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Introduction

Unifying Gravity and Quantum Mechanics

A branched spacetime manifold enables a unified description of gravity and quantum mechanics.

Background

The Rift in Physics

One of the largest problems in theoretical physics today is the discrepancy between general relativity and quantum mechanics.

General Relativity

In general relativity, gravity is curvature of spacetime.

Quantum Mechanics

Quantum mechanics describes a particle’s probability distribution using a complex wave function.

Unifying these two theories is a primary goal of theoretical physics.

In Knot Physics, a branched spacetime manifold enables a unified description of gravity and quantum mechanics.

Gravity

Spacetime is embedded in a larger space.

In Knot Physics, the spacetime manifold is embedded in a larger space.

More detail: The spacetime manifold is a 4-dimensional manifold embedded in a 6-dimensional Minkowski space.

Gravity is curvature of spacetime.

The geometry of the embedding may have curvature. As in general relativity, gravity is curvature of spacetime.

Fermions

A fermion is a knot in spacetime.

The spacetime manifold can be knotted. Knots in spacetime are elementary fermions—for example, electrons and quarks.

More detail: The constraints on the spacetime manifold allow it to pass through a singular state that produces a pair of topological defects. These topological defects are the fermions of Knot Physics, and we often refer to them as "knots." For more information, see Theory Summary: Fermions.

Knots have complex amplitudes.

Each knot has an angle relative to the spacetime manifold and a size. The knot’s angle and size can be described with a complex number, which we call the knot amplitude.

More detail: The spacetime manifold is embedded in a space that is larger by 2 dimensions. Knots on the spacetime manifold can rotate and change size in those 2 additional dimensions. The rotation and size of the knot can be described with an angle $$θ$$ and magnitude $$r$$. We can combine this information to get a complex number $$k=re^{iθ}$$, which is the knot amplitude.

Quantum Mechanics

Spacetime is branched, and one fermion has one knot on every branch.

In Knot Physics, spacetime is a branched manifold. Previously, a fermion was defined as simply a knot in spacetime, but branched spacetime necessitates a more precise definition: A fermion consists of one knot on each branch of spacetime. Each knot can move around on its own branch, independently of the knots on other branches.

The knot amplitudes on all branches produce the quantum wave function.

Spacetime consists of a large, but finite, number of branches. A fermion consists of a knot on each branch, and each knot has a knot amplitude. The knot amplitudes of the knots on all branches can be used to produce the quantum wave function for the fermion.

More detail: In Theory Summary: Assumptions, we introduce a branch weight $$w$$ that is defined at every point on the spacetime manifold. Multiplying the knot amplitude $$k$$ by the branch weight $$w$$ produces a complex number $$wk$$, which is the quantum amplitude for the knot. The sum of the quantum amplitudes of the knots on all branches is the quantum wave function for the fermion.

Summary

In Knot Physics, simple assumptions about the spacetime manifold result in both gravity and quantum mechanics. Curvature of spacetime results in gravity, while branches of spacetime result in quantum behavior.

More Detail: Quantum Interference

The spacetime manifold is a branched manifold. Each branch of the spacetime manifold can be considered a “history” in the sum-over-histories description of quantum mechanics. The branches split and recombine. Recombination of the branches causes recombination of knots. When two knots recombine, they recombine to a single knot which has a new size and angle and, therefore, a new quantum amplitude. The new quantum amplitude is the sum of the quantum amplitudes of the recombining knots. In this way, branch recombination produces quantum interference. For more information, see Theory Summary: Quantum Mechanics.

Unifying Gravity and Quantum Mechanics

A branched spacetime manifold enables a unified description of gravity and quantum mechanics.

Strong Force

A geometric theory of quarks results in asymptotic freedom, confinement, and gluons.

Electroweak

Electroweak unification is a consequence of including knot geometry in the description of the electromagnetic field.

Theory Summary

An overview of the entire theory, from simple assumptions about the spacetime manifold through particles, quantum mechanics, and forces