Andrzej M. Frydryszak

Massive relativistic particle model with spin from free two-twistor dynamics and its quantization

We consider a relativistic particle model in an enlarged relativistic phase space, M{sup 18}=(X{sub {mu}},P{sub {mu}},{eta}{sub {alpha}},{eta}{sub {alpha}},{sigma}{sub {alpha}},{sigma}{sub {alpha}},e,{phi}), which is derived from the free two-twistor dynamics. The spin sector variables ({eta}{sub {alpha}},{eta}{sub {alpha}},{sigma}{sub {alpha}},{sigma}{sub {alpha}}) satisfy two second class constraints and account for the relativistic spin structure, and the pair (e,{phi}) describes the electric charge sector. After introducing the Liouville one-form on M{sup 18}, derived by a nonlinear transformation of the canonical Liouville one-form on the two-twistor space, we analyze the dynamics described by the first and second class constraints. We use a composite orthogonal basis in four-momentum space to obtain the scalars defining the invariant spin projections. The first-quantized theory provides a consistent set of wave equations, determining the mass, spin, invariant spin projection and electric charge of the relativistic particle. The wave function provides a generating functional for free, massive higher spin fields.

EXTENSION OF THE SHIRAFUJI MODEL FOR MASSIVE PARTICLES WITH SPIN

We extend the Shirafuji model for massless particles with primary space–time coordinates and composite four-momenta to a model for massive particles with spin and electric charge. The primary variables in the model are the space–time four-vector, four scalars describing spin and charge degrees of freedom as well as a pair of Weyl spinors. The geometric description proposed in this paper provides an intermediate step between the free purely twistorial model in two-twistor space in which both space–time and four-momenta vectors are composite, and the standard particle model, where both space–time and four-momenta vectors are elementary. We quantize the model and find explicitly the first-quantized wave functions describing relativistic particles with mass, spin and electric charge. The space–time coordinates in the model are not commutative; this leads to a wave function that depends only on one covariant projection of the space–time four-vector (covariantized time coordinate) defining plane wave solutions.

Aspects of Pre-quantum Description of Deformed Theories

We discuss deformations of the classical systems in the phase space and show that one can use non-canonical transformations to relate regular and deformed models. On the other hand some of the models can be obtained as a classical limit of the deformed quantum models, i.e. as the result of the dequantization procedure. Nonrelativistic deformations are described.

N = 2 massive matter multiplet from quantization of extended classical mechanics

Abstract The recently proposed supersymmetric particle model invariant with respect to the N -extended superPoincare group ( N = 2, 4, ...) is quantized. For N = 2 the free supersymmetric quantized system is described by a scalar massive matter multiplet (two complex KG scalar fields and a Dirac spinor field, all with equal masses). In the proposed framework the Salam-Strathdee superfields describe the Schrodinger representation in first-quantized supersymmetric theory.

Qutrit geometric discord

Abstract Properties of the trace norm geometric discord of the system of two qutrits are studied. The geometric discord of qutrit Bell states, Werner states and bound entangled states is computed.

Measurement-induced qudit geometric discord

We study measurement-induced geometric discord based on the trace norm and generalize some properties known for qutrits to qudits. Previous preliminary results for bipartite qutrit systems (i.e. d = 3 systems) are here strictly proved for arbitrary d. The present study supports the observations, coming also from other approaches, that systems with show similar behaviour when quantum correlations are concerned, but there is pronounced difference between d = 2 and d = 3. Qubit systems are exceptionally simple. The underlying geometry of state spaces and related Lie groups is responsible for this.

NILPOTENT CLASSICAL MECHANICS

The formalism of nilpotent mechanics is introduced in the Lagrangian and Hamiltonian form. Systems are described using nilpotent, commuting coordinates η. Necessary geometrical notions and elements of generalized differential η-calculus are introduced. The so-called s-geometry, in a special case when it is orthogonally related to a traceless symmetric form, shows some resemblances to the symplectic geometry. As an example of an η-system the nilpotent oscillator is introduced and its supersymmetrization considered. It is shown that the R-symmetry known for the graded superfield oscillator also present here for the supersymmetric η-system. The generalized Poisson bracket for (η, p)-variables satisfies modified Leibniz rule and has nontrivial Jacobiator.

On the Grassmannian Special Functions I

A generalization of the notion of a special function to the case of anticommuting variables is presented. In particular, Grassmann-Hermite multinomials are obtained and their elementary properties are displayed.

NILPOTENT QUANTUM MECHANICS

We develop a generalized quantum mechanical formalism based on the nilpotent commuting variables (η-variables). In the nonrelativistic case such formalism provides natural realization of a two-level system (qubit). Using the space of η-wavefunctions, η-Hilbert space and generalized Schrodinger equation we study properties of pure multiqubit systems and also properties of some composed, hybrid models: fermion–qubit, boson–qubit. The fermion–qubit system can be truly supersymmetric, with both SUSY partners having identical spectra. It is a novel feature that SUSY transformations relate here only nilpotent object. The η-eigenfunctions of the Hamiltonian for the qubit–qubit system give the set of Bloch vectors as a natural basis.

Nilpotent classical mechanics:s-geometry

We introduce specific type of hyperbolic spaces. It is not a general linear covariant object, but is of use in constructing nilpotent systems. In the present work necessary definitions and relevant properties of configuration and phase spaces are indicated. As a working example we use aD=2 isotropic harmonic oscillator.