Understanding the second quantization of fermions in Clifford and in Grassmann space -- New way of second quantization of fermions, Part II
aa r X i v : . [ phy s i c s . g e n - ph ] O c t Understanding the second quantization of fermions in Clifford and inGrassmann spaceNew way of second quantization of fermionsPart II
N.S. Mankoˇc Borˇstnik and H.B.F. Nielsen University of Ljubljana, Slovenia Niels Bohr Institute, Denmark
We present in Part II the description of the internal degrees of freedom of fermions by thesuperposition of odd products of the Clifford algebra elements, either γ a ’s or ˜ γ a ’s [1–3], whichdetermine with their oddness the anticommuting properties of the creation and annihilationoperators of the second quantized fermions in even d -dimensional space-time, as we do inPart I of this paper by the Grassmann algebra elements θ a ’s and ∂∂θ a ’s. We discuss: i. Theproperties of the two kinds of the odd Clifford algebras, forming two independent spaces,both expressible with the Grassmann coordinates θ a ’s and their derivatives ∂∂θ a ’s [2, 7, 8]. ii. The freezing out procedure of one of the two kinds of the odd Clifford objects, enablingthat the remaining Clifford objects determine with their oddness in the tensor products ofthe finite number of the Clifford basis vectors and the infinite number of momentum basis,the creation and annihilation operators carrying the family quantum numbers and fulfillingthe anticommutation relations of the second quantized fermions: on the vacuum state, andon the whole Hilbert space defined by the sum of infinite number of ”Slater determinants”of empty and occupied single fermion states. iii.
The relation between the second quantizedfermions as postulated by Dirac [18–20] and the ones following from our Clifford algebracreation and annihilation operators, what offers the explanation for the Dirac postulates.
I. INTRODUCTION
In a long series of works we, mainly one of us N.S.M.B. ([1–3, 10–15] and the references therein),have found phenomenological success with the model named by N.S.M.B the spin-charge-family theory, with fermions, the internal space of which is describable as superposition of odd productsof the Clifford algebra elements γ a ’s in d = (13 + 1) (may be with d infinity), interacting with onlygravity. The spins of fermions from higher dimensions, d > (3 + 1), manifest in d = (3 + 1) ascharges of the standard model , the gravity originating in higher dimensions manifest as the standardmodel vector gauge fields and the scalar Higgs explaining the Yukawa couplings.There are two kinds of anticommuting algebras, the Grassmann algebra and the Clifford algebra,the later with two independent subalgebras. The Grassmann algebra, with elements θ a , and theirHermitian conjugated partners ∂∂θ a [3], describes fermions with the integer spins and charges inthe adjoint representations, the two Clifford algebras, we call their elements γ a and ˜ γ a , can each ofthem be used to describe half integer spins and charges in the fundamental representations. TheGrassmann algebra is expressible with the two Clifford 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 to the second quantized fermions, in the Grassmanncase and in the Clifford case, explaining therefore the postulates of Dirac for the second quantizedfermions.We learn in Part I of this paper, that in d -dimensional space 2 d − superposition of odd productsof d θ a ’s exist, chosen to be the eigenvectors of the Cartan subalgebra, Eq. (4) of Part I, andarranged in tensor products with the momentum space to be solutions of the equation of motionfor free massless “fermions”, Eq. (21) of Part I.The creation operators, defined as the tensor products of the superposition of the finite numberof ”basis vectors” in Grassmann space, guaranteeing the oddness of operators, and of the infinitebasis in momentum space, form — applied on the vacuum state — the second quantized states ofinteger spin ”Grassmann fermions”. The creation operators fulfill together with their Hermitianconjugated partners annihilation operators (based on the internal space of odd products of ∂∂θ a ’s)all the requirements of the anticommutation relations postulated by Dirac for fermions: i. on thesimple vacuum state | > (Eqs. (7,11) of Part I), ii. on the Hilbert space H (= Q ∞ ~p ⊗ N H ~p , withthe number of empty and occupied single fermion states for particular ~p equal to 2 d − ) of infinitemany ”Slater determinants” of all possible empty and occupied single fermion states (with theinfinite number of possibilities of moments for each of 2 d − internal degrees of freedom), Eqs. (25,34) of Part I.While the creation and annihilation operators, which are superposition of odd products of θ a ’s and ∂∂θ a ’s, respectively, anticommute on the vacuum state | φ o > = | > , Eq. (7,11), thesuperposition of even products of θ a ’s and ∂∂θ a ’s, respectively, commute, Eq. (16) of Part I.The superposition of odd products of γ a ’s and their Hermitian conjugated partners, as well asof odd products of ˜ γ a ’s and their Hermitian conjugated partners, on the corresponding vacuumstates, Eq. (18), anticommute. Since the tensor products of the ”basis vectors” determining theinternal space of Clifford fermions and of the basis in momentum space manifest oddness of theinternal space, no postulates of anticommutation relations as in the Dirac second quantizationproposal is needed also for Clifford fermions with the internal space described by one of the twoClifford objects (in Subsect. II B we make a choice of γ a ’s). The oddness of the ” basis vectors”,defining the internal space of fermions, transfers to the creation and annihilation operators formingthe second quantized single fermion states in the Clifford and the Grassmann space.The ”Grassmann fermions” have integer spins, and spins in the part with d ≥ d = (3 + 1), in adjoint representations, Table I in Part I. There is no operator whichwould connect different irreducible representations of the corresponding Lorentz group. There areno elementary fermions with integer spin observed so far either.The Clifford fermions, describing the internal space with γ a ’s, have half integer spins and spinsin the part with d ≥ d = (3 + 1) in fundamental representations [10–12, 15–17]. The operators ˜ S ab (= i { ˜ γ a ˜ γ b − ˜ γ b ˜ γ a ) } − ) connect, after the reduction of the Cliffordalgebra degrees of freedom by a factor of 2, Subsect. II B, different irreducible representations ofthe Lorentz group S ab (= i { γ a γ b − γ b γ a } − ) and determine “family” quantum numbers. All inagreement with the observed families of quarks and leptons.In Part II the properties of the two kinds of the Clifford algebras objects, γ a ’s and ˜ γ a ’s, arediscussed. Both are expressible with θ a ’s and ∂∂θ a ’s ( γ a = ( θ a + ∂∂θ a ), ˜ γ a = i ( θ a − ∂∂θ a ) [2, 7, 8]),and both are, up to a constant η aa = (1 , − , − , . . . , − γ a ’s or ˜ γ a ’s, respectively) has 2 d − members, together again 2 · d − members,the same as in the case of ”Grassmann fermions”.These two internal spaces, described by the two Clifford algebras, are independent, each of themwith their own generators of the Lorentz transformations, Eq. (3), and their corresponding Cartansubalgebras, Eq. (4).In each of these two internal spaces there exist 2 d − ”basis vectors” in 2 d − irreducible repre-sentations, chosen to be ‘’eigenvectors” of the corresponding Cartan subalgebra elements, Eq. (5),and having the properties of creation and annihilation operators (the Hermitian conjugated part-ners of the creation operators) on the vacuum state: i. The application of any creation operator onthe vacuum state, Eq. (18), gives nonzero contribution, while the application of any annihilationoperator on the vacuum state gives zero contribution. ii.
Within each of these two spaces all theannihilation operators anticommute among themselves and all the creation operators anticommuteamong themselves. iii.
The vacuum state is a superposition of products of the annihilation op-erators with their Hermitian conjugated partners creation operators, like in the Grassmann case.The Clifford vacuum states, Eq. (18), are not the identity like in the Grassmann case, Eq. (19) inPart I.However, there is not only the anticommutator of the creation operator and its Hermitian conju-gated partner, which gives the nonzero contribution on the vacuum state in each of the two spaces— what in the Grassmann algebra is the case, and what the postulates of Dirac require. There are,namely, the additional (2 d − −
1) members of the same irreducible representation, to which theHermitian conjugated partner of the creation operator belongs, giving the nonzero anticommutatorwith this creation operator on the vacuum state (Eq. (12) in Subsect. II A illustrates such a case).And, there is no operators, which would connect different irreducible representations in eachof the two Clifford algebras and correspondingly there is no “family” quantum number for eachirreducible representation, needed to describe the observed quarks and leptons. (Let the readerbe reminded that also the Grassmann algebra has no operators, which would connect differentirreducible representations. The Dirac’s second quantization postulates do not take care of chargesand families of fermions, both can be treated and incorporated into the second quantization pos-tulates as quantum numbers of additional groups as proposed by the standard model .) We solvethese problems with the requirement, presented in Eq. (12): ˜ γ a γ a = − i ( γ a ) , γ a ˜ γ a = i ( γ a ) .We present in the subsection I A of this section a short overview of steps, which lead to the secondquantized fermions in the Clifford space, offering the explanation for the Dirac’s postulates. In thesubsection I B we discuss our assumption, that the oddness of the ”basis vectors” in the internalspace transfer to the corresponding creation and annihilation operators determining the secondquantized single fermion states and correspondingly the Hilbert space of the second quantizedfermions, in a generalized way.We present in Sect. II the properties of the Clifford algebra ”basis vectors” in the space of dγ a ’s and in the space of d ˜ γ a ’s. In Subsect. II A we discuss properties of the ”basis vectors” of halfinteger spin. In Subsect. II B we discuss conditions, under which operators of one of these two kindsof the Clifford algebra objects demonstrate by themselves the anticommutation relations requiredfor the second quantized ”fermions”, manifesting the half integer spins, offering the explanationfor the spin and charges of the observed quarks and leptons and anti-quarks and anti-leptons andalso for their families [1–3, 10–16].In Subsect. II C we generate the basis states manifesting the family quantum numbers.In Subsect. II D the superposition of ”basis vectors”, solving the Weyl equation, are constructed,forming creation operators depending on the momenta and fulfilling with their Hermitian conju-gated partners the anticommutation relations for the second quantized fermions.We illustrate in Sect. II E properties of the Clifford odd ”basis vectors” in d = (5+1)-dimensionalspace, and extending the internal space in a tensor product to momentum space, we present alsothe superposition solving the Weyl equation, and correspondingly present creation and annihilationoperators depending on the momentum ~p .We present in Sect. III the Hilbert space H ~p of particular momentum ~p as ”Slater determinants”:i. with no ”fermions” occupying any of the 2 d − fermion states, ii. with one ”fermion” occupyingone of the 2 d − fermion states, iii. with two ”fermions” occupying the 2 d − fermion states,..., up tothe ”Slater determinant” with all possible fermion states of a particular ~p occupied by ”fermions”.The total Hilbert space H is then the tensor product Q ∞ ⊗ N of infinite number of H ~p . On H the tensor products of creation and annihilation operators (solving the equations of motion forfree massless fermions) manifest the anticommutation relations of second quantized ”fermions”without any postulates. We also illustrate the application of the tensor products of creation andannihillation operators on H in a simple toy model.In Subsect. III D the correspondence between our way and the Dirac way of second quantizedfermions is presented, demonstrating that our way does explain the Dirac’s postulates.In Sect. IV we note that the present work is the part of the project named the spin-charge-family theory of one of the two authors of this paper (N.S.M.B.).In Sect. V we comment on what we have learned from the second quantized integer spins”fermions”, with the internal degrees of freedom described with Grassmann algebra, manifesting(from the point of view of d = (3 + 1)) charges in the adjoint representations and compare theserecognitions with the recognitions, which the Clifford algebra is offering for the description offermions, appearing in families of the irreducible representations of the Lorentz group in the internal— Clifford — space, with half integer spins and charges and family quantum numbers in thefundamental representations [1–3, 10–15]. A. Steps leading to second quantized Clifford fermions
We claim that when the internal part of the single particle wave functions anticommute underthe Clifford algebra product ∗ A , then the wave functions with such internal part, extended witha tensor product to momentum space, anticommute as well, and so do anticommute the creationand annihillation operators, creating and annihilating the extended fermion states, assuming thatthe oddness of the algebra of the wave function extends to the creation and annihilation operatorsas presented in Subsect. I B.If the internal part commute with respect to ∗ A then the corresponding wave functions and thecreation operators commute as well.Let us present steps which lead to the second quantized Clifford fermions, when using the oddClifford algebra objects to define their internal space: i. The superposition of an odd number of the Clifford algebra elements, either of γ a ’s or of ˜ γ a ,each with 2 · (2 d − ) degrees of freedom, is used to describe the internal space of fermions in evendimensional spaces. ii. The ”basis vectors” — the superposition of an odd number of Clifford algebra elements — arechosen to be the ”eigenvectors” of the Cartan subalgebras, Eq. (4), of the corresponding Lorentzalgebras, Eq. (3), in each of the two algebras. iii.
There are two groups of 2 d − members of 2 d − irreducible representations of the correspondingLorentz group, for either γ a ’s or for ˜ γ a algebras, each member of one group has its Hermitianconjugated partner in another group.Making a choice of one group of ”basis vectors” (for either γ a ’s or for ˜ γ a ) to be creation operators,the other group of ”basis vectors” represents the annihilation operators. The creation operatorsanticommute among themselves and so do anticommute annihilation operators. iv. The vacuum state is then (for either γ a ’s or for ˜ γ a ’s algebras) the superposition of products ofannihilation × their Hermitian conjugated partners the creation operators.The application of the creation operators on the vacuum state forms the ”basis states” in eachof the two spaces. The application of the annihilation operators on the vacuum state gives zero,Subsect. I B. v. The requirement that application of ˜ γ a on γ a gives − iη aa , and the application of ˜ γ a onidentity gives iη aa and that only γ a ’s are used to determine the internal space of half integerfermions, Eq. (II B), reduces the dimension of the Clifford algebra for a factor of two, enablingthat the Cartan subalgebra of ˜ S ab ’s determines the ”family” quantum numbers of each irreduciblerepresentation of S ab ’s, Eq. (3), and correspondingly also of their Hermitian conjugated partners. vi. The tensor products of superposition of the finite number of members of the ”basis vectors” andthe infinite dimensional momentum basis, chosen to solve the Weyl equations for free massless halfinteger spin fermions, determine the creation and (their Hermitian conjugated partners) annihila-tion operators, which depend on the momenta ~p , while | p | = | ~p | ( p a = ( p , p , p , p , p , . . . , p d )),manifesting the properties of the observed fermions. These creation and annihilation operatorsfulfill on the Hilbert space all the requirements for the second quantized fermions, postulated byDirac, Eq. (28) [18–20]. vii. The second quantized Hilbert space H ~p of a particular ~p is a tensor product of creationoperators of a particular ~p , defining ”Slater determinants” with no single particle state occupied(with no creation operators applying on the vacuum state), with one single particle state occupied(with one creation operator applying on the vacuum state), with two single particle statesoccupied, and so on, defining in d -dimensional space 2 (2 d − ) dimensional space for each ~p . viii. Total Hilbert space is the infinite product ( ⊗ N ) of H ~p : H = Q ∞ ~p ⊗ N H ~p . The notation ⊗ N isto point out that odd algebraic products of the Clifford γ a ’s operators anticommute no matter forwhich ~p they define the orthonormalized superposition of ”basis vectors”, solving the equations ofmotion as the orthonormalized plane wave solutions with p = | ~p | and that the anticommutationcharacter keeps also in the tensor product of internal basis and momentum basis.Since the momentum space belonging to different ~p satisfy the ”orthogonality” relations, the cre-ation and annihilation operators determined by ~p anticommute with the creation and annihilationoperators determined by any other ~p ′ . This means that in what ever way the Hilbert space H is arranged, the sign is changed whenever a creation or an annihilation operator, applying onthe Hilbert space H , jumps over odd number of occupied states. No postulates for the secondquantized fermions are needed in our odd Clifford space with creation and annihilation operatorscarrying the family quantum numbers. x. 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.
B. Our main assumption and definitions (This subsection is the same as the one of Part I.)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 ~p, i defined in both spaces C ~p, i = | ~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 ~p,i | C ~p ′ ,j > = δ ( ~p~p ′ ) δ ij .Let us point out that either B i or C ~p, i apply algebraically on the vacuum state, B i ∗ A | ψ o > and C ~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 ~p,i ∗ A | ψ o > = | ψ ~p,i > , ˆ b † C ~p,i ∗ T | ψ ~p ′ ,j > = 0 , if ~p = ~p ′ and i = j , in all other cases it followsˆ b † C ~p,i ∗ T ˆ b † C ~p ′ ,j ∗ A | ψ o > = ∓ ˆ b † C ~p ′ ,j ∗ T ˆ b † C ~p,i ∗ A | ψ o > , with the sign ± depending on whether ˆ b † C ~p,i have both an odd character, the sign is − , or not, thenthe sign is +.To each creation operator ˆ b † C ~p,i its Hermitian conjugated partner represents the annihilationoperator ˆ b C ~p,i ˆ b C ~p,i = (ˆ b † C ~p,i ) † , with the propertyˆ b C ~p,i ∗ A | ψ o > = 0 , defining the vacuum state as | ψ o > : = X 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 ~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 ~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 ~p of ˆ b † C ~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 ~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 ~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 ~p,i applies on aparticular Slater determinant from the left hand side. Jumping over occupied states to the placewith its i and ~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 ~p,i jumps over odd numbers of occupied states. In the boson case thesign of the Slater determinant does not change.0When annihilation operator ˆ b C ~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, if 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 [1, 2, 7, 10] 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 CLIFFORD ALGEBRA IN EVEN DIMENSIONAL SPACES
We can learn in Part I that in d -dimensional space of anticommuting Grassmann coordinates(and of their Hermitian conjugated partners — derivatives), Eqs. (2,6) of Part I, there exist twokinds of the Clifford coordinates (operators) — γ a and ˜ γ a — both are expressible in terms of θ a and their conjugate momenta p θa = i ∂∂θ a [2]. γ a = ( θ a + ∂∂θ a ) , ˜ γ a = i ( θ a − ∂∂θ a ) ,θ a = 12 ( γ a − i ˜ γ a ) , ∂∂θ a = 12 ( γ a + i ˜ γ a ) , (1)1offering together 2 · d operators: 2 d of those which are products of γ a and 2 d of those which areproducts of ˜ γ a .Taking into account Eqs. (1,2) of Part I ( { θ a , θ b } + = 0, { ∂∂θ a , ∂∂θ b } + = 0, { θ a , ∂∂θ b } + = δ ab , θ a † = η aa ∂∂θ a and ( ∂∂θ a ) † = η aa θ a ) one finds { γ a , γ b } + = 2 η ab = { ˜ γ a , ˜ γ b } + , { γ a , ˜ γ b } + = 0 , ( a, b ) = (0 , , , , , · · · , d ) , ( γ a ) † = η aa γ a , (˜ γ a ) † = η aa ˜ γ a , (2)with η ab = diag { , − , − , · · · , − } .It follows for the generators of the Lorentz algebra of each of the two kinds of the Cliffordalgebra operators, S ab and ˜ S ab , that: S ab = i γ a γ b − γ b γ a ) , ˜ S ab = i γ a ˜ γ b − ˜ γ b ˜ γ a ) , S ab = S ab + ˜ S ab , { S ab , ˜ S ab } − = 0 , { S ab , γ c } − = i ( η bc γ a − η ac γ b ) , { ˜ S ab , ˜ γ c } − = i ( η bc ˜ γ a − η ac ˜ γ b ) , { S ab , ˜ γ c } − = 0 , { ˜ S ab , γ c } − = 0 , (3)where S ab = i ( θ a ∂∂θ b − θ b ∂∂θ a ), Eq. (3) of Part I.Let us make a choice of the Cartan subalgebra of the commuting operators of the Lorentzalgebra for each of the two kinds of the operators of the Clifford algebra, S ab and ˜ S ab , equivalentto the choice of Cartan subalgebra of S ab in the Grassmann case, Eq. (4) in Part I, S , S , S , · · · , S d − d , ˜ S , ˜ S , ˜ S , · · · , ˜ S d − d . (4)Representations of γ a and representations of ˜ γ a are independent, each with twice 2 d − membersin 2 d − irreducible representations of an odd Clifford character and with twice 2 d − members in2 d − irreducible representations of an even Clifford character in even dimensional spaces.We make a choice for the members of the irreducible representations of the two Lorentz groupsto be the ”eigenvectors” of the corresponding Cartan subalgebra of Eq. (4), taking into account2Eq. (2), S ab
12 ( γ a + η aa ik γ b ) = k γ a + η aa ik γ b ) ,S ab
12 (1 + ik γ a γ b ) = k ik γ a γ b ) , ˜ S ab
12 (˜ γ a + η aa ik ˜ γ b ) = k γ a + η aa ik ˜ γ b ) , ˜ S ab
12 (1 + ik ˜ γ a ˜ γ b ) = k ik ˜ γ a ˜ γ b ) . (5)The Clifford ”vectors” — nilpotents and projectors — of both algebras are normalized, up to aphase, with respect to Eq. (A1) of A. Both have half integer spins. The ”eigenvalues” of theoperator S , for example, for the ”vector” ( γ ∓ γ ) are equal to ± i , respectively, for the”vector” (1 ± γ γ ) are ± i , respectively, while all the rest ”vectors” have ”eigenvalues” ± . Onefinds equivalently for the ”eigenvectors” of the operator ˜ S : for ( ˜ γ ∓ ˜ γ ) the ”eigenvalues” ± i ,respectively, and for the ”eigenvectors” (1 ± ˜ γ ˜ γ ) the ”eigenvalues” k = ± i , respectively, whileall the rest ”vectors” have k = ± .To make discussions easier let us introduce the notation for the ”eigenvectors” of the two Cartansubalgebras, Eq. (4), Ref. [2, 7]. ab ( k ): = 12 ( γ a + η aa ik γ b ) , ab ( k ) † = η aa ab ( − k ) , ( ab ( k )) = 0 , ab [ k ]: = 12 (1 + ik γ a γ b ) , ab [ k ] † = ab [ k ] , ( ab [ k ]) = ab [ k ] , ab ˜( k ): = 12 (˜ γ a + η aa ik ˜ γ b ) , ab ˜( k ) † = η aa ab ˜( − k ) , ( ab ˜( k )) = 0 , ab ˜[ k ]: = 12 (1 + ik ˜ γ a ˜ γ b ) , ab ˜[ k ] † = ab ˜[ k ] , ( ab ˜[ k ]) = ab ˜[ k ] , (6)with k = η aa η bb . Let us notice that the “eigenvectors” of the Cartan subalgebras are eitherprojectors ( ab [ k ]) = ab [ k ] , ( ab ˜[ k ]) = ab ˜[ k ] , or nilpotents ( ab ( k )) = 0 , ( ab ˜( k )) = 0 . We pay attention on even dimensional spaces, d = 2(2 n + 1) or d = 4 n , n ≥ d either of nilpotents or of projectors or of both, are“eigenstates‘’ of all the members of the Cartan subalgebra, Eq. (4), of the corresponding Lorentzalgebra, forming 2 d − irreducible representations with 2 d − members in each of the two Cliffordalgebras cases.The ”basis vectors” of Eq. (7) are ”eigenvectors” of all the Cartan subalgebra members, Eq. (4),in d = 2(2 n + 1)-dimensional space of γ a ’s. The first one is the product of nilpotents only andcorrespondingly a superposition of an odd products of γ a ’s. The second one belongs to the sameirreducible representation as the first one, if it follows from the first one by the application of S ,for example. (+ i ) (+) · · · d − d (+) , [ − i ] [ − i ] (+) · · · d − d (+) , [ − i ] [ − ] · · · d − d [ − ] . (7)One finds for their Hermitian conjugated partners, up to a sign, ( − i ) ( − ) · · · d − d ( − ) , [ − i ] [ − i ] ( − ) · · · d − d ( − ) , [ − i ] [ − ] · · · · · · d − d [ − ] . The ”basis vectors” form an orthonormal basis within each of the irreducible representationsor among irreducible representations, like the product of the following annihilation and the corres-ponding creation operator: d − d ( − ) · · · ( − ) ( − i ) ∗ A (+ i ) (+) · · · d − d (+) = 1, while all the algebraic products, which do not relate theannihilation operators with their Hermitian conjugated creation operators, give zero.Usually the operators γ a ’s are represented as matrices. We use γ a ’s here to form the basis. Onecan find in Ref. [9] how does the application of γ a ’s on the basis defined in d = (3 + 1) look like. A. Clifford “basis vectors” with half integer spin
In the Grassmann case the 2 d − odd and 2 d − even Grassmann operators, which are superposi-tion of either odd or even products of θ a ’s, are well distinguishable from their 2 d − odd and 2 d − even Hermitian conjugated operators, which are superposition of odd and even products of ∂∂θ a ’s,Eq. (6) in Part I.In the Clifford case the relation between ”basis vectors” and their Hermitian conjugated partners(made of products of nilpotents ( ab ( k ) or ab ˜( k )) and projectors ( ab [ k ] or ab ˜[ k ]), Eq. (6), are less transparent(although still easy to be evaluated). This can be noticed in Eq. (6), since √ ( γ a + η aa i k γ b ) † is4 η aa √ ( γ a + η aa i ( − k ) γ b ), while ( √ (1 + ik γ a γ b )) † = √ (1 + ik γ a γ b ) is self adjoint. (This is the case alsofor representations in the sector of ˜ γ a ’s.)One easily sees that in even dimensional spaces, either in d = 2(2 n + 1) or in d = 4 n , theClifford odd ”basis vectors” (they are products of an odd number of nilpotents and an even num-ber of projectors) have their Hermitian conjugated partners in another irreducible representation,since Hermitian conjugation changes an odd number of nilpotents (changing at the same time thehandedness of the ”basis vectors”), while the generators of the Lorentz transformations change twonilpotents at the same time (keeping the handedness unchanged).The Clifford even ”basis vectors” have an even number of nilpotents and can have an odd or aneven number of projectors. Correspondingly an irreducible representation of an even ”basis vector”can be a product of projectors only and therefore is self adjoint.Let us recognize the properties of the nilpotents and projectors. The relations are taken fromRef. [10]. ab ( k ) ab ( k ) = 0 , ab ( k ) ab ( − k )= η aa ab [ k ] , ab [ k ] ab [ k ] = ab [ k ] , ab [ k ] ab [ − k ]= 0 , ab ( k ) ab [ k ] = 0 , ab [ k ] ab ( k )= ab ( k ) , ab ( k ) ab [ − k ] = ab ( k ) , ab [ k ] ab ( − k )= 0 . (8)The same relations are valid also if one replaces ab ( k ) with ab ˜( k ) and ab [ k ] with ab ˜[ k ], Eq. (6).Taking into account Eq. (8) one recognizes that the product of annihilation and the creationoperator from Eq. (7), ( − i ) ( − ) · · · d − d ( − ) ∗ A (+ i ) (+) · · · d − d (+) , applied on a vacuum state — definedas a sum of products of all annihilation × their Hermitian conjugated partner creation operatorsfrom all irreducible representations, [ − i ] [ − ] [ − ] · · · d − d [ − ] + [+ i ] [+] [ − ] · · · d − d [ − ] + [+ i ] [ − ] [+] [ − ] · · · d − d [ − ] + · · · , Eq. (18), gives a nonzero contribution, but is not the only one for a chosen creationoperator. There are several other choices, like [+ i ] [+] · · · d − d ( − ) ∗ A (+ i ) (+) · · · d − d (+) , [+ i ] ( − ) [+] · · · d − d ( − ) ∗ A (+ i ) (+) · · · d − d (+) , which also give nonzero contributions.Let us recognize:i. The two Clifford spaces, the one spanned by γ a ’s and the second one spanned by ˜ γ a ’s, areindependent vector spaces, each with 2 d ”vectors”.5ii. The Clifford odd ”vectors” (the superposition of products of odd numbers of γ a ’s or ˜ γ a ’s,respectively) can be arranged for each kind of the Clifford algebras into two groups of 2 d − mem-bers of 2 d − irreducible representations of the corresponding Lorentz group. The two groups areHermitian conjugated to each other.iii. Different irreducible representations are indistinguishable with respect to the ”eigenvalues”of the corresponding Cartan subalgebra members.iv. The Clifford even part (made of superposition of products of even numbers of γ a ’s and ˜ γ a ’s,respectively) splits as well into twice 2 d − · d − irreducible representations of the Lorentz group.One member of each Clifford even representation, the one which is the product of projectors only,is self adjoint. Members of one irreducible representation are with respect to the Cartan subalgebraindistinguishable from all the other irreducible representations.v. The 2 d − members of each of the 2 d − irreducible representations are orthogonal to oneanother and so are orthogonal their corresponding Hermitian conjugated partners.vi. Denoting ”basis vectors” by ˆ b m † f , (where f defines different irreducible representations and m a member in the representation f ), and their Hermitian conjugate partners by ˆ b mf = (ˆ b m † f ) † , letus start for d = 2(2 n + 1) with ˆ b m =1 † f =1 : = (+ i ) (+) · · · d − d (+) , (ˆ b m =1 † f =1 ) † = ˆ b m =1 f =1 : = d − d ( − ) · · · ( − ) ( − i ) , (9)and making a choice of the vacuum state | ψ oc > as a sum of all the products of ˆ b mf · ˆ b m † f for all f = (1 , , · · · , d − ), one recognizes for the ”basis vectors” of an odd Clifford character for each ofthe two Clifford algebras the properties ˆ b mf ∗ A | ψ oc > = 0 | ψ oc > , ˆ b m † f ∗ A | ψ oc > = | ψ mf > , { ˆ b mf , ˆ b m ′ f ′ } ∗ A + | ψ oc > = 0 | ψ oc > , { ˆ b m † f , ˆ b m ′ † f } ∗ A + | ψ oc > = | ψ oc > . (10) ∗ A represents the algebraic multiplication of ˆ b m † f and ˆ b m ′ f ′ among themselves and with the vacuumstate | ψ oc > of Eq.(18), which takes into account Eq. (2). All the products of Clifford algebraelements are up to now the algebraic ones and so are also the products in Eq. (10). Since we usehere anticommutation relations, we pointed out with ∗ A this algebraic character of the products, tobe later distinguished from the tensor product ∗ T , when the creation and annihilattion operators6are defined on an extended basis, which is the tensor product of the superposition of the ”basisvectors” of the Clifford space and of the momentum basis, applying on the Hilbert space of ”Slaterdeterminants”.Obviously, ˆ b m † f and ˆ b mf have on the level of the algebraic products, when applying on the vacuumstate | ψ oc > , almost the properties of creation and annihilation operators of the second quantizedfermions in the postulates of Dirac, as it is discussed in the next items. We illustrate properties of”basis vectors” and their Hermitian conjugated partners on the example of d = (5 + 1)-dimensionalspace in Subsect. II E.vii. a. There is, namely, the property, which the second quantized fermions should fulfill inaddition to the relations of Eq. (10). The anticommutation relations of creation and annihilationoperators should be: { ˆ b mf , ˆ b m ′ † f ′ } ∗ A + | ψ oc > = δ mm ′ δ ff ′ | ψ oc > . (11)For any ˆ b mf and any ˆ b m ′ † f ′ this is not the case; besides ˆ b m =1 f =1 = d − d ( − ) · · · ( − ) ( − ) ( − i ), for example, alsoˆ b m ′ f ′ = d − d ( − ) · · · ( − ) [+] [+ i ] , and several others give, when applied on ˆ b m =1 † f =1 , nonzero contributions. There are namely 2 d − − the familiesshould exist .vii. c. The operators should exist, which connect one irreducible representation of fermionswith all the other irreducible representations.vii. d. Two independent choices for describing the internal degrees of freedom of the observedquarks and leptons are not in agreement with the observed properties of fermions.We solve these problems, cited in vii. a., vii. b., vii. c. and vii. d., by reducing the degrees offreedom offered by the two kinds of the Clifford algebras, γ a ’s and ˜ γ a ’s, making a choice of one — γ a ’s — to describe the internal space of fermions, and using the other one — ˜ γ a ’s — to describethe ”family” quantum number of each irreducible representation of S ab ’s in space defined by γ a ’s.7 B. Reduction of the Clifford space by the postulate
The creation and annihilation operators of an odd Clifford algebra of both kinds, of either γ a ’sor ˜ γ a ’s, would obviously obey the anticommutation relations for the second quantized fermions,postulated by Dirac, at least on the vacuum state, which is a sum of all the products of anni-hilation times, ∗ A , the corresponding creation operators, provided that each of the irreduciblerepresentations would carry a different quantum number.But we know that a particular member m has for all the irreducible representations the samequantum numbers, that is the same ”eigenvalues” of the Cartan subalgebra (for the vector spaceof either γ a ’s or ˜ γ a ’s), Eq. (6). The only possibility to ”dress” each irreducible representation of one kind of the two independentvector spaces with a new, let us say ”family” quantum number, is that we ”sacrifice” one of thetwo vector spaces, let us make a choice of ˜ γ a ’s, and use ˜ γ a ’s to define the ”family” quantumnumber for each irreducible representation of the vector space of γ a ’s, while keeping the relationsof Eq. (2) unchanged: { γ a , γ b } + = 2 η ab = { ˜ γ a , ˜ γ b } + , { γ a , ˜ γ b } + = 0, ( γ a ) † = η aa γ a , (˜ γ a ) † = η aa ˜ γ a ,( a, b ) = (0 , , , , , · · · , d ).We therefore postulate :Let ˜ γ a ’s operate on γ a ’s as follows [2, 3, 8, 14, 15]˜ γ a B = ( − ) B i Bγ a , (12)with ( − ) B = −
1, if B is (a function of) an odd product of γ a ’s, otherwise ( − ) B = 1 [8].After this postulate the vector space of ˜ γ a ’s is correspondingly ”frozen out”. No vector spaceof ˜ γ a ’s needs to be taken into account any longer, in agreement with the observed properties offermions. This solves the problems vii.a - vii. d. of Subsect. II A.Taking into account Eq. (12) we can check that: a. Relations of Eq. (2) remain unchanged [53]. b. Relations of Eq. (3) remain unchanged [54]. c. The eigenvalues of the operators S ab and ˜ S ab on nilpotents and projectors of γ a ’s are after the8reduction of Clifford space S ab ab ( k )= k ab ( k ) , ˜ S ab ab ( k )= k ab ( k ) ,S ab ab [ k ]= k ab [ k ] , ˜ S ab ab [ k ]= − k ab [ k ] , (13)demonstrating that the eigenvalues of S ab on nilpotents and projectors of γ a ’s differ from theeigenvalues of ˜ S ab , so that ˜ S ab can be used to denote irreducible representations of S ab with the”family” quantum number, what solves the problems vii. b. and vii. c. of Subsect. II A. d. We further recognize that γ a transform ab ( k ) into ab [ − k ], never to ab [ k ], while ˜ γ a transform ab ( k ) into ab [ k ], never to ab [ − k ] γ a ab ( k )= η aa ab [ − k ] , γ b ab ( k )= − ik ab [ − k ] ,γ a ab [ k ]= ab ( − k ) , γ b ab [ k ]= − ikη aa ab ( − k ) , ˜ γ a ab ( k )= − iη aa ab [ k ] , ˜ γ b ab ( k )= − k ab [ k ] , ˜ γ a ab [ k ]= i ab ( k ) , ˜ γ b ab [ k ]= − kη aa ab ( k ) . (14) e. One finds, using Eq. (12), ab ˜( k ) ab ( k ) = 0 , ab ˜( − k ) ab ( k )= − i η aa ab [ k ] , ab ˜( k ) ab [ k ] = i ab ( k ) , ab ˜( k ) ab [ − k ]= 0 , ab ˜[ k ] ab ( k ) = ab ( k ) , ab ˜[ − k ] ab ( k )= 0 , ab ˜[ k ] ab [ k ] = 0 , ab ˜[ − k ] ab [ k ]= ab [ k ] . (15) f. From Eq. (14) it follows S ac ab ( k ) cd ( k ) = − i η aa η cc ab [ − k ] cd [ − k ] , ˜ S ac ab ( k ) cd ( k ) = i η aa η cc ab [ k ] cd [ k ] ,S ac ab [ k ] cd [ k ] = i ab ( − k ) cd ( − k ) , ˜ S ac ab [ k ] cd [ k ] = − i ab ( k ) cd ( k ) ,S ac ab ( k ) cd [ k ] = − i η aa ab [ − k ] cd ( − k ) , ˜ S ac ab ( k ) cd [ k ] = − i η aa ab [ k ] cd ( k ) ,S ac ab [ k ] cd ( k ) = i η cc ab ( − k ) cd [ − k ] , ˜ S ac ab [ k ] cd ( k ) = i η cc ab ( k ) cd [ k ] . (16)9 g. Each irreducible representation has now the ”family” quantum number, determined by ˜ S ab ofthe Cartan subalgebra of Eq. (4). Correspondingly the creation and annihilation operators fulfillalgebraically the anticommutation relations of Dirac second quantized fermions: Different irre-ducible representations carry different ”family” quantum numbers and to each ”family” quantummember only one Hermitian conjugated partner with the same ”family” quantum number belong.Also each summand of the vacuum state, Eq. (18), belongs to a particular ”family”. This solvesthe problem vii. a. of Subsect. II A.The anticommutation relations of Dirac fermions are therefore fulfilled on the vacuum state,Eq. (18), on the algebraic level, without postulating them. They follow by themselves from thefact that the creation and annihilation operators are superposition of odd products of γ a ’s. Statement 1:
The oddness of the products of γ a ’s guarantees the anticommuting propertiesof all objects which include odd number of γ a ’s.We shall show in Subsect. II D of this section, and in Sect. III, that the same relations are validalso on the Hilbert space of all the second quantized fermions states, with the creation operatorsdefined on the tensor product of ”basis vectors” of the Clifford algebra and on the basis of themomentum space, where the Hilbert space is defined with the creation operators of all possiblemomenta of all possible ”Slater determinants” applying on | ψ oc > .Let us write down the anticommutation relations of Clifford odd ”basic vectors”, representingthe creation operators and of the corresponding annihilation operators again. { ˆ b mf , ˆ b m ′ † f ′ } ∗ A + | ψ oc > = δ mm ′ δ ff ′ | ψ oc > , { ˆ b mf , ˆ b m ′ f ′ } ∗ A + | ψ oc > = 0 · | ψ oc > , { ˆ b m † f , ˆ b m ′ † f ′ } ∗ A + | ψ oc > = 0 · | ψ oc > , ˆ b m † f ∗ A | ψ oc > = | ψ mf > , ˆ b mf ∗ A | ψ oc > = 0 · | ψ oc > , (17)with ( m, m ′ ) denoting the ”family” members and ( f, f ′ ) denoting ”families”, ∗ A represents thealgebraic multiplication of ˆ b mf with the vacuum state | ψ oc > of Eq.(18) and among themselves,taking into account Eq. (2). h. The vacuum state for the vector space determined by γ a ’s remains unchanged | ψ oc > , Eq. (80)of Ref. [3], it is a sum of the products of any annihilation operator with its Hermitian conjugated0partner of any family. | ψ oc > = [ − i ] [ − ] [ − ] · · · d − d [ − ] + [+ i ] [+] [ − ] · · · d − d [ − ]+ [+ i ] [ − ] [+] · · · d − d [ − ] + · · · | > , for d = 2(2 n + 1) , | ψ oc > = [ − i ] [ − ] [ − ] · · · d − d − [ − ] d − d [+]+ [+ i ] [+] [ − ] · · · d − d − [ − ] d − d [+] + · · · | > , for d = 4 n , (18) n is a positive integer. i. Taking into account the relation among θ a in Eq. (1) and Eq. (12), requiring that ˜ γ a · iγ a ,it follows algebraically θ a = γ a , ∂∂θ a = 0 . (19)The Hermitian conjugated part of the space in the Grassmann case is ”freezed out” together withthe ”vector” space of ˜ γ a ’s. C. Clifford fermions with families
Let us make a choice of the starting creation operator ˆ b † of an odd Clifford character and ofits Hermitian conjugated partner in d = 2(2 n + 1) and d = 4 n , respectively, as followsˆ b † : = (+ i ) (+) (+) · · · d − d − (+) d − d (+) , (ˆ b † ) † = ˆ b : = d − d ( − ) d − d − ( − ) · · · ( − ) ( − ) ( − i ) ,d = 2(2 n + 1) , ˆ b † : = (+ i ) (+) (+) · · · d − d − (+) d − d [+] , (ˆ b † ) † = ˆ b : = d − d [+] d − d − ( − ) · · · ( − ) ( − ) ( − i ) ,d = 4 n . (20)All the rest ”vectors”, belonging to the same Lorentz representation, follow by the application ofthe Lorentz generators S ab ’s.The representations with different ”family” quantum numbers are reachable by ˜ S ab , since, ac-cording to Eq. (16), we recognize that ˜ S ac transforms two nilpotents ab ( k ) cd ( k ) into two projectors1 ab [ k ] cd [ k ], without changing k ( ˜ S ac transforms ab [ k ] cd [ k ] into ab ( k ) cd ( k ), as well as ab [ k ] cd ( k ) into ab ( k ) cd [ k ]). All the”family” members are reachable from one member of a new family by the application of S ab ’s.In this way, by starting with the creation operator ˆ b † , Eq. (20), 2 d − ”families”, each with2 d − ”family” members follow.Let us find the starting member of the next ”family” to the ”family” of Eq. (20) by the appli-cation of ˜ S ˆ b † : = [+ i ] [+] (+) · · · d − d − (+) d − d (+) , ˆ b : = d − d ( − ) d − d − ( − ) · · · ( − ) [+] [+ i ] . (21)The corresponding annihilation operators, that is the Hermitian conjugated partners of 2 d − ”families”, each with 2 d − ”family” members, following from the starting creation operator ˆ b † bythe application of S ab ’s — the family members — and the application of ˜ S ab — the same familymember of another family — can be obtained by Hermitian conjugation. The creation and annihilation operators of an odd Clifford character, expressed by nilpotentsand projectors of γ a ’s, obey anticommutation relations of Eq. (17), without postulating the secondquantized anticommutation relations as we explain in Subsect. II B.The even partners of the Clifford odd creation and annihilation operators follow by either theapplication of γ a on the creation operators, leading to 2 d − ”families”, each with 2 d − members,or with the application of ˜ γ a on the creation operators, leading to another group of the Cliffordeven operators, again with the 2 d − ”families”, each with 2 d − members.It is not difficult to recognize, that each of the Clifford even ”families”, obtained by the ap-plication of γ a or by ˜ γ a on the creation operators, contains one selfadjoint operator, which is theproduct of projectors only, contributing as a summand to the vacuum state, Eq. (18). D. Action for free massless Clifford fermions with half integer spin and solutions of Weylequations
To relate the creation operators, expressed with the Clifford odd ”basis vectors”, and the cre-ation operators, creating the second quantized fermions, we define the tensor products of the finitenumber of odd Clifford ”basis vectors” and infinite basis of momentum space. To compare prop-erties of our creation operators of the second quantized fermions with those of Dirac, the solutionof the equations of motion of the Weyl (for massless free fermions) or of the Dirac equations areappropriate.2The Lorentz invariant action for a free massless fermion in Clifford space is well known A = Z d d x
12 ( ψ † γ γ a p a ψ ) + h.c. , (22) p a = i ∂∂x a , leading to the equation of motion γ a p a | ψ > = 0 , (23)and to the Klein-Gordon equation γ a p a γ b p b | ψ > = p a p a | ψ > = 0 ,γ appears in the action to take care of the Lorentz invariance of the action.Our Clifford algebra ”basis vectors” offer the description of only the internal degrees of freedomof fermions (in d = (3 + 1) the ”basis vectors” offers the description of only the spin and familydegrees of freedom, in d ≥ d − × d − —of basis vectors of the odd products of γ a ) to the momentum or coordinate space with (infinitenumber of) basis. Statement 2:
For deriving the anticommutation relations for the Clifford fermions, to becompared with the anticommutation relations of the second quantized fermions, we need to definethe tensor product of the Clifford odd ”basis vectors” and the momentum space basis ( p a ,γ a ) = | p a > ∗ T | γ a > . The new state vector space is the tensor product of the internal space of fermions and the spaceof momenta or coordinates. All states have an odd Clifford character due to oddness of the internalspace.Solutions of Eq. (23) for free massless fermions of momentum p a , a = (0 , , , , , . . . , d ) aresuperposition of ”basis vectors” ˆ b m † f , expressed by operators γ a , where f denotes a ”family” and m a ”family” member quantum number, Eqs. (20, 21), and of plane waves in the case of free, inour case, massless fermions. The equations of motion require that | p | = | ~p | . Correspondingly it3follows < x | ψ sf ( ˜p , p ) > | p = | ˜p | = Z dp δ ( p − | ~p | ) ˆ b sf † ( ~p ) e − ip a x a ∗ A | ψ oc > = (ˆ b sf † ( ~p ) · e − i ( p x − ε~p · ~x ) ) | p = | ~p | ∗ A | ψ oc > , where we define , ˆ b sf † ( ~p ) | p = | ~p | def = X m c sf m ( ~p, | p | = | ~p | ) ˆ b m † f , | ψ sf ( ˜x , x ) > = Z + ∞−∞ d d − p ( √ π ) d − (ˆ b sf † ( ~p ) e − i ( p x − ε~p · ~x ) | p = | ~p | ∗ A | ψ oc > , (24) s represents different orthonormalized solutions of the equations of motion, ε = ±
1, dependingon handedness and spin of solutions, c sf m ( ~p, | p | = | ~p | ) are coefficients, depending on momentum | ~p | with | p | = | ~p | , while ∗ A denotes the algebraic multiplication of the ”basis vectors” ˆ b m † f on thevacuum state | ψ oc > , Eq. (17).An illustration of ˆ b sf † ( ~p ) is presented in Subsect. II E.Since the “basis vectors” in internal space of fermions are orthogonal according to Eq. (10)( { ˆ b mf ∗ A , ˆ b m ′ † f ′ ∗ A } + | ψ oc > = ˆ b mf ∗ A ˆ b m ′ † f ′ ∗ A | ψ oc > ),ˆ b mf ∗ A ˆ b m ′ † f ′ ∗ A | ψ oc > = δ mm ′ δ ff ′ | ψ oc > , it follows for particular ~p , p = | ~p | , that X m c sf ∗ m ( ~p, | p | = | ~p | ) c s ′ f ′ m ( ~p, | p | = | ~p | ) = δ ss ′ δ ff ′ , leading to Z d d − x ( √ π ) d − < ψ s ′ f ′ ( ~p ′ , p ′ = | ~p ′ | ) | x > < x || ψ sf ( ~p, p = | ~p | ) > = Z d d − x ( √ π ) d − e ip ′ a x a | p ′ = | ~p ′ | e − ip a x a | p | = | ~p | · < ψ oc | (ˆ b s ′ f ′ ( ~p ′ ) ˆ b sf † ( ~p )) ∗ A | ψ oc > = δss ′ δ ff ′ δ ( ~p − ~p ′ ) , (25)while we take into account that R d d − x ( √ π ) d − e ip ′ a x a e − ip a x a = δ ( ~p − ~p ′ ).Let us now evaluate the scalar product < ψ sf ( ~x, x ) | ψ s ′ f ′ ( ~x ′ , x ) > , taking into account thatthe scalar product is evaluated at a time x and correspondingly using the relation < ψ sf ( ~x, x ) | ψ s ′ f ′ ( ~x ′ , x ) > = δ ss ′ δ ff ′ δ ( ~x − ~x ′ ) = Z dp √ π Z dp ′ √ π δ ( p − p ′ ) Z + ∞−∞ d d − p ′ ( √ π ) d − Z + ∞−∞ d d − p ( √ π ) d − δ ( p − | ~p | ) δ ( p ′ − | ~p ′ | ) < ψ oc | (ˆ b s ′ f ′ ( ~p ′ , p ′ ) ˆ b sf † ( ~p, p )) ∗ A | ψ oc > e ip ′ a x ′ a e − ip a x a = Z dp √ π Z + ∞−∞ d d − p ′ ( √ π ) d − δ ( p − | ~p ′ | ) Z + ∞−∞ d d − p ( √ π ) d − δ ( p − | ~p | ) < ψ oc | (ˆ b sf ( ~p, p ) ˆ b s ′ f ′ † ( ~p ′ , p )) ∗ A | ψ oc > e i ( p x − ~p · ~x ) e − i ( p x ′ − ~p ′ · ~x ) . (26)4The scalar product < ψ sf ( ~x, x ) | ψ s ′ f ′ ( ~x ′ , x ) > has obviously the desired properties of the secondquantized states.Let us define the creation operators ˆ b sf † tot ( ~p ), which determine, when applying on the vacuumstate, the fermion states, Eq. (24),ˆ b sf † tot ( ~p ) def = ˆ b sf † ( ~p ) e − i ( p x − ~p · ~x ) , ˆ b sftot ( ~p ) = (ˆ b sf † tot ( ~p )) † = ˆ b sf ( ~p ) e i ( p x − ~p · ~x ) , ˆ b sf † tot ( ~p ) | ψ oc > = | ψ sf ( ~p, p = | ~p | ) > . (27)In Eq. (27) ˆ b sf † tot ( ~p ) creates on the vacuum state | ψ oc > the single fermion states. We can mul-tiply, using the tensor product ∗ T multiplication this time, an arbitrary number of such singleparticle states, what means that we multiply an arbitrary number of creation operators ˆ b sf † tot ( ~p ) ∗ T ˆ b s ′ f ′ † tot ( ~p ′ ) ∗ T · · · ∗ T ˆ b s ′′ f ′′ † tot ( ~p ′′ ), applying on | ψ oc > , which gives nonzero contributions, provided thatthey distinguish among themselves in at least one of the properties ( s, f, ~p ), in the internal spacequantum numbers ( s, f ) or in momentum part ~p , due to the orthonormal property of plane waves.The space of all such functions, which one can form - including the identity - represents thesecond quantized Hilbert space. We present these tensor products as ”Slater determinants” ofoccupied and empty states in Section III.Due to anticommutation relations of any two of creation operators { ˆ b sf † ( ~p ) , ˆ b s ′ f ′ ( ~p ) } + | ψ oc > = δ ff ′ δ ss ′ | ψ oc > , Eqs. (17, 24), while plane waves form the orthonormal basis in the momentum rep-resentation, Eq. (25), the new creation operators ˆ b sf † tot ( ~p ), which are are generated on the tensorproducts of both spaces, internal and momentum, fulfill the anticommutation relations when ap-plied on | ψ oc > . { ˆ b sftot ( ~p ) , ˆ b sf † tot ( ~p ′ ) } + ∗ T | ψ oc > = δ ss ′ δ ff ′ δ ( ~p − ~p ′ ) | ψ oc > , { ˆ b sftot ( ~p ) , ˆ b s ′ f ′ tot ( ~p ′ ) } + ∗ T | ψ oc > = 0 · | ψ oc > , { ˆ b sf † tot ( ~p ) , ˆ b s ′ f ′ † tot ( ~p ′ ) } + ∗ T | ψ oc > = 0 · | ψ oc > , ˆ b sf † tot ( ~p ) ∗ T | ψ oc> = | ψ sf ( ~p ) > , ˆ b sftot ( ~p ) ∗ T | ψ oc > = 0 · | ψ oc > , | p | = | ~p | . (28)It is not difficult to show that ˆ b sftot ( ~p ) and ˆ b sf † tot ( ~p ) manifest the same anticommutation relations alsoon tensor products of an arbitrary chosen set of single fermion states, what we discuss in Sect. III.5Therefore, with the choice of the Clifford odd ”basis states” to describe the internal space offermions (we can proceed equivalently in the Grassmann case) and using the tensor product of theinternal space and the momentum or coordinate space to solve the equations of motion, we derivethe anticommutation relations among creation operators ˆ b sf † tot ( ~p ) and their Hermitian conjugatedpartners annihilation operators (ˆ b sf † tot ( ~p )) † = ˆ b sf ( ~p ) e i ( p x − ~p, · ~x ) = ˆ b sftot ( ~p ), with | p | = | ~p | . Whileapplication of ˆ b sf † tot ( ~p ) on | ψ oc > generates the single fermion state, the application of ˆ b sftot ( ~p ) giveszero.We shall demonstrate in Sect. III that there is { ˆ b s ′ f ′ tot ( ~p ′ ) , ˆ b sf † tot ( ~p ) } + , which when applied onthe Hilbert space of the second quantized fermions (that is on tensor products of all single fermionstates, or equivalently on all possible ”Slater determinants”), gives zero when at least one of( s ′ , f ′ , ~p ′ ) differ from ( s, f, ~p ), while { ˆ b sftot ( ~p ) , ˆ b sf † tot ( ~p ) } + applied on the Hilbert space, gives thewhole Hilbert space back.Taking into account the last line of Eq. (24) and Eqs. (26,27), the creation operators Ψ † follow,which determine, when applying on the vacuum state | ψ oc > , the fermion fields | ψ sf ( ˜x , x ) > ,depending on coordinates at particular time x Ψ sf † ( ~x, x ) def = Z + ∞−∞ d d − p ( √ π ) d − ˆ b sf † tot ( ~p ) | p | = | ~p | , Ψ sf † ( ~x, x ) | ψ oc > = | ψ sf ( ~x, x ) > , { Ψ sf † ( ~x, x ) , Ψ s ′ f ′ ( ~x ′ , x ) } + | ψ oc > = δ ss ′ δ ff ′ δ ( ~x − ~x ′ ) | ψ oc > , { Ψ sf ( ~x, x ) , Ψ s ′ f ′ ( ~x ′ , x ) } + | ψ oc > = 0 . { Ψ sf † ( ~x, x ) , Ψ s ′ f ′ † ( ~x ′ , x ) } + | ψ oc > = 0 , (29)where Ψ † ( ~x, x ) and Ψ sf ( ~x, x ) are creation and annihilation partners, respectively, Hermitianconjugated to each other, in the coordinate representation, presenting the creation and annihilationoperators of the second quantized fields.The application of the creation operators ˆ b sf † tot ( ~p ) | p | = | ~p | and Ψ † ( ~x, x ) and their Hermitianconjugated partners on the Hilbert space of fermion fields will be discussed in Sect. III.Dirac uses the Lagrange and Hamilton formalism for fermion fields and assuming that the secondquantized states should anticommute to describe fermions, he derives the anticommuting creationand annihillation operators. In Subsect. III D we compare the Dirac anticommutation relationswith our way of deriving anticommutation relations for second quantized fields in details.In Subsect. II E the properties of creation and annihilation operators, ˆ b sf † tot ( ~p ) and ˆ b sftot ( ~p ), re-spectively, described by the odd Clifford algebra objects in d = (5 + 1)-dimensional space are6discussed. E. Illustration of Clifford fermions with families in d = (5 + 1) -dimensional space We illustrate properties of the Clifford odd, and correspondingly anticommuting, creation and theirHermitian conjugated partners annihilation operators, belonging to 2 − = 4 ”families”, each with 2 − = 4members in d = (5 + 1)-dimensional space. The spin in the fifth and the sixth dimension manifests as thecharge in d = (3 + 1).In Table I the ”basis vectors” of odd and even Clifford character are presented. They are ”eigenvectors”of the Cartan subalgebras, Eq. (4).Half of the Clifford odd ”basis vectors” are (chosen to be) creation operators ˆ b m † f , denoted in table by oddI , appearing in four ”families”, f = (1( a ) , b ) , c ) , d )). The rest half of the Clifford odd ”basis vectors”are their Hermitian conjugated partners ˆ b mf , presented in odd II part and denoted with the corresponding”family” and family members ( a m , b m , c m , d m ) quantum numbers.The normalized vacuum state is the product of ˆ b mf · ˆ b m † f — this product is the same for each member ofa particular family and different for different families — summed over four families | ψ oc > = 1 p − ( [ − i ] [ − [ −
1] + [+ i ] [+1] [ − [+ i ] [ − [+1] + [ − i ] [+1] [+1]) . (30)One easily checks, by taking into account Eq. (15), that the creation operators ˆ b m † f and the annihilationoperators ˆ b mf fulfill the anticommutation relations of Eq (17).The summands of the vacuum state | ψ oc > appear among selfadjoint members of even I part of Table I,each of summands belong to different ”family” [55].All the Clifford even ”families” with ”family” members of Table I can be obtained as algebraic products, ∗ A , of the Clifford odd ”vectors” of the same table.Let us find the solutions of the Weyl equation, Eq. (23), taking into account four basis creation operatorsof the first family, f = 1( a ), in Table I. Assuming that moments in the fifth and the sixth dimensions arezero, p a = ( p , p , p , p , , p = | ~p | , can befound, two with the positive charge and with spin S either equal to or to − , and two with thenegative charge − and again with S either or − . TABLE I: 2 d = 64 ”eigenvectors” of the Cartan subalgebra, Eq. (4), of the Clifford odd and even algebras in d = (5 + 1) are presented, divided into four groups, each group with four ”families”, each ”family” with four”family” members. Two of four groups are sums of an odd number of γ a ’s. The ”basis vectors”, ˆ b m † f , Eqs. (20,21), in odd I group, belong to four ”families” ( f = 1( a ) , b ) , c ) , d )) with four members ( m = 1 , , , b mf , among ”basis vectors” of the odd II part, denoted by thecorresponding ”family” and ”family” members ( a m , b m , c m , d m ) quantum numbers. The ”family” quantumnumbers, the eigenvalue of ( ˜ S , ˜ S , ˜ S ), of ˆ b m † f are written above each ”family”. The two groups with theeven number of γ a ’s, even I and even II , have their Hermitian conjugated partners within their own groupeach. There are members in each group, which are products of projectors only. Numbers —
03 12 56 —denote the indexes of the corresponding Cartan subalgebra members ( ˜ S , ˜ S , ˜ S ), Eq. (4). In the columns(7 , ,
9) the eigenvalues of the Cartan subalgebra members ( S , S , S ), Eq. (4), are presented. The lasttwo columns tell the handedness of d = (5 + 1), Γ (5+1) , and of d = (3 + 1), Γ (3+1) , respectively, defined inEq.(B2). odd I m f = 1( a ) f = 2( b ) f = 3( c ) f = 4( d ) S S S Γ (5+1) Γ (3+1) ( i , , ) ( − i , − , ) ( − i , , − ) ( i , − , − )
03 12 56 03 12 56 03 12 56 03 12 56 (+ i ) (+) (+) [+ i ] [+] (+) [+ i ] (+) [+] (+ i ) [+] [+] i − i ][ − ](+) ( − i )( − )(+) ( − i )[ − ][+] [ − i ]( − )[+] − i −
12 12 − i ](+)[ − ] ( − i )[+][ − ] ( − i )(+)( − ) [ − i ][+]( − ) − i − −
14 (+ i )[ − ][ − ] [+ i ]( − )[ − ] [+ i ][ − ]( − ) (+ i )( − )( − ) i − − − odd II S S S Γ (5+1) Γ (3+1)03 12 56 fm
03 12 56 fm
03 12 56 fm
03 12 56 fm ( − i )(+)(+) d [ − i ][+](+) d [ − i ](+)[+] d ( − i )[+][+] d − i − − i ][ − ](+) c (+ i )( − )(+) c (+ i )[ − ][+] c [+ i ]( − )[+] c i −
12 12 − − i ](+)[ − ] b (+ i )[+][ − ] b (+ i )(+)( − ) b [+ i ][+]( − ) b i − − − i )[ − ][ − ] a [ − i ]( − )[ − ] a [ − i ][ − ]( − ) a ( − i )( − )( − ) a − i − − − even I m S S S Γ (5+1) Γ (3+1) ( i , , ) ( − i , − , ) ( i , − , − ) ( − i , , − )
03 12 56 03 12 56 03 12 56 03 12 56 − i ](+)(+) ( − i )[+](+) [ − i ][+][+] ( − i )(+)[+] − i − −
12 (+ i )[ − ](+) [+ i ]( − )(+) (+ i )( − )[+] [+ i ][ − ][+] i −
12 12 − −
13 (+ i )(+)[ − ] [+ i ][+][ − ] (+ i )[+]( − ) [+ i ](+)( − ) i − − − i ][ − ][ − ] ( − i )( − )[ − ] [ − i ]( − )( − ) ( − i )[ − ]( − ) − i − − − even II m S S S Γ (5+1) Γ (3+1) ( − i , , ) ( i , − , ) ( − i , − , − ) ( i , , − )
03 12 56 03 12 56 03 12 56 03 12 56 i ](+)(+) (+ i )[+](+) [+ i ][+][+] (+ i )(+)[+] i − i )[ − ](+) [ − i ]( − )(+) ( − i )( − )[+] [ − i ][ − ][+] − i −
12 12 − i )(+)[ − ] [ − i ][+][ − ] ( − i )[+]( − ) [ − i ](+)( − ) − i − −
14 [+ i ][ − ][ − ] (+ i )( − )[ − ] [+ i ]( − )( − ) (+ i )[ − ]( − ) i − − − Clifford odd creation operators in d = (5 + 1) p = | p | , S = 12 , Γ (3+1) = 1 , ˆ b † tot ( ~p ) : = β (cid:18) (+ i ) (+) | (+) + p + ip | p | + | p | [ − i ] [ − ] | (+) (cid:19) ! · e − i ( | p | x − ~p · ~x ) , ˆ b † tot ( ~p ) : = β ∗ (cid:18) [ − i ] [ − ] | (+) − p − ip | p | + | p | (+ i ) (+) | (+) (cid:19) ! · e − i ( | p | x + ~p · ~x ) , Clifford odd creation operators in d = (5 + 1) p = | p | , S = − , Γ (3+1) = − , ˆ b † tot ( ~p ) : = − β (cid:18) [ − i ] (+) | [ − ] + p + ip | p | + | p | (+ i ) [ − ] | [ − ] (cid:19) ! · e − i ( | p | x + ~p · ~x ) , ˆ b † tot ( ~p ) : = − β ∗ (cid:18) (+ i ) [ − ] | [ − ] − p − ip | p | + | p | [ − i ] (+) | [ − ] (cid:19) ! · e − i ( | p | x − ~p · ~x ) , (31)Index s =(1 , , , counts different solutions of the Weyl equations, index f =1 denotes the family quantumnumber, all solutions belong to the same family, while β ∗ β = | p | + | p | | p | takes care that the correspondingstates are normalized.All four superposition of ˆ b sf † tot ( ~p ) | p = | ~p | = P m c sf =1 m ( ~p, | p | = | ~p | ) ˆ b m † f =1 e − i ( p x − ε~p · ~x ) , with m = (1 , m = (3 ,
4) for the second two states, Table I, s = (1 , , , III. HILBERT SPACE OF CLIFFORD FERMIONS
The Clifford odd creation operators ˆ b sf † tot ( ~p ), with | p | = | ~p | , are defined in Eq. (27) on thetensor products of the (2 d − ) ”basis vectors” (describing the internal space of fermion fields)and of the (continuously) infinite number of basis in the momentum space. The solutions of theWeyl equation, Eq. (23), are plane waves of particular momentum ~p and with energy related tomomentum, | p | = | ~p | .The creation operator ˆ b sf † tot ( ~p ) defines, when applied on the vacuum state | ψ oc > , the s th of the2 d − plane wave solutions of a particular momentum ~p belonging to the f th of the 2 d − ”families”.They fulfill together with the Hermitian conjugated partners annihilation operators ˆ b sftot ( ~p ) theanticommutation relations of Eq. (28).9These creation operators form the Hilbert space of ”Slater determinants”, defining for each”Slater determinant” the ”space” for any of the single particle fermion states of an odd Cliffordcharacter, due to the oddness of the ”basis vector” of an odd Clifford character. Each of these”spaces” can be empty or occupied. Correspondingly there is the ”Slater determinant” with allthe ”spaces” empty, the ”Slater determinants” with only one of the ”spaces” occupied, any one,and all the rest empty, the ”Slater determinants” with two ”spaces” occupied, any two, and all therest empty, and so on.These ”Slater determinant” of all possible occupied and empty states can be explained as wellif introducing the tensor multiplication of single fermion states of any quantum number and anymomentum, with the constant included. Statement 3 : Introducing the tensor product multiplication ∗ T of any number of Clifford oddfermion states of all possible internal quantum numbers and all possible momenta (that is of anynumber of ˆ b s f † tot ( ~p )) of any ( s, f, ~p ) we generate the Hilbert space of Clifford fermions.The Hilbert space of a particular momentum ~p , H ~p , contains the finite number of ”Slaterdeterminants”. The number of ”Slater determinants” is in d -dimensional space equal to N H ~p = 2 d − . (32)The total Hilbert space of anticommuting fermions is the product ⊗ N of the Hilbert spaces ofparticular ~p H = ∞ Y ~p ⊗ N H ~p . (33)The total Hilbert space H is correspondingly infinite and contains N H ”Slater determinants” N H = ∞ Y ~p d − . (34)Before starting to comment the application of the creation operators ˆ b sf † tot ( ~p ) and annihilationˆ b sftot ( ~p ) operators on the Hilbert space H (described with all possible ”Slater determinants” of allpossible occupied and empty fermion states of all possible ( s, f, ~p ), or by the tensor products of allpossible single fermion states of all possible ( s, f, ~p ), with the identity included) let us discuss prop-erties of creation and annihilation operators, the anticommutation relations of which are presentedin Eq. (28).The creation operators ˆ b sf † tot ( ~p ) and the annihilation operators ˆ b s ′ f ′ tot ( ~p ′ ), having an odd Clifford0character, anticommute, manifesting the properties as followsˆ b sf † tot ( ~p ) ∗ T ˆ b s ′ f ′ † tot ( ~p ′ ) = − ˆ b s ′ f ′ † tot ( ~p ′ ) ∗ T ˆ b sf † tot ( ~p ) , ˆ b sftot ( ~p ) ∗ T ˆ b s ′ f ′ tot ( ~p ′ ) = − ˆ b s ′ f ′ tot ( ~p ′ ) ∗ T ˆ b sftot ( ~p ) , ˆ b sftot ( ~p ) ∗ T ˆ b s ′ f ′ † tot ( ~p ′ ) = − ˆ b s ′ f ′ † tot ( ~p ′ ) ∗ T ˆ b sftot ( ~p ) , if at least one of ( s, f, ~p ) is different from ( s ′ , f ′ , ~p ′ ) , ˆ b sf † tot ( ~p ) ∗ T ˆ b sf † tot ( ~p ) = 0 , ˆ b sftot ( ~p ) ∗ T ˆ b sftot ( ~p ) = 0 , ˆ b sftot ( ~p ) ∗ T ˆ b sf † tot ( ~p ) = 1 (identity) , ˆ b sftot ( ~p ) | ψ oc > = 0 . (35)The above relations, leading from the commutation relations of Eq. (28), determine the rules ofthe application of creation and annihilation operators on ”Slater determinants”: i. The creation operator ˆ b sf † tot ( ~p ) jumps over the creation operators determining the occupied stateof another kind (that is over the occupied state distinguishing from the jumping creation one inany of the internal quantum numbers ( s, f ) or in ~p ) up to the last step when it comes to its ownempty state with the quantum numbers ( f, s ) and ~p , occupying this empty state, or, if this stateis already occupied, gives zero. Whenever ˆ b sf † tot ( ~p ) jumps over an occupied state changes the signof the ”Slater determinant”. ii. The annihilation operator changes the sign whenever jumping over the occupied state carryingdifferent internal quantum numbers ( s, f ) or ~p , unless it comes to the occupied state with its owninternal quantum numbers ( s, f ) and its own ~p , emptying this state, or, if this state is empty, giveszero.Let us point out that the Clifford odd creation operators, ˆ b sf † tot ( ~p ), and annihilation operators,ˆ b s ′ f ′ tot ( ~p ′ ), fulfill the anticommutation relations of Eq. (28) for any ~p and any ( s, f ) due to theanticommuting character (the Clifford oddness) of the ”basis vectors”, ˆ b m † f and their Hermitianconjugated partners ˆ b mf , Eqs. (20, 21), what means that the anticommuting character of creationand annihilation operators is not postulated.The total Hilbert space H has infinite number of degrees of freedom (of ”Slater determinants”)due to the infinite number of Hilbert spaces H ~p of particular ~p , H = Q ∞ ~p ⊗ N H ~p , while the Hilbertspace H ~p of particular momentum ~p has the finite dimension 2 d − .In Subsects. III A, III B, III C the properties of Hilbert spaces are discussed in more details.1 TABLE II: The four creation operators of the irreducible representation odd I from Table I, d = (5 + 1), f = 1( a ). together with their Hermitian conjugated partners are presented (up to a phase). i f = 1( a ) Her . con . f = 1( a )1 (+ i ) (+) (+) ( − i ) ( − ) ( − )2 [ − i ] [ − ] (+) [ − i ] [ − ] ( − )3 [ − i ] (+) [ − ] [ − i ] ( − ) [ − ]4 (+ i ) [ − ] [ − ] ( − i ) [ − ] [ − ] A. Application of ˆ b sf † tot ( ~p ) and ˆ b sftot ( ~p ) on Hilbert space of Clifford fermions of particular ~p The 2 d − Clifford odd creation operators of particular momentum ~p , ˆ b sf † tot ( ~p, p ), with the prop-erty | p | = | ~p | , each representing the s th solution of Eq. (23) for a particular family f , fulfill togetherwith the (Hermitian conjugated partners) annihilation operators ˆ b sftot ( ~p ) the anticommutation re-lations of Eq. (28), the application of which on the Hilbert space of ”Slater determinants” arediscussed in Eq. (35) and in the text below this equation.The Hilbert space H ~p of a particular momentum ~p consists correspondingly of 2 d − “Slaterdeterminants”. Let us write down explicitly these 2 d − contributions to the Hilbert space H ~p of a particular momentum ~p , using the notation that sf ~p represents the unoccupied state | ψ sf ( ~p, p ) > | | p | = | ~p | = ˆ b sf † tot ( ~p ) | | p | = | ~p | | ψ oc > of the s th solution of the equations of motion forthe f th family and the momentum | p | = | ~p | ), Eq. (24), while sf ~p represents the correspondingoccupied state.The number operator is defined as N sf~p = ˆ b sf † tot ( ~p ) ∗ T ˆ b sftot ( ~p ) ,N sf~p | ψ oc > = 0 · | ψ oc > , N sf~p ∗ T sf ~p = 0 ,N sf~p ∗ T sf ~p = 1 · sf ~p , N sf~p ∗ T N sf~p ∗ T sf ~p = 1 · sf ~p . (36)One can check the above relations on the example of d = (5 + 1), with the ”basis vectors” for f = 1presented in Table II and with the solution for Weyl equation, Eq. (23), presented in Eq. (31).Let us write down the Hilbert space of second quantized fermions H ~p , using the simplifiednotation as in Part I, Sect. III.A., counting for f = 1 empty states as rp , and occupied states as rp , with r = (1 , . . . , d − ), for f = 2 we count r = 2 d − + 1 , · · · , d − . Correspondingly we can2represent H ~p as follows | , , , . . . , d − p > | , | , , , . . . , d − p > | , | , , , . . . , d − p > | , | , , , . . . , d − p > | , ... | , , , . . . , d − p > | d − +2 , ... | , , , . . . , d − p > | d − , (37)with a part with none of states occupied ( N rp = 0 for all r = 1 , . . . , d − ), with a part with onlyone of states occupied ( N rp = 1 for a particular r = (1 , . . . , d − ), while N r ′ p = 0 for all the others r ′ = r ), with a part with only two of states occupied ( N rp = 1 and N r ′ p = 1, where r and r ′ runfrom (1 , . . . , d − ), and so on. The last part has all the states occupied.It is not difficult to see that the creation and annihilation operators, when applied on thisHilbert space H ~p , fulfill the anticommutation relations for the second quantized Clifford fermions. { ˆ b sftot ( ~p ) , ˆ b s ′ f ′ † tot ( ~p ) } ∗ T + H ~p = δ ss ′ δ ff ′ H ~p , { ˆ b sftot ( ~p ) , ˆ b s ′ f ′ tot ( ~p ) } ∗ T + H ~p = 0 · H ~p , { ˆ b sf † tot ( ~p ) , ˆ b s ′ f ′ † tot ( ~p ) } ∗ T + H ~p = 0 · H ~p . (38)The proof for the above relations easily follows if one takes into account that whenever the creationor annihilation operator jumps over an odd products of occupied states the sign of the ”Slaterdeterminant” changes due to the oddness of the occupied states, while states, belonging to different ~p are orthogonal [56], see Eq. (35) and the text below this equation. Then one sees that thecontribution of the application of ˆ b sf † tot ( ~p ) ∗ T ˆ b s ′ f ′ tot ( ~p ) ∗ T on H ~p has the opposite sign than thecontribution of ˆ b s ′ f ′ tot ( ~p ) ∗ T ˆ b sf † tot ( ~p ) ∗ T on H ~p .If the creation and annihilation operators are Hermitian conjugated to each other, the resultfollows ( ˆ b sftot ( ~p ) ∗ T ˆ b sf † tot ( ~p ) + ˆ b sf † tot ( ~p ) ∗ T ˆ b sftot ( ~p ) ) ∗ T H ~p = H ~p , manifesting that this application of H ~p gives the whole H ~p back. Each of the two summandsoperates on their own half of H ~p . Jumping together over an even number of occupied states,3ˆ b sftot ( ~p ) and ˆ b sf † tot ( ~p ) do not change the sign of the particular “Slater determinant”. (Let us addthat ˆ b sftot ( ~p ) reduces for the particular s and f the Hilbert space H ~p for the factor , and so doesˆ b sf † tot ( ~p ). The sum of both, applied on H ~p , reproduces the whole H ~p .)Let us repeat that the number of ”Slater determinants” in the Hilbert space of particularmomentum ~p , H ~p , in d -dimensional space is finite and equal to N H ~p = 2 d − . B. Application of ˆ b sf † tot ( ~p ) and ˆ b sftot ( ~p ) on total Hilbert space H of Clifford fermions The total Hilbert space of anticommuting fermions is the infinite product of the Hilbert spacesof particular ~p , Eq. (33), H = Q ∞ ~p ⊗ N H ~p .Due to the Clifford odd character of creation and annihilation operators, Eq. (28), and theorthogonality of the plane waves belonging to different momenta ~p , it follows that ˆ b sf † tot ( ~p ) ∗ T ˆ b sf † tot ( ~p ′ ) ∗ T H 6 = 0, ~p = ~p ′ , while { ˆ b sf † tot ( ~p ) ∗ T ˆ b sf † tot ( ~p ′ )+ ˆ b sf † tot ( ~p ′ ) ∗ T ˆ b sf † tot ( ~p ) } ∗ T H = 0, ~p = ~p ′ . This can be proven if taking into account Eq. (35). For “plane wave solutions” of equa-tions of motion in a box the momentum ~p is discretized, otherwise is continuous. The numberof “Slater determinants” in the Hilbert space H in d -dimensional space is infinite (in both cases) N H = Q ∞ ~p d − .Since the creation operators ˆ b sf † tot ( ~p ) and the annihilation operators ˆ b s ′ f ′ tot ( ~p ′ ) fulfill for particular ~p the anticommutation relations on H ~p , Eq. (38), and since the momentum states, the plane wavesolutions, are orthogonal, and correspondingly the creation and annihilation operators definedon the tensor products of the internal basis and the momentum basis, representing fermions,anticommute, Eq. (28) (the Clifford odd objects ˆ b sf † tot ( ~p ) demonstrate their oddness also with respectto ˆ b sf † tot ( ~p ′ )), the anticommutation relations follow also for the application of ˆ b sf † tot ( ~p ) and ˆ b sftot ( ~p )on H { ˆ b sftot ( ~p ) , ˆ b s ′ f ′ † tot ( ~p ′ ) } ∗ T + H = δ ss ′ δ ff ′ δ ( ~p − ~p ′ ) H , { ˆ b sf † tot ( ~p ) , ˆ b s ′ f ′ † tot ( ~p ′ ) } ∗ T + H = 0 · H , { ˆ b sf † tot ( ~p ) , ˆ b s ′ f ′ † tot ( ~p ′ ) } ∗ T + H = 0 · H . (39) C. Illustration of H in d = (1 + 1)Let us illustrate the properties of H and the application of the creation operators on H in d = (1 + 1)dimensional space in a toy model with two discrete momenta ( p , p ). Generalization to many momenta isstraightforward. The internal space of fermions contains only one creation operator, one “basis vector” ˆ b = (+ i ), onefamily member m = 1 of the only family f = 1. Correspondingly the creation operators ˆ b † tot ( ~p i ) | | p | = | p i | := (+ i ) e − i ( p x − p i x ) | | p i | = | p i | distinguish only in momentum space of the fermion degrees of freedom. TheirHermitian conjugated annihilation operators are ˆ b tot ( ~p i ) | p | = | p i | , while the vacuum state is | ψ oc > = ( − i ) · (+ i )= [ − i ].The whole Hilbert space for this toy model has correspondingly four ”Slater determinants”, numeratedby | > i , i = (1 , , , | p p > | , | p p > | , | p p > | , | p p > | ) , p represents an empty state and p the occupied state. Let us evaluate the application of { ˆ b tot ( ~p ) , ˆ b † tot ( ~p ) } ∗ T + on the Hilbert space H . It follows { ˆ b tot ( ~p )) , ˆ b † tot ( ~p ) } ∗ T + H =ˆ b tot ( ~p ) ∗ T ( | p p > | → − | p p > | → ) +ˆ b † tot ( ~p ) ∗ T ( | p p > → , + | p p > → ) =( −| p p > → → + | p p > → → ) = 0 . D. Relation between second quantized fermions of Dirac and second quantized fermionsoriginated in odd Clifford algebra
The Clifford odd creation operators ˆ b sf † tot ( ~p ) and their Hermitian conjugated partners annihi-lation operators ˆ b sftot ( ~p ) obey the anticommutation relations of Eq. (39) — on the vacuum state | ψ oc > , Eq. (18), and on the whole Hilbert space H , Eq. (39). Creation operators, ˆ b sf † tot ( ~p ), oper-ating on a vacuum state, as well as on the whole Hilbert space, define second quantized fermionstates.Let us relate here the Dirac’s second quantization relations and the relations between creationoperators ˆ b sftot ( ~p ) and their Hermitian conjugated partners annihilation operators, without payingattention on the charges and family quantum numbers, since Dirac’s creation operators do not payattention on these two kinds of quantum numbers. We shall relate vectors in d = (3 + 1) of bothorigins.In the Dirac case the second quantized field operators are in d = (3 + 1) dimensions postulatedas follows Ψ hs † ( ~x, x ) = X m,~p k ˆ a h † m ( ~p k ) v hsm ( ~p k ) . (40)5 v hsm ( ~p k ) = u hsm ( ~p k ) e − i ( p x − ε~p k · ~x ) are the two left handed (Γ (3+1) = − h ) and the two righthanded (Γ (3+1) = 1 = h , Eq. (B.3)) two-component column matrices, m = (1 , s of the Weyl equation for free massless fermions of particular momentum | ~p k | = | p k | ([19], Eqs. (20-49) - (20-51)), the factor ε = ± a h † m ( ~p k ) are by Dirac postulated creation operators, which together with the annihilation oper-ators ˆ a hm ( ~p k ), fulfill the anticommutation relations ([19], Eqs. (20-49) - (20-51)) { ˆ a h † m ( ~p k ) , ˆ a h ′ † n ( ~p l ) } ∗ T + = 0 = { ˆ a hm ( ~p k ) , ˆ a h ′ n ( ~p l ) } ∗ T + , { ˆ a hm ( ~p k ) , ˆ a h ′ † n ( ~p l ) } ∗ T + = δ mn δ hh ′ δ ~p k ~p l (41)in the case of discretized momenta for a fermion in a box. (Massive fermions are represented byfour vectors which are the superposition of both handedness.)Let us present the two ”basis vectors” ˆ b h † m , m = (1 , , h representing left and right handedness,in the internal space of fermions in d = (3 + 1), described by the Clifford odd algebra, representingthe creation operators of one particular family ( f not shown in this case), without charges, ofone handedness and with spins ± , respectively, operating on the vacuum state | ψ oc > = [+ i ] [ − ] : ˆ b h † = [+ i ] (+) and ˆ b h † = ( − i ) [ − ], Eq. (20, 21) [57], with h = 1, representing the right handedness.These two ”basis vectors” should be compared with the two vectors, one corresponding to the spin and the other to the spin − in the Dirac case.Since Dirac did not postulate such creation operators on the level of ˆ b h † m , let us postulate themnow on the level of ˆ b h † m , to be able to compare in this paper presented creation operators for thisparticular case, ˆ a h †↑ and ˆ a h †↓ , of right handedness h and spin up and down ( ↑ , ↓ ) as followsˆ b h † = [+ i ]) (+) to be related to ˆ a h †↑ , ˆ b h † = ( − i )) [ − ] to be related to ˆ a h †↓ . One should repeat this also for left handedness h = −
1. But these creation operators ˆ a h † m , m =(1 ,
2) = ( ↑ , ↓ ), still can not be compared with the Dirac’s ones.Let us make the superposition of both creation operators of particular handedness h , ˆa hs † ( ~p k ) := α hs ↑ ( ~p k ) ˆ a h †↑ + α hs ↓ ( ~p k ) ˆ a h †↓ , with the coefficients α hs ↑ ( ~p k ) and α hs ↓ ( ~p k ) chosen so that ˆa hs † tot ( ~p k ): = ˆa hs † ( ~p k ) e − i ( p x − ~p · ~x ) solves the equations of motion, Eq. (23) [58], for a plane wave e iε~p k · ~x for | ~p k | = | p k | , then it follows ˆa hs † tot ( ~p k ) := ( α hs ↑ ( ~p k ) ˆ a h †↑ + α hs ↓ ( ~p k ) ˆ a h †↓ ) e − i ( p x − ~p k · ~x ) = X m ˆ a h † m ( ~p k ) v hsm ( ~p k ) , (42)where the summation runs over m up and down spin m of the chosen handedness h .6Since v hsm ( ~p k ) = u hsm ( ~p k ) e − i ( p x − ~p k · ~x ) it follows also that ˆa hs † ( ~p k ) = P m u hsm ˆ a h † m , and u hsm ( ~p k ) = α hsm ( ~p k ). We conclude that ˆa hs † tot ( ~p k ) obviously determine ˆ a h † m ( ~p k ) v hsm ( ~p k ) =ˆ a h † m ( ~p k ) u hsm ( ~p k ) e − i ( p x − ~p k · ~x ) .Anticommutation relations of Eq. (41), postulated by Dirac, ensure the equivalent anticommu-tation relations also for ˆa hs † ( ~p k ) and ˆa hs ( ~p k ).Now we are able to relate creation and annihilation operators in both cases, the Dirac case andour case of using the odd Clifford algebra to represent the internal space of fermions.ˆ b h † = [+ i ] (+) to be related to ˆ a h †↑ , ˆ b h † = ( − i ) [ − ] to be related to ˆ a h h †↓ ˆ b h = − [+ i ] ( − ) to be related to ˆ a h ↑ , ˆ b h = (+ i ) [ − ] to be related to ˆ a h ↓ , (43)both sides representing the creation operators, with S = and handedness Γ (3+1) = 1, Eq. (B2),in the first row, and with S = − and handedness Γ (3+1) = 1 = h , in the second row [59].None of the creation operators, ˆ a h † m , m = ( ↑ , ↓ ) and ˆ b h † m , m = (1 , ˆa hs † ( ~p k ) and ˆb sf † ( ~p k ) as well as ˆa hs † tot ( ~p k ) and ˆb sf † tot ( ~p k ) do depend on momenta.The creation operators ˆ a s † tot ( ~p k ) fulfill the anticommutation relations of Eqs. (28, 38, 39), thesame as ˆ b sf † tot ( ~p ) do. We can just replace ˆ a s † tot ( ~p k ) by ˆ b sf † tot ( ~p ) for any of families (for plane wavessolutions with continuous ~p ).We can conclude: ˆ a hs † tot ( ~p ) is to be related to ˆ b hs † tot ( ~p ) , ˆ a h † m , m = ( ↑ , ↓ ) is to be related to ˆ b h † m m = (1 , , (44)with h representing the handedness. This can be done for any chosen family in the Clifford case.In all the relations with ˆ b hs † tot ( ~p ) the handedness is not written explicitly and is included in theindex m and in the index s , while the index f represents the family quantum number. Only in thischapter we introduce handedness in addition to clarify the relations.hIn the Clifford case the charges origin in spins d ≥
6. In d = (13 + 1) all the charges of quarksand leptons and antiquarks and antileptons can be explained, as well as the families of quarks andleptons and antiquarks and antileptons. In the Dirac case charges come from additional groupsand so do families.Let us add: The odd Clifford algebra influences the algebra of the associated creation andannihilation operators acting on the second quantized Hilbert space H ; Due to oddness of the7Clifford algebra, which determines internal degrees of freedom of fermions, the creation operatorsand their Hermitian conjugated annihilation partners, determined on the tensor products of internaland momentum space, make the creation and annihilation operators to anticommute.We conclude: The by Dirac postulated creation operators, ˆ a h † m ( ~p ), and their annihilation part-ners, ˆ a hm ( ~p ), Eqs. (40, 42), related in Eq. (44) to the Clifford odd creation and annihilation op-erators, manifest that the odd Clifford algebra offers the explanation for the second quantizationpostulates of Dirac. IV. CREATION AND ANNIHILATION OPERATORS IN d = (13 + 1) -DIMENSIONALSPACE The standard model offered an elegant new step in understanding elementary fermion and bosonfields by postulating: i. Massless family members of (coloured) quarks and (colourless) leptons, the left handed fermionsas the weak charged doublets and the weak chargeless right hand members, the left handed quarksdistinguishing in the hyper charge from the left handed leptons, each right handed member havinga different hyper charge. All fermion charges are in the fundamental representation of the corre-sponding groups. Antifermions carry the corresponding anticharges and opposite handedness. Themassless families to each family member exist. ii.
The existence of the massless vector gauge fields to the observed charges of quarks and leptons,carrying charges in the corresponding adjoint representations. iii.
The existence of a massive scalar Higgs, gaining at some step of the expanding universe thenonzero vacuum expectation value, responsible for masses of fermions and heavy bosons and forthe Yukawa couplings. The Higgs carries a half integer weak charge and hyper charge. iv.
Fermions and bosons are second quantized fields.The standard model assumptions have in the literature several explanations, mostly with manynew not explained assumptions. The most successful seem to be the grand unifying theories [21–35, 37–39], if postulating in addition the family group and the corresponding gauge scalar fields.The spin-charge-family theory, the project of one of the authors of this paper (N.S.M.B. [1–3, 10–16]), is offering the explanation for all the assumptions of the standard model , unifying in d = (13 + 1)-dimensional space not only charges, but also charges and spins and families [2, 7],explaining the appearance of families [8, 10, 15], the appearance of the vector gauge fields [12, 14],of the scalar field and the Yukawa couplings [13]. Theory offers the explanation for the dark8matter [4, 5], for the matter-antimatter asymmetry [11], and makes several predictions [4, 6, 11].The spin-charge-family theory is a kind of the Kaluza-Klein like theories [16, 41, 43–49] due tothe assumption that in d ≥ spin-charge-family theory d ≥ (13 + 1)) fermions interactwith the gravity only (vielbeins and two kinds of the spin connection fields). Correspondingly thistheory shares with the Kaluza-Klein like theories their weak points, at least: a. Not yet solved the quantization problem of the gravitational field. b. The spontaneous break of the starting symmetry, which would at low energies manifest theobserved almost massless fermions [43]. c. The appearance of gravitational anomalies, what makes the theory not well defined [52], but inthe low energy limit the fields manifest in d = (3 + 1) properties of the observed vector and scalargauge fields. d. And other problems.In the spin-charge-family theory fermions interact in d = (13 + 1) with the gravity only: with thespin connections (the gauge fields of S ab and of ˜ S ab ) and vielbeins (the gauge fields of momenta),with fermions as a condensate present, breaking the symmetry (and with no other gauge fieldspresent), manifesting at low energies in d = (3 + 1) as the ordinary gravity and all the observedvector gauge fields.It is proven in Refs. [50, 51], that one can have massless spinors even after breaking the startingsymmetry. Ref. [12] proves, that at low enough energies, after breaking the staring symmetry, thetwo spin connections manifest in d = (3 + 1) as the observed vector gauge fields, as well as thescalar fields, which offer the explanation for the Higgs and the Yukawa couplings. Ref. [11] offersthe explanation for the matter-antimatter asymmetry due to the existence of the scalar fields withthe “colour charges” in the fundamental representations. In Ref. [16] the spin-charge-family theoryexplains the standard model triangle anomaly cancellation better than the SO (10) theory [22].The working hypotheses of the authors of this paper (in particular of N.S.M.B.) is, since thehigher dimensions used in the spin-charge-family theory offer in an elegant (simple) way explana-tions for the so many observed phenomena, that they should not be excluded by the renormalizationand anomaly arguments. At least the low energy behavior of the spin connections and vielbeins asvector and scalar gauge fields manifest as the known and more or less well defined theories.In this paper we present that using the half of the odd Clifford algebra objects to explainthe internal degrees of freedom of fermions (the other half represent the Hermitian conjugatedpartners), as suggested by the spin-charge-family theory, leads to the second quantized fermionswithout postulates of Dirac [60].9 V. CONCLUSIONS
We present in Part I and Part II of this paper that the description of the internal space offermions with the odd elements of the anticommuting algebra defines the creation and annihilationoperators, which anticommute when applied on the corresponding vacuum state. The internalspace extends its oddness to creation and annihilation operators generated on the tensor productsof the internal basis with finite numbers of elements and the momentum basis with infinite numberof elements. The application of these creation and annihilation operators on the Hilbert space,determined by the tensor multiplication of all possible creation operators of any numbers, mani-fests the same anticommutation relations as the creation and annihilation of the second quantizedfermions, explaining therefore the Dirac postulates of the second quantized fermion fields.In the subsection I B 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 generalized point of view.We learn in Part I of this paper, that odd products of superposition of θ a ’s, Eqs. (8-11,13,22) inPart I, exist forming the odd algebra ”basis vectors” in the internal space of ”Grassmann fermions”with integer spin, which together with their Hermitian conjugated partners fulfill on the algebraiclevel on the vacuum state all the requirements for the anticommutation relations for the Diracfermions. The creation and annihilation operators, defined on the tensor products of the superpo-sition of the Grassmann odd algebra ”basis vectors” and the momentum space basis, and manifest-ing correspondingly the oddness of the ”basis vectors”, fulfill the anticommutation relations of thesecond quantized Dirac’s fermions on the vacuum state, as well as on the ”Slater determinants” ofall possibilities of occupied and empty single particle ”Grassmann fermion” states of integer spinsof any number. These ”Slater deerminants”, representing the Hilbert space of second quantized”Grassmann fermions”, can be represented as well with the tensor product multiplication of anypossible choice of single ”Grasmann fermion states” of all possible numbers of states, started withnone (that is with the identity), distinguishing at least either in one of the quantum numbers ofthe ”basic vectors” or in momentum basis.In Part II we learn, that the creation and annihilation operators exist in the Clifford odd algebra,defining the internal space of half integer fermions, which applying on the vacuum state fulfill theanticommutation relations postulated by Dirac. Creation operators, defined on the tensor productsof the superposition of the finite ”basis vectors” of the internal space described with the Clifford0algebra and of the infinite momentum basis, fulfill as well together with their Hermitian conjugatedannihilation operators the anticommutation relations postulated by Dirac, on the vacuum stateand on the Hilbert space of infinite number of the single particle fermion states, N H = Q ∞ ~p d − ,Eqs. (33, 34), creating ”Slater determinants” (Eqs. (37, 39)), but only after the reduction of thedegrees of freedom of the Clifford algebras for a factor of two, Eq. (12).The reduction of the Clifford algebras for the factor of two leaves the anticommutation relationsof Eqs. (2, 3) unchanged, enabling the appearance of family quantum numbers. The Cliffordfermions carry half integer spins, families and charges in fundamental representations, Eq. (5).The reduction of Clifford space causes the reduction also in Grassmann space, what leads tothe disappearance of integer spin fermions, Eq. (19).The Clifford algebra oddness of the ”basis vectors”, describing the internal space of fermions,makes odd also the corresponding fermion states defined on the tensor products of the internal andmomentum space. Correspondingly any two states fulfill the anticommutation relations and so doany tensor products of odd numbers of fermion states, forming the Hilbert second quantized space.The creation operators, defined on the tensor products of ”basis vectors” and plane waves, solvethe equations of motion, in our case for free massless fermions, Eq. (23), leading from the action,Eq. (22).Anticommutation relations are not postulated, as it is in the Dirac case, they follow from theoddness of the Clifford objects, and correspondingly explain the second postulates of Dirac.The relation between the Dirac’s creation and annihilation operators and the ones offered by theodd Clifford algebra, discussed in In Subsect. III D demonstrates that the basic differences betweenthese two descriptions is on the level of the single particle creation operators: While the oddClifford algebra offers the creation and annihilation operators, which fulfill the anticommutationrelations, already on the level of the ”basis vectors” determining the internal space of fermions,Eq. (17), when applied on the vacuum state, Dirac postulates the anticommutation relations onthe level of second quantized objects, following the procedure of Lagrange and Hamilton.The final result is in both cases equivalent, leading to the Hilbert space of second quantizedfields. However, our way not only explains the Dirac postulates but demonstrates in addition,that also the single particle states in the first quantization do anticommute due to the oddnessof the ”basis vectors” defining the internal space. The oddness of the Clifford objects of creationand annihilation operators is transmitted from the ”basis vectors” of internal space to the tensorproducts of the superposition of the ”basis vectors” and the momentum or coordinate space.Correspondingly the odd Clifford algebra, equipped with the family quantum numbers, and1fulfilling the anticommutation relations already on the level of the single particle creation opera-tors applying on the vacuum state, as well as on the level of the whole Hilbert space, offers theexplanation for the anticommutation relations, postulated by Dirac.The Hilbert space of all ”Slater determinants” with any number of occupied or empty states ofan odd character, follows in all three cases, the Dirac one (with postulated creation and annihilationoperators and offering no families and no charges), the Grassmann one (offering spins and charges inadjoint representations, and no families) and the Clifford one (offering spins, families and charges),in an equivalent way: due to the anticommuting creation and annihilation operators, representing”basis vectors” and their Hermitian conjugated partners. One can see this in Sect. III D.Let us repeat: Internal space contributes the final number of states, the infinity of number ofstates is due to momentum/coordinate space [61].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 [2] of two functions of the Grassmanncoordinates < B | θ >< C | θ > , < B | θ > = < θ | B > † , < b | θ > = d X k =0 b a ...a k θ a · · · θ a k , by requiring { dθ a , θ b } + = 0 , Z dθ a = 0 , Z dθ a θ a = 1 , Z d d θ θ θ · · · θ d = 1 ,d d θ = dθ d . . . dθ , ω = Π dk =0 ( ∂∂θ k + θ k ) , (A1)with ∂∂θ a θ c = η ac . We shall use the weight function [2] ω = Π dk =0 ( ∂∂θ k + θ k ) to define the scalarproduct in Grassmann space < B | C >< B | C > = Z d d θ a ω < B | θ > < θ | C > = d X k =0 b ∗ b ...b k c b ...b k . (A2)To define norms in Clifford space Eq. (A1) can be used as well.2 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 = b = i γ a γ b and ˜ S ab | a = b = i ˜ γ a ˜ γ b , as followsΓ ( d ) : = ( i ) d/ Y 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] N. Mankoˇc Borˇstnik, ”Spin connection as a superpartner of a vielbein”,
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Phys. Rev. D , 083534 (2009), 1-16.[5] G. Bregar, N.S. Mankoˇc Borˇstnik, ”Can we predict the fourth family masses for quarks and leptons?”,Proceedings (arXiv:1403.4441) to the 16 th Workshop ”What comes beyond the standard models”,Bled, 14-21 of July, 2013, Ed. N.S. Mankoˇc Borˇstnik, H.B. Nielsen, D. Lukman, DMFA Zaloˇzniˇstvo,Ljubljana December 2013, p. 31-51, [arXiv:1212.4055]. [6] G. Bregar, N.S. Mankoˇc Borˇstnik, ”The new experimental data for the quarks mixing matrix are inbetter agreement with the spin-charge-family theory predictions”, Proceedings to the 17 th Workshop”What comes beyond the standard models”, Bled, 20-28 of July, 2014, Ed. N.S. Mankoˇc Borˇstnik, H.B.Nielsen, D. Lukman, DMFA Zaloˇzniˇstvo, Ljubljana December 2014, p.20-45 [ arXiv:1502.06786v1][arXiv:1412.5866].[7] N.S. Mankoˇc Borˇstnik, H.B.F. Nielsen,
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J. Phys. A: Math. Theor. { ˜ γ a , ˜ γ b } + = 2 η ab remains valid when applied on B , if B is either an odd or an even product of γ a ’s: { ˜ γ a , ˜ γ b } + γ c = − i (˜ γ a γ c γ b + ˜ γ b γ c γ a ) = − i i γ c ( γ b γ a + γ a γ b ) = 2 η ab γ c ,while { ˜ γ a , ˜ γ b } + γ c γ d = i (˜ γ a γ c γ d γ b + ˜ γ b γ c γ d γ a ) = i ( − i ) γ c γ d ( γ b γ a + γ a γ b ) = 2 η ab γ c γ d . The relationis valid for any γ c and γ d , even if c = d .[54] One easily checks that ˜ γ a † γ c = − iγ c γ a † = − iη aa γ c γ a = η aa ˜ γ a γ c = − iη aa γ c γ a .[55] If we would make a choice for creation operators the ”families” with the “family” members of odd II of Table I, instead of ”families” with the “family” members of odd I , then their Hermitian conjugatedpartners would be the ”families” with the “family” members in odd I . The vacuum state would be thesum of products of annihilation operators of odd I times the creation operators of odd II and would bethe sum of selfadjoint members appearing in even II .[56] The orthogonality of the states are even easier to be visualized since the two delta functions at ~x andat ~x ′ , ~x = ~x ′ are obviously orthogonal.[57] We choose in the Clifford case the first two members of the third family in Table I, since they manifestin d = (3 + 1) the Clifford odd character.[58] The equations of motion read in the Dirac case: { ˆ p +( − iS i ˆ p i ) } ( α s ( ~p k ) ˆa † + α s ( ~p k ) ˆa † ) e − i ( p x − ~p k · ~x ) =0. To solve them we need to recognize that the matrices in the chiral representation S i , i = (1 , ˆa † into ˆa † , and opposite.[59] The vacuum state is on the left hand side equal to [+ i ] [ − ], while on the right hand side the correspondingvacuum state can be defined, if we follow our way of defining the vacuum state, to be proportional to(ˆ a ↑ ˆ a †↑ + ˆ a ↓ ˆ a †↓ ).[60] The authors of Ref. [42] let us know that their path integral formulation enabled them to see a great dealof what we present in this paper. We went through their paper noting that they did a lot concerningpath integral formulation of quantum mechanics, offering ways to treat anomalies. But we couldn’trecognize that they propose some replacement for the Dirac postulates of creation and annihilationoperators. We also could not found whether our proposal for explaining the Dirac postulates wouldbring any new light on path integral formulations and anomalies cancelations. To clarify this topics thediscussions with authors would be needed.[61] Let us add that the single particle vacuum state is the sum of products of annihilation ××