This paper arises from the observation by Dirac [1] that an exact measurement of the velocity of an electron should give one of the instantaneous values ± c. The velocity v < c with which the electron moves bodily is in some statistical sense an average. This aspect of the quantum states of motion of a relativistic particle has frequently been mentioned [2], [3] and, as is wellknown, is connected with the self energy problem. Such deductions from quantum mechanics have led the author to consider the possibility of constructing from purely statistical ideas an entity having the physical properties of a relativistic elementary particle. One might reasonably hope to build up characteristics such as rest mass and spin in such a scheme. Properties such as energy and momentum will arise as averages over fluctuating random quantities. In this paper we shall concentrate on rest mass, energy and momentum. We shall show that a zero rest mass object which is regarded as being imbedded in a statistical structure emulates the characteristic of rest mass. The average energy and momentum of this statistical system are then shown to satisfy the usual classical relativistic relation between energy and momentum. The statistical structure chosen is that of an object with a randomly orientated velocity vector, c, which has a fixed magnitude c … |c| where c is the velocity of light.This paper arises from the observation by Dirac [1] that an exact measurement of the velocity of an electron should give one of the instantaneous values ± c. The velocity v < c with which the electron moves bodily is in some statistical sense an average. This aspect of the quantum states of motion of a relativistic particle has frequently been mentioned [2], [3] and, as is wellknown, is connected with the self energy problem. Such deductions from quantum mechanics have led the author to consider the possibility of constructing from purely statistical ideas an entity having the physical properties of a relativistic elementary particle. One might reasonably hope to build up characteristics such as rest mass and spin in such a scheme. Properties such as energy and momentum will arise as averages over fluctuating random quantities. In this paper we shall concentrate on rest mass, energy and momentum. We shall show that a zero rest mass object which is regarded as being imbedded in a statistical structure emulates the characteristic of rest mass. The average energy and momentum of this statistical system are then shown to satisfy the usual classical relativistic relation between energy and momentum. The statistical structure chosen is that of an object with a randomly orientated velocity vector, c, which has a fixed magnitude c … |c| where c is the velocity of light.

Journal of Applied ProbabilityAug 1, 19674 citations

Elementary excitations of S =1 antiferromagnetic Heisenberg chains with bond alternation

Shoji Yamamoto

A world-line quantum Monte Carlo method is employed to study the elementary excitations of S=1 antiferromagnetic Heisenberg chains with bond alternation described by the Hamiltonian, scrH=J ${\mathit{tsum}}_{\mathit{i}}$ [1-(-1 ${)}^{\mathit{i}}$ \ensuremath{\delta}] ${\mathrm{S}}_{\mathit{i}}$ \ensuremath{\cdot} ${\mathrm{S}}_{\mathit{i}+1}$ . Imaginary-time correlation functions of this system at low-enough temperatures exhibit almost single-exponential decay and thus the lowest excitation spectrum can be successfully extracted from them. As \ensuremath{\delta} increases from 0 to 1, the system encounters a continuous phase transition at \ensuremath{\delta}\ensuremath{\simeq}0.25, where the energy(E)-momentum(q) relation is fitted well to the form E(q)=( ${\mathit{v}}^{2}$ ${\mathrm{sin}}^{2}$ q+${\mathrm{\ensuremath{\Delta}}}^{2}$ ${)}^{1/2}$ with the effective light velocity v=2.46\ifmmode\pm\else\textpm\fi{}0.08 and the energy gap \ensuremath{\Delta} which vanishes in the long chain limit. The central charge c at the massless critical point is estimated to be 1.02\ifmmode\pm\else\textpm\fi{}0.09, which suggests the universality class of the Gaussian model.A world-line quantum Monte Carlo method is employed to study the elementary excitations of S=1 antiferromagnetic Heisenberg chains with bond alternation described by the Hamiltonian, scrH=J ${\mathit{tsum}}_{\mathit{i}}$ [1-(-1${)}^{\mathit{i}}$\ensuremath{\delta}]${\mathrm{S}}_{\mathit{i}}$\ensuremath{\cdot}${\mathrm{S}}_{\mathit{i}+1}$. Imaginary-time correlation functions of this system at low-enough temperatures exhibit almost single-exponential decay and thus the lowest excitation spectrum can be successfully extracted from them. As \ensuremath{\delta} increases from 0 to 1, the system encounters a continuous phase transition at \ensuremath{\delta}\ensuremath{\simeq}0.25, where the energy(E)-momentum(q) relation is fitted well to the form E(q)=(${\mathit{v}}^{2}$ ${\mathrm{sin}}^{2}$q+${\mathrm{\ensuremath{\Delta}}}^{2}$${)}^{1/2}$ with the effective light velocity v=2.46\ifmmode\pm\else\textpm\fi{}0.08 and the energy gap \ensuremath{\Delta} which vanishes in the long chain limit. The central charge c at the massless critical point is estimated to be 1.02\ifmmode\pm\else\textpm\fi{}0.09, which suggests the universality class of the Gaussian model.

Physical review. B, Condensed matterJun 1, 199526 citations

A world-line quantum Monte Carlo method is employed to study the elementary excitations of S=1 antiferromagnetic Heisenberg chains with bond alternation described by the Hamiltonian, scrH=J

{\mathit{tsum}}_{\mathit{i}}

[1-(-1

{)}^{\mathit{i}}

\ensuremath{\delta}]

{\mathrm{S}}_{\mathit{i}}

\ensuremath{\cdot}

{\mathrm{S}}_{\mathit{i}+1}

. Imaginary-time correlation functions of this system at low-enough temperatures exhibit almost single-exponential decay and thus the lowest excitation spectrum can be successfully extracted from them. As \ensuremath{\delta} increases from 0 to 1, the system encounters a continuous phase transition at \ensuremath{\delta}\ensuremath{\simeq}0.25, where the energy(E)-momentum(q) relation is fitted well to the form E(q)=(

{\mathit{v}}^{2}

{\mathrm{sin}}^{2}

q+${\mathrm{\ensuremath{\Delta}}}^{2}$

{)}^{1/2}

with the effective light velocity v=2.46\ifmmode\pm\else\textpm\fi{}0.08 and the energy gap \ensuremath{\Delta} which vanishes in the long chain limit. The central charge c at the massless critical point is estimated to be 1.02\ifmmode\pm\else\textpm\fi{}0.09, which suggests the universality class of the Gaussian model.