arXiv subjects

Timothy D. Andersen

Publications and source records attributed to Timothy D. Andersen.

14 recordsLinked to original sources

arXiv2021

Chaotic deterministic quantization in a 5D general relativity

How to quantize gravity is a major outstanding open question in quantum physics. While many approaches assume Einstein's theory is an effective low-energy theory, another possibility is that standard methods of quantization are the problem. In this paper, I analyze a quantization mechanism based on chaotic dynamics of 5D general relativity (with imaginary time) with BKL dynamics in the mixmaster universe as an example. I propose that the randomness of quantum mechanics as well as its other properties such as nonlocality derive from chaotic flow of 4D spacetime through a 5th dimension, with the metric tensor under Wick rotation to Euclidean space acting as a heat bath for other quantum fields. This is done by showing that the theory meets mixing conditions such that it is chaotically self-quantizing and quantizes other fields to which it is coupled, such that in the limit taking chaotic dynamics scale to zero the quantization is equivalent to a stochastic quantization. A classical stability analysis shows this dimension is likely spacelike.

gr-qc↗

arXiv2020

A Dynamic Histories Interpretation of Quantum Theory

The problem of how to interpret quantum mechanics has persisted for a century. The disconnect between the wavefunction state vector and what is observed in experimental apparati has had no shortage of explanations. But all explanations so far fall short of a compelling and complete interpretation. In this letter, I present a novel interpretation called dynamic histories. I show mathematically how quantum mechanics can be reinterpreted as deterministically evolving dynamical world lines in a 5D universe. Quantum probabilities can be then be reinterpreted as stemming from ignorance of the state of our own world line. Meanwhile, the lack of observed superposition in experimental apparati is explained in that we only live on a single history with a definite set of properties. Hence, superposition is not an actual state of a particle but a model of ignorance as in classical probability theory. This explains nonlocal effects without superluminal communication. I also discuss how this relates to 5D Kaluza-Klein theory.

physics.gen-ph↗

arXiv2019

Quantization of Fields by Averaging Classical Evolution Equations

This paper extends the formalism for quantizing field theories via a microcanonical quantum field theory and Hamilton's principle to classical evolution equations. These are based on the well-known correspondence under a Wick rotation between quantum field theories and 4-D statistical mechanical theories. By placing quantum field theories on a 4+1-D under Wick rotation to 5-D, expectations of observables are calculated for a microcanonical field theory averaging Hamiltonian flow over a fifth spacelike dimension, a technique common in lattice gauge simulations but not in perturbation theory. In a novel demonstration, averaging pairs of external lines in the classical Feynman diagrams over the fifth dimension generates diagrams with loops and vacuum fluctuations identical to Standard Model diagrams. Because it is microcanonical, this approach, while equivalent for standard quantum fields theories in the Standard Model, is able to quantize theories that have no canonical quantization. It is also unique in representing expectations as averages over solutions to an ordinary, classical PDE rather than a path integral or operator based approaches. Hence, this approach draws a clear connection between quantum field theory and classical field theory in higher dimensions which has implications towards how quantum effects are interpreted. In particular, it raises questions about how violations of the ergodic hypothesis could influence quantum measurements even in standard, non-statistical quantum field theory.

hep-th↗

arXiv2014

On the absence of black hole event horizons: a test of De Sitter Yang-Mills Theory

De Sitter Quantum Gravity is a Yang-Mills theory based on the de Sitter or SO(4,1) group and a promising candidate for a quantum theory of gravity. In this paper, an exact, static, spherically symmetric solution of the classical equations is derived. I show that when the Schwarzchild radius to distance ratio is at post-Newtonian order the theory agrees with general relativity for all parameters but that, once the ratio becomes closer to unity, they differ. At the Schwarzchild radius from a black hole singularity, general relativity predicts an event horizon, which has become a controversial topic in quantum gravity because of information preservation issues. In the De Sitter theory I show, however, that time-like escape paths exist for any mass black hole until the singularity itself is reached. Since an event horizon has never been directly observed and there is currently no observation on which the two theories disagree, this provides a powerful test of the De Sitter theory.

physics.gen-ph↗

arXiv2013

SO(4,1) Yang-Mills theory of quantum gravity

The search for a quantum theory of gravity has become one of the most well-known problems in theoretical physics. Problems quantizing general relativity because it is not renormalizable have led to a search for a new theory of gravity that, while still agreeing with measured observations, is renormalizable. In this paper, a spin-1 Yang-Mills force theory with a SO(4,1) or {\em de Sitter} group symmetry is developed. By deriving the standard geodesic equation and the first post-Newtonian approximation equations, it is shown that this theory, coupled to Dirac fields, predicts all N-body and light observations of gravitational phenomena to within experimental accuracy. Furthermore, because of the separation of gauge covariance from coordinate diffeomorphism, the theory satisfies the strong equivalence principle while maintaining a Minkowski coordinate metric. Cosmology is also briefly addressed: Vacuum energy is the most common explanation for the accelerating expansion of the universe but suffers from the drawback that any reasonable prediction of it is 120 orders of magnitude too large. The de Sitter solution to the Einstein Field Equations is an alternative to vacuum energy as an explanation for the accelerating expansion of the universe but only if the universe is approximately a vacuum. The proposed gauge theory, however, avoids both these problems and, cosmologically, the accelerating expansion of the universe is shown as a consequence of the de Sitter group Lie algebra. In addition, with quantized mass, because it is a generic massless, semi-simple Yang-Mills theory, it is mathematically proved to be a perturbatively renormalizable quantum theory of gravity.

math-ph↗

arXiv2013

An exact integration of a ϕ^4 quantum field theory

Most quantum field theories are not exactly solvable. In this paper show the statistical equivalence of the standard exponential path integral to products of Heaviside functions, i.e. a product of specially tuned uniform distributions. This allows exact integrations of certain quantum field theories. I apply the equivalence to calculate the exact, non-perturbative path integral for a 3+1-D scalar (real) phi-4 field theory.

physics.gen-ph↗

arXiv2013

Exact solutions for the 2d one component plasma

The 2d one component gas of pointlike charges in a uniform neutralizing background interacting with a logarithmic potential is a common model for plasmas. In its classical equilibrium statistics at fixed temperature (canonical ensemble) it is formally related to certain types of random matrices with Gaussian distribution and complex eigenvalues. In this paper, I present an exact integration of this ensemble for $N$ such particles (or alternatively $N\times N$ matrices) for all even non-negative temperatures, a significant open problem in statistical physics for several decades. I achieve this exact integration via an exact integration of a related ensemble, the two-dimensional Selberg integral.

math-ph↗

arXiv2012

Lorentz Covariant Lattice Gauge Theory

Lattice gauge theory's discretization of spacetime suffers from a drawback in that Lorentz covariance is lost because the axes of the lattice create preferred directions in spacetime. Smaller and smaller lattice spacings decrease the effect but fail to eliminate it completely. It has been argued recently that detecting such a set of preferred directions or similar constraints would indicate whether the universe itself has an underlying lattice, i.e. the digital universe hypothesis. In this paper, I demonstrate a technique for accomplishing lattice gauge theory simulations while maintaining exact Lorentz covariance by replacing the lattice with a lattice graph such that the metric is defined as a discrete, Lorentz covariant matrix potential over the graph rather than a metric over an underlying manifold. This technique eliminates the symmetry violation of standard lattice gauge theory and suggests that, even in a digital universe, Lorentz covariance can still hold.

math-ph↗

arXiv2012

Exact solution to the 1d one component Coulomb gas at fixed energy

The one dimensional one component plasma has applications to one dimensional particle systems with logarithmic interactions such as charges in a single channel wire or vortex filaments in a fluid convection stream. The exact integral of this plasma in the canonical ensemble with a gaussian confining potential has already been computed. In this paper, I compute the exact volume of the phase space of the plasma of N particles at fixed energy without a confining potential using a microcanonical ensemble and show that, as in the two-dimensional case, it has negative temperature states, suggesting that one dimensional turbulence can occur from vortex/electron clustering.

math-ph↗

arXiv2010

Runaway expansion in confined quasi-2D plasmas and vortex fluids

The confined, quasi-two-dimensional guiding center plasma and a system of interacting line vortices in an ideal fluid are examples of Hamiltonian systems with infinite interaction distances. The existence of metastable states with negative specific is investigated by standard entropy maximization of the thermodynamic limit of vortices as they become infinitesimal and form a continuous field. We find metastable states and suggest that these imply a runaway reaction leading to a rapid expansion of a confined plasma or fluid similar to the rapid collapse of globular clusters in astrophysics.

math-ph↗

arXiv2008

Exact solution to a nearly parallel vortex filament mean-field theory

Nearly parallel vortex filaments are a generalization of point vortices and describe many phenomena under conservation of angular momentum including vortices forming in deep ocean convection, magnetically confined plasmas, and the solar atmosphere. While point vortices represent perfectly straight, parallel lines of circulation, nearly parallel vortex filaments have some curvature due to internal viscosity. They interact logarithmically but have a kinetic self-energy as well. In this letter, I present an exact solution of a system of these filaments under angular momentum conservation in a mean-field theory. I show that the filaments have an infinite order phase transition not present in the point vortex model.

cond-mat.stat-mech↗

arXiv2007

Trapped slender vortex filaments in statistical equilibrium

The statistical mechanics of nearly parallel vortex filaments confined in the unbounded plane by angular momentum, first studied by Lions and Majda (2000), is investigated using a mean-field approximation to interaction and a spherical constraint to develop an explicit formula for the mean square vortex position or length scale of the system, $R$, verified with Path Integral Monte Carlo simulations. We confirm that 3D filaments resist confinement in a different way than 2D point vortices and that this results in a profound shift at high-densities for the length scale of quasi-2D versus strictly-2D models of vorticity fields in which angular momentum is conserved. Our analytical results correspond well with those of the Monte Carlo simulations and show a 3D effects contributing significantly to determination of the length scale.

math-ph↗

arXiv2006

Explicit mean-field radius for nearly parallel vortex filaments in statistical equilibrium

Geophysical research has focused on flows, such as ocean currents, as two dimensional. Two dimensional point or blob vortex models have the advantage of having a Hamiltonian, whereas 3D vortex filament or tube systems do not necessarily have one, although they do have action functionals. On the other hand, certain classes of 3D vortex models called nearly parallel vortex filament models do have a Hamiltonian and are more accurate descriptions of geophysical and atmospheric flows than purely 2D models, especially at smaller scales. In these ``quasi-2D'' models we replace 2D point vortices with vortex filaments that are very straight and nearly parallel but have Brownian variations along their lengths due to local self-induction. When very straight, quasi-2D filaments are expected to have virtually the same planar density distributions as 2D models. An open problem is when quasi-2D model statistics behave differently than those of the related 2D system and how this difference is manifested. In this paper we study the nearly parallel vortex filament model of Klein, Majda, Damodaran in statistical equilibrium. We are able to obtain a free-energy functional for the system in a non-extensive thermodynamic limit that is a function of the mean square vortex position $R^2$ and solve \emph{explicitly} for $R^2$. Such an explicit formula has never been obtained for a non-2D model. We compare the results of our formula to a 2-D formula of \cite{Lim:2005} and show qualitatively different behavior even when we disallow vortex braiding. We further confirm our results using Path Integral Monte Carlo (Ceperley (1995)) \emph{without} permutations and that the Klein, Majda, Damodaran model's asymptotic assumptions \emph{are valid} for parameters where these deviations occur.

math-ph↗

arXiv2006

Statistical Equilibrium of trapped slender vortex filaments - a continuum model

Systems of nearly parallel, slender vortex filaments in which angular momentum is conserved are an important simplification of the Navier-Stokes equations where turbulence can be studied in statistical equilibrium. We study the canonical Gibbs distribution based on the Klein-Majda-Damodaran (KMD) model and find a divergence in the mean square vortex position from that of the point vortex model of Onsager at high temperature. We subsequently develop a free energy equation based on the non-interacting case, with a spherical constraint, which we approximate using the method of Kac-Berlin, adding a mean field term for logarithmic interaction. This free energy equation, we use to predict the Monte Carlo results.

cond-mat.stat-mech↗