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Introduction

Unifying Gravity and Quantum Mechanics

Knot Physics uses a geometric model of spacetime to provide a unified description of Unifying Gravity and Quantum Mechanics.

Background

The Rift in Physics


Physics is currently divided into two theories:

Curvature of the spacetime manifold

GENERAL RELATIVITY

In general relativity, gravity is curvature of spacetime.

Probability distribution of a particle

QUANTUM MECHANICS

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

Curvature of the spacetime manifold

GENERAL RELATIVITY

In general relativity, gravity is curvature of spacetime.

Probability distribution of a particle

QUANTUM MECHANICS

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

Unifying these two theories is an important goal of physics.



In Knot Physics, geometric model of spacetime results in quantum mechanics and gravity.

1. Gravity


The spacetime manifold embedded in a 6-dimensional space

SPACETIME IS INSIDE A LARGER SPACE.

In Knot Physics, spacetime is inside of a larger space, and spacetime can bend and move around inside that larger space.

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



Curvature of the spacetime manifold

GRAVITY IS SPACETIME CURVATURE.

As in general relativity, gravity is curvature of spacetime.



2. Particles


Knot on the spacetime manfold

A PARTICLE IS A KNOT IN SPACETIME.

Spacetime can be knotted. Knots in spacetime are elementary particles—for example, electrons and quarks. (On this page, particles and knots refer specifically to elementary fermions.)

More detail: An n-dimensional manifold can be knotted only if it is in an n+2-dimensional space.



Complex amplitudes of knots on the spacetime manifold

KNOTS HAVE COMPLEX AMPLITUDES.

Each knot has an angle relative to spacetime 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 knot has a radius \(r\). Also, because the manifold is in a space that is bigger by 2 dimensions, the knot can rotate in those 2 dimensions at an angle \(θ\). We can combine this information to get a complex number \(k=re^{iθ}\), which is the knot amplitude.



3. Quantum Mechanics


The quantum wave function of a single knot

THE SIMPLEST QUANTUM WAVE FUNCTION IS THE AMPLITUDE OF ONE KNOT.

The quantum wave function describes a particle’s probability distribution. In Knot Physics, a single knot amplitude gives the simplest possible quantum wave function.

More detail: The simplest wave function \(ψ\) is:
$$ψ = \begin{cases} k, & \text{at the knot’s location} \\ 0, & \text{everywhere else.} \end{cases}$$




Quantum wave function of knots on several branches

SPACETIME IS BRANCHED, AND ONE PARTICLE HAS ONE KNOT ON EVERY BRANCH.

In Knot Physics, spacetime consists of multiple branches. Previously, a particle was defined as simply a knot in spacetime; however, branched spacetime necessitates a more precise definition of a particle: one particle has one knot on every branch of spacetime. Each knot can move around on its own branch, independently of the knots on other branches. The single particle—which consists of all those knots—is effectively in multiple places at the same time. This allows “quantum superposition” in which one particle seems to be in multiple places at the same time.



Quantum wave function of knots on many branches

THE QUANTUM WAVE FUNCTION IS THE SUM OF THE KNOT AMPLITUDES ON ALL BRANCHES.

Spacetime can have many branches, and a particle has a knot on each branch. The sum of all the knot amplitudes produces the quantum wave function for the particle.

More detail: 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 knot amplitude. The new knot amplitude is the sum of the knot amplitudes of the recombining knots. This implies that recombination makes knot amplitudes additive in the same way that quantum superposition makes wave functions additive. In this way, branch recombination produces quantum interference.



Unification


Branched spacetime manifold

In Knot Physics, gravity results from the bending of spacetime, and quantum mechanics results from branching. Both bending and branching can occur simultaneously and consistently. In this way, Knot Physics unifies Unifying Gravity and Quantum Mechanics within a single theory.



Learn More


Branched spacetime manifold

Unifying Gravity and Quantum Mechanics

A geometric model of spacetime provides a unified description of gravity and quantum mechanics.

Linked knots

Strong Force

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

Electromagnetic field of two knots

Electroweak

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