International Journal of Geometric Methods in Modern Physics

Starting from a short review of spaces of generalized sections of vector bundles, we give a concise systematic description, in precise geometric terms, of Leray densities, principal value densities, propagators and elementary solutions of field equations in flat spacetime. We then sketch a partly original geometric presentation of free quantum fields and show how propagators arise from their graded commutators in the boson and fermion cases.

Let $M$ be a Riemannian manifold and ${𝒫}M$ be the space of all smooth paths on $M$. We describe geodesics on path space ${𝒫}M$. Normal neighborhoods on ${𝒫}M$ have been discussed. We identify paths on $M$ under “back-track” equivalence. Under this identification, we show that if $M$ is complete, then geodesics on the path space yield a double category. This double category has a natural interpretation in terms of the worldsheets generated by freely moving (without any external force) strings.

The main purpose of this paper is to present a new approach to logic or what we will call superlogic. This approach constitutes a new way of looking at the connection between quantum mechanics and logic. It is a geometrization of the quantum logic. Note that this superlogic is not distributive reflecting a good propriety to describe quantum mechanics, non-commutative spaces and contains a nilpotent element.

We consider the $𝔞𝔣𝔣(1)$-module structure on the spaces of differential operators acting on the spaces of weighted densities. We compute the first differential cohomology of the Lie superalgebra $𝔞𝔣𝔣(1)$ with coefficients in differential operators acting on the spaces of weighted densities. We study also the super analogue of this problem getting the same results.

A quick review on the analytical aspects of holographic superconductors (HSCs) with Weyl corrections has been presented. Mainly, we focus on matching method and variational approaches. Different types of such HSC have been investigated — s-wave, p-wave and Stúckelberg ones. We also review the fundamental construction of a p-wave type, in which the non-Abelian gauge field is coupled to the Weyl tensor. The results are compared from numerics to analytical results.

Within the geometrical framework developed in [Geometric constrained variational calculus. I: Piecewise smooth extremals, Int. J. Geom. Methods Mod. Phys. 12 (2015) 1550061], the problem of minimality for constrained calculus of variations is analyzed among the class of differentiable curves. A fully covariant representation of the second variation of the action functional, based on a suitable gauge transformation of the Lagrangian, is explicitly worked out. Both necessary and sufficient conditions for minimality are proved, and reinterpreted in terms of Jacobi fields.

It is described how the Extended Relativity Theory in $C$-spaces (Clifford spaces) allows a unified formulation of point particles, strings, membranes and $p$-branes, moving in ordinary target spacetime backgrounds, within the description of a single polyparticle moving in $C$-spaces. The degrees of freedom of the latter are provided by Clifford polyvector-valued coordinates (antisymmetric tensorial coordinates). A correspondence between the $p$-brane ($p$-loop) “Schrödinger-like” equations of Ansoldi–Aurilia–Spallucci and the polyparticle wave equation in $C$-spaces is found via the polyparticle/$p$-brane correspondence. This correspondence might provide another unexplored avenue to quantize $p$-branes (a notoriously difficult and unsolved problem) from the more straightforward quantization of the polyparticle in $C$-spaces, even in the presence of external interactions. We conclude with comments about the compositeness nature of the polyvector-valued coordinate operators in terms of ordinary $p$-brane coordinates via the evaluation of $n$-ary commutators.

Integrable Heisenberg ferromagnetic equations are an important subclass of integrable systems. The M-XCIX equation is one of a generalizations of the Heisenberg ferromagnetic equation and are integrable. In this paper, the Darboux transformation of the M-XCIX equation is constructed. Using the DT, a 1-soliton solution of the M-XCIX equation is presented.

We prove that warped product manifolds with $p$-dimensional base, $p=1,2,$ satisfy some pseudosymmetry type curvature conditions. These conditions are formed from the metric tensor $g$, the Riemann–Christoffel curvature tensor $R$, the Ricci tensor $S$ and the Weyl conformal curvature $C$ of the considered manifolds. The main result of the paper states that if $p=2$ and the fiber is a semi-Riemannian space of constant curvature (when $n$ is greater or equal to 5) then the $(0,6)$-tensors $R\cdot R-Q(S,R)$ and $C\cdot C$ of such warped products are proportional to the $(0,6)$-tensor $Q(g,C)$ and the tensor $C$ is a linear combination of some Kulkarni–Nomizu products formed from the tensors $g$ and $S$. We also obtain curvature properties of this kind of quasi-Einstein and 2-quasi-Einstein manifolds, and in particular, of the Goedel metric, generalized spherically symmetric metrics and generalized Vaidya metrics.

A semiclassical formulation of the spin Hall effect for physical systems satisfying Dirac-like equation is introduced. We demonstrate that the main contribution to the spin Hall conductivity is given by the spin Chern number whether the spin is conserved or not at the quantum level. We illustrated the formulation within the Kane–Mele model of graphene in the absence and in the presence of the Rashba spin-orbit coupling term.

One problem that appears in theories with compact extra dimensions is the shadowing phenomenon, which makes it difficult to identify the topology and the geometry of the extra-dimensional manifold, if it exists. In this paper, we address this problem from the perspective of Newton’s law modifications caused by compact extra dimensions. After providing the modifications cause by some of the most phenomenologically interesting compact manifolds, we qualitatively study and compare the modifications that these manifolds cause to Newton’s law. As we demonstrate, the shadowing phenomenon persists in this approach too, and it seems that the shadowing phenomenon is grouped in “shadowing zones”. At these points, it is impossible to distinguish which compact space causes the modification of Newton’s law. However, away from the shadowing zones, the effects of the different compact manifolds are distinguishable. It is possible that these effects can be observed in the next generation of Newton’s law experiments.

We construct the gravitational energy–momentum tensor in general relativity through the Noether theorem. In particular, we explicitly demonstrate that the constructed quantity can vary as a tensor under the general coordinate transformation. Furthermore, we verify that the energy–momentum conservation is satisfied because one of the two indices of the energy–momentum tensor should be in the local Lorentz frame. It is also shown that the gravitational energy and the matter one cancel out in certain space-times.

The ultrarelativistic boosting procedure had been applied in the literature to map the metric of Schwarzschild–de Sitter spacetime into a metric describing de Sitter spacetime plus a shock-wave singularity located on a null hypersurface. This paper evaluates the Riemann curvature tensor of the boosted Schwarzschild–de Sitter metric by means of numerical calculations, which make it possible to reach the ultrarelativistic regime gradually by letting the boost velocity approach the speed of light. Thus, for the first time in the literature, the singular limit of curvature, through Dirac’s $\delta$ distribution and its derivatives, is numerically evaluated for this class of spacetimes. Moreover, the analysis of the Kretschmann invariant and the geodesic equation shows that the spacetime possesses a “scalar curvature singularity” within a 3-sphere and it is possible to define what we here call “boosted horizon”, a sort of elastic wall where all particles are surprisingly pushed away, as numerical analysis demonstrates. This seems to suggest that such “boosted geometries” are ruled by a sort of “antigravity effect” since all geodesics seem to refuse to enter the “boosted horizon” and are “reflected” by it, even though their initial conditions are aimed at driving the particles toward the “boosted horizon” itself. Eventually, the equivalence with the coordinate shift method is invoked in order to demonstrate that all ${\delta }^2$ terms appearing in the Riemann curvature tensor give vanishing contribution in distributional sense.

In this paper, we establish a general Kastler–Kalau–Walze type theorem for any dimensional manifolds with boundary which generalizes the results in [Y. Wang, Lower-dimensional volumes and Kastler–Kalau–Walze type theorem for manifolds with boundary, Commun. Theor. Phys. 54 (2010) 38–42]. This solves a problem of the referee of [J. Wang and Y. Wang, A Kastler–Kalau–Walze type theorem for five-dimensional manifolds with boundary, Int. J. Geom. Meth. Mod. Phys. 12(5) (2015), Article ID: 1550064, 34 pp.], which is a general expression of the lower dimensional volumes in terms of the geometric data on the manifold.

It is commonly claimed in the recent literature that certain solutions to wave equations of positive energy of Dirac-type with internal variables are characterized by a non-thermal spectrum. As part of that statement, it was said that the transformations and symmetries involved in equations of such type corresponded to a particular representation of the Lorentz group. In this paper, we give the general solution to this problem emphasizing the interplay between the group structure, the corresponding algebra and the physical spectrum. This analysis is completed with a strong discussion and proving that: (i) the physical states are represented by coherent states; (ii) the solutions in [Yu. P. Stepanovsky, Nucl. Phys. B (Proc. Suppl.) 102 (2001) 407–411; 103 (2001) 407–411] are not general, (iii) the symmetries of the considered physical system in [Yu. P. Stepanovsky, Nucl. Phys. B (Proc. Suppl.) 102 (2001) 407–411; 103 (2001) 407–411] (equations and geometry) do not correspond to the Lorentz group but to the fourth covering: the Metaplectic group $Mp(n)$.

We use the idea of the symmetry between the spacetime coordinates $x^{\mu }$ and the energy–momentum $p^{\mu }$ in quantum theory to construct a momentum space quantum gravity geometry with a metric $s_{\mu \nu }$ and a curvature tensor ${P^{\lambda }}_{\mu \nu \rho }$. For a closed maximally symmetric momentum space with a constant 3-curvature, the volume of the $p$-space admits a cutoff with an invariant maximum momentum $a$. A Wheeler–DeWitt-type wave equation is obtained in the momentum space representation. The vacuum energy density and the self-energy of a charged particle are shown to be finite, and modifications of the electromagnetic radiation density and the entropy density of a system of particles occur for high frequencies.