Literature

Explore the literature of Knot Physics, from its foundational account of branched spacetime to recent papers developing its mathematical structure, physical applications, and testable implications.

Research papers

Published papers and preprints presenting mathematical and physical developments in Knot Physics.

Finite Path Integrals on Stochastic Branched Structures

Roukaya Dekhil, Clifford Ellgen, & Bruno Klajn (Mar 2026)

Published in J. Phys. A: Math. Theor. (2026) • DOI: 10.1088/1751-8121/ae513a (opens in a new tab)

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In this paper, we present a statistical model of spacetime trajectories based on a finite collection of paths organized into a branched manifold. For each configuration of the branched manifold, we define a Shannon entropy. Given the variational nature of both the action in physics and the entropy in statistical mechanics, we explore the hypothesis that the classical action is proportional to this entropy. Under this assumption, we derive a Wick-rotated version of the path integral that remains finite and exhibits both quantum interference at the microscopic level and classical determinism at the macroscopic scale. In effect, this version of the path integral differs from the standard one because it assigns weights of non-uniform magnitude to different paths. The model suggests that wave function collapse can be interpreted as a consequence of entropy maximization. Although still idealized, this framework provides a possible route toward unifying quantum and classical descriptions within a common finite-entropy structure.

Geometric Inflation and Late-Time Cosmic Acceleration from Embedded Spacetime Dynamics

Ali Nayeri & Clifford Ellgen (Jan 2026)

Preprint

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Abstract: We develop a cosmological framework in which spacetime is treated as a four-dimensional manifold dynamically embedded in a higher-dimensional flat Minkowski background. Ultrarelativistic motion of the embedded manifold induces strong time-dilation effects between embedding time and proper time, generating a genuine phase of inflation with strict exponential expansion for comoving observers, without invoking an inflaton field or scalar potential. The inflationary phase satisfies the defining kinematic criteria, including a shrinking comoving Hubble radius, and admits a natural graceful exit as time dilation weakens. At late times, large-scale embedding dynamics give rise to a geometric expansion attractor that yields sustained cosmic acceleration without a bare cosmological constant. More generally, the attractor can be quasi-stationary, allowing a slow weakening of the effective acceleration rate while remaining non-phantom. Small deviations from uniform embedding motion excite long-wavelength co-dimensional modes that generate subdominant oscillatory corrections to the expansion rate. We derive the structure of linear perturbations arising from embedding fluctuations and show that they naturally produce nearly scale-invariant curvature perturbations with a suppressed tensor-to-scalar ratio. This framework provides a unified geometric origin for inflation, primordial structure, and late-time acceleration, without new fields or fine tuning.
Incorporating Gravity into the Path Integral of Quantum Mechanics Using the Thermodynamics of Spacetime

Garrett Biehle, Clifford Ellgen, Bassem Sabra, & Sebastian Zając (Nov 2022)

Preprint

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We use principles from the thermodynamics of spacetime to modify the path integral of quantum mechanics. Entropy of the vacuum is interpreted as microstates that correspond to the measure of the path integral. The result is a contribution to the action that is proportional to the Einstein-Hilbert action. Because the contribution is real, not imaginary, it is likely to result in convergence in many cases. Paths that minimize the Einstein-Hilbert action make the largest contribution to the path integral, implying that the maximum likelihood paths are solutions of the Einstein equation.

Foundational texts

The primary reference brings Knot Physics together in one self-contained account. The materials that follow develop particular foundations, calculations, and physical applications.

Primary Reference

Knot Physics: A Complete Description of the Theory

This self-contained document presents the complete Knot Physics research program, including its quantitative results, testable predictions, limitations, and open questions. Use this book as the primary reference for technical evaluation.

Clifford Ellgen (Aug 2026)

Research Manuscript

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This document is a complete description of the Knot Physics research program: a theory in which spacetime is a branched 4-dimensional manifold embedded in a flat 6-dimensional Minkowski space, matter consists of topological defects in that manifold, and both quantum mechanics and gravity emerge from the statistical mechanics of the branches. It is written to be evaluated, not merely read. Every load-bearing derivation appears in full, in one consistent notation, so that a careful reader—human or machine—can check the mathematics directly rather than take the claims on trust.

Preface continues in the book.

Physics on a Branched Knotted Spacetime Manifold

Clifford Ellgen & Garrett Biehle (Aug 2021)

Research Manuscript

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This paper reproduces the dynamics of quantum mechanics with a four-dimensional spacetime manifold that is branched and embedded in a six-dimensional Minkowski space. Elementary fermions are represented by knots in the manifold, and these knots have the properties of the familiar particles. We derive a continuous model that approximates the behavior of the manifold's discrete branches. The model produces dynamics on the manifold that corresponds to the gravitational, strong, and electroweak interactions.

(Minor updates made on 8/23/2021)
Knot Physics: Deriving the Fine Structure Constant

Clifford Ellgen (Sep 2015)

Research Manuscript

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Knot Physics describes the geometry of particles and fields. In a previous paper we described the topology and geometry of an electron. From the geometry of an electron we can construct a mathematical model relating its charge to its spin angular momentum. From experimental data, the spin angular momentum is \(ℏ/2.\) Therefore the mathematical model provides a comparison of electron charge to Planck's constant, which gives the fine structure constant \(\alpha\). We find that using only electromagnetic momentum to derive the fine structure constant predicts a value for \(\alpha^{-1}\) that is about two orders of magnitude too small. However, the equations of Knot Physics imply that the electromagnetic field cusp must be compensated by a geometric field cusp. The geometric cusp is the source of a geometric field. The geometric field has momentum that is significantly larger than the momentum from the electromagnetic field. The angular momentum of the two fields together predicts a fine structure constant of \(\alpha^{-1} \approx 136.85.\) Compared to the actual value of \(\alpha^{-1} \approx 137.04,\) the error is \(0.13 \% .\) Including the effects of virtual particles may reduce the error further.

Calculations as Mathematica files:

Knot Physics: Entanglement and Locality

Clifford Ellgen (Jul 2016)

Research Manuscript

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We describe entanglement and locality in Knot Physics. In Knot Physics, spacetime is a branched manifold. The quantum information of a system is encoded in the branches of the manifold. We show how that quantum information can persist despite the continual recombination of the branches of the manifold. We also note that the quantum collapse of state of the branches is non-local. That non-locality allows for non-local effects of entanglement without additional assumptions. We apply this description to the EPR paradox.
Knot Physics: Dark Energy

Clifford Ellgen (Jan 2023)

Research Manuscript

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We assume that spacetime is embedded in a Minkowski space and the metric on spacetime is induced by the Minkowski metric. Expansion of spacetime causes a redshift that corresponds to the usual cosmological redshift of general relativity. Changing expansion velocity also affects the redshift and introduces an additional term that is not included in the redshift effect attributed to general relativity. This extra contribution may explain the difference between astronomical data and the redshift predictions of general relativity.

Scope of this result: the additional redshift contribution by itself is too small to explain late-time acceleration. Knot Physics instead models cosmic acceleration using effective rigidity terms and a geometric expansion attractor; deriving those effective terms from the microscopic branch model remains open.

Knot Physics: Dark Matter

Clifford Ellgen (Jul 2016)

Research Manuscript

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We describe dark matter in Knot Physics. Knot Physics assumes that spacetime is a branched 4-manifold embedded in a Minkowski 6-space. The theory has three metrics. The Minkowski space has the standard Minkowski metric \(\eta_{\mu\nu}.\) The second metric, \(\bar \eta_{\mu\nu},\) is just the restriction of \(\eta_{\mu\nu}\) to the spacetime manifold. In a previous work, we showed how mass and energy affect the curvature of \(\bar \eta_{\mu\nu}\) , reproducing results of general relativity. The third metric, \(g_{\mu\nu},\) is used to constrain the branches of the spacetime manifold. In this paper, we derive an approximate relationship between \(\bar \eta_{\mu\nu}\) and \(g_{\mu\nu}.\) The relationship implies \(\bar \eta_{\mu\nu}\) can have non-zero Ricci curvature without a massive source particle. We show how this result has many of the characteristics of dark matter.
Knot Physics: Neutrino Helicity

Clifford Ellgen (Mar 2015)

Research Manuscript

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We use the assumptions of Knot Physics to prove that a collection of interacting neutrinos and antineutrinos maximize their quantum probability when all neutrinos are of the same helicity and all antineutrinos are of the opposite helicity. In a previous paper we showed that the geometry of gravity spontaneously breaks symmetry. We show here that the geometry of gravity couples the neutrino linear momentum to its quantum phase. Likewise, the quantum phase of an interacting neutrino couples to its spin angular momentum. Therefore, the symmetry breaking of gravity couples the linear momentum of an interacting neutrino to its spin angular momentum, producing consistent helicity.
Knot Physics: Vacuum Geometry

Clifford Ellgen & Bassem Sabra (May 2022)

Research Manuscript—Draft

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In a previous work, we describe a branched four-dimensional spacetime manifold embedded in a six-dimensional Minkowski space. In this paper, we provide additional information about the geometry of the vacuum. In this description, the classical vacuum can be described as Lorentz invariant, and the quantum vacuum is best described as Lorentz isotropic. We provide evidence that the vacuum has properties corresponding to a vacuum energy and a vacuum temperature.

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