International Journal of Geometric Methods in Modern Physics

The main purpose of this paper is to study some properties of bicomplex-holomorphic 4-manifolds endowed with a holomorphic double Walker–Norden metric.

In this study, Fermi–Walker derivative, Fermi–Walker parallelism, non-rotating frame, Fermi–Walker termed Darboux vector concepts are given for Lie groups in E⁴. First, we get any Frénet curve and any vector field along the Frénet curve in a Lie group. Then, the Fermi–Walker derivative is defined for the Lie group. Fermi–Walker derivative and Fermi–Walker parallelism are analyzed in Lie groups. Finally, the necessary conditions for Fermi–Walker parallelism are explained.

We present the Dirac equation in a geometry with torsion and non-metricity balancing generality and simplicity as much as possible. In doing so, we use the vielbein formalism and the Clifford algebra. We also use an index-free formalism which allows us to construct objects that are totally invariant. It turns out that the previous apparatuses not only make possible a simple deduction of the Dirac equation but also allow us to exhibit some details that is generally obscure in the literature.

This study proposes to improve the Lagrangian energy equations for instructed complex jet bundles on Kähler manifolds. The coordinates on the bundle structure of Kähler manifolds have been given for real and imaginary dimensions. For given bundle structures, all fundamental geometrical properties have been investigated and applications to complex bundle structures are carried out. The energy equations have been applied to the numerical example in order to test its performance. Moreover, velocity and time dimensions for energy movement equations have been presented as a new concept. This study shows some physical applications of those equations and interpretations are made. Results show that the imaginary coordinates are same as real coordinates. One of the interesting conclusion of this study is that the Lagrangian energy of a movement particle is static when the particle moves in a large velocity.

Although the standard generally covariant Dirac equation is unique in a topologically simple spacetime, it has been shown that it leads to non-uniqueness problems for the Hamiltonian and energy operators, including the non-uniqueness of the energy spectrum. These problems should be solved by restricting the choice of the Dirac gamma field in a consistent way. Recently, we proposed to impose the value of the rotation rate of the tetrad field. This is not necessarily easy to implement and works only in a given reference frame. Here, we propose that the gamma field should change only by constant gauge transformations. To get that situation, we are naturally led to assume that the metric can be put in a space-isotropic diagonal form. When this is the case, it distinguishes a preferred reference frame. We show that by defining the gamma field from the "diagonal tetrad" in a chart in which the metric has that form, the uniqueness problems are solved at once for all reference frames. We discuss the physical relevance of the metric considered and our restriction to first-quantized theory.

In this paper, we generalize Schwinger realization of the 𝔰𝔲(2) algebra to construct a two-mode realization for deformed 𝔰𝔲(2) algebra on a sphere. We obtain a nonlinear (f-deformed) Schwinger realization with a deformation function corresponding to the curvature of sphere that in the flat limit tends to unity. With the use of this nonlinear two-mode algebra, we construct the associated two-mode coherent states (CSs) on the sphere and investigate their quantum entanglement. We also compare the quantum statistical properties of the two modes of the constructed CSs, including anticorrelation and antibunching effects. Particularly, the influence of the curvature of the physical space on the nonclassical properties of two modes is clarified.

The aim of this paper is to construct and investigate a Finsler structure within the framework of a Generalized Absolute Parallelism (GAP)-space. The Finsler structure is obtained from the vector fields forming the parallelization of the GAP-space. The resulting space, which we refer to as a Finslerized absolute parallelism (parallelizable) space, combines within its geometric structure the simplicity of GAP-geometry and the richness of Finsler geometry, hence is potentially more suitable for applications and especially for describing physical phenomena. A study of the geometry of the two structures and their interrelation is carried out. Five connections are introduced and their torsion and curvature tensors derived. Some special Finslerized parallelizable spaces are singled out. One of the main reasons to introduce this new space is that both absolute parallelism and Finsler geometries have proved effective in the formulation of physical theories, so it is worthy to try to build a more general geometric structure that would share the benefits of both geometries.

A pure Dirac's canonical analysis, defined in the full phase space for the Husain–Kuchar (HK) model is discussed in detail. This approach allows us to determine the extended action, the extended Hamiltonian, the complete constraint algebra and the gauge transformations for all variables that occur in the action principle. The complete set of constraints defined on the full phase space allow us to calculate the Dirac algebra structure of the theory and a local weighted measure for the path integral quantization method. Finally, we discuss briefly the necessary mathematical structure to perform the canonical quantization program within the framework of the loop quantum gravity approach.

Special Lagrangian sub-manifolds of Calabi–Yau (CY) 3-folds are used to describe membrane instanton solutions of N = 1, 11-dimensional super-gravity theories. Super-symmetry is the essential ingredient that relates super-gravity branes to special Lagrangian sub-manifolds [K. Becker, M. Becker and A. Strominger, Fivebranes, membranes and non-perturbative string theory, Nucl. Phys. B456(1–2) (1995) 135–152]. In this note we would like to explain this relation, more explicitly and with more details than in the current literature.

We present an embedding of the Tsallis entropy into the three-dimensional Heisenberg group, in order to understand the meaning of generalized independence as encoded in the Tsallis entropy composition property. We infer that the Tsallis entropy composition induces fractal properties on the underlying Euclidean space. Using a theorem of Milnor/Wolf/Tits/Gromov, we justify why the underlying configuration/phase space of systems described by the Tsallis entropy has polynomial growth for both discrete and Riemannian cases. We provide a geometric framework that elucidates Abe's formula for the Tsallis entropy, in terms the Pansu derivative of a map between sub-Riemannian spaces.

Banach Dirac bundles are introduced within the framework of infinite-dimensional Banach manifolds and their basic properties are derived. Banach Lie algebroids are geometrical objects recently introduced by the first author. It is the main goal of the present paper to reveal the most important links between Banach Dirac bundles and Banach Lie algebroids, respectively.

We study the pseudoduality transformations in two-dimensional N = (2, 2) sigma models on Kähler manifolds. We show that structures on the target space can be transformed into the pseudodual manifolds by means of (anti)holomorphic preserving mapping. This map requires that torsions related to individual spaces and riemann connection on pseudodual manifold must vanish. We also consider holomorphic isometries which puts additional constraints on the pseudoduality.

We study the renormalization group flow for the Higgs self-coupling in the presence of gravitational correction terms. We show that the resulting equation is equivalent to a singular linear ODE, which has explicit solutions in terms of hypergeometric functions. We discuss the implications of this model with gravitational corrections on the Higgs mass estimates in particle physics models based on the spectral action functional.

The JLO character formula due to Jaffe–Lesniewski–Osterwalder [Quantum K-theory: the Chern character, Commun. Math. Phys.112 (1988) 75–88] assigns to each Fredholm module a cocycle in entire cyclic cohomology. It descends to define a cohomological Chern character on K-homology. This paper extends the definition of the JLO character formula for Breuer–Fredholm modules, the modules that represent type II noncommutative geometry; and shows that the JLO character formula coincides with the Connes character formula [see M. Benameur and T. Fack, Type II noncommutative geometry. I. Dixmier trace in von Neumann algebras, Adv. Math.199 (2006) 29–87] at the level of entire cyclic cohomology.