Real Clifford algebras for arbitrary number of space and time dimensions as well as their representations in terms of spinors are reviewed and discussed. The Clifford algebras are classified in terms of isomorphic matrix algebras of real, complex or quaternionic type. Spinors are defined as elements of minimal or quasi-minimal left ideals within the Clifford algebra and as representations of the pin and spin groups. Two types of Dirac adjoint spinors are introduced carefully. The relation between mathematical structures and applications to describe relativistic fermions is emphasized throughout.
Classification of quantum spinor fields according to quantum bilinear covariants is introduced in a context of quantum Clifford algebras on Minkowski spacetime. Once the bilinear covariants are expressed in terms of algebraic spinor fields, the duality between spinor and quantum spinor fields is thus discussed. Hence, by endowing the underlying spacetime with an arbitrary bilinear form with a antisymmetric part in addition to a symmetric spacetime metric, quantum algebraic spinor fields and deformed bilinear covariants can be constructed. They are therefore compared to the classical (non quant…
It is shown that since the geometric spinors are elements of Clifford algebras, they must have the same transformation properties as any other Clifford number. In general, a Clifford number $Φ$ transforms into a new Clifford number $Φ'$ according to $Φ\to Φ' = {\rm{R}}\,Φ\,{\rm{S}}$, i.e., by the multiplication from the left and from the right by two Clifford numbers ${\rm R}$ and ${\rm S}$. We study the case of $Cl(1,3)$, which is the Clifford algebra of the Minkowski spacetime. Depending on choice of ${\rm R}$ and ${\rm S}$, there are various possibilities, including the transformations of v…
In this paper, we show that via a novel construction every rank-3 root system induces a root system of rank 4. Via the Cartan-Dieudonné theorem, an even number of successive Coxeter reflections yields rotations that in a Clifford algebra framework are described by spinors. In three dimensions these spinors themselves have a natural four-dimensional Euclidean structure, and discrete spinor groups can therefore be interpreted as 4D polytopes. In fact, we show that these polytopes have to be root systems, thereby inducing Coxeter groups of rank 4, and that their automorphism groups include two fa…
Clifford number formalism for Maxwell equations is considered. The Clifford imaginary unit for space-time is introduced as coordinate independent form of fully antisymmetric fourth-rank tensor. The representation of Maxwell equations in massless Dirac equation form is considered; we also consider two approaches to the invariance of Dirac equation in respect of the Lorentz transformations. According to the first approach, the unknown column is invariant and according to the second approach it has the transformation properties known as spinorial ones. Clifford number representation for nonlinear…
We examine the structure of the Clifford algebra associated with a Hermitian bilinear form and apply the result to a dynamical model of the relativistic point particle. The dynamics of the particle is described by a Dirac spinor with components in a Clifford algebra. This spinor determines, through the Clifford algebra, both the space-time coordinates and their conjugate momenta and satisfies a first order equation of motion which leads to the usual space-time canonical equations of motion. The constraints appear as the equations of motion for the einbein and spin connection which are needed t…
In this paper we examine a new class of five dimensional (5D) exact solutions in extra dimension gravity possessing Lie algebroid symmetry. The constructions provide a motivation for the theory of Clifford nonholonomic algebroids elaborated in Ref. hep-th/0501217. Such Einstein-Dirac spacetimes are parametrized by generic off--diagonal metrics and nonholonomic frames (vielbeins) with associated nonlinear connection structure. They describe self-consistent propagations of (3D) Dirac wave packets in 5D nonholonomically deformed Taub NUT spacetimes and have two physically distinct properties: Fis…
We present a Mathematica package for doing computations with gamma matrices, spinors, tensors and other objects, in any dimension and signature. The approach we use is based on defining the commutation relations of the relevant matrices, and is thus general and flexible. As examples, we reproduce torsion conditions for $AdS_3$ compactification of type IIB supergravity from literature, and check the vanishing of supersymmetry variations of 10-dimensional supersymmetric Yang-Mills theory.
We study briefly some properties of real Clifford algebras and identify them as matrix algebras. We then show that the representation space on which Clifford algebras act are spinors and we study in details matrix representations. The precise structure of these matrices gives rise to the type of spinors one is able to construct in a given space-time dimension: Majorana or Weyl. Properties of spinors are also studied. We finally show how Clifford algebras enable us to construct supersymmetric extensions of the Poincaré algebra. A special attention to the four, ten and eleven-dimensional space-t…
$CPT$ groups for spinor fields in de Sitter and anti-de Sitter spaces are defined in the framework of automorphism groups of Clifford algebras. It is shown that de Sitter spaces with mutually opposite signatures correspond to Clifford algebras with different algebraic structure that induces an essential difference of $CPT$ groups associated with these spaces. $CPT$ groups for charged particles are considered with respect to phase factors on the various spinor spaces related with real subalgebras of the simple Clifford algebra over the complex field (Dirac algebra). It is shown that $CPT$ group…
In this paper we study Dirac-Hestenes spinor fields (DHSF) on a four-dimensional Riemann-Cartan spacetime (RCST). We prove that these fields must be defined as certain equivalence classes of even sections of the Clifford bundle (over the RCST), thereby being certain particular sections of a new bundle named Spin-Clifford bundle (SCB). The conditions for the existence of the SCB are studied and are shown to be equivalent to the famous Geroch's theorem concerning to the existence of spinor structures in a Lorentzian spacetime. We introduce also the covariant and algebraic Dirac spinor fields and…
We propose a new framework for constructing geometric and physical models on nonholonomic manifolds provided both with Clifford -- Lie algebroid symmetry and nonlinear connection structure. Explicit parametrizations of generic off-diagonal metrics and linear and nonlinear connections define different types of Finsler, Lagrange and/or Riemann-Cartan spaces. A generalization to spinor fields and Dirac operators on nonholonomic manifolds motivates the theory of Clifford algebroids defined as Clifford bundles, in general, enabled with nonintegrable distributions defining the nonlinear connection. …
Recent work has shown that every 3D root system allows the construction of a correponding 4D root system via an `induction theorem'. In this paper, we look at the icosahedral case of $H_3\rightarrow H_4$ in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem, giving a simple construction of the Pin and Spin covers. Using this connection with $H_3$ via the induction theorem sheds light on geometric aspects of the $H_4$ root system (the $600$-cell) as well as other related po…
Let $(V,g)$ be a $2n$-dimensional hyperbolic space and $C(V,g)$ its Clifford algebra. $C(V,g)$ has a $\mathbb Z$-grading, $C^k $, and an algebra isomorphism $C(V,g)\cong End(S)$, $S$ the space of spinors. É. Cartan defined operators $L_k: End(S) \to C^k$ which are involved in the definition of pure spinors. We shall give a more refined study of the operator $L_2$, in fact, obtain explicit formulae for it in terms of spinor inner products and combinatorics, as well as the matrix of it in a basis of pure spinors. Using this information we give a construction of the exceptional Lie algebras $\mat…
Clifford algebras and Majorana conditions are analyzed in any spacetime. An index labeling inequivalent $Γ$-structures up to orthogonal conjugations is introduced. Inequivalent charge-operators in even-dimensions, invariant under Wick rotations, are considered. The hermiticity condition on free-spinors lagrangians is presented. The constraints put by the Majorana condition on the free-spinors dynamics are analyzed. Tables specifying which spacetimes admit lagrangians with non-vanishing kinetic, massive or pseudomassive terms (for both charge-operators in even dimensions) are given. The admissi…
Using a technique \cite{holgernorma2002} to construct a basis for spinors and ``families'' of spinors in terms of Clifford algebra objects, we define other Clifford algebra objects, which transform the state of one ''family'' of spinors into the state of another ''family'' of spinors, changing nothing but the ''family'' number. The proposed transformation works - as does the technique - for all dimensions and any signature and might open a path to understanding families of quarks and leptons\cite{norma92,norma93,normaixtapa2001,pikanorma2002}.
Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4, as well as those of E_8, have been used extensively in the literature. The present paper analyses such Coxeter groups in the Clifford Geometric Algebra framework, which affords a simple way of performing reflections and rotations whilst exposing more clearly the underlying geometry. The Clifford approach shows that the quaternionic representations in fact have very simple geometric interpretations. The representations of the groups A_1 x A_1 x A_1, A_3, B_3 and H_3 of rank 3 in terms of pure quaternions are shown to …
We formulate Dirac-Kaehler fermion action by introducing a new Clifford product with noncommutative differential form on a lattice. Hermiticity of the Dirac-Kaehler action requires to choose the lattice structure having both orientabilities on a link. The Kogut-Susskind fermion and the staggered fermion actions are derived directly from the Dirac-Kaehler fermion formulated by the Clifford product. The lattice QCD action with Dirac-Kaehler matter fermion is also derived via an inner product defined by the Clifford product.
Because of the isomorphism ${C \kern -0.1em \ell}_{1,3}(\Bbb{C})\cong{C \kern -0.1em \ell}_{2,3}(\Bbb{R})$, it is possible to complexify the spacetime Clifford algebra ${C \kern -0.1em \ell}_{1,3}(\Bbb{R})$ by adding one additional timelike dimension to the Minkowski spacetime. In a recent work we showed how this treatment provide a particular interpretation of Dirac particles and antiparticles in terms of the new temporal dimension. In this article we thoroughly study the structure of the real Clifford algebra ${C \kern -0.1em \ell}_{2,3}(\Bbb{R})$ paying special attention to the isomorphism …
In this paper, we make the case that Clifford algebra is the natural framework for root systems and reflection groups, as well as related groups such as the conformal and modular groups: The metric that exists on these spaces can always be used to construct the corresponding Clifford algebra. Via the Cartan-Dieudonné theorem all the transformations of interest can be written as products of reflections and thus via `sandwiching' with Clifford algebra multivectors. These multivector groups can be used to perform concrete calculations in different groups, e.g. the various types of polyhedral grou…
This is the second part of an article about q-deformed analogs of spinor calculus. The considerations refer to quantum spaces of physical interest, i.e. q-deformed Euclidean space in three or four dimensions as well as q-deformed Minkowski space. The Clifford algebras corresponding to these quantum spaces are treated. Especially, their commutation relations and their Hopf structures are written down. Bases of the four-dimensional Clifford algebras are constructed and their properties are discussed. Matrix representations of the Clifford algebras lead to q-deformed Dirac-matrices for the four-d…
We suggest Clifford algebra as a useful simplifying language for present quantum dynamics. Clifford algebras arise from representations of the permutation groups as they arise from representations of the rotation groups. Aggregates using such representations for their permutations obey Clifford statistics. The vectors supporting the Clifford algebras of permutations and rotations are plexors and spinors respectively. Physical spinors may actually be plexors describing quantum ensembles, not simple individuals. We use Clifford statistics to define quantum fields on a quantum space-time, and to …
We investigate the construction and properties of Clifford algebras by a similar manner as our previous construction of the octonions, namely as a twisting of group algebras of Z_2^n by a cocycle. Our approach is more general than the usual one based on generators and relations. We obtain in particular the periodicity properties and a new construction of spinors in terms of left and right multiplication in the Clifford algebra.
It is shown how the vector space $V_{8,8}$ arises from the Clifford algebra $Cl(1,3)$ of spacetime. The latter algebra describes fundamental objects such as strings and branes in terms of their $r$-volume degrees of freedom, $x^{μ_1 μ_2 ...μ_r}$ $\equiv x^M$, $r=0,1,2,3$, that generalizethe concept of center of mass. Taking into account that there are sixteen $x^M$, $M=1,2,3,...,16$, and in general $16 \times 15/2 = 120$ rotations of the form $x'^M = {R^M}_N x^N$, we can consider $x^M$ as components of a vector $X=x^M q_M$, where $q_M$ are generators of the Clifford algebra $Cl(8,8)$. The vect…
Dedicated to Ludwig Faddeev on his 80th birthday. Ludwig exemplifies perfectly a mathematical physicist: significant contribution to mathematics (algebraic properties of integrable systems) and physics (quantum field theory). In this note I present an exercise which bridges mathematics (restricted Clifford algebra) to physics (Majorana fermions).
We investigate the constraint equations of the Lounesto spinor fields classification and show that it can be used to completely characterize all the singular classes, which are potential accommodations for further mass dimension one fermions, beyond the well known Elko spinor fields. This result can be useful for two purposes: besides a great abridgement in the classification of a given spinor field, we provide a general form of each class of spinor fields, which can be used furthermore to search for a general classification of spinors dynamics.
Quaternionic and octonionic realizations of Clifford algebras and spinors are classified and explicitly constructed in terms of recursive formulas. The most general free dynamics in arbitrary signature space-times for both quaternionic and octonionic spinors is presented. In the octonionic case we further provide a systematic list of results and tables expressing, e.g., the relations of the octonionic Clifford algebras with the $G_2$ cosets over the Lorentz algebras, the identities satisfied by the higher-rank antisymmetric octonionic tensors and so on. Applications of these results range from…
A real representation theory of real Clifford algebra has been studied in further detail, especially in connection with Fierz identities. As its application, we have constructed real octonion algebras as well as related octonionic triple system in terms of 8-component spinors associated with the Clifford algebras $C(0,7)$ and $C(4,3)$.
Spinor structure is understood as a totality of tensor products of biquaternion algebras, and the each tensor product is associated with an irreducible representation of the Lorentz group. A so-defined algebraic structure allows one to apply modulo 8 periodicity of Clifford algebras on the system of real and quaternionic representations of the Lorentz group. It is shown that modulo 8 periodic action of the Brauer-Wall group generates modulo 2 periodic relations on the system of representations, and all the totality of representations under this action forms a self-similar fractal structure. So…
Spinors are central to physics: all matter (fermions) is made of spinors, and all forces arise from symmetries of spinors. It is common to consider the geometric (Clifford) algebra as the fundamental edifice from which spinors emerge. This paper advocates the alternative view that spinors are more fundamental than the geometric algebra. The algebra consisting of linear combinations of scalars, column spinors, row spinors, multivectors, and their various products, can be termed the supergeometric algebra. The inner product of a row spinor with a column spinor yields a scalar, while the outer pr…
We further explore the idea that physics takes place in Clifford space which should be considered as a generalization of spacetime. Following the old observation that spinors can be represented as members of left ideals of Clifford algebra, we point out that the transformations which mix bosons and fermions could be represented by means of operators acting on Clifford algebra-valued (polyvector) fields. A generic polyvector field can be expanded either in terms of bosonic, or in terms of fermionic fields. In particular, a scalar field can transform into a mixture of bosonic and/or fermionic fi…
We prove in this paper that the elliptic $R$--matrix of the eight vertex free fermion model is the intertwiner $R$--matrix of a quantum deformed Clifford--Hopf algebra. This algebra is constructed by affinization of a quantum Hopf deformation of the Clifford algebra.
Classical anti-commuting spinor fields and their dynamics are derived from the geometry of the Clifford bundle over spacetime via the BRST formulation. In conjunction with Kaluza-Klein theory, this results in a geometric description of all the fields and dynamics of the standard model coupled to gravity and provides the starting point for a new approach to quantum gravity.
Clifford geometric algebras of multivectors are treated in detail. These algebras are build over a graded space and exhibit a grading or multivector structure. The careful study of the endomorphisms of this space makes it clear, that opposite Clifford algebras have to be used also. Based on this mathematics, we give a fully Clifford algebraic account on generating functionals, which is thereby geometric. The field operators are shown to be Clifford and opposite Clifford maps. This picture relying on geometry does not need positivity in principle. Furthermore, we propose a transition from opera…
We propose a Clifford algebra approach to chiral symmetry breaking and fermion mass hierarchies in the context of composite Higgs bosons. Standard model fermions are represented by algebraic spinors of six-dimensional binary Clifford algebra, while ternary Clifford algebra-related flavor projection operators control allowable flavor-mixing interactions. There are three composite electroweak Higgs bosons resulted from top quark, tau neutrino, and tau lepton condensations. Each of the three condensations gives rise to masses of four different fermions. The fermion mass hierarchies within these t…
It is well-known that the Clifford algebra Cl(2n) can be given a description in terms of creation/annihilation operators acting in the space of inhomogeneous differential forms on C^n. We refer to such inhomogeneous differential forms as polyforms. The construction proceeds by choosing a complex structure J on R^(2n). Spinors are then polyforms on one of the two totally-isotropic subspaces C^n that arise as eigenspaces of J. There is a similar description in the split signature case Cl(n,n), with differential forms now being those on R^n. In this case the model is constructed by choosing a par…
Each isometric complex structure on a 2$\ell$-dimensional euclidean space $E$ corresponds to an identification of the Clifford algebra of $E$ with the canonical anticommutation relation algebra for $\ell$ ( fermionic) degrees of freedom. The simple spinors in the terminology of E.~Cartan or the pure spinors in the one of C. Chevalley are the associated vacua. The corresponding states are the Fock states (i.e. pure free states), therefore, none of the above terminologies is very good.
The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the non-associativity and non-commutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octonionic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest $\perm_3 \times SO(8)$ structure in this framework.
The physical fields (electromagnetic and electron fields) considered in the framework of Clifford algebras $\C_2$ and $\C_4$. The electron field described by the algebra $\C_4$ which in spinor representation is realized by well-known Dirac $γ$-matrices, and by force of isomorphism $\C_{4}\cong\C_{2}\otimes \C_{2}$ is represented as a tensor product of two photon fields. By means of this introduced a system of electron field equations, which in particular cases is coincide with Dirac's and Maxwell's equations.
A model of Yang-Mills interactions and gravity in terms of the Clifford algebra Cl(0,6) is presented. The gravity and Yang-Mills actions are formulated as different order terms in a generalized action. The feebleness of gravity as well as the smallness of the cosmological constant and theta terms are discussed at the classical level. The invariance groups, including the de Sitter and the Pati-Salam SU(4) subgroups, consist of gauge transformations from either side of an algebraic spinor. Upon symmetry breaking via the Higgs fields, the remaining symmetries are the Lorentz SO(1,3), color SU(3),…
We present in Part II the description of the internal degrees of freedom of fermions by the superposition of odd products of the Clifford algebra elements, either $γ^a$'s or $\tildeγ^a$'s, which determine with their oddness the anticommuting properties of the creation and annihilation operators of the second quantized fermion fields in even $d$-dimensional space-time, as we do in Part I of this paper by the Grassmann algebra elements $θ^a$'s and $\frac{\partial}{\partial θ_a}$'s. We discuss: {\bf i.} The properties of the two kinds of the odd Clifford algebras, forming two independent spaces, …
In this paper we approach the issue of Clifford algebra basis deformation, allowing for bilinear covariants associated to Elko spinors which satisfy the Fierz-Pauli-Kofink identities. We present a complete analysis of covariance, taking into account the involved dual structure associated to Elko. Moreover, the possible generalizations to the recently presented new dual structure are performed.
In a long series of works the author has demonstrated that the model named the {\it spin-charge-family} theory offers the explanation for all in the {\it standard model} assumed properties of the fermion and boson fields, as well as for many of their so far observed properties if the space-time is $\ge (13 +1)$ while fermions interact with gravity only. In this paper, I briefly report on the so far achievements of the theory. The main contribution demonstrates the offer of the Clifford odd and even objects for the description of the internal spaces of fermion (Clifford odd) and boson (Clifford…
This is an extension of quantum spinor construction in \cite{DF2}. We define quantum affine Clifford algebras based on the tensor category and the solutions of q-KZ equations, construct quantum spinor representations of $U_q(\hat{\frak gl}(n))$ and explain classical and quantum boson-fermion correspondence.
In the review article in Progress in Particle and Nuclear Physics (vol.121(2021) 103890)) the authors present the achievements so far of the spin-charge-family theory, which offers the explanation for all the so far observed properties of elementary fermion and boson fields, if the space-time is higher than d=(3+1), it must be $d\ge (13+1)$. Fermions interact with gravity only. Ref. PPNP (vol.121(2021) 103890)) presents, in addition to a rather detailed review of all the achievements of this theory so far, also an explanation for the postulates of the second quantization for fermionic fields: …
Both algebras, Clifford and Grassmann, offer "basis vectors" for describing the internal degrees of freedom of fermions. The oddness of the "basis vectors", transferred to the creation operators, which are tensor products of the finite number of "basis vectors" and the infinite number of momentum basis, and to their Hermitian conjugated partners annihilation operators, offers the second quantization of fermions without postulating the conditions proposed by Dirac, enabling the explanation of the Dirac's postulates. But while the Clifford fermions manifest the half integer spins -- in agreement…
In a long series of works, it has been demonstrated, that the {\it spin-charge-family} theory offers the explanation for all in the {\it standard model} assumed properties of the second quantized fermion and boson fields, offering several predictions as well as explanations for several of the observed phenomena. The theory assumes a simple starting action in even dimensional spaces with $d \ge (13 +1)$ with massless fermions interacting with gravity only. The internal spaces of fermion and boson fields are described by the Clifford odd and even objects, respectively. This contribution discusse…
A topological interpretation of the electron--electron interaction under FQHE conditions gives rise to the existence of some kind of composite fermions. The interaction term is similar to the Coulomb interaction and produces punctures in the configuration space which change its topology. Now the relevant configuration space is not simply connected and as a consequence, ``new'' fermions, so called ``exotic spinors'', appear. We argue that composite fermions may be interpreted as exotic spinors. The denominator of the filling factor is calculated to be an odd number. We point out further possibl…
An alternative, pedagogically simpler derivation of the allowed physical wave fronts of a propagating electromagnetic signal is presented using geometric algebra. Maxwell's equations can be expressed in a single multivector equation using 3D Clifford algebra (isomorphic to Pauli algebra spinorial formulation of electromagnetism). Subsequently one can more easily solve for the time evolution of both the electric and magnetic field simultaneously in terms of the fields evaluated only on a 3D hypersurface. The form of the special "characteristic" surfaces for which the time derivative of the fiel…
This paper discusses the Higgs and spinor fermion terms of the D4-D5-E6 model of a series of papers (hep-ph/9301210, hep-th/9302030, hep-th/9306011, and hep-th/9402003) an 8-dimensional spacetime is reduced to 4-dimensions. The gauge boson terms give SU(3)xSU(2)xU(1) for the color, weak, and electromagnetic forces and gravity of the MacDowell-Mansouri type, which has recently been shown by Nieto, Obregon, and Socorro (gr-qc/9402029) to be equivalent, up to a Pontrjagin topological term, to the Ashtekar formulation.
It is well known that instantons are classical topological solutions existing in the context of quantum field theories that lie behind the standard model of particles. To provide a better understanding for the dynamical nature of spinor-type instanton solutions, conformal invariant pure spinor fermionic models that admit particle-like solutions for the derived classical field equations are studied in this work under cosine wave forcing. For this purpose the effects of external periodic forcing on two systems having different dimensions and quantum spinor numbers and have been obtained under th…
This contribution presents properties of the second quantized not only fermion fields but also boson fields, if the second quantization of both kinds of fields origins in the description of the internal space of fields with the ''basis vectors'' which are the superposition of odd (when describing fermions) or even (when describing bosons) products of the Clifford algebra operators $γ^a$'s. The tensor products of the ''basis vectors'' with the basis in ordinary space forming the creation operators manifest the anticommutativty (of fermions) or commutativity (of bosons) of the ''basis vectors'',…
Stress-energies for Fermi fields are derived from the principle of general covariance. This is done by developing a notion of Lie derivatives of spinors along arbitrary vector fields. A substantial theory of such derivatives was first introduced by Kosmann; here I show how an apparent conflict in the literature on this is due to a difference in the definitions of spinors, and that tracking the Lie derivative of the Infeld-van der Waerden symbol, as well as the spinor fields under consideration, gives a fuller picture of the geometry and leads to the Fermion stress-energy. The differences in th…
We construct a mass dimension one fermionic field associated with flag-dipole spinors. These spinors are related to Elko (flag-pole spinors) by a one-parameter matrix transformation $\mathcal{Z}(z)$ where $z$ is a complex number. The theory is non-local and non-covariant. While it is possible to obtain a Lorentz-invariant theory via $τ$-deformation, we choose to study the effects of non-locality and non-covariance. Our motivation for doing so is explained. We show that a fermionic field with $|z|\neq1$ and $|z|=1$ are physically equivalent. But for fermionic fields with more than one value of …
It is a well known fact from the group theory that irreducible tensor representations of classical groups are suitably characterized by irreducible representations of the symmetric groups. However, due to their different nature, vector and spinor representations are only connected and not united in such description. Clifford algebras are an ideal tool with which to describe symmetries of multi-particle systems since they contain spinor and vector representations within the same formalism, and, moreover, allow for a complete study of all classical Lie groups. In this work, together with an acco…
We show that the massive (or massless) neutrinos can be described as special states of Heisenberg nonlinear spinors. As a by-product of this decomposition a particularly attractive consequence appears: the possibility of relating the existence of only three species of mass-less neutrinos to such internal non-linear structure. At the same time it allows the possibility that neutrino oscillation can occurs even for massless neutrinos.
