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# Papers

These papers cover theory fundamentals as well as a variety of topics, including entanglement and dark matter.

## Fundamentals

This paper describes the core of the theory, and provides necessary background for the rest of the papers.

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.

C. Ellgen & G. Biehle

C. Ellgen & G. Biehle

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## Topics

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 \(\alpha^{-1} \approx 136.85\), the error is \(0.13 \% \).
Including the effects of virtual particles may reduce the error further.

C. Ellgen

C. Ellgen

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- Fine structure constant calculation: knot_fine_structure_calculate.nb
- Calculation function verification: fine_structure_function_verification.nb

Calculations as Mathematica files:

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.

C. Ellgen

C. Ellgen

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We describe features of cosmology in Knot Physics that have the characteristics of dark
energy. Knot Physics assumes that spacetime is a branched 4-manifold embedded in a
Minkowski 6-space. The cosmology of an embedded spacetime manifold is described by the
expansion and contraction of the manifold in the embedding space. We show that the
motion of the manifold in the embedding space contributes to the redshift of photons on
the manifold. In this way, the embedded manifold model provides an alternative
explanation for the redshift data that has been used as evidence for dark energy.

C. Ellgen

C. Ellgen

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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.

C. Ellgen

C. Ellgen

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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.

C. Ellgen

C. Ellgen

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