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Introduction

# Strong Force

In Knot Physics, a geometric theory of quarks results in asymptotic freedom, confinement, and gluons.

Background

## Strong Force in the Standard Model

In the Standard Model of physics, particles like protons and neutrons consist of quarks, which are bound together by the strong force. Quarks that are bound by the strong force obey properties known as asymptotic freedom and confinement.

## Strong Force in Knot Physics

Knot Physics uses geometry to describe how quarks bind to each other.

FERMIONS ARE KNOTS IN SPACETIME.

In Knot Physics, spacetime is inside of a larger space and can be knotted. Knots in spacetime are the elementary fermions—for example, quarks and electrons.

More detail: The spacetime manifold $$M$$ is a 4-dimensional manifold embedded in a 6-dimensional Minkowski space. An n-dimensional manifold can be knotted only if it is in an n+2-dimensional space. For more information on spacetime embedding and knots, see Gravity and Quantum Mechanics.

Knots in spacetime can link to each other, and linked knots are quarks. For example, a proton consists of three linked knots.

More detail: An elementary fermion is a topological defect with homeomorphism class $$\mathbb{R}^3 \# (S^1 \times P^2)$$, referred to here as a knot. Embeddings of these topological defects can link such that they cannot be separated from each other. This linking can only occur because spacetime is embedded in an n+2-dimensional space.

One property of the strong force is asymptotic freedom: quarks that are close to each other do not exert much force on each other.

In Knot Physics, asymptotic freedom occurs because linked knots do not exert force on each other when they are closer to each other than the knot radius.

Another property of the strong force is confinement: quarks cannot be separated from each other.

In Knot Physics, linked knots cannot be separated.

GLUONS ARE THE FORCE BETWEEN LINKED KNOTS.

The force-carrier boson of the strong force is called the gluon. The gluon is responsible for the force between quarks.

In Knot Physics, when linked knots are pulled away from each other, they are pulled back by the other linked knots. This pulling force performs the same function as gluons do in the Standard Model.

## Summary

Fermions are knots in spacetime, and linked knots are quarks. Linking implies asymptotic freedom, confinement, and gluons—the properties of strong force. Strong force is therefore a consequence of the linking of knots.

More detail

### QCD

Each quark is displaced from the center of the particle by a displacement vector $$q$$. Because spacetime is embedded in 5 spatial dimensions, each displacement vector $$q$$ is a 5-vector. Let the center of the particle be the origin of coordinates, then the quark displacement vectors sum to zero. For example, a baryon has three quarks with quark vectors $$a$$, $$b$$, $$c$$, and the quark vectors must sum to zero, which is to say $$a + b + c = 0$$. Because the quarks must remain within a certain distance of the center of the particle, the magnitude of each $$q$$ must be less than a certain distance. In arbitrary units, each $$q$$ must have magnitude less than or equal to 1; that is, $$|q| \le 1$$. This is equivalent to adding a non-physical sixth coordinate to each $$q$$ to make a $$q'$$ such that $$|q'| \le 1$$. In other words, convert $$q = (q_1, q_2, q_3, q_4, q_5)$$ to $$q' = (q_1, q_2, q_3, q_4, q_5, q_6)$$, choosing $$q_6$$ such that $$|q'| = 1$$. Then convert $$q'$$ from a real 6-vector to a complex 3-vector $$q’’ = (q_1 + iq_2, q_3 + iq_4, q_5 + iq_6)$$. Now each quark has a complex 3-vector of unit magnitude. The addition of the sixth coordinate implies that the vectors may not sum exactly to zero, but it will be a good estimate.

The linked knots can therefore be approximately described with complex 3-vectors of unit magnitude that sum to zero, which is the description of color charge. Furthermore, the quark displacements have no effect on the Lagrangian as long as the quarks are close enough to each other, which is guaranteed by $$|q’’|=1$$ . A gauge group can therefore be used. For complex 3-vectors, it suffices to use the gauge group $$\rm{SU}(3)$$. This reproduces the description of quantum chromodynamics.