Quantization of torsion in a space of zero Riemannian curvature

Starting from the general form of the Lagrangian (quadratic with respect to the curvature tensor), the free torsion field and its interaction with spinor matter is considered for the particular case of a zero Riemann—Christoffel tensor. It is shown that the torsion field is equivalent to the superposition of the wave functions of massive axial-vector, massless and massive pseudoscalar particles (torsions). In addition to this, an axial-vector torsion is generated by the divergenceless part of the spin pseudovector of matter, and pseudoscalar torsions by the divergence of this pseudovector.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 116–121, January, 1975.In conclusion, the author wishes to thank Professor D. D. Ivanenko most sincerely for presenting this problem and constantly following its progress, as well as making a number of valuable comments.     

Particles with curvature and torsion in three-dimensional pseudo-Riemannian space forms

We consider the motion of relativistic particles described by an action which is a function of the curvature and torsion of the particle path. The Euler–Lagrange equations and the dynamical constants of the motion are expressed in a simple way in terms of a suitable coordinate system. The moduli spaces of solutions in a three-dimensional pseudo-Riemannian space form are completely exhibited.     

Quantization on a two-dimensional phase space with a constant curvature tensor

Some properties of the star product of the Weyl type (i.e., associated with the Weyl ordering) are proved. Fedosov construction of the *-product on a two-dimensional phase space with a constant curvature tensor is presented. Eigenvalue equations for momentum p and position q on a two-dimensional phase space with constant curvature tensors are solved.     

Topology of zero riemann curvature flat space in supergravity

The equation of motion of the gravitino is investigated in a flat space reduced from the Kerr geometry by taking the massless limit of the gravitational source. We adopt the ansatz ψ_μ(x)=δ_μoψ(x), i.e., the Coulomb potential analogue to the Rarita-Schwinger Majorana field of the gravitino. A non-trivial exact classical solution of ψ(x) is obtained and it is interpreted as the source of the intrinsic topology of the background flat space. This flat space is spanned by two Minkowski sheets interconnected through a disk of radius a (the angular momentum parameter in the Kerr geometry).     

A maximally symmetric space with torsion

A maximally symmetric space, i.e., homogeneous and isotropic at every point, possessing totally antisymmetric torsion is dealt with. It is found that maximum symmetry restricts the dimension of the space to three. The three-curvature tensor for the space is obtained and from its form a three-metric is then constructed. The three-space is then allowed to evolve in time so that a four-metric of the formds ²= –dt ²+ (3)g _ij dx ⁱ dx ^j is possible. From this an equation of motion is obtained which predicts an initial- and final-state singularity.Part of this work was done as a doctoral thesis requirement at Queen Mary College, University of London.     

Field equations for a Riemann space with torsion

It is shown that a unique system of equations suitable for the purposes of combining gravitation and electromagnetism and differing by the fact that its compatibility is assured by two identities of different order in differentiation exists in the Einstein theory of absolute parallelism. The system turns out to be irregular in the case of the usual dimensionality of space. The possibility is noted of solutions with a zero electromagnetic field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 83–87, September, 1990.     

Spinors in quantum space with torsion

Dirac spinors are considered in quantized fiber-bundled spaces. It is shown that the spin connection has the same internal structure as in the Riemann-Cartan space as well as the quantized one. It is also assumed that the neutrino oscillation mechanism can be linked to the quantized and fibered character of the space at small distances.     

Some potentials for the curvature tensor on three-dimensional manifolds

We study equations of Riemann–Lanczos type on three dimensional manifolds. Obstructions to global existence for global Lanczos potentials are pointed out. We check that the imposition of the original Lanczos symmetries on the potential leads to equations which do not have a determined type, leading to problems when trying to prove global existence. We show that elliptic equations can be obtained by relaxing those symmetry requirements in at least two different ways, leading to global existence of potentials under natural conditions. A second order potential for the Ricci tensor is introduced.     

Quadratic Lagrangians in a space with torsion

A general expression is given for a quadratic (with respect to the curvature tensor) Lagrangian in a space with torsion. The given formalism is discussed in application to generalized Einstein-Cartan gravitation theory and the theory of the spin gauge field introduced in localization of the tetrad Lorentz transformation group.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 7–10, December, 1977.The author is indebted to Prof. D. D. Ivanenko, V. N. Ponomarev, and P. L. Poznanin for a useful and stimulating discussion.     

Thermal fluctuations of elastic filaments with spontaneous curvature and torsion

We study the effects of thermal fluctuations on thin elastic filaments with spontaneous curvature and torsion. We derive analytical expressions for the orientational correlation functions and for the persistence length of helices and find that this length varies nonmonotonically with the strength of thermal fluctuations. In the weak fluctuation regime, the persistence length of a spontaneously twisted helix has three resonance peaks as a function of the twist rate. In the limit of strong fluctuations, all memory of the helical shape is lost.     

Noether symmetries and conserved quantities for spaces with a section of zero curvature

In an earlier paper (Feroze, 2010 [21]), the existence of new conserved quantities (Noether invariants) for spaces of different curvatures was discussed. There, it was conjectured that the number of new conserved quantities for spaces with an m$m$-dimensional section of zero curvature is m$m$. Here, along with the proof of this conjecture, the form of the new conserved quantities is also presented. For the illustration of the theorem, an example of conformally flat spacetime is constructed which also demonstrates that the conformal Killing vectors (CKVs), in general, are not symmetries of the Lagrangian for the geodesic equation.     

Irreducible tensor form of three-particle operator for open-shell atoms

A three-particle operator in a second quantized form is studied systematically and comprehensively. The operator is transformed into irreducible tensor form. Possible coupling schemes, identified by the classes of symmetric group S6, are presented. Recoupling coefficients that make it possible to transform a given scheme into another are produced by using the angular momentum theory combined with quasispin formalism. The classification of the three-particle operator which acts on n = 1, 2,..., 6 open shells of equivalent electrons of atom is considered. The procedure to construct three-particle matrix elements are examined.     

The Casimir operator of a metric connection with skew-symmetric torsion

For any triple (Mⁿ,g,) consisting of a Riemannian manifold and a metric connection with skew-symmetric torsion we introduce an elliptic, second-order operator Ω acting on spinor fields. In case of a naturally reductive space and its canonical connection, our construction yields the Casimir operator of the isometry group. Several non-homogeneous geometries (Sasakian, nearly Kähler, cocalibrated G₂-structures) admit unique connections with skew-symmetric torsion. We study the corresponding Casimir operator and compare its kernel with the space of -parallel spinors.     

Spinor and electromagnetic fields in space with torsion

A combined system of Einstein-Cartan, Ivanenko-Heisenberg, and Maxwell equations is reduced to a combined system of Einstein, Dirac, and Maxwell equations in Riemann space. A solution in which metric singularities are absent is found.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 31–35, November, 1978.     

Physical interpretation of a curvilinear space with torsion

An extension of the general theory of relativity which makes it possible to incorporate into the geometric scheme the classical electromagnetic field based on the four-dimensional curvilinear metric space with torsion is proposed. This scheme is used to determine the structure of elementary particles, viewed as the characteristic states of the curvilinear space. The equations describing these states are derived, and the static spherically symmetric solution for the gravitational and electric fields that can be regarded as a stable single-particle state is found. It is shown that in order to describe correctly the structure of a stable elementary particle the magnetic field must be taken into account.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 59–62, October, 1990.     

The cosserat continuum in a space with torsion

A variational principle is formulated for the Cosserat material continuum in a space with torsion.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 11–13, December, 1977.     

Stochastic physical origin of the quantum operator algebra and phase space interpretation of the Hilbert space formalism: The relativistic spin zero case

Based on a recent association of quantum observable algebra with stochastic processes in the frame of the causal stochastic interpretation of quantum mechanics, a relativistic Hilbert space is defined for the Klein-Gordon case. It is demonstrated that unitary transformations in Hilbert space reflect canonical transformations in the associated phase space, manifesting thus an underlying symplectic structure.