Spinor structure of space-time

The spinor structure on space-time manifold is investigated in the frame of Crumeyrolle's approach. Some of his theorems are simplified. The equivalence of this approach to the Milnor and Lichnerowicz one is shown using topological properties of the group space of ₀. The equivalence of any two spinor structures on simply connected space-time is established.Partly supported by the Polish Government under the Research Program MR I.7.     

On the structure of space-time caustics

Caustics formed by timelike and null geodesics in a space-timeM are investigated. Care is taken to distinguish the conjugate points in the tangent space (T-conjugate points) from conjugate points in the manifold (M-conjugate points). It is shown that most nonspacelike conjugate points are regular, i.e. with all neighbouring conjugate points having the same degree of degeneracy. The regular timelikeT-conjugate locus is shown to be a smooth 3-dimensional submanifold of the tangent space. Analogously, the regular nullT-conjugate locus is shown to be a smooth 2-dimensional submanifold of the light cone in the tangent space. The smoothness properties of the null caustic are used to show that if an observer sees focusing in all directions, then there will necessarily be a cusp in the caustic. If, in addition, all the null conjugate points have maximal degree of degeneracy (as in the closed Friedmann-Robertson-Walker universes), then the space-time is closed.     

The structure of the b-completion of space-time

Structured space, as a natural generalization of the manifold concept, is defined to be a topological space with a sheaf of real function algebras which are suitably localized and closed with respect to composition with smooth Euclidean functions. Vector fields, differential forms, linear connection and curvature are introduced on structured spaces. It is shown that structured spaces correctly model space-times with singularities. Schmidt's b-boundary of space-time is constructed in the category of structured spaces, and well known difficulties with the b-boundaries of the closed Friedman and Schwarzschild space-times are disentangled. It is argued that the b-boundary of space-time, when considered in the category of structured spaces, can serve as a good definition of classical singularities.     

The structure of space-time transformations

LetT be a one-to-one mapping ofn-dimensional space-timeM onto itself. IfT maps light cones onto light cones and dimM3, it is shown thatT is, up to a scale factor, an inhomogeneous Lorentz transformation. Thus constancy of light velocity alone implies the Lorentz group (up to dilatations). The same holds ifT andT ^–1 preserve (x–y)²>0. This generalizes Zeeman's Theorem. It is then shown that ifT maps lightlike lines onto (arbitrary) straight lines and if dimM3, thenT is linear. The last result can be applied to transformations connecting different reference frames in a relativistic or non-relativistic theory.     

Real spin-Clifford bundle and the spinor structure of space-time

By analyzing the conditions for the existence on a space-time of a global algebraic spinor field, we prove the following result, known as Geroch's theorem: A necessary and sufficient condition for to admit a spinor structure is that the orthonormal frame bundleF ₀() have a global section. Our proof, which does not use in any stage the complexification of _1,3 (the space-time Clifford algebra), is simple, requiring only the explicit construction of the algebraic spinor and the spinorial metric within _1,3 and elementary facts about associated bundles and the bundle reduction process. This is to be compared with the original proof, which uses the full algebraic topology machinery. We also clarify the relation of the covariant spinor structure and Graf'se-spinor structure.     

Probing the space-time structure of multiparticle production processes

It is shown that the models for highenergy multiparticle production in which the notion of a hadronicformation time plays a central role are in disagreement with recently published compilations of the rescatter probabilities in 100 GeV/c \(\bar p\) d and 200 GeV/c p d collisions.     

The observer-dependence of cellular space-time structure

It is argued that space-time has a cellular structure, the exact structure being observer-dependent and consistent with the amount of energy he has available for refining his measuring apparatus. The usual concept of a single distance in continuous space is replaced by the concept of a distance set between cells, the elements of each set depending on the cellular structures of both the space and the measuring rod that is used in the measurement. The idea that there are many different ways of measuring the same observable is abandoned: instead, the definition of the original observable becomes split by the different measuring processes used, and the results of a measurement of each new observable defined by this splitting are predicted from the eigenvalues of a common operator by using an observer-dependent construction. Transformations between observers with different cellular structures are considered. The transformation is not as exact as in the continuous case, with at best a cell of one space being associated with a set of cells in the other. This transformation is determined by information being exchanged by the observers concerning the locations in their two spaces of a finite number of common events. The transformation becomes more exact as more information is exchanged.     

Intrinsic algebraic characterization of space-time structure

Starting from a partially ordered set ofC ^*-algebras _i representing algebras of observables of physical subsystems, we derive a topological Hausdorff space as a candidate for some generalized space-time with the help of which one can define a net , of algebras. This opens a way to define a physical theory without an underlying metaphysical manifold, an aspect which may be relevant for the unification of general relativity and quantum field theory.     

Some consequences of a non-commutative space-time structure

The existence of a fundamental length (or fundamental time) has been conjectured in many contexts. Here we discuss some consequences of a fundamental constant of this type, which emerges as a consequence of deformation-stability considerations leading to a non-commutative space-time structure. This mathematically well defined structure is sufficiently constrained to allow for unambiguous experimental predictions. In particular we discuss the phase-space volume modifications and their relevance for the calculation of the Greisen-Zatsepin-Kuz’min sphere. The (small) corrections to the spectrum of the Coulomb problem are also computed.Received: 23 December 2004, Revised: 18 April 2005, Published online: 8 July 2005PACS: 13.60.-r; 03.65.Bz; 98.70.Sa     

K-causal structure of space-time in general relativity

Using K-causal relation introduced by Sorkin and Woolgar [1], we generalize results of Garcia-Parrado and Senovilla [2,3] on causal maps. We also introduce causality conditions with respect to K-causality which are analogous to those in classical causality theory and prove their inter-relationships. We introduce a new causality condition following the work of Bombelli and Noldus [4] and show that this condition lies in between global hyperbolicity and causal simplicity. This approach is simpler and more general as compared to traditional causal approach [5,6] and it has been used by Penrose et al [7] in giving a new proof of positivity of mass theorem. C ⁰-space-time structures arise in many mathematical and physical situations like conical singularities, discontinuous matter distributions, phenomena of topology-change in quantum field theory etc.      

Connections between abstract quantum theory and space-time structure

By ur-theoretic and general relativistic arguments, a new cosmological model is introduced which avoids most well-known cosmoiogical problems.     

The structure of anomalies for arbitrary dimension of the space-time

The construction of anomalies in Ward identities is investigated for arbitrary dimension of the space-time. If the internal symmetry group is semi-simple and compact we find that for even dimension of the space-time the anomalous term is unique and that no anomaly is constructible for odd dimension.     

Establishment of the Riemannian structure of space-time by classical means

Efforts at providing a physical-axiomatic foundation of the space-time structure of the general theory of relativity have led, when based on simple empirical facts about freely falling particles and light signals, in a satisfying manner only to a Weyl space-time. By adding postulates based on quantum theory, however, the usual pseudo-Riemannian space-time can be reached. We present a newclassical postulate which provides the same results. It is based upon the notion of the radar distance between freely falling particles and demands the approximate equality of the growth of the radar distance for particle pairs of equal, small initial velocities. We show that given this, a property results, as found in earlier work by the author, that distinguishes between Weyl and Lorentz space-times. The property refers to a special metric and decides whether its metric connection has the given free-fall worldlines as geodesics or not. It consists in the vanishing of the mixed spatiotemporal componentsg _i4 of this metric in suitable coordinates along the worldline of the freely falling observer, as the rest system of which the coordinates are constructed.