Narrow Results

In this paper, we focus on the Janus symplectic group. We explore its various symmetries and its action on the elements of the dual of its Lie algebra, called torsors. Special attention is given to the charge symmetry, which highlights the matter–antimatter duality within both sets of components.

A careful ab initio construction of the finite-mass (1/2, 1/2) representation space of the Lorentz group reveals it to be a spin-parity multiplet. In general, it does not lend itself to a single-spin interpretation. We find that the (1/2, 1/2) representation space for massive particles naturally bifurcates into a triplet and a singlet of opposite relative intrinsic parties. The textbook separation into spin-1 and spin-0 states occurs only for certain limited kinematical settings. We construct a wave equation for the (1/2, 1/2) multiplet, and show that the particles and antiparticles in this representation space do not carry a definite spin but only a definite relative intrinsic parity. In general, both spin-1 and spin-0 are covariantly inseparable inhabitants of massive vector fields. This last observation suggests that scalar particles, such as the Higgs, are natural inhabitants of massive (1/2, 1/2) representation space.

This paper aims to give a coordinate-based introduction to the so-called Lounesto spinorial classification scheme. Among other results, it has evinced classes of spinors which fail to satisfy Dirac equation. The underlying idea and the central aspects of such spinorial categorization are introduced in an argumentative basis, after which we delve into a commented account on recent results obtained from (and within) this branch of research.

The capabilities of some approaches to the relativistic description of hadronic states with any rest spin are analysed. The key feature in the Wigner's construction of irreducible representations of the Poincaré group, which makes this construction fruitless in particle physics, is picked out. A realization of unitary irreducible representations of the Poincaré group of the standard type, which has not yet been considered, is discussed. The viability of the description of hadrons by the Poincaré group representations of the standard type in the space of the infinite-component ISFIR-class fields is pointed out.

Super-Poincaré algebra in $D=6$ space–time dimensions has been studied in terms of quaternionic representation of Lorentz group. Starting the connection of quaternion Lorentz group with $SO(1,5)$ group, the $SL(2,ℍ)$ spinors for Dirac and Weyl representations of Poincaré group are described consistently to extend the Poincaré algebra to super-Poincaré algebra for $D=6$ space–time.

Studies of scattering amplitudes for electric and magnetic charges have identified previously overlooked multiparticle representations of the Poincaré group in four dimensions. Such representations associate nontrivial quantum numbers (known as pairwise helicities) with asymptotically separated pairs of particles, and thus cannot be described as tensor products of one-particle states. We extend this construction to sources and spacetimes of higher dimension. We first establish the dynamical origin of pairwise helicity in $p$-form electrodynamics coupled to mutually nonlocal branes. We then interpret this pairwise helicity as a quantum number under an SO(2) pairwise little group associated with pairs of distinct branes. We further characterize the “higher” little groups that could in principle be used to induce multiparticle or multi-brane representations of the Lorentz group.

We present a Master equation for description of fermions and bosons for special relativities with two invariant scales, , (c and λ_P). We introduce canonically-conjugate variables (χ⁰,χ) to (∊,π) of Judes-Visser. Together, they bring in a formal element of linearity and locality in an otherwise non-linear and non-local theory. Special relativities with two invariant scales provide all corrections, say, to the standard model of the high energy physics, in terms of one fundamental constant, λ_P. It is emphasized that spacetime of special relativities with two invariant scales carries an intrinsic quantum-gravitational character. In an addenda, we also comment on the physical importance of a phase factor that the whole literature on the subject has missed and present a brief critique of . In addition, we remark that the most natural and physically viable shall require momentum-space and spacetime to be non-commutative with the non-commutativity determined by the spin content and C, P, and T properties of the examined representation space. Therefore, in a physically successful , the notion of spacetime is expected to be deeply intertwined with specific properties of the test particle.

We derive source-free Maxwell-like equations in flat space–time for any helicity j by comparing the transformation properties of the 2(2j+1) states that carry the manifestly covariant representations of the inhomogeneous Lorentz group with the transformation properties of the two helicity j states that carry the irreducible representations of this group. The set of constraints so derived involves a pair of curl equations and a pair of divergence equations. These reduce to the free-field Maxwell equations for j = 1 and the analogous equations coupling the gravito-electric and the gravito-magnetic fields for j = 2.

The invariant metrizability problem for affine connections on a manifold, formulated by Tanaka and Krupka for connected Lie groups actions, is considered in the particular cases of Lorentz and Poincaré (inhomogeneous Lorentz) groups. Conditions under which an affine connection on the open submanifold $ℝ\times ({ℝ}^3\backslash \{(0,0,0)\})$ of the Euclidean space ${ℝ}^4$ coincides with the Levi-Civita connection of some $\mathrm{\text{SO}}(3,1)$, respectively $({ℝ}^4{\times }_s\mathrm{\text{SO}}(3,1))$-invariant metric field are studied. We give complete description of metrizable Lorentz-invariant connections. Explicit solutions (metric fields) of the invariant metrizability equations are found and their properties are discussed.

For hyperbolic systems of conservation laws in one space dimension endowed with a mathematical entropy, we define the notion of entropy velocity and we give sufficient conditions for such a system to be covariant under the action of a group of space-time transformations. These conditions naturally introduce a representation of the group in the space of states. We construct such hyperbolic systems from the knowledge of data on the manifold of null velocity. We apply these ideas for Galileo, Lorentz, and circular groups and, in particular, focus on nontrivial examples of $2\times 2$ systems of conservation laws.

We study the discrete symmetries (P, C and T) on the kinematical level within the extended Poincaré Group. On the basis of the Silagadze research, we investigate the question of the definitions of the discrete symmetry operators both on the classical level, and in the secondary-quantization scheme. We study the physical contents within several bases: light-front formulation, helicity basis, angular momentum basis, and so on, on several practical examples. We analize problems in construction of the neutral particles in the the (1/2, 0) + (0, 1/2) representation, the (1, 0) + (0, 1) and the (1/2, 1/2) representations of the Lorentz Group. As well known, the photon has the quantum numbers 1⁻, so the (1, 0)+(0, 1) representation of the Lorentz group is relevant to its description. We have ambiguities in the definitions of the corresponding operators P, C, T, which lead to different physical consequences. It appears that the answers are connected with the helicity basis properties, and commutations/anticommutations of the corresponding operators, P, C, T, and C², P², (CP)² properties.

We construct self/anti-self charge conjugate (Majorana-like) states in the (1/2, 0) ⊕ (0, 1/2) representation of the Lorentz group, and their analogs for higher spins within the quantum field theory. The problem of the basis rotations and that of the selection of phases in the Dirac-like and Majorana-like field operators are considered. The discrete symmetries properties (P, C, T) are studied. The corresponding dynamical equations are presented. In the (1/2, 0) ⊕ (0, 1/2) representation they obey the Dirac-like equation with eight components, which has been first introduced by Markov. Thus, the Fock space for corresponding quantum fields is doubled (as shown by Ziino). The particular attention has been paid to the questions of chirality and helicity (two concepts which are frequently confused in the literature) for Dirac and Majorana states. We further review several experimental consequences which follow from the previous works of M.Kirchbach et al. on neutrinoless double beta decay, and G.J.Ni et al. on meson lifetimes.