arXiv subjects

John Mashford

Publications and source records attributed to John Mashford.

10 recordsLinked to original sources

arXiv2025

Computation of the masses of the elementary particles

An approach to gauge theory in the context of locally conformally flat space-time is described. It is discussed how there are a number of natural principal bundles associated with any given locally conformally flat space-time $X$. The simplest of these principal bundles is the bundle $P_X(G)$ with structure group $G=U(2,2)$. An 11 dimensional bundle $Q$ with structure group a certain 7 dimensional group $K$ is constructed by a method involving a reduction of structure group for the bundle $P_X(G)$. It is shown how the gauge groups $U(1)$, $SU(2)$ and $SU(3)$ can be derived from the geometry of locally conformally flat space-time. Fock spaces of multiparticle states for the fields of the standard model are constructed in the context of bundles with these groups as structure groups. Scattering and other particle interaction processes are defined in terms of linear maps between multiparticle state spaces. A technique for computing analytically and/or computationally the masses of the elementary particles is described. This method involves the computation of a certain quantity called the integral mass spectrum for a given family of particles and then the masses of the particles in the family are determined to be the locations of the peaks of the integral mass spectrum. The method is applied successfully in the electroweak sector to the cases of the charged leptons, $μ$ and $τ$, and the Z$^0$ particle.

math-ph↗

arXiv2025

Computation of the one-loop spectral QCD running coupling using covariant spectral regularization

Methods described in the literature for the computation of the QCD running coupling are essentially all defined with respect to the renormalization group equations and these equations are associated with the method of renormalization for dealing with infinities in Feynman integrals. The problem with the renormalization group equations is their prediction of the unphysical Landau pole which, for QCD occurs at an energy of the order of a few hundred MeV. The models described in the literature generally interlace high energy renormalization group predictions with modified low energy formulations. It would be desirable to have a method for the computation of the running strong coupling which is not {\em ad hoc} but is unified over the whole range of energies and is based on a single mathematically rigorous formulation which is guided by physical principles. In this paper we describe a method using the regularization technique called covariant spectral regularization for which renormalization is not required. The densities associated with the quark and gluon bubbles are computed without requiring renormalization and hence the spectral QCD vacuum polarization tensor is determined. It is found that the position space spectral QCD running coupling is an analytic function which does not manifest a Landau pole, but instead it manifests what might be called a ``Landau peak", and has the property known as ``freezing of $α_s$" in the infrared. We determine a spectral bare strong coupling constant of $α_b\approx(411)^{-1}$ which can be used in higher order QCD computations (this is to be compared with QCD using renormalization where the bare strong coupling is infinite). Thus it seems that one can conclude that, when analyzed using covariant spectral regularization, QCD is perturbative at all energies.

physics.gen-ph↗

arXiv2022

UV and IR divergence-free calculation of the vertex function at arbitrary values of its arguments

The vertex function is analyzed using covariant spectral regularization without encountering any divergence, either UV or IR. The mathematics of covariant spectral regularization for covariant matrix valued measures with one Lorentz index on open subsets of Minkowski space is described. This is then applied to the case of the vertex function and expressions for the densities associated with the vertex function in the t channel and the s channel with respect to Lebesgue measure on Minkowski space are obtained. These densities are well defined, non-divergent and analytic over their domains of definition and are obtained without using renormalization or needing to consider final state radiation. The limit of the expression for the vertex function in the t channel at low energy and low momenta is computed resulting in the classical result for the leading order (LO) contribution to the anomalous magnetic moment of the electron. Also the density for the vertex function in the s channel is used to compute the LO vertex correction contribution to the high energy limit of the cross section for the process $e^{+}e^{-}\rightarrowμ^{+}μ^{-}$.

physics.gen-ph↗

arXiv2022

Calculation without IR divergence of the soft photon high energy limit of final state radiation for the process $e^{+}e^{-}\rightarrowμ^{+}μ^{-}γ$

In this paper it is shown that the final state radiation process $e^{+}e^{-}\rightarrowμ^{+}μ^{-}γ$ at tree level is not associated with any IR divergence in the soft photon high energy limit if the calculation is done using a careful treatment of a certain distributional object (in fact, a measure) arising from the Feynman amplitude for the process. Thus there need be no "infrared catastrophe" associated with the process. It is shown that, in fact, the cross section for the final state photons for the the process vanishes in the soft photon high energy limit.

physics.gen-ph↗

arXiv2021

Divergence free quantum field theory using a spectral calculus of Lorentz invariant measures

This paper presents a spectral calculus for computing the spectrum of a causal Lorentz invariant Borel complex measure on Minkowski space, thereby enabling one to compute the density for such a measure with respect to Lebesque measure. It is proved that the convolution of arbitrary causal Lorentz invariant Borel measures exists and the product of such measures exists in a wide class of cases. Techniques for their computation are presented. Divergent integrals in quantum field theory (QFT) are shown to have a well defined existence as Lorentz covariant measures. The case of vacuum polarization is considered and the spectral vacuum polarization function is shown to have very close agreement with the vacuum polarization function obtained using dimensional regularization / renormalization in the timelike domain. Using the spectral vacuum polarization function the exact Uehling potential function is derived. The spectral running coupling constant is computed and is shown to converge for all energies while the integral defining the running coupling constant obtained using dimensional regularization / renormalization is shown to diverge for all non-zero energies.

physics.gen-ph↗

arXiv2021

Second quantized quantum field theory based on invariance properties of locally conformally flat space-times

Well defined quantum field theory (QFT) for the electroweak force including quantum electrodynamics (QED) and the weak force is obtained by considering natural unitary representations of a group $K\subset U(2,2)$, where $K$ is locally isomorphic to $SL(2,{\bf C})\times U(1)$, on a state space of Schwartz spinors, a Fock space ${\mathcal F}$ of multiparticle states and a space ${\mathcal H}$ of fermionic multiparticle states which forms a Grassmann algebra. These algebras are defined constructively and emerge from the requirement of covariance associated with the geometry of space-time. (Here $K$ is the structure group of a certain principal bundle associated with a given Möbius structure modeling space-time.) Scattering processes are associated with intertwining operators between various algebras, which are encoded in an associated bundle of kernel algebras. Supersymmetry emerges naturally from the algebraic structure of the theory. Kernels can be generated using $K$ covariant matrix valued measures given a suitable definition of covariance. It is shown how Feynman propagators, fermion loops and the electron self energy can be given well defined interpretations as measures covariant in this sense. An example of the methods described in the paper is given in which the first order Feynman amplitude of electro-electron scattering ($ee\rightarrow ee$) is derived from a simple order (2,2) kernel. A second example is given explaining muon decay which is a manifestation of the weak force.

math-ph↗

arXiv2020

Computation of the leading order contributions to the Lamb shift for the H atom using spectral regularization

The Uehling contribution to the Lamb shift can be computed exactly in terms of the Uehling potential function. However derivations of this function are complex involving avoiding divergences using intricate techniques from early quantum field theory (QFT) or else more modern approaches using charge and mass renormalization. In the present paper we derive the Uehling potential function in a fairly conceptually straightforward way not involving renormalization in which the vacuum polarization tensor is viewed as a Lorentz invariant 2-tensor valued measure on Minkowski space. Furthermore we compute a complex matrix valued potential function for the electron self-energy contribution to the Lamb shift. The resulting potential function is derived in a conceptually simple way not involving renormalization and can be used for higher order computations in QFT involving multiple loops.

physics.gen-ph↗

arXiv2020

A Neural Markovian Multiresolution Image Labeling Algorithm

This paper describes the results of formally evaluating the MCV (Markov concurrent vision) image labeling algorithm which is a (semi-) hierarchical algorithm commencing with a partition made up of single pixel regions and merging regions or subsets of regions using a Markov random field (MRF) image model. It is an example of a general approach to computer vision called concurrent vision in which the operations of image segmentation and image classification are carried out concurrently. While many image labeling algorithms output a single partition, or segmentation, the MCV algorithm outputs a sequence of partitions and this more elaborate structure may provide information that is valuable for higher level vision systems. With certain types of MRF the component of the system for image evaluation can be implemented as a hardwired feed forward neural network. While being applicable to images (i.e. 2D signals), the algorithm is equally applicable to 1D signals (e.g. speech) or 3D signals (e.g. video sequences) (though its performance in such domains remains to be tested). The algorithm is assessed using subjective and objective criteria with very good results.

cs.CV↗

arXiv2020

An approach to classical quantum field theory based on the geometry of locally conformally flat space-time

This paper gives an introduction to certain classical physical theories described in the context of locally Minkowskian causal structures (LMCSs). For simplicity of exposition we consider LMCSs which have locally Euclidean topology (i.e. are manifolds) and hence are Möbius structures. We describe natural principal bundle structures associated with Möbius structures. Fermion fields are associated with sections of vector bundles associated with the principal bundles while interaction fields (bosons) are associated with endomorphisms of the space of fermion fields. Classical quantum field theory (the Dirac equation and Maxwell's equations) is obtained by considering representations of the structure group $K\subset U(2,2)$ of a principal bundle associated with a given Möbius structure where $K$, while being a subset of $U(2,2)$, is also isomorphic to $SL(2,{\bf C})\times U(1)$. The analysis requires the use of an intertwining operator between the action of $K$ on ${\bf R}^4$ and the adjoint action action of $K$ on $u(2,2)$ and it is shown that the Feynman slash operator, in the chiral representation for the Dirac gamma matrices, has this intertwining property.

gr-qc↗

arXiv2018

Stochastic temporal data upscaling using the generalized k-nearest neighbor algorithm

Three methods of temporal data upscaling, which may collectively be called the generalized k-nearest neighbor (GkNN) method, are considered. The accuracy of the GkNN simulation of month by month yield is considered (where the term yield denotes the dependent variable). The notion of an eventually well distributed time series is introduced and on the basis of this assumption some properties of the average annual yield and its variance for a GkNN simulation are computed. The total yield over a planning period is determined and a general framework for considering the GkNN algorithm based on the notion of stochastically dependent time series is described and it is shown that for a sufficiently large training set the GkNN simulation has the same statistical properties as the training data. An example of the application of the methodology is given in the problem of simulating yield of a rainwater tank given monthly climatic data.

stat.ME↗