Abstracts:We briefly review models of relativistic particles with spin. Departuring from the oldest attempts to describe the spin within the lagrangian framework we pass throught various non supersymmetric models. Then the component and superfield formulations of the spinning particle and superparticle models are reviewed. Our focus is mainly on the classical side of the problem, but some quantization questions are mentioned as well.

We quantify the geometric measure of entanglement in terms of mean values of observables of entangled system. For pure states we find the relation of geometric measure of entanglement with the mean value of spin one-half for the system composed of spin and arbitrary quantum system. The geometric measure of entanglement for mixed states of rank-2 is studied as well. We find the explicit expression for geometric entanglement and the relation of entanglement in this case with the values of spin correlations. These results allow to find experimentally the value of entanglement by measuring a value of the mean spin and the spin correlations for pure and mixed states, respectively. The obtained results are applied for calculation of entanglement during the evolution in spin chain with Ising interaction , two-spin Ising 1 model in transverse fluctuating magnetic field, Schrödinger cat in fluctuating magnetic field.

We study quantum brachistochrone problem for the spin-1 system in a magnetic field of a constant absolute value. Such system gives us a possibility to examine in detail the statement of papers [A. Carlini et al., Phys. Rev. Lett. 96, 060503 (2006), D. C. Brody, D. W. Hook, J. Phys. A 39, L167, (2006)] that the state vectors realizing the evolution with the minimal time of passage evolve along the subspace spanned by the initial and final state vectors. Using explicit example we show the existence of quantum brachistochrone with minimal possible time, but the state vector of which, during the evolution leaves the subspace spanned by the initial and final state vectors. This is the result of the choice of more constrained Hamiltonian then assumed in the general quantum brachistochrone problem, but what is worth noting, despite that such evolution is more complicated it is still time optimal. This might be important for experiment, where general Hamiltonian with the all allowed parameters is difficult to implement, but constrained one depending on magnetic field can be realized. However for pre-constrained Hamiltonian not all final states are accessible. Present result does not contradict general statement of the quantum brachistochrone problem, but gives new insight how time optimal passage can be realized.Recently Carlini et al.[1] generalized the classical brachistochrone problem to the quantum case. The quantum brachistochrone problem can be formulated in the following way:

We consider a relativistic particle model in an enlarged relativistic phase space, M 18 = (Xµ, Pµ, ηα, ηα, σα, σα, e, φ), which is derived from the free two-twistor dynamics. The spin sector variables (ηα, ηα, σα, σα) satisfy two second class constraints and account for the relativistic spin structure, and the pair (e, φ) describes the electric charge sector. After introducing the Liouville one-form on M 18 , derived by a non-linear 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 wavefunction provides a generating functional for free, massive higher spin fields.

We extend the Shirafuji model for massless particles with primary spacetime coordinates and composite four-momenta to a model for massive particles with spin and electric charge. The primary variables in the model are the spacetime 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 spacetime and four-momenta vectors are composite, and the standard particle model, where both spacetime and four-momenta vectors are elementary. We quantize the model and find explicitly the first-quantized wavefunctions describing relativistic particles with mass, spin and electric charge. The spacetime coordinates in the model are not commutative; this leads to a wavefunction that depends only on one covariant projection of the spacetime fourvector (covariantized time coordinate) defining plane wave solutions.