Relativistic wavefunctions on the Poincaré group

Adv. Appl. Clifford Algebras

2014

Supergroups are defined in the framework of Z 2 -graded Clifford algebras over the fields of real and complex numbers, respectively. It is shown that cyclic structures of complex and real supergroups are defined by Brauer-Wall groups related with the modulo 2 and modulo 8 periodicities of the complex and real Clifford algebras. Particle (fermionic and bosonic) representations of a universal covering (spinor group Spin + (1, 3)) of the proper orthochronous Lorentz group are constructed via the Clifford algebra formalism. Complex and real supergroups are defined on the representation system of Spin + (1, 3). It is shown that a cyclic (modulo 2) structure of the complex supergroup is equivalent to a supersymmetric action, that is, it converts fermionic representations into bosonic representations and vice versa. The cyclic action of the real supergroup leads to a much more high-graded symmetry related with the modulo 8 periodicity of the real Clifford algebras. This symmetry acts on the system of real representations of Spin + (1, 3).

Section: Let Us Consider the Operatorsmentioning

confidence: 99%

Cyclic Structures of Cliffordian Supergroups and Particle Representations of Spin+(1, 3)

Adv. Appl. Clifford Algebras

2014

“…hold. The formula (24) gives a relation between the roots of polynomial D and eigenvalues of the matrix Γ 0 . It is easy to see that along with the each non-null eigenvalue λ the matrix Γ 0 has an eigenvalue −λ of the same multiplicity.…”

Section: Relativistic Wave Equationsmentioning

confidence: 99%

Spinor Structure and Internal Symmetries

2015

Int J Theor Phys

Spinor structure and internal symmetries are considered within one theoretical framework based on the generalized spin and abstract Hilbert space. Complex momentum is understood as a generating kernel of the underlying spinor structure. It is shown that tensor products of biquaternion algebras are associated with the each irreducible representation of the Lorentz group. Space-time discrete symmetries $P$, $T$ and their combination $PT$ are generated by the fundamental automorphisms of this algebraic background (Clifford algebras). Charge conjugation $C$ is presented by a pseudoautomorphism of the complex Clifford algebra. This description of the operation $C$ allows one to distinguish charged and neutral particles including particle-antiparticle interchange and truly neutral particles. Spin and charge multiplets, based on the interlocking representations of the Lorentz group, are introduced. A central point of the work is a correspondence between Wigner definition of elementary particle as an irreducible representation of the Poincar\'{e} group and $SU(3)$-description (quark scheme) of the particle as a vector of the supermultiplet (irreducible representation of $SU(3)$). This correspondence is realized on the ground of a spin-charge Hilbert space. Basic hadron supermultiplets of $SU(3)$-theory (baryon octet and two meson octets) are studied in this framework. It is shown that quark phenomenologies are naturally incorporated into presented scheme. The relationship between mass and spin is established. The introduced spin-mass formula and its combination with Gell-Mann--Okubo mass formula allows one to take a new look at the problem of mass spectrum of elementary particles.Comment: 44 page

“…The fields (1/2, 0) ⊕ (0, 1/2) and (1, 0) ⊕ (0, 1) (Dirac and Maxwell fields) are particular cases of fields of the type (l, 0) ⊕ (0, l). Wave equations for such fields and their general solutions were found in the works [54,56,59]. It is easy to see that the interlocking scheme, corresponded to the Maxwell field, contains the field of tensor type:…”

Section: Proposition 1 ([55]mentioning

confidence: 99%

CPT Groups of Higher Spin Fields

2011

Int J Theor Phys

CP T groups of higher spin fields are defined in the framework of automorphism groups of Clifford algebras associated with the complex representations of the proper orthochronous Lorentz group. Higher spin fields are understood as the fields on the Poincaré group which describe orientable (extended) objects. A general method for construction of CP T groups of the fields of any spin is given. CP T groups of the fields of spin-1/2, spin-1 and spin-3/2 are considered in detail. CP T groups of the fields of tensor type are discussed. It is shown that tensor fields correspond to particles of the same spin with different masses.