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Physical Review A

Klauder’s continuous representations and fuzzy coordinate variables. II

J. P. Hsu, C. B. Whan

Published Sep 1, 1988• 5 citations
Publisher

Abstract

Based on Klauder's continuous representations in Hilbert space, we discuss a fundamental fuzziness of the coordinate and momentum variables for a quantum particle. These fuzzy dynamical variables are intimately related to a very small radical length R(\ensuremath{\lesssim}${10}^{\mathrm{\ensuremath{-}}18}$ cm) and a very large cosmological length S (\ensuremath{\gtrsim}${10}^{10}$ light years). The uncertainty \ensuremath{\Delta}q for the measurement of a particle's position alone is assumed to be restricted by S\ensuremath{\gtrsim}\ensuremath{\Delta}q\ensuremath{\gtrsim}R. Physical properties of fuzzy base states (p\ensuremath{\Vert} and (q\ensuremath{\Vert} and their general cosmological and very-high-energy implications are discussed. In particular, a new density of states and its possible effects on decay rates and scattering cross sections at large momenta p\ensuremath{\gtrsim}\ensuremath{\Elzxh}/R\ensuremath{\sim}${10}^{4}$ GeV/c (c is the velocity of light) are considered. We also discuss the superposition of waves in the presence of R and S.

DOI
10.1103/physreva.38.2248

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Based on Klauder's continuous representations in Hilbert space, we discuss a fundamental fuzziness of the coordinate and momentum variables for a quantum particle. These fuzzy dynamical variables are intimately related to a very small radical length R(\ensuremath{\lesssim}10\ensuremath−18{10}^{\mathrm{\ensuremath{-}}18}10\ensuremath−18 cm) and a very large cosmological length S (\ensuremath{\gtrsim}1010{10}^{10}1010 light years). The uncertainty \ensuremath{\Delta}q for the measurement of a particle's position alone is assumed to be restricted by S\ensuremath{\gtrsim}\ensuremath{\Delta}q\ensuremath{\gtrsim}R. Physical properties of fuzzy base states (p\ensuremath{\Vert} and (q\ensuremath{\Vert} and their general cosmological and very-high-energy implications are discussed. In particular, a new density of states and its possible effects on decay rates and scattering cross sections at large momenta p\ensuremath{\gtrsim}\ensuremath{\Elzxh}/R\ensuremath{\sim}104{10}^{4}104 GeV/c (c is the velocity of light) are considered. We also discuss the superposition of waves in the presence of R and S.
Klauder's continuous representation naturally allows a basic length scale. It is assumed to be fundamental for continuous dynamical variables Q and P in a general quantum theory. We explore new properties of a particle's coordinates at short distances by constructing continuous base states in the coordinate space rather than in the phase space. The results show that the position operator Q has neither eigenvalues nor eigenvectors. There emerges a picture of ``fuzzy point particles''---point particles whose positions can never be measured with an accuracy smaller than \ensuremath{\Delta}qmin=R{q}_{\mathrm{min}=\mathrm{R}}qmin=R​, where R is a very small length scale for the coordinate variable. In the limit R\ensuremath{\rightarrow}0, Klauder's continuous representation reduces to Dirac's eigenbras and kets. The property \ensuremath{\Delta}q\ensuremath{\ge}R implies that waves with arbitrarily short wavelengths do not exist in the physical world.
In a generalized quantum-mechanical framework with fuzzy dynamical variables, there exists a small radical length R${10}^{\mathrm{\ensuremath{-}}17}$ cm, which is related to the physical impossibility of unlimited accuracy in measuring a particle's position. It sheds light on the problem of a detailed and gradual transition, as the mass increases, from the probabilistic behavior of microscopic objects (mcR\ensuremath{\ll}\ensuremath{\Elzxh}) to the deterministic behavior of macroscopic objects (mcR\ensuremath{\gg}\ensuremath{\Elzxh}), a problem unaddressable within the framework of ordinary quantum mechanics. A generalized equation of motion for a particle with an arbitrary mass m is postulated. It possesses a new local gauge symmetry and reduces to the Schr\"odinger and the Hamilton-Jacobi equations in the limits of small m and large m, respectively. For the double-slit experiment, the theory predicts that, as the mass increases with the momentum fixed, the interference pattern will have roughly half the usual intensity and two bright spots when mcR/\ensuremath{\Elzxh}\ensuremath{\sim}1.
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 expressing quantum mechanical expectation values as averages with respect to nonnegative density functions on phase space, by analogy with classical mechanics, is reexamined in the light of some earlier work on fuzzy phase spaces. It is shown that such phase space representations are possible if ordinary phase space is replaced by a so-called fuzzy phase space, on which the usual marginal distribution functions are redefined to conform to the fact that arbitrarily precise simultaneous measurements on position and momentum are not compatible with quantum mechanics. In the process a generalization of Wigner’s theorem on the nonexistence of phase space representations of quantum mechanics, which also satisfy the standard (classical) marginality conditions in position and momentum, is obtained. It is shown that a (continuous) representation of quantum mechanics exists on a given fuzzy phase space if an only if the corresponding confidence functions for position and momentum measurements satisfy the Heisenberg uncertainty relations.
Candidate theories of quantum gravity predict the presence of a minimal measurable length at high energies. Such feature is in contrast with the Heisenberg Uncertainty Principle. Therefore, phenomenological approaches to quantum gravity introduced models spelled as modifications of quantum mechanics including a minimal length. The effects of such modification are expected to be relevant at large energies/small lengths. One first consequence is that position eigenstates are not included in such models due to the presence of a minimal uncertainty in position. Furthermore, depending on the particular modification of the position–momentum commutator, when such models are considered from momentum space, the position operator is changed, and a measure factor appears to let the position operator be self-adjoint. As a consequence, the (quasi-)position representation acquires numerous issues. For example, the position operator is no longer a multiplicative operator, and the momentum of a free particle does not correspond directly to its wave number. Here, we will review such issues, clarifying aspects of minimal length models, with particular reference to the representation of the position operator. Furthermore, we will show how such a (quasi-)position description of quantum mechanical models with a minimal length affects results concerning simple systems.