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.     

External fields as intrinsic geometry

There is an interesting dichotomy between a space-time metric considered as external field in a flat background and the same considered as an intrinsic part of the geometry of space-time. We shall describe and compare two other external fields which can be absorbed into an appropriate redefinition of the geometry, this time a noncommutative one. We shall also recall some previous incidences of the same phenomena involving bosonic field theories. It is known that some such theories on the commutative geometry of space-time can be re-expressed as abelian-gauge theory in an appropriate noncommutative geometry. The noncommutative structure can be considered as containing extra modes all of whose dynamics are given by the one abelian action. Received: 1 December 2000 / Published online: 23 January 2001     

A Five-Dimensional Model of the Gravitational Interaction of an Electromagnetic Field in an Affine-Metric Space

A geometric model for the gravitational interaction of an electromagnetic field in an affine-metric space with torsion and nonmetricity is proposed which describes the dynamics of an empty 5-dimensional affine-metric space. The gravitational and the electromagnetic field are presented in terms of the metric tensor of a 5-dimensional space-time. The equations of the theory are deduced from the variation principle with the use of the (4 + 1)-splitting formalism. Exact spherically symmetrical solutions have been obtained for the system of equations of the presented theory, and their possible astrophysical consequences have been investigated.     

The Kerr space-time near the ring singularity

We investigate the geometry of the Kerr space-time near the ring singularity. A systematic study of the mathematical and physical structure of this region reveals that the singularity in the Kerr space-time is naturally understood in terms of a subset of the immersion of the set defined byr=0 (in Boyer-Lindquist coordinates) in the Kerr space-time. It is well known that the Kerr space-time is not a differentiable manifold (due to the curvature singularity) or a topological manifold, but a well defined topological space with a structure that is manifested by the constrast in taking limits along the hypersurface atr=0 and the equatorial plane which approach singularity. We find that the ring singularity is either an edge or a self-intersection of the topological space depending on which extension of the metric throughr=0 is implemented. A major result of this analysis is the extrapolation to the general accelerating case of Carter's proof that the only nonspacelike geodesics which can reach the ring singularity are restricted to the equatorial plane. For finite magnitudes of proper acceleration, it is shown that only lightlike trajectories that asymptotically approach the null generator of the ring singularity can reach it from above or below the equatorial plane.     

Geometric algebra and star products on the phase space

Superanalysis can be deformed with a fermionic star product into a Clifford calculus that is equivalent to geometric algebra. With this multivector formalism it is then possible to formulate Riemannian geometry and an inhomogeneous generalization of exterior calculus. Moreover, it is shown here how symplectic and Poisson geometry fit in this context. The application of this formalism together with the bosonic star product formalism of deformation quantization leads then on space and space-time to a natural appearance of spin structures and on phase space to BRST structures that were found in the path integral formulation of classical mechanics. Furthermore it will be shown that Poincaré and Lie-Poisson reduction can be formulated in this formalism.     

The static metrics with cylindrical symmetry describing a model of cosmic strings

We discuss a set of static metrics with cylindrical symmetry describing the interior and exterior space-time of a model of cosmic strings considered recently in cosmology. The interior metric depends on one arbitrary function and the exterior on one constant. We find the relation between this constant and the linear mass density of the cosmic string. A cosmic string can be also treated as a line source in the framework of the distribution-valued curvature formalism which allows us to obtain again the same relation.     

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.     

Scalar torsion and a new symmetry of general relativity

We reformulate the general theory of relativity in the language of Riemann–Cartan geometry. We start from the assumption that the space-time can be described as a non-Riemannian manifold, which, in addition to the metric field, is endowed with torsion. In this new framework, the gravitational field is represented not only by the metric, but also by the torsion, which is completely determined by a geometric scalar field. We show that in this formulation general relativity has a new kind of invariance, whose invariance group consists of a set of conformal and gauge transformations, called Cartan transformations. These involve both the metric tensor and the torsion vector field, and are similar to the well known Weyl gauge transformations. By making use of the concept of Cartan gauges, we show that, under Cartan transformations, the new formalism leads to different pictures of the same gravitational phenomena. We illustrate this fact by looking at the one of the classical tests of general relativity theory, namely the gravitational spectral shift. Finally, we extend the concept of space-time symmetry to Riemann–Cartan space-times with scalar torsion and obtain the conservation laws for auto-parallel motions in a static spherically symmetric vacuum space-time in a Cartan gauge, whose orbits are identical to Schwarzschild orbits in general relativity.     

Quantization of the Interior Schwarzschild Black Hole

We study a Hamiltonian quantum formalism of a spherically symmetric space-time which can be identified with the interior of a Schwarzschild black hole. The phase space of this model is spanned by two dynamical variables and their conjugate momenta. It is shown that the classical Lagrangian of the model gives rise the interior metric of a Schwarzschild black hole. We also show that the mass of such a system is a Dirac observable and then by quantization of the model by Wheeler-DeWitt approach and constructing suitable wave packets we get the mass spectrum of the black hole.     

A Viable Model of Locally Anisotropic Space-Time and the Finslerian Generalization of the Relativity Theory

The work is devoted to the construction of a viable model of locally anisotropic, i.e. Finslerian, space-time and to the generalization, on this basis, of the relativistic theory of gravitation. Arguments in favour of this model are considered. From physical considerations the concrete form of the Finslerian metric has been reconstructed and within the framework of the correspondence principle a formalism of the theory has been developed. The approach suggested is aimed at developing unified gauge theories of all fundamental interactions. Much attention is given to the nontrivial physical manifestations of local space anisotropy and to the possibility of its experimental detection.     

Classical mechanics in phase space revisited

A Bose type of classical Hamilton algebra, i.e., the algebra of the canonical formalism of classical mechanics, is represented on a linear space of functions of phase space variables. The symplectic metric of the phase space and possible algorithms of classical mechanics (which include the standard one) are derived. It is shown that to each of the classical algorithms there is a corresponding one in the phase space formulation of quantum mechanics.     

Gravity in non-commutative geometry

We study general relativity in the framework of non-commutative differential geometry. As a prerequisite we develop the basic notions of non-commutative Riemannian geometry, including analogues of Riemannian metric, curvature and scalar curvature. This enables us to introduce a generalized Einstein-Hilbert action for non-commutative Riemannian spaces. As an example we study a space-time which is the product of a four dimensional manifold by a two-point space, using the tools of non-commutative Riemannian geometry, and derive its generalized Einstein-Hilbert action. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.Dedicated to H. ArakiSupported in part by the Swiss National Foundation (SNF)     

ROTATING RINDLER SPACE TIME WITH CONSTANT ANGULAR VELOCITY

A new space time metric is derived from Kerr metric if its mass and location approach to infinite in an appropriate way. The new space-time is an infinitesimal neighborhood nearby one of the two horizon poles of an infinite Kerr black hole. In other words, it is the second order infinitesimal neighborhood nearby one of the two horizon poles of a Kerr black hole. It is flat and has event horizon and infinite red shift surface. We prove that it is a rotating Rindler space time with constant angular velocity.     

Quantized space-time,torsion, and magnetic monopoles

Within the tetrad formalism we introduce quantized space-time in the curvilinear case by using general coordinate transformations with noncommuting terms. Fermion and boson fields are studied and the affine connection is also defined in this space. It is shown that space-time torsion and magnetic monopoles appear as consequences of the theory with quantized space-time at small distances. This method may open a new way of understanding topological structure of space-time.     

Hamilton formalism in non-cummutative geometry

We study the Hamilton formalism for Connes-Lott models, i.e. for Yang-Mills theory in non-commutative geometry. The starting point is an associative *-algebra A which is of the form A = C (I, A_s), where A_s is itself an associative *-algebra. With appropriate choice of a K-cycle over A it is possible to identify the time-like part of the generalized differential algebra constructed out of A. We define the non-commutative analogue of integration on space-like surfaces via the Dixmier trace restricted to the representation of the space-like part A_s of the algebra. Due to this restriction it is possible to define the Lagrange function resp. Hamilton function also for Minkowskian space-time. We identify the phase-space and give a definition of the Poisson bracket for Yang-Mills theory in non-commutative geometry. This general formalism is applied to a model on a two-point space and to a model on Minkowski space-time x two-point space.     

Geometry of the Parameter Space of a Quantum System: Classical Point of View

The local geometry of the parameter space of a quantum system is described by the quantum metric tensor and the Berry curvature, which are two fundamental objects that play a crucial role in understanding geometrical aspects of condensed matter physics. Classical integrable systems are considered and a new approach is reported to obtain the classical analogs of the quantum metric tensor and the Berry curvature. An advantage of this approach is that it can be applied to a wide variety of classical systems corresponding to quantum systems with bosonic and fermionic degrees of freedom. The approach used arises from the semiclassical approximation of the Berry curvature and the quantum metric tensor in the Lagrangian formalism. This semiclassical approximation is exploited to establish, for the first time, the relation between the quantum metric tensor and its classical counterpart. The approach described is illustrated and validated by applying it to five systems: the generalized harmonic oscillator, the symmetric and linearly coupled harmonic oscillators, the singular Euclidean oscillator, and a spin-half particle in a magnetic field. Finally, some potential applications of this approach and possible generalizations that can be of interest in the field of condensed matter physics are mentioned.