Narrow Results

Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard Feynman–Kac formula is taken here by subordinate Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which an L^p-L^q bound of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.

We consider a two-magnon systems in an ν-dimensional isotropic non-Heisenberg ferromagnet with spin value S = 3/2 and nearest-neighbor interactions. Spectrum and bound states (BS) of the system for all values of full quasi-momentum Λ, and for arbitrary value of lattice dimensionality ν, and for all values of Hamiltonian parameters are investigated. We show that (i) for arbitrary ν ≥ 2 and for full quasi-momentum in the form Λ = (Λ₁; Λ₂; … ;Λ_ν) = (Λ₀;Λ₀; …; Λ₀) the change of energy spectrum of the system is similar to that observed in the case of ν = 1. In this case the operator with J + J₁ - 23J₂ ≠ 0 has only one additional BS. (ii) The energy z of this additional BS is degenerate ν - 1 times. (iii) If Λ ≠ (Λ₀;Λ₀;…;Λ₀), we show the existence no more 2ν + 1 bound states in the system in ν-dimensional lattice.

In the gauge theory of gravity based on the Poincaré group (the semidirect product of the Lorentz group and the spacetime translations) the mass (energy–momentum) and the spin are treated on an equal footing as the sources of the gravitational field. The corresponding spacetime manifold carries the Riemann–Cartan geometric structure with the nontrivial curvature and torsion. We describe some aspects of the classical Poincaré gauge theory of gravity. Namely, the Lagrange–Noether formalism is presented in full generality, and the family of quadratic (in the curvature and the torsion) models is analyzed in detail. We discuss the special case of the spinless matter and demonstrate that Einstein's theory arises as a degenerate model in the class of the quadratic Poincaré theories. Another central point is the overview of the so-called double duality method for constructing of the exact solutions of the classical field equations.

The main result of the paper is a new representation of the Weyl Lagrangian (massless Dirac Lagrangian). As the dynamical variable, we use the coframe, i.e. an orthonormal tetrad of covector fields. We write down a simple Lagrangian — wedge product of axial torsion with a lightlike element of the coframe — and show that variation of the resulting action with respect to the coframe produces the Weyl equation. The advantage of our approach is that it does not require the use of spinors, Pauli matrices or covariant differentiation. The only geometric concepts we use are those of a metric, differential form, wedge product and exterior derivative. Our result assigns a variational meaning to the tetrad representation of the Weyl equation suggested by J. B. Griffiths and R. A. Newing.

The use of geometrical quantities and geometrical methods in the calculation and interpretation of physical systems have been very successful. Among others it has been shown how non-relativistic spin one half and massless relativistic spin one half with the corresponding Pauli matrices can be phrased geometrically. An analogous formulation for massive and relativistic spin one half was however missing. By the use of geometrical angular coordinates we construct a concrete realization of relativistic and massive spin one half. A Lagrange formulation is found, equations of motion are derived, and Lorentz invariance is discussed.

A covariant Hamiltonian formalism for the dynamics of compact spinning bodies in curved space-time in the test-particle limit is described. The construction allows a large class of Hamiltonians accounting for specific properties and interactions of spinning bodies. The dynamics for a minimal and a specific non-minimal Hamiltonian is discussed. An independent derivation of the equations of motion from an appropriate energy–momentum tensor is provided. It is shown how to derive constants of motion, both background-independent and background-dependent ones.

This paper derives the elements of classical Einstein–Cartan theory (EC) from classical general relativity (GR) in two ways. (I) Derive discrete versions of torsion (translational holonomy) and the spin-torsion field equation of EC from one Kerr solution in GR. (II) Derive the field equations of EC as the continuum limit of a distribution of many Kerr masses in classical GR. The convergence computations employ “epsilon-delta” arguments, and are not as rigorous as convergence in Sobolev norm. Inequality constraints needed for convergence restrict the limits from continuing to an infinitesimal length scale. EC enables modeling exchange of intrinsic and orbital angular momentum, which GR cannot do. Derivation of EC from GR strengthens the case for EC and for new physics derived from EC.

This paper presents theoretical arguments that certain virtual bound states carry the trace component of affine torsion. The motivation for this work is that Einstein–Cartan theory, which extends general relativity by including torsion to model intrinsic angular momentum, is becoming more credible. We are not aware of any situation for which there is evidence or substantial argument for the presence of torsion trace, except in the continuum theory of edge dislocations in crystals. The main evidence for the hypothesis consists of analogies between the structure of virtual bound states and (a) geometry of dislocations in crystal lattices, which are modeled with torsion; and (b) modeling of intrinsic angular momentum by torsion in Einstein–Cartan theory and the theory of micro-elasticity. The work focuses on conjectured presence of torsion in para-positronium, which intermediates annihilation of an electron and a positron with opposite z-spins. If the virtual bound state carries torsion, then the local law of conservation of angular momentum can hold over the spacelike separation during annihilation.