A classification of space-time structures

The paper contains a review of various bundles which may be associated to the bundle of linear frames and used to describe properties of space relevant to physics. Restrictions, extensions, prolongations and reductions are defined in terms of morphisms of principal bundles. It is shown that the holonomic prolongation of a G-structure exist iff the corresponding structure function vanishes. G-connections are related to restrictions of the bundle of second-order frames. It is shown that these restrictions may be used to classify theories of space-time and gravitation. A distinction is made between a projective connection and a geodetic structure. In the framework of the Einstein-Cartan theory, the projective connection of a space-time is compatible with its metric tensor iff the spin density is bivector-valued. As an example, we mention a new theory of gravitation and electromagnetism based on the Weyl-Cartan structure of space-time and on the Yang quadratic Lagrangian.     

Background Independent Quantum Mechanics, Classical Geometric Forms and Geometric Quantum Mechanics—I

The geometry of the symplectic structures and Fubini-Study metric is discussed. Discussion in the paper addresses geometry of Quantum Mechanics in the classical phase space. Also, geometry of Quantum Mechanics in the projective Hilbert space has been discussed for the chosen Quantum states. Since the theory of classical gravity is basically geometric in nature and Quantum Mechanics is in no way devoid of geometry, the explorations pertaining to more and more geometry in Quantum Mechanics could prove to be valuable for larger objectives such as understanding of gravity.     

Projective spinor geometry and prespace

A method originally conceived by Bohm for abstracting key features of the metric geometry from an underlying spinor ordering is generalized to the projective geometry. This allows the introduction of the spinor into a projective context and the definition of an associated geometric algebra. The projective spinor may then be regarded as defining a pregeometry for the projective space.This article is dedicated to Professor David Bohm on the occasion of his 70th birthday. I would like to take this opportunity to offer him my warmest congratulations and also to express to him my sincerest appreciation for this kindness, guidance, and inspiration over many years.     

Noncommutative coordinate picture of the quantum phase space

We illustrate an isomorphic representation of the observable algebra for quantum mechanics in terms of the functions on the projective Hilbert space, and its Hilbert space analog, with a noncommutative product in terms of explicit coordinates and discuss the physical and dynamical picture. The isomorphism is then used as a base for the translation of the differential symplectic geometry of the infinite dimensional manifolds onto the observable algebra as a noncommutative geometry. Hence, we obtain the latter from the physical theory itself. We have essentially an extended formalism of the Schr̎odinger versus Heisenberg picture which we describe mathematically as like a coordinate map from the phase space, for which we have presented argument to be seen as the quantum model of the physical space, to the noncommutative geometry coordinated by the six position and momentum operators. The observable algebra is taken essentially as an algebra of formal functions on the latter operators. The work formulates the intuitive idea that the noncommutative geometry can be seen as an alternative, noncommutative coordinate, picture of familiar quantum phase space, at least so long as the symplectic geometry is concerned.     

Maxwell, Mechanism, and the Nature of Electricity

Maxwell aimed to reduce electromagnetism to the mechanics of an ether and even proposed a detailed ether model of electromagnetic phenomena that could accommodate light waves. I argue in this paper that Maxwell's undoubted successes in electromagnetism came about in spite, rather than because of his attempts to reduce electromagnetism to mechanics. By the end of the nineteenth century it had become clear that electric charge and the electromagnetic field were primitives on a par with, and not to be reduced to mechanical entities such as mass.     

Electrodynamics from a metric

A metric is given that produces a space in which the geodesic equation is identical with the Lorentz equation of motion for a charged particle. The gravitational field equations in the same space indicate a geometric origin for the electromagnetic energy-momentum tensor. A comparison is made with Kaluza-Klein theories and it is determined that the present theory is distinct from them because it corresponds to a timelike, noncompact fifth dimension. Since the metric is velocity-dependent, it is actually a Finsler space rather than a Riemannian space metric. Its special form, however, allows computations to be done in terms of Riemannian geometry.     

Electrodynamics and Spacetime Geometry: Foundations

We explore the intimate connection between spacetime geometry and electrodynamics. This link is already implicit in the constitutive relations between the field strengths and excitations, which are an essential part of the axiomatic structure of electromagnetism, clearly formulated via integration theory and differential forms. We review the foundations of classical electromagnetism based on charge and magnetic flux conservation, the Lorentz force and the constitutive relations. These relations introduce the conformal part of the metric and allow the study of electrodynamics for specific spacetime geometries. At the foundational level, we discuss the possibility of generalizing the vacuum constitutive relations, by relaxing the fixed conditions of homogeneity and isotropy, and by assuming that the symmetry properties of the electro-vacuum follow the spacetime isometries. The implications of this extension are briefly discussed in the context of the intimate connection between electromagnetism and the geometry (and causal structure) of spacetime.     

Moduli of ADHM sheaves and the local Donaldson-Thomas theory

The ADHM construction establishes a one-to-one correspondence between framed torsion free sheaves on the projective plane and stable framed representations of a quiver with relations in the category of complex vector spaces. This paper studies the geometry of moduli spaces of representations of the same quiver with relations in the abelian category of coherent sheaves on a smooth complex projective curve X. In particular it is proven that this moduli space is virtually smooth and related by relative Beilinson spectral sequence to the curve counting construction via stable pairs of Pandharipande and Thomas. This yields a new conjectural construction for the local Donaldson-Thomas theory of curves as well as a natural higher rank generalization.     

Time,the grand illusion

The reconcilability of gravitational with electromagnetic clocks suggests that a rigorous analysis of time will provide understanding of the unity of gravity and electromagnetism. Time is found to be fundamentally a property of elementary particles, only derivatively a property of clocks. A declaration is made: that the flow of an elementary particle's timeis the change of its radius, that time is therefore illusory. The de Sitter expanding universe is derived from this principle by treating elementary particles as spheres in Euclidean space. The hyperspheres of de Sitter space call up a five-dimensional metric manifold whose geometry models gravity, electromagnetism, and other phenomena tied to the structure of matter; neutrinos are provided for. Distance in this manifold is related to a secondary time, not correlated to primary time, but just as illusory. A particle's inertial rest mass is the relative rate of its two proper times; mass and charge are jointly, not individually, conserved.     

Geometry of financial markets—Towards information theory model of markets

Most parameters used to describe states and dynamics of financial market depend on proportions of the appropriate variables rather than on their actual values. Therefore, projective geometry seems to be the correct language to describe the theater of financial activities. We suppose that the objects of interest of agents, called here baskets, form a vector space over the reals. A portfolio is defined as an equivalence class of baskets containing assets in the same proportions. Therefore portfolios form a projective space. Cross ratios, being invariants of projective maps, form key structures in the proposed model. Quotation with respect to an asset Ξ (i.e. in units of Ξ) is given by linear maps. Among various types of metrics that have financial interpretation, the min-max metric on the space of quotations can be introduced. This metric has an interesting interpretation in terms of rates of return. It can be generalized so that to incorporate a new numerical parameter (called temperature) that describes agent's lack of knowledge about the state of the market. In a dual way, a metric on the space of market quotation is defined. In addition, one can define an interesting metric structure on the space of portfolios/quotation that is invariant with respect to hyperbolic (Lorentz) symmetries of the space of portfolios. The introduced formalism opens new interesting and possibly fruitful fields of research.     

Wesson’s Induced Matter Theory with a Weylian Bulk

The foundations of Wesson’s induced matter theory are analyzed. It is shown that the empty—without matter—5-dimensional bulk must be regarded as a Weylian space rather than as a Riemannian one. Revising the geometry of the bulk, we have assumed that a Weylian connection vector and a gauge function exist in addition to the metric tensor. The framework of a Weyl–Dirac version of Wesson’s theory is elaborated and discussed. In the 4-dimensional hypersurface (brane), one obtains equations describing both fields, the gravitational and the electromagnetic. The result is a geometrically based unified theory of gravitation and electromagnetism with mass and current induced by the bulk. In special cases on obtains on the brane the equations of Einstein–Maxwell, or these of the original induced matter theory.     

Integrable System and Motion of Curves in Projective and Similarity Geometries

Based on the natural frame in the projective geometry, motions of curves in projective geometry are studied. It is shown that several integrable equations including Sawada-Kotera and KK equations arise from motion of plane curves in projective geometries. Motion of space curves described by acceleration field and governed by endowing an extra space variable in similarity geometry P³ is also studied.     

A remark on coefficients of connection with two gauge fields

Recently a geometrical gravitational theory based on the coefficients of connection with two gauge fields was proposed. The objective of the present paper is to show that the geometry based on the coefficients of connection with two gauge fields can be obtained by the successive operations of conformal and projective transformations on Riemannian metric.     

A hundred years of special theory of relativity

A special theory of relativity is considered here as an episode of non-Euclidean geometry. Special attention is drawn to the fact that the replacement of the fifth Euclidean postulate by the Lobachevsky postulate of parallel straight lines in the space of velocities of a material point leads to the replacement of the postulate of the same time rate by the postulate of the same velocity of light in all inertial reference systems.     

Naturalness of the space of states in quantum mechanics

We show how certain constructions of quantum mechanics, like monopoles, instantons, and the Schrödinger-von Neumann equation, are related to geometric functors which are representable. We study the differential geometry of the projective bundle associated with an infinite-dimensional separable Hilbert space, and we construct a universal connection which, is described as a subspace of skwe-Hermitian operators. This connection is responsible for the Berry phase.     

Hypercomplex quantum mechanics

The fundamental axioms of the quantum theory do not explicitly identify the algebraic structure of the linear space for which orthogonal subspaces correspond to the propositions (equivalence classes of physical questions). The projective geometry of the weakly modular orthocomplemented lattice of propositions may be imbedded in a complex Hilbert space; this is the structure which has traditionally been used. This paper reviews some work which has been devoted to generalizing the target space of this imbedding to Hilbert modules of a more general type. In particular, detailed discussion is given of the simplest generalization of the complex Hilbert space, that of the quaternion Hilbert module.     

Scalar curvature and projective compactness

Consider a manifold with boundary, and such that the interior is equipped with a pseudo-Riemannian metric. We prove that, under mild asymptotic non-vanishing conditions on the scalar curvature, if the Levi-Civita connection of the interior does not extend to the boundary (because for example the interior is complete) whereas its projective structure does, then the metric is projectively compact of order 2; this order is a measure of volume growth towards infinity. This implies a host of results including that the metric satisfies asymptotic Einstein conditions, and induces a canonical conformal structure on the boundary. Underpinning this work is a new interpretation of scalar curvature in terms of projective geometry. This enables us to show that if the projective structure of a metric extends to the boundary then its scalar curvature also naturally and smoothly extends.