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}${q}_{\mathrm{min}=\mathrm{R}}$, 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.

Physical Review AMar 1, 19884 citations

Variational solution of the Schrödinger equation

Kálmán Varga, Joseph A. Driscoll

Cambridge University Press eBooksApr 14, 20110 citations

Klauder’s continuous representations and fuzzy coordinate variables. II

J. P. Hsu, C. B. Whan

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.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.

Physical Review ASep 1, 19885 citations

Variational Method Applied to Schrödinger-Like Equation

Elso Drigo Filho, Regina Maria Ricotta, Natália Fávaro Ribeiro

Springer eBooksJan 1, 20190 citations

{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.

{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.