International Journal of Geometric Methods in Modern Physics

We discuss some dynamical and geometrical properties of an autonomous underwater vehicle with a fixed control authority.

Let Λ ⊂ ℂ be the ℤ-module generated by 1 and , where τ is a positive real number. Let Z := ℂ/Λ be the corresponding complex torus of dimension one. Our aim here is to give a general construction of holomorphic principal Z-bundles over a complex manifold X.

Let θ₁ and θ₂ be two C^∞ real closed two-forms on X such that the Hodge type (0, 2) component of the form vanishes, and the elements in H²(X, ℂ) represented by θ₁ and θ₂ are contained in the image of H²(X, ℤ). For such a pair we construct a holomorphic principal Z-bundle over X. Conversely, given any holomorphic principal Z-bundle E_Z over X, we construct a pair of closed differential forms on X of the above type.

Fractional calculus has recently attracted considerable attention. In particular, various fractional differential equations are used to model nonlinear wave theory that arises in many different areas of physics such as Josephson junction theory, field theory, theory of lattices, etc. Thus one may expect fractional calculus, in particular fractional differential equations, plays an important role in quantum field theories which are expected to satisfy fractional generalization of Klein–Gordon and Dirac equations. Until now, in high-energy physics and quantum field theories the derivative operator has only been used in integer steps. In this paper, we want to extend the idea of differentiation to arbitrary non-integers steps. We will address multi-dimensional fractional action-like problems of the calculus of variations where fractional field theories and fractional differential Dirac operators are constructed.

In this paper we give some examples of almost para-hyperhermitian structures on the tangent bundle of an almost product manifold, on the product manifold M × ℝ, where M is a manifold endowed with a mixed 3-structure and on the circle bundle over a manifold with a mixed 3-structure.

We analyze the role played by local translational symmetry in the context of gauge theories of fundamental interactions. Translational connections and fields are introduced, with special attention being paid to their universal coupling to other variables, as well as to their contributions to field equations and to conserved quantities.

In this paper, we consider a dynamical system related to the Yang–Mills system for a field with gauge group SU(2). We solve this system in terms of genus two hyperelliptic functions. The corresponding invariant surface defined by the two constants of motion can be completed as a cyclic double cover of an abelian surface (the jacobian of a genus 2 curve) and we show that this system is algebraic completely integrable in the generalized sense. Also we show that this system is part of an algebraic completely integrable system in five unknowns having three constants of motion.

Building on a variation of 't Hooft's harmonic function ansatz for SU(2) instantons on ℝ⁴, we provide new explicit nonsingular solutions of the Yang–Mills anti-self-duality equations on Euclidean spacetime with gauge group SL(2, ℂ) and SL(3, ℝ).

We investigate canonical aspects concerning the relation between symmetries and conservation laws in gauge-natural field theories. In particular, we find that a canonical spinor connection can be selected by the simple requirement of the global existence of canonical superpotentials for the Lagrangian describing the coupling of gravitational and Fermionic fields. In fact, the naturality of a suitably defined variational Lagragian implies the existence of an associated energy-momentum conserved current. Such a current defines a Hamiltonian form in the corresponding phase space; we show that an associated Hamiltonian connection is canonically defined along the kernel of the generalized gauge-natural Jacobi morphism and uniquely characterizes the canonical spinor connection.

In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schrödinger framework from this perspective and provide a description of the Weyl–Wigner construction. Finally, after reviewing the basics of the geometric formulation of quantum mechanics, we apply the methods presented to the most interesting cases of finite dimensional Hilbert spaces: those of two, three and four level systems (one qubit, one qutrit and two qubit systems). As a more practical application, we discuss the advantages that the geometric formulation of quantum mechanics can provide us with in the study of situations as the functional independence of entanglement witnesses.