International Journal of Geometric Methods in Modern Physics

The embedding of a manifold M into a Hilbert-space ℍ induces, via the pull-back, a tensor field on M out of the Hermitian tensor on ℍ. We propose a general procedure to compute these tensors in particular for Manifolds admitting a Lie-Group structure.

We continue to study the Chern–Simons E₈ Gauge theory of Gravity developed by the author which is a unified field theory (at the Planck scale) of a Lanczos–Lovelock Gravitational theory with a E₈ Generalized Yang–Mills (GYM) field theory, and is defined in the 15D boundary of a 16D bulk space. The Exceptional E₈ Geometry of the 256-dim slice of the 256 × 256-dimensional flat Clifford (16) space is explicitly constructed based on a spin connection , that gauges the generalized Lorentz transformations in the tangent space of the 256-dim curved slice, and the 256 × 256 components of the vielbein field , that gauge the nonabelian translations. Thus, in one-scoop, the vielbein encodes all of the 248 (nonabelian) E₈ generators and 8 additional (abelian) translations associated with the vectorial parts of the generators of the diagonal subalgebra [Cl(8) ⊗ Cl(8)]_diag ⊂ Cl(16). The generalized curvature, Ricci tensor, Ricci scalar, torsion, torsion vector and the Einstein–Hilbert–Cartan action is constructed. A preliminary analysis of how to construct a Clifford Superspace (that is far richer than ordinary superspace) based on orthogonal and symplectic Clifford algebras is presented. Finally, it is shown how an E₈ ordinary Yang–Mills in 8D, after a sequence of symmetry breaking processes E₈ → E₇ → E₆ → SO(8, 2), and performing a Kaluza–Klein–Batakis compactification on CP², involving a nontrivial torsion, leads to a (Conformal) Gravity and Yang–Mills theory based on the Standard Model in 4D. The conclusion is devoted to explaining how Conformal (super) Gravity and (super) Yang–Mills theory in any dimension can be embedded into a (super) Clifford-algebra-valued gauge field theory.

We study the class of higher-dimensional Kundt metrics admitting a covariantly constant null vector, known as CCNV spacetimes. We pay particular attention to those CCNV spacetimes with constant (polynomial) curvature invariants (CSI). We investigate the existence of an additional isometry in CCNV spacetimes, by studying the Killing equations for the general form of the CCNV metric. In particular, we list all CCNV spacetimes allowing an additional non-spacelike isometry for all values of the light-cone coordinate v, which are of interest due to the invariance of the metric under a translation in v. As an application we use our results to find all CSICCNV spacetimes with an additional isometry as well as the subset of these spacetimes in which the isometry is non-spacelike for all values v.

In this paper we are analyzing the stability of equilibria and also the existence of periodic solutions of three dimensional Hamiltonian systems using Darboux–Weinstein coordinates.

Dual-helicity eigenspinors of the charge conjugation operator (ELKO spinor fields) belong — together with Majorana spinor fields — to a wider class of spinor fields, the so-called flagpole spinor fields, corresponding to the class-(5), according to Lounesto spinor field classification based on the relations and values taken by their associated bilinear covariants. There exists only six such disjoint classes: the first three corresponding to Dirac spinor fields, and the other three respectively corresponding to flagpole, flag-dipole and Weyl spinor fields. Using the mapping from ELKO spinor fields to the three classes Dirac spinor fields, it is shown that the Einstein–Hilbert, the Einstein–Palatini, and the Holst actions can be derived from the Quadratic Spinor Lagrangian (QSL), as the prime Lagrangian for supergravity. The Holst action is related to the Ashtekar's quantum gravity formulation. To each one of these classes, there corresponds a unique kind of action for a covariant gravity theory. Furthermore we consider the necessary and sufficient conditions to map Dirac spinor fields (DSFs) to ELKO, in order to naturally extend the Standard Model to spinor fields possessing mass dimension one. As ELKO is a prime candidate to describe dark matter and can be obtained from the DSFs, via a mapping explicitly constructed that does not preserve spinor field classes, we prove that — in particular — the Einstein–Hilbert, Einstein–Palatini, and Holst actions can be derived from the QSL, as a fundamental Lagrangian for supergravity, via ELKO spinor fields. The geometric meaning of the mass dimension-transmuting operator — leading ELKO Lagrangian into the Dirac Lagrangian — is also pointed out, together with its relationship to the instanton Hopf fibration.

We present a group of transformations in the space of generalized connections that contains the set of transformations generated by the flux variables of loop quantum gravity. This group is labeled by certain SU(2)-valued functions on the bundle of directions in the spatial manifold. A further generalization is obtained by considering functions that depend on germs of analytic curves, rather than just on directions.

We study the interplays between paracontact geometry and the theory of bi-Legendrian manifolds. We interpret the bi-Legendrian connection of a bi-Legendrian manifold M as the paracontact connection of a canonical paracontact structure induced on M and then we discuss many consequences of that result both for bi-Legendrian and for paracontact manifolds, as a classification of paracontact metric structures. Finally new classes of examples of paracontact manifolds are presented.

It has been argued that, underlying any given quantum-mechanical model, there exists at least one deterministic system that reproduces, after prequantization, the given quantum dynamics. For a quantum mechanics with a complex d-dimensional Hilbert space, the Lie group SU(d) represents classical canonical transformations on the projective space ℂℙ^d-1 of quantum states. Let R stand for the Ricci flow of the manifold SU(d-1) down to one point, and let P denote the projection from the Hopf bundle onto its base ℂℙ^d-1. Then the underlying deterministic model we propose here is the Lie group SU(d), acted on by the operation PR. Finally we comment on some possible consequences that our model may have on a quantum theory of gravity.

We construct second-quantized (field) theories on coset spaces of pseudo-unitary groups U(N₊,N_-). The existence of degenerated quantum vacua (coherent states of zero modes) leads to a breakdown of the original pseudo-unitary symmetry. The action of some spontaneously broken symmetry transformations destabilize the vacuum and make it to radiate. We study the structure of this thermal radiation for curved phase spaces of constant curvature: complex projective spaces ℂP^{N - 1} = SU(N)/U(N - 1) and open complex balls ℂD^{N - 1} = SU(1,N - 1)/U(N - 1). Positive curvature is related to generalized Fermi–Dirac (FD) statistics, whereas negative curvature is connected with generalized Bose–Einstein (BE) statistics, the standard cases being recovered for N = 2. We also make some comments on the contribution of the vacuum (dark) energy to the cosmological constant and the phenomenon of inflation.

Using the idea of the degree of a smooth mapping between two manifolds of the same dimension we present here the topological (homotopical) classification of the mappings between spheres of the same dimension, vector fields, monopole and instanton solutions. Starting with a review of the elements of Riemannian geometry we also present an original elementary proof of the Gauss–Bonnet theorem and also one of the Poincaré–Hopf theorem.