Relativistic scalar equations for a particle with an arbitrary spin

Relativistic Poincaré-invariant wave equations for zero-mass and heavy particles with an arbitrary spin are constructed on the basis of special infinite-dimensional representation of the Lorentz group. The equations form a compatible system of linear differential equations for an unknown scalar function and contain spin s as a parameter (arbitrary complex number). It is also shown that the equations obtained in this way include the well-known finite-component wave equations as a special case of half-integral or integral spin.State University, Omsk. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 39–44, June, 1992.     

Foldy-Wouthuysen transformations for any spin

We give the generalized Foldy-Wouthuysen transformation for any 2(2J + 1)-component Poincaré-invariant Hamiltonian theory that describes free massive spin-J particles and that is subject to the conditions: (a) every observable $\text{O}$ is either Hermitian ($\text{O}$ = $\text{O}$⁺) or pseudo-Hermitian ($\text{O}$ = ?₃$\text{O}$⁺?₃) and (b) the theory is invariant under the discrete symmetries. The requirement that the Hamiltonian be defined in the rest frame specifies one and only one boost generator that is also defined at p = 0.     

Relativistic and separable classical hamiltonian particle dynamics

We show within the Hamiltonian formalism the existence of classical relativistic mechanics of N scalar particles interacting at a distance which satisfies the requirements of Poincaré invariance, separability, world-line invariance and Einstein causality. The line of approach which is adopted here uses the methods of the theory of systems with constraints applied to manifestly covariant systems of particles. The study is limited to the case of scalar interactions remaining weak in the whole phase space and vanishing at large space-like separation distances of the particles. Poincaré invariance requires the inclusion of many-body, up to N-body, potentials. Separability requires the use of individual or two-body variables and the construction of the total interaction from basic two-body interactions. Position variables of the particles are constructed in terms of the canonical variables of the theory according to the world-line invariance condition and the subsidiary conditions of the non-relativistic limit and separability. Positivity constraints on the interaction masses squared of the particles ensure that the velocities of the latter remain always smaller than the velocity of light.     

Relativistic Bogoljubov equations for superconductivity

From a nonlinear bispinor Lagrange density relativistic Bogoljubov equations for super-conductivity are derived.The author would like to thank Dr. N.Salié for helpful discussions.     

Relativistic oscillator model with spin for nucleon resonances

The relativistic three-body problem is approached via the extension of the SL(2, C) group to the Sp(4, C) one. In terms of Sp(4, C) spinors, a Dirac-like equation with three-body kinematics is composed. After introducing the linear in coordinates interaction, it describes the spin-1/2 oscillator. For this system, the exact energy spectrum is derived and then applied to fit the Regge trajectories of baryon N-resonances in the (E ², J) plane. The model predicts linear trajectories at high total energy E with some form of nonlinearity at low E.     

自旋粒子的相对论波函数

从相对论波动方程和Lorentz变换理论出发,讨论了自旋粒子的相对论波函数,并给出了求相对论粒子高自旋态的方法。     

Relativistic polarization density matrix for particles of arbitrary spin

The polarization density matrix for particles of arbitrary spin can be expressed as a linear function of the expectation values of the generators of the polarization symmetry group of the corresponding wave equations.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 84–88, April, 1984.     

Spacetime path formalism for massive particles of any spin

Earlier work presented spacetime path formalism for relativistic quantum mechanics arising naturally from the fundamental principles of the Born probability rule, superposition, and spacetime translation invariance. The resulting formalism can be seen as a foundation for a number of previous parametrized approaches to relativistic quantum mechanics in the literature. Because time is treated similarly to the three-space coordinates, rather than as an evolution parameter, such approaches have proved particularly useful in the study of quantum gravity and cosmology. The present paper extends the foundational spacetime path formalism to include massive, non-scalar particles of any (integer or half-integer) spin. This is done by generalizing the principle of translational invariance used in the scalar case to the principle of full Poincaré invariance, leading to a formulation for the non-scalar propagator in terms of a path integral over the Poincaré group. Once the difficulty of the non-compactness of the component Lorentz group is dealt with, the subsequent development is remarkably parallel to the scalar case. This allows the formalism to retain a clear probabilistic interpretation throughout, with a natural reduction to non-relativistic quantum mechanics closely related to the well-known generalized Foldy-Wouthuysen transformation.     

Relativistic generalization of quasi-Chaplygin equations

A relativistic generalization of quasi-Chaplygin (quasi-gas) equations describing the evolution of unstable media with negative compressibility is proposed. Examples of the media whose dynamics can be described by the proposed equations are considered. An analytic solution to these nonlinear equations is obtained for the 1D case.     

Relativistic bound-state equations for fermions with instantaneous interactions

In thispaper three types of relativistic bound-state equations for a fermion pair with instantaneous interaction are studied, viz., the instantaneous Bethe-Salpeter equation, the quasi-potential equation, and the two-particle Dirac equation. General forms for the equations describing bound states with arbitrary spin, parity, and charge parity are derived. For the special case of spinless states bound by interactions with a Coulomb-type potential the properties of the ground-state solutions of the three equations are investigated both analytically and numerically. The coupling-constant spectrum turns out to depend strongly on the spinor structure of the fermion interaction. If the latter is chosen such that the nonrelativistic limits of the equations coincide, an analogous spectrum is found for the instantaneous Bethe-Salpeter and the quasi-potential equations, whereas the two-particle Dirac equation yields qualitatively different results.     

A new hierarchy of coupled degenerate hamiltonian equations

A new hierarchy of pairs of coupled nonlinear evolution equations is proposed. The first pair of coupled equations can be reduced to the single equation w_tt - w_tx ± 4e^w = 0. It is shown that the equations in this hierarchy possess an infinite number of conserved densities, and that a degenerate symplectic structure can be introduced to the whole hierarchy.     

The Maxwell-Vlasov equations as a continuous hamiltonian system

The well-known Maxwell-Vlasov equations that describe a collisionless plasma are cast into hamiltonian form. The dynamical variables are the physical although noncanonical variables E, B and f. We present a Poisson bracket which acts on these variables and the energy functional to produce the equations of motion.     

Relativistic spin on the Poincaré group

Classical spinning particles are interpreted in terms of an underlying geometric theory. They are described by trajectories on the Poincaré group. Upon quantization an eleven-dimensional Kaluza-Klein type theory is obtained which incorporates spin and isospin in a local SL(2, C)×U(1)×SU(2) gauge theory, unifying gravity and the pre-Higgs standard model. The relation to parametrized relativistic quantum theory is discussed.     

Spin hamiltonian for which the chiral spin liquid is the exact ground state

We construct a Hamiltonian that singles out the chiral spin liquid on a square lattice with periodic boundary conditions as the exact and, apart from the twofold topological degeneracy, unique ground state.