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SOME CONSEQUENCES OF SPACE TIME FUZZINESS

Kourosh Nozari, Behnaz Fazlpour

Published Jan 1, 2007• 9 citations
OpenAlex

Abstract

Finite resolution of spacetime at Planck scale gives it a fuzzy structure (the so-called foamy or fractal spacetime). This fuzzy structure of spacetime is a consequence of quantum fluctuation of geometry itself and can be described within non-commutative geometry and some alternative approaches to quantum gravity. In this paper, some consequences of spacetime fuzziness are studied. Due to this fuzzy structure, some basic notions of ordinary quantum mechanics such as position space representation, wave packet broadening during its propagation and coherent states of quantum mechanical systems should be re-examined.

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Finite resolution of spacetime at Planck scale gives it a fuzzy structure (the so-called foamy or fractal spacetime). This fuzzy structure of spacetime is a consequence of quantum fluctuation of geometry itself and can be described within non-commutative geometry and some alternative approaches to quantum gravity. In this paper, some consequences of spacetime fuzziness are studied. Due to this fuzzy structure, some basic notions of ordinary quantum mechanics such as position space representation, wave packet broadening during its propagation and coherent states of quantum mechanical systems should be re-examined.
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Preface Space-Time Geometry & Quantum Events The Transaction as a Quantum Concept Stochastic Foundation of Quantum Mechanics & the Origin of Particle Spin Fractal Space-Time as an Underlying Structure of the Standard Model Relativity of Scales, Fractal Space & Quantum Potentials Three Possible Implications of Space-Time Discreteness Quantum Computing Space-Time Metric Gauge Fields in Deformed Special Relativity Non-Local Granular Space-Time Foam as an Ultimate Arena at the Planck Scale Structural Approach to the Elementary Particle Theory Forming Physical Fields & Pseudometric & Metric Manifolds. Noncommutativity & Discrete Structures in Classical & Quantum Physics Index.
Fuzzy geometry considered as the possible mathematical framework for reformulation of quantum-mechanical formalism in geometric terms. In this approach the states of massive particle m correspond to elements of fuzzy manifold called fuzzy points. In 1-dimensional case, due to manifold ultraweak (fuzzy) topology, m space coordinate x acquires principal uncertainty dx and described by positive, normalized density w(x,t). Analogous uncertainties appear for fuzzy point on 3-dimensional manifold. It's shown that m states on such manifold are equivalent to vectors (rays) on complex Hilbert space, their evolution correspond to Shroedinger dynamics of nonrelativistic quantum particle.
Commutative fuzzy geometry considered as the possible mathematical framework for reformulation of quantum-mechanical formalism in strictly geometric terms. In that approach the states of massive particle m correspond to elements of fuzzy manifold called fuzzy points. Due to manifold weak topology, in 1-dimensional case m space coordinate x acquires principal uncertainty σ x and described by positive, normalized density ω ( x,t ), the analogous uncertainties appear for 3-dimensional manifold. It's shown that m states on such manifold are equivalent to Dirac vectors of complex Hilbert space, their evolution correspond to Shroedinger dynamics of massive quantum particle.
The problem of time in quantum gravity is often presented as a consequence of applying quantum theory to general relativity. These notes adopt a broader perspective. We first examine how time enters classical mechanics, relativity, quantum theory, and quantum field theory, distinguishing three principal aspects: causal order, temporal duration, and the present. This analysis shows that the conceptual difficulties associated with time do not arise only when gravity is quantised. Tensions between causal structure, clocks, observables, and measurement are already present in the theories from which quantum gravity is constructed. We then survey responses to the problem of time across the principal approaches to quantum gravity. These include canonical and Wheeler-DeWitt quantisation, path-integral and histories formulations, perturbative and background-dependent approaches, causal sets, and twistor theory. The programmes are compared according to whether they retain an external time, render time problematic through quantisation, or incorporate causal or temporal structure into their foundations. Particular emphasis is placed on questions arising from quantum foundations and relativistic quantum information. We conclude by examining linearised gravity and weak-gravity quantum systems. This regime shows that the problem of time is not confined to Planck-scale physics, but arises already in attempts to provide a predictive and operationally meaningful quantum description of spacetime near flat geometry.
The quantum spacetime and the phase space with fuzzy structure are investigated as the possible quantization formalism. In this theory, the state of the nonrelativistic particle m corresponds to the element of fuzzy ordered set (Foset), i.e. the fuzzy point. Due to Foset partial (weak) ordering, the m space coordinate x acquires principal uncertainty σ x . It is shown that Schrödinger formalism of quantum mechanics can be completely derived from consideration of m evolution in fuzzy phase space with minimal number of axioms.