Gravitational interaction of a vector field with account of the nonmetricity of space-time

The gravitational interaction of a vector field is investigated in a space with the nonmetricity described by the Weyl vector. The analogue of the Coulomb law for the electrostatic field of a point charge is found in such a space. It is shown that taking account of the nonmetricity of space-time leads to the appearance of a nonlinearity in a massive vector field, resulting in the sine-Gordon and shine-Gordon equations. The screening of the vector-field mass as a consequence of its interaction with the nonmetricity is clarified. The solution for the Reissner-Nordström problem in a Weyl space is obtained, which asymptotically coincides with the solution of the same problem in general relativity, but nowhere does it possess singularities apart from at the origin. The obtained results show that it is reasonable to take account of the nonmetricity when describing the gravitational interaction of a vector field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 50–54, August, 1984.     

Spinor structure of space-time

The spinor structure on space-time manifold is investigated in the frame of Crumeyrolle's approach. Some of his theorems are simplified. The equivalence of this approach to the Milnor and Lichnerowicz one is shown using topological properties of the group space of ₀. The equivalence of any two spinor structures on simply connected space-time is established.Partly supported by the Polish Government under the Research Program MR I.7.     

Waves of a nonmetricity field and large-scale inhomogeneity of matter in the Weyl-Cartan space-time

The equation of nonmetricity field fluctuations is derived for the Weyl-Cartan space. A correlation is found between densities of various kinds of matter in the inflating Universe. The mass of the nonmetricity field quantum defined as the quantum of interaction with matter having nonzero dilaton charge is calculated. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 39–43, April, 2006.     

Spinor waves in a space-time lattice (II)

In a previous note, an exceptional space-time lattice was found by a roundabout heuristic process. This process was far from convincing; here a more translucent characterization of the lattice is presented. A cornerstone is the consideration of pairs of reciprocal lattices, together with the basic symmetry (S ₄) of the metric tensor. The basic requirement is that one member of a pair of reciprocal lattices contains the other as a sublattice. One preferred lattice is discussed in some detail; it contains three copies of its reciprocal lattice, and it is the simplest example satisfying the requirements. In the expression of the metric tensor in terms of the lattice generators a possible topology on the lattice is suggested. By means of this topology, propagation of spinor waves can be formulated. This proposed—the simplest—propagation mechanism is inhibited, though, by the fact that the three sublattices are required to carry the two types of spinors alternatively. This inhibition can be lifted by introducing a second type of elementary propagation, to next nearest neighbors. If this inhibition is only feebly lifted, this would result in particles with mass small as compared to the inverse of the lattice constant, presumably the Planck mass. Including the propagation to next nearest neighbors leads to spinor waves with six components, two components for each sublattice. In the long-wavelength limit four of them obey a massive Dirac equation, while the remaining two obey a Weyl equation. These considerations conceivably provide a root for the lack of parity invariance in nature, and for the joint occurrence of pairs of massive and massless spinor waves. The construction, furthermore, allows one to accommodate just three different families of spinor waves of this type. Extension of the above arguments outside the realm of the long-wavelength limit forcibly makes the lattice concept independent of the original continuous Minkowski spacetime: the latter is no longer a unique embedding space for the lattice, but appears as anapproximate interpolation, valid near the long-wavelength limit. This may be the minimal requirement to be imposed on a lattice theory in the light of the empirical evidence, if the scale of the lattice structure is, compared to the empirical scales, as small as the Planck scale.     

Scalar field equation in Robertson-Walker space-time

The quantization of the scalar field is reconsidered in some of its basic elements in the context of the Robertson-Walker space-time. The integration of the generalized Klein-Gordon equation is performed by preliminary separation of the equation with the usual separation method. The orthonormal mode solutions are determined by the explicit integration of the resulting angular and radial equations and by standard properties of the time equation. The time evolution given by the standard cosmological model is briefly discussed.     

Spinor algebras

We consider supersymmetry algebras in space–times with arbitrary signature and minimal number of spinor generators. The interrelation between super Poincaré and super conformal algebras is elucidated. Minimal super conformal algebras are seen to have as bosonic part a classical semisimple algebra naturally associated to the spin group. This algebra, the Spin(s,t)-algebra, depends both on the dimension and on the signature of space–time. We also consider maximal super conformal algebras, which are classified by the orthosymplectic algebras.     

Quantum field theory in curved space-time as thermo field dynamics

The strong analogy between states defined in the context of quantum field theory in curved space-time (QFT-CST) and the ones defined in the thermo field dynamics (TFD) of Takahashi and Umezawa [1] is shown. This analogy is useful in order to introduce the entropy operator in CST in the same way as in TFD. When the extremum condition in the thermodynamical potential is imposed, a family of Bogoliubov transformations that give us a planckian spectrum is found, even in pathological cases such as the minimally coupled scalar field.     

Spinor brane

The thick brane model supported by a nonlinear spinor field is constructed. The different cases with the various values of the cosmological constant ${\Lambda \left( {l} < \\ =\\ > \right) 0}${\Lambda \left( \begin{array}{l} < \\ =\\ > \end{array} \right) 0} are investigated. It is shown that regular analytical spinor thick brane solutions with asymptotically Minkowski (at Λ = 0) or anti-de Sitter spacetimes (at Λ <  0) do exist.     

Long-range scalar field in conformally flat space-time

Consequences of a massless scalar field in conformally flat space-time are studied. Then a wide class of solutions of the scalar field is obtained.     

An electromagnetic field in the Plebanski space-time

The electromagnetic wave equation in the Plebanski space-time has been reduced to a one-dimensional wave equation with real short-range potential. We then find that the superradiance phenomenon occurs.     

Electromagnetic field in two-dimensional locally flat space-time

It is shown that an infinite-dimensional symmetry is present in two-dimensional electromagnetic field theory. The generators of the ensuing Virasoro algebra are explicitly calculated both for periodic and antiperiodic fields.     

Spinor matter in a particular 5-dimensional projective unified field theory

After presenting the foundation and the basic equations of a new 5-dimensional projective unified field theory, the problem of incorporating spinor fields into this framework is investigated. Apart from Pauli's method, we propose a new approach which leads to a consistent 5-dimensional spinor theory with a series of physical consequences (variability of the 4-dimensional rest mass, instability of 4-dimensional stationary states, etc.).Dedicated to Prof. Peter Bergmann on the occasion of this 70th birthday.     

Self-interaction and massive vector field in the Schwarzschild space-time

Using a multipole expansion, we determine formally the massive vector field generated by a point source held fixed in the Schwarzschild space-time. We prove that its limit, when the mass of the Proca field goes to zero, does not tend to the corresponding massless vector field. In this limit we evaluate the expression of the self-force acting on the particle. The result is in accordance with that of Vilenkin, without the assumption that the point source is at a large distance from the horizon, that we extend to the case of a Reissner-Nordström space-time. We also investigate a further case: a point source within a spherical shell of matter for which the Proca field tends to the corresponding massless vector field.     

Some remarks on nonlocal field theory and space-time quantization

From the point of view that the charge and mass of an electron is of dynamical origin and quantization of charge in units ofe is related to the space-time quantization as developed in an earlier paper, we here show that it is possible to consider that the internal space within the elementary domain of the quantized space-time world is not governed by Lorentz invariance. This helps us to develop a consistent theory of nonlocal fields for extended particles where the infinite mass degeneracy is avoided. Moreover, this ensures the convergence of nonlocal field theories and suggests that massless particles like photons and neutrinos, though they may be taken to be of extended structure, will appear only as point particles in the physical world. In this picture, Lorentz invariance appears to be a consequence of the distribution of matter and energy in the Universe, and this may be taken to be another interpretation of Mach's principle.     

Geometric quantization and quantum field theory in curved space-time

The Kostant-Souriau geometric quantization theory is applied to the problem of constructing a generally covariant quantum field theory. The occupation number formalism for a scalar field is introduced as a semiclassical approximation which is valid in low curvature regions of space-time and which depends on making a particular choice of polarization in the classical phase space of a single massive particle. The application of the formalism to particle creation problems is outlined.     

Spinor geometry

In this paper the construction of the geometry begins with the assignment of a spinor (spinor ether) and the coordinates x are constructed as a spinor product. It is shown that the corresponding space is a Friedmann space and the coordinates x are Friedmann coordinates. The system of gravitational and field equations is closed. The theory contains eight real functions which specify both the reference system and the coordinate grid. The theory admits quantization of space-time and is free of the difficulties associated with inertia and the absolute character of flat space-time.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 7–12, February, 1977.     

Spinor Relativity

Spinor relativity is a unified field theory, which derives gravitational and electromagnetic fields as well as a spinor field from the geometry of an eight-dimensional complex and ‘chiral’ manifold. The structure of the theory is analogous to that of general relativity: it is based on a metric with invariance group GL(ℂ²), which combines the Lorentz group with electromagnetic U(1), and the dynamics is determined by an action, which is an integral of a curvature scalar and does not contain coupling constants. The theory is related to physics on spacetime by the assumption of a symmetry-breaking ground state such that a four-dimensional submanifold with classical properties arises. In the vicinity of the ground state, the scale of which is of Planck order, the equation system of spinor relativity reduces to the usual Einstein and Maxwell equations describing gravitational and electromagnetic fields coupled to a Dirac spinor field, which satisfies a non-linear equation; an additional equation relates the electromagnetic field to the polarization of the ground state condensate.