International Journal of Geometric Methods in Modern Physics

It is shown that the equations of relativistic Bohmian mechanics for multiple bosonic particles have a dual description in terms of a classical theory of conformally "curved" space-time. This shows that it is possible to formulate quantum mechanics as a purely classical geometrical theory. The results are further generalized to interactions with an external electromagnetic field.

Using the almost product structure given by Druta, we introduce a metrical framed f(3, -1)-structure on the tangent bundle of a Riemannian manifold. Then by restricting this metrical framed f(3, -1)-structure to the tangent sphere bundle, we obtain an almost metrical paracontact structure on the tangent sphere bundle.

A description of the space of G-connections using the tangent groupoid is given. As the tangent groupoid parameter is away from zero, the G-connections act as convolution operators on a Hilbert space. The gauge action is examined in the tangent groupoid description of the G-connections. Tetrads are formulated as Dirac type operators. The connection variables and tetrad variables in Ashtekar's gravity are presented as operators on a Hilbert space.

In the present paper we obtain, in a covariant form, and in their full generality, the field equations in a relativistic general Kaluza–Klein space. This is done by using the Riemannian horizontal connection defined in [3], and some 4D horizontal tensor fields, as for instance: horizontal Ricci tensor, horizontal Einstein gravitational tensor field, horizontal electromagnetic energy–momentum tensor field, etc. Also, we present some inter-relations between STM theory and brane-world theory. This enables us to introduce in brane theory some electromagnetic potentials constructed by means of the warp function.

We continue with the study of Clifford-space (C-space) gravity and analyze further the C-space generalized gravitational field equations which are obtained from a variational principle based on the generalization of the Einstein–Hilbert–Cartan action. One of the main features is that the C-space connection requires torsion in order to have consistency with the Clifford algebraic structure associated with the curved C-space basis generators. Hence no spin matter is required to induce torsion since it already exists in the vacuum. The field equations in C-spaces associated to a Clifford algebra in D dimensions are not equivalent to the ordinary gravitational equations with torsion in higher 2^D dimensions. The most physically relevant conclusion, besides the presence of torsion in the vacuum, is the contribution of the higher grade metric components g^{μ₁μ₂ν₁ν₂}, g^{μ₁μ₂μ₃ν₁ν₂ν₃},… of the C-space metric to dark energy/dark matter.

We consider states defined on non-trivial topologies of Torus, Mobius and Torus–Mobius. Adequate operators leading to the construction of coherent states, two-mode coherent states and entangled states are derived and we show a class of entangled states on Torus that are related to the entanglement of photons by orbital angular momentum states.

In this paper, some possible candidates for the study of gravity are proposed in terms of complex Finsler geometry. These mainly concern the complex Hermitian versions of weakly gravitational metric and Schwarzschild metric. For the weakly gravitational fields, we state few interesting geometrical and physical aspects such as the conditions under which a complex Finsler metrics are projectively related to the weakly gravitational metric. In the Kähler case, the geodesic curves of the weakly gravitational metric are obtained. Some applications concerning the deformations of the weakly gravitational Hermitian metric to a complex Randers metric are described. Another candidate for gravity is given by so-called Hermitian Schwarzschild metric for which some geodesic curves are highlighted. The last part of the paper is devoted to a generalization of the complex Klein–Gordon equations, in terms of Quantum field theory on a curved space.

Families of objects appear in several contexts, like algebraic topology, theory of deformations, theoretical physics, etc. An unified coordinate-free algebraic framework for families of geometrical quantities is presented here, which allows one to work without introducing ad hoc spaces, by using the language of differential calculus over commutative algebras. An advantage of such an approach, based on the notion of sliceable structures on cylinders, is that the fundamental theorems of standard calculus are straightforwardly generalized to the context of families. As an example of that, we prove the universal homotopy formula.

In this paper, we start from an extension of the notion of holonomy on diffeological bundles, reformulate the notion of regular Lie group or Frölicher Lie groups, state an Ambrose–Singer theorem that enlarges the one stated in [J.-P. Magnot, Structure groups and holonomy in infinite dimensions, Bull. Sci. Math.128 (2004) 513–529], and conclude with a differential geometric treatment of KP hierarchy. The examples of Lie groups that are studied are principally those obtained by enlarging some graded Frölicher (Lie) algebras such as formal q-series of the quantum algebra of pseudo-differential operators. These deformations can be defined for classical pseudo-differential operators but they are used here on formal pseudo-differential operators in order to get a differential geometric framework to deal with the KP hierarchy that is known to be completely integrable with formal power series. Here, we get an integration of the Zakharov–Shabat connection form by means of smooth sections of a (differential geometric) bundle with structure group, some groups of q-deformed operators. The integration obtained by Mulase [Complete integrability of the Kadomtsev–Petviashvili equation Adv. Math.54 (1984) 57–66], and the key tools he developed, are totally recovered on the germs of the smooth maps of our construction. The tool coming from (classical) differential geometry used in this construction is the holonomy group, on which we have an Ambrose–Singer-like theorem: the Lie algebra is spanned by the curvature elements. This result is proved for any connection a diffeological principal bundle with structure group a regular Frölicher Lie group. The case of a (classical) Lie group modeled on a complete locally convex topological vector space is also recovered and the work developed in [J.-P. Magnot, Difféologie du fibré d'Holonomie en dimension infinie, Math. Rep. Canadian Roy. Math. Soc.28(4) (2006); J.-P. Magnot, Structure groups and holonomy in infinite dimensions, Bull. Sci. Math. 128 (2004) 513–529] is completed.

The Riemann curvature in Riemann–Finsler geometry can be regarded as a collection of linear operators on the tangent spaces. The algebraic properties of these operators may be linked to the geometry and the topology of the underlying space. The principal curvatures of a Finsler space (M, F) at a point x are the eigenvalues of the Riemann curvature operator at x. They are real functions κ on the slit tangent manifold TM₀. A principal curvature κ(x, y) is said to be isotropic (respectively, quadratic) if κ(x, y)/F(x, y) is a function of x only (respectively, κ(x, y) is quadratic with respect to y). On the other hand, the Randers metrics are the most popular and prominent metrics in pure and applied disciplines. Here, it is proved that if a Randers metric admits an isotropic principal curvature, then F is of isotropic S-curvature. The same result is also established for F to admit a quadratic principal curvature. These results extend Shen's verbal results about Randers metrics of scalar flag curvature K = K(x) as well as those Randers metrics with quadratic Riemann curvature operator. The Riemann curvature may be broken into two operators and . The isotropic and quadratic principal curvature are characterized in terms of the eigenvalues of and .

In this paper, a complete classification of completely irreducible closed linear bundles of Lie algebras is given. We also apply the so-called "Raïs formula" to compute the index of each of the Lie bundles of square matrices.

We review the theory of nonlocal symmetries of nonlinear partial differential equations and, as examples, we present infinite-dimensional Lie algebras of nonlocal symmetries of the Fokas–Qiao and Kaup–Kupershmidt equations. Then, we consider nonlocal symmetries of a family which contains the Korteweg–de Vries (KdV) and (a subclass of) the Rosenau–Hyman compacton-bearing K(m, n) equations. We find that the only member of the family which possesses nonlocal symmetries (of a kind specified in Sec. 3 below) is precisely the KdV equation. We take this fact as an indication that the K(m, n) equations are not integrable in general, and we use the formal symmetry approach of Shabat to check this claim: we prove that the only integrable cases of the full K(m, n) family are the KdV and modified KdV equations.

This work concerns the definition and analysis of a new class of Lie systems on Poisson manifolds enjoying rich geometric features: the Lie–Hamilton systems. We devise methods to study their superposition rules, time independent constants of motion and Lie symmetries, linearizability conditions, etc. Our results are illustrated by examples of physical and mathematical interest.