International Journal of Geometric Methods in Modern Physics

It has been argued that, underlying the M-theoretic dualities, there should exist a symmetry relating the semiclassical and the strong-quantum regimes of a given action integral. On the other hand, a field-theoretic exchange between long and short distances (similar in nature to the T-duality of strings) has been shown to provide a starting point for quantum gravity, in that this exchange enforces the existence of a fundamental length scale on spacetime. In this paper, we prove that the above semiclassical vs. strong-quantum symmetry is equivalent to the exchange of long and short distances. Hence the former symmetry, as much as the latter, also enforces the existence of a length scale. We apply these facts in order to classify all possible duality groups of a given action integral on spacetime, regardless of its specific nature and of its degrees of freedom.

From the modern viewpoint and by the geometric method, this paper provides a concise foundation for the quantum theory of massless spin-3/2 field in Minkowski spacetime, which includes both the one-particle's quantum mechanics and the many-particle's quantum field theory. The explicit result presented here is useful for the investigation of spin-3/2 field in various circumstances such as supergravity, twistor programme, Casimir effect, and quantum inequality.

Hopf structure of the prototype realizations of the W₂-algebra and also N = 1 superconformal algebra are obtained using the bosonic and also fermionic Feigin–Fuchs type of free massless scalar fields in the operator product expansion (OPE) language.

The stability problem, the existence of periodic solutions, the Lax formulation and the numerical integration via the Lie–Trotter integrator of the Kowalevski top dynamics are discussed, and some of their properties are pointed out.

The usual action of the Yang–Mills theory is given by the quadratic form of curvatures of a principal G bundle defined on four-dimensional manifolds. The nonlinear generalization which is known as the Born–Infeld action has been given. In this paper we give another nonlinear generalization on four-dimensional manifolds and call it a universal Yang–Mills action. The advantage of our model is that the action splits automatically into two parts consisting of self-dual and anti-self-dual directions, that is, we have automatically the self-dual and anti-self-dual equations without solving the equations of motion as in usual case. Our method may be applicable to recent non-commutative Yang–Mills theories studied widely.

It is known that superpotentials of so-called higher-order theories of gravitation and Einstein theories with a cosmological constant are different in the "Palatini" formulation. However, these theories share a number of solutions, the energy of which should be defined out of superpotentials. We shall check that using augmented variational principles, recently introduced by us, the energy of a solution is defined regardless of the theory, and hence regardless of the superpotential used.

The eigenvalue problem for the square integrable solutions is studied usually for elliptic equations. In this paper we consider such a problem for the hyperbolic Klein–Gordon equation on Lorentzian manifolds. The investigation could help to answer the question why elementary particles have a discrete mass spectrum. An infinite family of square integrable solutions for the Klein–Gordon equation on the Friedman type manifolds is constructed. These solutions have a discrete mass spectrum and a finite action. In particular the solutions on de Sitter space are investigated.

ℤ₂-gradings of Clifford algebras are reviewed and we shall be concerned with an α-grading based on the structure of inner automorphisms, which is closely related to the spacetime splitting, if we consider the standard conjugation map automorphism by an arbitrary, but fixed, splitting vector. After briefly sketching the orthogonal and parallel components of products of differential forms, where we introduce the parallel [orthogonal] part as the space [time] component, we provide a detailed exposition of the Dirac operator splitting and we show how the differential operator parallel and orthogonal components are related to the Lie derivative along the splitting vector and the angular momentum splitting bivector. We also introduce multivectorial-induced α-gradings and present the Dirac equation in terms of the spacetime splitting, where the Dirac spinor field is shown to be a direct sum of two quaternions. We point out some possible physical applications of the formalism developed.

Some aspects of the geometry of (1, 1) tensor fields T with vanishing Nijenhuis torsion are discussed; in particular, the cohomolgy of the exterior differential associated with T is analyzed and some geometric properties of the shape operator are derived.

Many problems of physics or biology involve very irregular objects like the rugged surface of a malignant cell nucleus or the structure of space-time at the atomic scale. We define and study non-differentiable deformations of the classical Cartesian space ℝⁿ which can be viewed as the basic bricks to construct irregular objects. They are obtained by taking the topological product of n-graphs of nowhere differentiable real valued functions. Our point of view is to replace the study of a non-differentiable function by the dynamical study of a one-parameter family of smooth regularization of this function. In particular, this allows us to construct a one-parameter family of smooth coordinates systems on non-differentiable deformations of ℝⁿ, which depend on the smoothing parameter via an explicit differential equation called a scale law. Deformations of ℝⁿ are examples of a new class of geometrical objects called scale manifolds which are defined in this paper. As an application, we derive rigorously the main results of the scale-relativity theory developed by Nottale in the framework of a scale space-time manifold.

The Hamilton–Jacobi problem is revisited bearing in mind the consequences arising from a possible bi-Hamiltonian structure. The problem is formulated on the tangent bundle for Lagrangian systems in order to avoid the bias of the existence of a natural symplectic structure on the cotangent bundle. First it is developed for systems described by regular Lagrangians and then extended to systems described by singular Lagrangians with no secondary constraints. We also consider the example of the free relativistic particle, the rigid body and the electron–monopole system.