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}${q}_{\mathrm{min}=\mathrm{R}}$, 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.
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, 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.
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}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 GeV/c (c is the velocity of light) are considered. We also discuss the superposition of waves in the presence of R and S.
It is well-known that the coordinate as a continuous variable, consisting of a set of all points between 0 and L contradicts the observability of measurement. In other words there might exist a fundamental length in nature, such as the Planck length λP, so that it is not possible to measure a position coordinate with accuracy smaller than this fundamental length. It is therefore necessary to investigate the formulation of quantum mechanics using only discrete variables as coordinates. To investigate all of quantum mechanics or any branch of physics from this approach is of course a daunting task and thus it is worthwhile to consider a specific simple problem in order to formulate the basic ideas. In this note we compare the solutions of Schrodinger equation for one-dimensional free particle under the usual space-time continuum with those that are obtained when space-time is assumed to be quantized using a simple model. For this purpose, we replace the derivatives occurring in Schrodinger equation with the corresponding discrete derivatives. We compute the probability density (and the probability current) under the two scenarios; they turn out to be quite different in the two cases. We also obtain the operator identity for the commutator [p,x]q under the assumption of quantized space-time and contrast it with the usual commutator [p,x].
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.
We present a continuous representation theory based on the affine group. This theory is applicable to a mechanical system which has one or more of its classical canonical coordinates restricted to a smaller range than − ∞ to ∞. Such systems are especially troublesome in the usual quantization approach since, as is well known from von Neumann's work, the relation [P, Q] = −iI implies that P and Q must have a spectrum from − ∞ to ∞ if they are to be self-adjoint. Consequently, if the spectrum of either P or Q is restricted, at least one of the operators, say Q, is not self-adjoint and does not have a spectral resolution. Thus Q cannot generate a coordinate representation. This leads us to consider a different pair of operators, P and B, both of which are self-adjoint and which obey [P, B] = −iP. The Lie group corresponding to this latter algebra is the affine group, which has two unitarily inequivalent, irreducible representations, one in which the spectrum of P is positive. Using the affine group as our kinematical group, we have developed continuous representations analogous to those Klauder and McKenna developed for the canonical group, and have shown that the former representations have almost all the desirable properties of the latter.
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.
The concept of position is a basic one in our construction of a model of thenphysical world: yet the theory of localisation in quantum mechanics is fraughtnwith difficulties, especially in the relativistic case. A strictly localised state forna nonrelativistic particle cannot be described by an ordinary function. In thenrelativistic case the problems are manifold. For a free electron, using the Diracnposition operator x, we cannot construct positive-energy states which localisenthe particle within a compact region. Furthermore it has been suggested that anconsistent one-particle description of localisation on a scale small compared tonthe Compton wavelength of the particle is ruled out because pair-productionnwould take place. These matters are dicussed at greater length in Chaptern1, where several proposed localisation schemes (that is, sets of definitions ofnlocalised states and position operators) are reviewed: none of these is deemednwholly satisfactory. Introductory remarks are also made in this chapter aboutnthe remarkable phenomenon of 'probability backflow', which is discussed further below, and which is relevant to our discussion of localisation.In Chapter 2 we show that while strict localisation of the free Dirac particle is not describable within the usual mathematical formalism, in terms of Dirac's position operator x it is possible to describe sequences of positive-energy states whose spread Dx = ((x -- x0)2) about a point X0 approaches zero.In Chapter 3 the component states of these sequences are used as the basis of a full causally well-behaved solution to the localisation problem for the free electron, with natural space-time transformation properties. It is shown that although x is not an observable in the usual sense, the four-vector density (p(x, t), j(x, t)/c) is observable, and can be localised arbitrarily precisely about any point in space, at any instant of time, using only positive-energy states. A suitable spin operator can be diagonalised at the same time.n n