International Journal of Geometric Methods in Modern Physics

We review the current status of Finsler–Lagrange geometry and generalizations. The goal is to aid non-experts on Finsler spaces, but physicists and geometers skilled in general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. We also would like to orient mathematicians working in generalized Finsler and Kähler geometry and geometric mechanics how they could perform their results in order to be accepted by the community of "orthodox" physicists.

Although the bulk of former models of Finsler–Lagrange spaces where elaborated on tangent bundles, the surprising result advocated in our works is that such locally anisotropic structures can be modeled equivalently on Riemann–Cartan spaces, even as exact solutions in Einstein and/or string gravity, if nonholonomic distributions and moving frames of references are introduced into consideration.

We also propose a canonical scheme when geometrical objects on a (pseudo) Riemannian space are nonholonomically deformed into generalized Lagrange, or Finsler, configurations on the same manifold. Such canonical transforms are defined by the coefficients of a prime metric and generate target spaces as Lagrange structures, their models of almost Hermitian/Kähler, or nonholonomic Riemann spaces.

Finally, we consider some classes of exact solutions in string and Einstein gravity modeling Lagrange–Finsler structures with solitonic pp-waves and speculate on their physical meaning.

The most general operator of D-differentiation is proved to be expressible as a combination of covariant differentiation and a tensor field. The torsion, the curvature and the non-metricity of an arbitrary D-operator are consequently given in terms of those of covariant differentiation. The scalar curvature is also obtained.

We discuss some geometrical and dynamical properties of Rabinovich systems.

The contour argument was introduced by Peierls for two dimensional Ising model. Peierls benefited from the particular symmetries of the Ising model. For non-symmetric models the argument was developed by Pirogov and Sinai. It is very general and rather difficult. Intuitively clear that the Peierls argument does work for any symmetric model. But contours defined in Pirogov–Sinai theory do not work if one wants to use Peierls argument for more general symmetric models. We give a new definition of contour which allows relatively easier proof to the main result of the Pirogov–Sinai theory for symmetric models. Namely, our contours allow us to apply the classical Peierls argument (with contour removal operation).

In certain neighborhood U of an arbitrary point of a symplectic manifold M we construct a Fedosov-type star-product *_L such that for an arbitrary leaf ℘ of a given polarization the vanishing ideal of ℘ ∩ U in the commutative algebra C^∞(U)[[h]] is a left ideal in the deformed algebra (C^∞(U)[[h]],*_L). With certain additional assumptions on M, *_L becomes a so-called star-product with separation of variables.

Three-dimensional Lorentzian manifolds with commuting curvature operators are studied. A complete description is given at the algebraic level. Consequences are obtained at the differentiable setting for manifolds which additionally are assumed to be locally symmetric or homogeneous.

We present an account of negative differential forms within a natural algebraic framework of differential graded algebras, and explain their relationship with forms on path spaces.

The differential forms on the jet bundle J^∞E of a bundle E → M over a compact n-manifold M of degree greater than n determine differential forms on the space Γ(E) of sections of E. The forms obtained in this way are called local forms on Γ(E), and its cohomology is called the local cohomology of Γ(E). More generally, if a group acts on E, we can define the local -invariant cohomology. The local cohomology is computed in terms of the cohomology of the jet bundle by means of the variational bicomplex theory. A similar result is obtained for the local -invariant cohomology. Using these results and the techniques for the computation of the cohomology of invariant variational bicomplexes in terms of relative Gelfand–Fuchs cohomology introduced in [4], we construct non trivial local cohomology classes in the important cases of Riemannian metrics with the action of diffeomorphisms, and connections on a principal bundle with the action of automorphisms.

We consider an action of the group of curves in GL(2,ℝ) on the set of linear systems and therefore on the set of Schrödinger equations in full similarity with the action of the group of curves in SL(2,ℝ) on the set of Riccati equations considered in previous articles. We also consider the transformations defined by a first-order differential expression which carry solutions of a Schrödinger equation into solutions of another one. We find then two non-trivial situations: transformations which can be described by the previous transformation group, generalizing previous work by us, and transformations which are singular. We show that both situations appear, e.g., in the usual problem of partner Hamiltonians in quantum mechanics.

We show that the difference Bäcklund algorithm, both in the finite and confluent versions, can be understood in terms of the above mentioned transformation group, the case of two exactly equal factorization energies being an instance of the singular case. We apply the generalized theorem relating three eigenfunctions of three different Hamiltonians to the generation of new potentials with a known (excited state) eigenfunction, starting from potentials of Coulomb, Morse and Rosen–Morse type. The potentials found are new and non-trivial.

Recently, black hole and brane production at CERN's Large Hadron Collider (LHC) has been widely discussed. We suggest that there is a possibility to test causality at the LHC. We argue that if the scale of quantum gravity is of the order of few TeVs, proton-proton collisions at the LHC could lead to the formation of time machines (spacetime regions with closed timelike curves) which violate causality. One model for the time machine is a traversable wormhole. We argue that the traversable wormhole production cross section at the LHC is of the same order as the cross section for the black hole production. Traversable wormholes assume violation of the null energy condition (NEC) and an exotic matter similar to the dark energy is required. Decay of the wormholes/time machines and signatures of time machine events at the LHC are discussed.