48) † tot ( (cid:126)p ) ∗ T ˆ b θk stot ( (cid:126)p ) H (cid:126)p .If the creation and annihilation operators are Hermitian conjugated to each other, the result of { ˆ b θk stot ( (cid:126)p ) ∗ T ˆ b θk s † tot ( (cid:126)p ) + ˆ b θk s † tot ( (cid:126)p ) ∗ T ˆ b θk stot ( (cid:126)p ) }H (cid:126)p = H (cid:126)p is the whole H (cid:126)p back. Each of the two summands operates on its own half of H (cid:126)p . Jumping togetherover even number of occupied states ˆ b θk stot ( (cid:126)p ) and ˆ b θk s † tot ( (cid:126)p ) do not change the sign of particular“Slater determinant”. (Let us add that ˆ b θk stot ( (cid:126)p ) reduces for particular k and s the Hilbert space H (cid:126)p for a factor , and so does ˆ b θk s † tot ( (cid:126)p ). The sum of both, applied on H (cid:126)p , reproduces the whole H (cid:126)p .)4 B. ”Slater determinants” of Hilbert space of anticommuting integer spin “fermions”
The total Hilbert space of anticommuting ”fermions” is the infinite product of the Hilbert spacesof particular (cid:126)p , H = (cid:81) ∞ (cid:126)p ⊗ N H (cid:126)p , Eq. (29), represented by infinite numbers of ”Slater determinants” N H = (cid:81) ∞ (cid:126)p d − , Eq. (30). The notation ⊗ N is to point out that creation operators ˆ b θk s † tot ( (cid:126)p ), whichorigin in superposition of odd number of θ a ’s, keep their odd character also in the tensor productsof the internal and momentum space, as well as in the ”Slater determinants”, in which creationoperators determine the occupied states.The application of creation operators ˆ b θk s † tot ( (cid:126)p ) and their Hermitian conjugated annihilationoperators ˆ b θk stot ( (cid:126)p ) on the Hilbert space H has the property, manifested in Eq. (27), leading to theconclusion that the application of ˆ b θk s † tot ( (cid:126)p ) ∗ T ˆ b θk (cid:48) s (cid:48) † tot ( (cid:126)p (cid:48) ) ∗ T H is not zero if at least one of ( k, s, (cid:126)p )is not equal to ( k (cid:48) , s (cid:48) , (cid:126)p (cid:48) ), while ˆ b θk s † tot ( (cid:126)p ) ∗ T ˆ b θk (cid:48) s (cid:48) † tot ( (cid:126)p (cid:48) ) ∗ T H + ˆ b θk (cid:48) s (cid:48) † tot ( (cid:126)p (cid:48) ) ∗ T ˆ b θk s † tot ( (cid:126)p ) ∗ T H = 0 forany ( k, s, (cid:126)p ) and any ( k (cid:48) , s (cid:48) , (cid:126)p (cid:48) ), what is not difficult to prove when taking into account Eq. (27).One can easily show that the creation operators ˆ b θk s † tot ( (cid:126)p ) and the annihilation operators ˆ b θk stot ( (cid:126)p (cid:48) )fulfill equivalent anticommutation on the whole Hilbert space of infinity many ”Slater determi-nants” as they do on the Hilbert space H (cid:126)p . { ˆ b θk stot ( (cid:126)p ) , ˆ b θk s † tot ( (cid:126)p (cid:48) ) } ∗ T + H = δ kk (cid:48) δ ss (cid:48) δ ( (cid:126)p − (cid:126)p (cid:48) ) H , { ˆ b θk stot ( (cid:126)p ) , ˆ b θk s † tot ( (cid:126)p (cid:48) ) } ∗ T + H = 0 · H , { ˆ b θk s † tot ( (cid:126)p ) , ˆ b θk (cid:48) s (cid:48) † tot ( (cid:126)p (cid:48) ) } ∗ T + H = 0 · H . (34)Creation operators, ˆ b sf † tot ( (cid:126)p ), operating on a vacuum state, as well as on the whole Hilbert space,define the second quantized fermion states. C. Relations between creation operators ˆ b θ k s † tot ( (cid:126)p ) in the Grassmann odd algebra and thecreation operators postulated by Dirac Creation operators ˆ b θ k s † tot ( (cid:126)p ) define the second quantized ”fermion” fields of integer spins.Since the second quantized Dirac fermions have the half integer spin, the ”Grassmann fermions”,the internal degrees of which is described by the Grassmann odd algebra, have the integer spin. Thecomparison between the second quantized fields of Dirac and those presented in this Part I of thepaper can only be done on a rather general level. We leave therefore the detailed comparison of thecreation and annihilation operators for fermions with half integer spins between those postulatedby Dirac and the ones following from the Clifford odd algebra presented in Part II to Subsect. 3.4of Part II.5Here we discuss only the relations among appearance of the creation and annihilation operatorsoffered by the Grassmann odd algebra and those postulated by Dirac. In both cases we treat only d = (3 + 1)-dimensional space, that is we solve the equations of motion for p a = ( p , p , p , p )(in the case that d > d = (3 + 1), when p a =( p , p , p , p , , , . . . , d = (3 + 1) dimensions postulated as follows Ψ s † ( x , (cid:126)x ) = (cid:88) i,(cid:126)p k ˆ a † i ( (cid:126)p k ) u si ( (cid:126)p k ) e − i ( p x − ε(cid:126)p · (cid:126)x ) . (35) v si ( (cid:126)p k ) (= u si e − i ( p x ) − ε(cid:126)p · (cid:126)x ) are the two left handed (Γ (3+1) = −
1) and the two right handed(Γ (3+1) = 1, Eq. (B.3)) two-component column matrices, representing the four solutions s of theWeyl equation for free massless fermions of particular momentum | (cid:126)p k | = | p k | ([2], Eqs. (20-49) -(20-51)), the factor ε = ± a † i ( (cid:126)p k ) are by Dirac postulated creation operators, which together with annihilation operatorsˆ a i ( (cid:126)p k ), fulfill the anticommutation relations ([2], Eqs. (20-49) - (20-51)), { ˆ a † i ( (cid:126)p k ) , ˆ a † j ( (cid:126)p l ) } ∗ T + = 0 = { ˆ a i ( (cid:126)p k ) , ˆ a j ( (cid:126)p l ) } ∗ T + , { ˆ a i ( (cid:126)p k ) , ˆ a † j ( (cid:126)p l ) } ∗ T + = δ ij δ (cid:126)p k (cid:126)p l , (36)in the case of discretized momenta for a fermion in a box. Creation operators and annihilation op-erators, ˆ a † i ( (cid:126)p k ) and ˆ a i ( (cid:126)p k ), are postulated to have on the Hilbert space of all ”Slater determinants”these anticommutation properties.To be able to relate the creation operators of Dirac ˆ a † i ( (cid:126)p k ) with ˆ b θks † tot ( (cid:126)p k ) from Eq. (34), let usremind the reader that ˆ b θks † tot ( (cid:126)p k ) is a superposition of basic vectors ˆ b θk † i with the coefficients c ksi ( (cid:126)p ),which depend on the momentum (cid:126)p , Eq. (22) (ˆ b θk s † ( (cid:126)p ) = (cid:80) i c ksi ( (cid:126)p ) ˆ b θk † i ), so that ˆ b θks † tot ( (cid:126)p k ) (= (cid:80) i c ksi ( (cid:126)p ) ˆ b θk † i e − i ( p x − ε(cid:126)p · (cid:126)x ) ) solves the equations of motion for free massless ”Grassmann fermions”for plane waves, while | p | = | (cid:126)p | .We treat in this subsection the Grassmann case in (3+1)-dimensional space only, without takingcare on different irreducible representations k as well as on charges, in order to be able to relatethe creation and annihilation operators in Grassmann space with the Dirac’s ones. In this case theodd Grassmann creation operators are expressible with the ”basic vectors”, which are fourplets,6presented in Table I on the 7 th up to the 10 th lines, the same on both decuplets, neglecting θ θ contribution. (They have handedness in d = (3 + 1) equal zero.)Let us rewrite creation operators in the Dirac case so that their expressions resemble the ex-pression for the creation operators ˆ b θs † tot ( (cid:126)p k ) = (cid:80) i c si ( (cid:126)p ) ˆ b θ † i e − i ( p x − ε(cid:126)p · (cid:126)x ) , leaving out the index ofthe irreducible representation. ˆa s † tot ( (cid:126)p k ) def = (cid:88) i ˆ a † i ( (cid:126)p k ) u si ( (cid:126)p k ) e − i ( p x − ε(cid:126)p · (cid:126)x ) def = (cid:88) i α si ( (cid:126)p k ) ˆ a † i e − i ( p x − ε(cid:126)p · (cid:126)x ) to be compared withˆ b θs † tot ( (cid:126)p k ) = (cid:88) i c si ( (cid:126)p ) ˆ b θ † i e − i ( p x − ε(cid:126)p · (cid:126)x ) . (37)We define in the Dirac case two creation operators: ˆa s † tot ( (cid:126)p k ) and ˆ a † i . Since Ψ s † ( x , (cid:126)x ) = (cid:80) (cid:126)p k ˆa s † tot ( (cid:126)p k ), Eq. (35), we realize that the two expressions u si ( (cid:126)p k ) ˆ a † i ( (cid:126)p k ) and α si ( (cid:126)p k ) ˆ a † i describethe same degrees of freedom.These new creation operators ˆa s † tot ( (cid:126)p k ) can not be related directly to ˆb θs † tot ( (cid:126)p k ), since the first onesdescribe the second quantized fields of the half integer spin fermions, while the later describe thesecond quantized integer spin ”fermion” fields. However, both fulfill the anticomutation relationsof Eq. (34).The reader can notice that the creation operators ˆ a † i do not depend on (cid:126)p as also ˆ b θ † i do not,both describing the internal degrees of freedom, while α si ( (cid:126)p k ) ˆ a † i and α si ( (cid:126)p k ) ˆ b θ † i do.The creation and annihilation operators of Dirac fulfill obviously the anticommutation relationsof Eq. (34). To see this we only have to replace ˆ b θh s † tot ( (cid:126)p ) by ˆ a h s † tot ( (cid:126)p ) by taking into account relationof Eq. (37).Creation and annihillation operators of the Dirac second quantized fermions with half integerspins are in Part II, in Subsect. III.D, related to the corresponding ones, offered by the Cliffordalgebra. Relating the creation and annihilation operators offered by the Clifford algebra objectswith the Dirac’s ones ensures us that the Clifford odd algebra explains the Dirac’s postulates. IV. CONCLUSIONS
We learn in this Part I paper, that in d -dimensional space the superposition of odd productsof θ a ’s exist, Eqs. (8, 10, 9), chosen to be the eigenvectors of the Cartan subalgebra, Eq. (5),which together with their Hermitian conjugated partners, odd products of ∂∂θ a ’s, Eqs. (2, 8, 6),fulfill on the algebraic level on the vacuum state | φ o > = | > , Eq. (25), the requirements for theanticommutation relations for the Dirac’s fermions.7The creation operators defined on the tensor products of internal space of ”Grassmann basisvectors” (of finite number of basis states) and of momentum space (with infinite number of basisstates), arranged to be solutions of the equation of motion for free massless ”Grassmann fermions”,Eq. (21), form the infinite dimensional Hilbert space of ”Slater determinants” of (continuous) infi-nite number of momenta, with 2 d − possibilities for each momentum (cid:126)p , Eq. (34)). These creationoperators and their Hermitian conjugated partners fulfill on the Hilbert space the anticommutationrelations postulated by Dirac for second quantized fermion fields.We demonstrate the way of deriving second quantized integer fermion fields.In the subsection I A we clarify the relation between our description of the internal space offermions with ”basis vectors”, manifesting oddness and transferring the oddness to the corre-sponding creation and annihilation operators of second quantized fermions, to the ordinary secondquantized creation and annihilation operators from a slightly different point of view.Since the creation and annihilation operators, which are superposition of odd products of θ a ’sand ∂∂θ a ’s, respectively, anticommute algebraically when applying on the vacuum state, Eq. (11,12) (while the corresponding even products of θ a ’s and ∂∂θ a ’s commute, Eq. (16)), it follows thatalso creation operators, defined on tensor products of the finite number of ”basis vectors” (describ-ing the internal degrees of freedom of ”Grassmann fermions”) and on infinite basis of momentumspace, together with their Hermitian conjugated partners annihilation operators, fulfill the anti-commutation relations of Eq. (34). The use of the Grassmann odd algebra to describe the internalspace of ”Grassmann fermions” offers the anticommutation relations without postulating them:on the (simple) vacuum state as well as on the Hilbert space of infinite number of ”Slater deter-minants” of all possible single particle states, empty or occupied, of the second quantized integerspin ”fermion” fields. Correspondingly we second quantized ”fermion fields” without postulatingcommutation relations of Dirac.The internal ”basis vectors” are chosen to be eigenvectors of the Cartan subalgebra operatorsin the way that the symmetry agrees with the properties of usual Dirac’s creation and annihilationoperators of second quantized fermions — in the Clifford case for half integer spin, while in the”Grassmann fermions” for the integer spins.The ”Grassmann fermions” carry the spin and charges, originated in d ≥
5, in the adjointrepresentations. ”Grassmann fermions” offer no families, what means that there is no availableoperators, which would connect different irreducible representations of the Lorentz group (withoutbreaking symmetries).No elementary ”Grassmann fermions” with the spins and charges in the adjoint representations8have been observed, and since the observed quarks and leptons and anti-quarks and anti-leptonshave half integer spins, charges in the fundamental representations and appear in families, it doesnot seem possible for the future observation of the integer spin elementary ”Grassmann fermions”,especially not since Eq. (19) in Part II demonstrates that the reduction of space in Clifford case,needed for the appearance of second quantized half integer fermions, reduces also the Grassmannspace, leaving no place for second quantized ”Grassmann fermions” with the integer spin.In Part II two kinds of operators are studied; There are namely two kinds of the Clifford algebraobjects, γ a = ( θ a + ∂∂θ a ) and ˜ γ a = i ( θ a − ∂∂θ a ), which anticommute, { γ a , ˜ γ a } + = 0 ( { γ a , γ b } + =2 η ab = { ˜ γ a , ˜ γ b } + ), and offer correspondingly two kinds of independent representations.Each of these two kinds of independent representations can be arranged into irreducible rep-resentations with respect to the two Lorentz generators — S ab = i ( γ a γ b − γ b γ a ) and ˜ S ab = i (˜ γ a ˜ γ b − ˜ γ b ˜ γ a ). All the Clifford irreducible representations of any of the two kinds of algebras areindependent and disconnected.The two Dirac’s actions in d -dimensional space for free massless fermions ( A = (cid:82) d d x ( ψ † γ γ a p a ψ ) + h.c. and ˜ A = (cid:82) d d x ( ψ † ˜ γ ˜ γ a p a ψ ) + h.c. ) lead to the equations ofmotion, which have the solutions in both kinds of algebras for an odd Clifford character (they aresuperposition of an odd products of γ a ’s and ˜ γ a ’s, respectively), forming on the tensor product offinite number of ”basis vectors” describing the internal space and of the infinite number of basisof momentum space, the creation and annihilation operators, which only ”almost” anticommute,while the Grassmann odd creation and annihilation operators do anticommute. Although ”vec-tors” of one irreducible representation of an odd Clifford algebra character, anticommute amongthemselves and so do their Hermitian conjugated partners in each of the two kinds of the Cliffordalgebras, the anticommutation relations among creation and annihilation operators in each of thetwo Clifford algebras separately, do not fulfill the requirement, that only the anticommutator of acreation operator and its Hermitian conjugated partner gives a nonzero contribution.The decision, the postulate, Eq. (12), that only one kind of the Clifford algebra objects — wemake a choice of γ a — describes the internal space of fermions, while the second kind — ˜ γ a inthis case — does not, and consequently determine “family” quantum numbers which distinguishamong irreducible representations of S ab , solves the problems: a. Creation operators and their Hermitian conjugated partners, which are odd products of super-positions of γ a , applied on the vacuum state, fulfill on the algebraic level the anticommutation re-lations, and the creation and annihilation operators creating the second quantized Clifford fermionfields fulfill all the requirements, which Dirac postulated for fermions.9 b. Different irreducible representations with respect to S ab carry now different ”family” quantumnumbers determined by d commuting operators among ˜ S ab . c. The operators of the Lorentz algebra, which do not belong to the Cartan subalgebra, connectdifferent irreducible representations of S ab .The above mentioned decision, Eq. (19) in Part II, obviously reduces the degrees of freedom ofthe odd (and even) Clifford algebra, while opening the possibility for the appearance of ”families”,as well as for the explanation for the Dirac’s second quantization postulates. This decision, reducingas well the degrees of freedom of Grassmann algebra, disables the existence of the integer spin”fermions” as elementary particles.Let us point out again at the end that when the internal part of the single particle wave functionanticommute under the algebra product ∗ A , then this implies that the wave functions with suchinternal part anticommute under the extension of ∗ A to the (full) single particle wave functionsand so do anticommute the corresponding creation and annihilation operators what manifests alsoon the properties of the Hilbert space.The anticommuting single fermion states manifest correspondingly the oddness already on thelevel of the first quantization. Appendix A: Norms in Grassmann space and Clifford space
Let us define the integral over the Grassmann space [6] of two functions of the Grassmanncoordinates < B | θ >< C | θ > , < B | θ > = < θ | B > † , < b | θ > = d (cid:88) k =0 b a ...a k θ a · · · θ a k , by requiring { dθ a , θ b } + = 0 , (cid:90) dθ a = 0 , (cid:90) dθ a θ a = 1 , (cid:90) d d θ θ θ · · · θ d = 1 ,d d θ = dθ d . . . dθ , ω = d (cid:89) k =0 ( ∂∂θ k + θ k ) , (A1)0with ∂∂θ a θ c = η ac . We shall use the weight function [6] ω = (cid:81) dk =0 ( ∂∂θ k + θ k ) to define the scalarproduct in Grassmann space < B | C >< B | C > = (cid:90) d d θ a ω < B | θ > < θ | C > = d (cid:88) k =0 (cid:90) b ∗ b ...b k c b ...b k . (A2)To define norms in Clifford space Eq. (A1) can be used as well. Appendix B: Handedness in Grassmann and Clifford space
The handedness Γ ( d ) is one of the invariants of the group SO ( d ), with the infinitesimal generatorsof the Lorentz group S ab , defined asΓ ( d ) = αε a a ...a d − a d S a a · S a a · · · S a d − a d , (B1)with α , which is chosen so that Γ ( d ) = ± S ab is defined in Eq. (3), while in the Clifford case Eq. (B1) simplifies,if we take into account that S ab | a (cid:54) = b = i γ a γ b and ˜ S ab | a (cid:54) = b = i ˜ γ a ˜ γ b , as followsΓ ( d ) : = ( i ) d/ (cid:89) a ( √ η aa γ a ) , if d = 2 n . (B2) Acknowledgments
The author N.S.M.B. thanks Department of Physics, FMF, University of Ljubljana, Societyof Mathematicians, Physicists and Astronomers of Slovenia, for supporting the research on the spin-charge-family theory, the author H.B.N. thanks the Niels Bohr Institute for being allowed tostaying as emeritus, both authors thank DMFA and Matjaˇz Breskvar of Beyond Semiconductorfor donations, in particular for sponsoring the annual workshops entitled ”What comes beyond thestandard models” at Bled. [1] P.A.M. Dirac
Proc. Roy. Soc. (London) , A 117 (1928) 610.[2] H.A. Bethe, R.W. Jackiw, ”Intermediate quantum mechanics”, New York : W.A. Benjamin, 1968.[3] S. Weinberg, ”The quantum theory of fields”, Cambridge, Cambridge University Press, 2015. [4] J. de Boer, B. Peeters, K. Skenderis, P. van Nieuwenhuizen, ”Loop calculations in quantum-mechanicalnon-linear sigma models sigma models with fermions and applications to anomalies”, Nucl.Phys. B459(1996) 631-692 [arXiv:hep-th/9509158].[5] N. Mankoˇc Borˇstnik, ”Spin connection as a superpartner of a vielbein”, Phys. Lett.
B 292 (1992) 25-29.[6] N. Mankoˇc Borˇstnik, ”Spinor and vector representations in four dimensional Grassmann space”,
J. ofMath. Phys. (1993) 3731-3745.[7] N.S. Mankoˇc Borˇstnik, ”Spin-charge-family theory is offering next step in understanding elementaryparticles and fields and correspondingly universe”, J. Phys.: Conf. Ser. 845 012017 [arXiv:1409.4981,arXiv:1607.01618v2].[8] N.S. Mankoˇc Borˇstnik, ”Matter-antimatter asymmetry in the spin-charge-family theory”, Phys. Rev.
D 91 (2015) 065004 [arXiv:1409.7791].[9] N.S. Mankoˇc Borˇstnik, ”The spin-charge-family theory explains why the scalar Higgs carries the weakcharge ± and the hyper charge ∓ ”, Proceedings to the 17 th Workshop ”What comes beyond thestandard models”, Bled, 20-28 of July, 2014, Ed. N.S. Mankoˇc Borˇstnik, H.B. Nielsen, D. Lukman,DMFA Zaloˇzniˇstvo, Ljubljana December 2014, p.163-82 [ arXiv:1502.06786v1] [arXiv:1409.4981].[10] N.S. Mankoˇc Borˇstnik N S, ”The spin-charge-family theory is explaining the origin of families, of theHiggs and the Yukawa couplings”,
J. of Modern Phys. (2013) 823[arXiv:1312.1542].[11] N.S. Mankoˇc Borˇstnik, ”The explanation for the origin of the Higgs scalar and for the Yukawa couplingsby the spin-charge-family theory”, J.of Mod. Physics (2015) 2244 [arXiv:1409.4981].[12] N.S. Mankoˇc Borˇstnik and H.B. Nielsen, ”Why nature made a choice of Clifford and not Grassmanncoordinates”, Proceedings to the 20 th Workshop ”What comes beyond the standard models”, Bled, 9-17 of July, 2017, Ed. N.S. Mankoˇc Borˇstnik, H.B. Nielsen, D. Lukman, DMFA Zaloˇzniˇstvo, Ljubljana,December 2017, p. 89-120 [arXiv:1802.05554v4].[13] N.S. Mankoˇc Borˇstnik, H.B.F. Nielsen,
J. of Math. Phys. , 5782 (2002) [arXiv:hep-th/0111257].[14] N.S. Mankoˇc Borˇstnik, H.B.F. Nielsen, J. of Math. Phys. Phys. Rev.
D 91 (2015) 065004 [arXiv:1409.7791].[17] N.S. Mankoˇc Borˇstnik, D. Lukman, ”Vector and scalar gauge fields with respect to d = (3 + 1) inKaluza-Klein theories and in the spin-charge-family theory ”, Eur. Phys. J. C (2017) 231.[18] N.S. Mankoˇc Borˇstnik, [arXiv:1502.06786v1] [arXiv:1409.4981].[19] N.S. Mankoˇc Borˇstnik, J. of Modern Phys. (2013) 823 [arXiv:1312.1542].[20] N.S. Mankoˇc Borˇstnik, J.of Mod. Physics (2015) 2244 [arXiv:1409.4981].[21] N.S. Mankoˇc Borˇstnik, H.B.F. Nielsen, Fortschritte der Physik, Progress of Physics (2017) 1700046.[22] Relations among operators and their Hermitian conjugated partners in both kinds of the Cliffordalgebra objects are more complicated than in the Grassmann case, where the Hermitian conjugatedoperators follow by taking into account Eq. (2). In the Clifford case ( γ a + η aa i k γ b ) † is proportional to ( γ a + η aa i ( − k ) γ b ), while √ (1 + ik γ a γ b ) are self adjoint. This is the case also for representations in the sector of ˜ γ a ’s.[23] We shall see in Part II that the vacuum states are in the Clifford case, similarly as in the Grassmanncase, for both kinds of the Clifford algebra objects, γ a ’s and ˜ γ a ’s, sums of products of the annihilation × its Hermitian conjugated creation operators, and correspondingly self adjoint operators, but theyare not the identity.[24] Taking into account the relations γ a = ( θ a + ∂∂θ a ), ˜ γ a = i ( θ a − ∂∂θ a ), γ G = − iη aa γ a ˜ γ a the Lagrangedensity can be rewritten as L G = − i φ † γ G ˜ γ a (ˆ p a φ ) = − i { φ † γ G ˜ γ a ˆ p a φ − ˆ p a φ † γ G ˜ γ a φ } .[25] Varying the action with respect to φ † and φ it follows: ∂ L G ∂φ † − ˆ p a ∂ L G ∂ ˆ p a φ † = 0 = − i γ G ˜ γ a ˆ p a φ , and ∂ L G ∂φ − ˆ p a ∂ L G ∂ (ˆ p a φ ) = 0 = i ˆ p aa
D 91 (2015) 065004 [arXiv:1409.7791].[17] N.S. Mankoˇc Borˇstnik, D. Lukman, ”Vector and scalar gauge fields with respect to d = (3 + 1) inKaluza-Klein theories and in the spin-charge-family theory ”, Eur. Phys. J. C (2017) 231.[18] N.S. Mankoˇc Borˇstnik, [arXiv:1502.06786v1] [arXiv:1409.4981].[19] N.S. Mankoˇc Borˇstnik, J. of Modern Phys. (2013) 823 [arXiv:1312.1542].[20] N.S. Mankoˇc Borˇstnik, J.of Mod. Physics (2015) 2244 [arXiv:1409.4981].[21] N.S. Mankoˇc Borˇstnik, H.B.F. Nielsen, Fortschritte der Physik, Progress of Physics (2017) 1700046.[22] Relations among operators and their Hermitian conjugated partners in both kinds of the Cliffordalgebra objects are more complicated than in the Grassmann case, where the Hermitian conjugatedoperators follow by taking into account Eq. (2). In the Clifford case ( γ a + η aa i k γ b ) † is proportional to ( γ a + η aa i ( − k ) γ b ), while √ (1 + ik γ a γ b ) are self adjoint. This is the case also for representations in the sector of ˜ γ a ’s.[23] We shall see in Part II that the vacuum states are in the Clifford case, similarly as in the Grassmanncase, for both kinds of the Clifford algebra objects, γ a ’s and ˜ γ a ’s, sums of products of the annihilation × its Hermitian conjugated creation operators, and correspondingly self adjoint operators, but theyare not the identity.[24] Taking into account the relations γ a = ( θ a + ∂∂θ a ), ˜ γ a = i ( θ a − ∂∂θ a ), γ G = − iη aa γ a ˜ γ a the Lagrangedensity can be rewritten as L G = − i φ † γ G ˜ γ a (ˆ p a φ ) = − i { φ † γ G ˜ γ a ˆ p a φ − ˆ p a φ † γ G ˜ γ a φ } .[25] Varying the action with respect to φ † and φ it follows: ∂ L G ∂φ † − ˆ p a ∂ L G ∂ ˆ p a φ † = 0 = − i γ G ˜ γ a ˆ p a φ , and ∂ L G ∂φ − ˆ p a ∂ L G ∂ (ˆ p a φ ) = 0 = i ˆ p aa φ † γ G ˜ γ aa