A few ideas from knot physics
In knot physics we use a topological approach to describe particles. To do this we embed a 4-dimensional spacetime manifold in a Minkowski 6-space. From this simple assumption, we can derive many physical results.
Something easier to imagine:
Let's begin with a simple example. Suppose we have a spacetime manifold that has 1 dimension of space and 1 dimension of time. We would embed that manifold in 4 dimensions. A constant time slice of the spacetime manifold looks like an infinitely long piece of string embedded in 3 dimensions. By taking consecutive constant time slices, we can watch the string move around. We require that the string cannot pass through itself. This means that a knot in the string cannot be removed. If there are multiple knots in the string then they can move around and interact with each other. In this description, the string is like space and the knots are like particles. This spacetime manifold with 1 space dimension and 1 time dimension has its own physics. Can a similar type of physics also apply to a spacetime with more dimensions?
This is a constant time slice of a spacetime manifold with 1 dimension of space and 1 dimension of time. This constant time slice is knotted.
More like our universe:
Suppose we have a spacetime manifold with n dimensions of space and 1 dimension of time. Is it still possible to have knots on the spacetime manifold? We can consider the constant time slices. Each constant time slice is a n-dimensional manifold. From topology, we know that a n-dimensional manifold can have knots if and only if it is embedded in a n+2-dimensional space. We assume that the spacetime manifold has 3 dimensions of space and 1 dimension of time. Therefore, knots in the spacetime manifold require that the manifold is embedded in a space with 3+2+1=6 dimensions.
Distinguising time and space:
We now have a 4-dimensional manifold embedded in a 6-dimensional space. What distinguishes a "time" direction from a "space" direction? The manifold is embedded in a 6-dimensional space and we assume that the 6-space is Minkowski. The metric on the Minkowski 6-space is diag(1,-1,-1,-1,-1,-1). The metric on the spacetime manifold comes from the metric on the Minkowski 6-space. This gives the spacetime manifold a metric which is Lorentzian almost everywhere. The directions on the spacetime manifold that have positive metric signature are timelike.
How can this make physics?
We know that we need a spacetime manifold to explain physics. Relativity describes the manifold as Lorentzian and gravity results when the metric is not constant. The spacetime manifold we describe here is embedded in a 6-space and it does not need to be flat. When it is not flat, it has curvature. We show in the papers that the equations of its curvature match the equations of general relativity.
But what else can we explain using just the manifold? If particles are knots on the manifold then the properties of those knots should generate the properties of particles. Particle topology affects the manifold geometry. We show in the papers that the affect on manifold geometry explains the observed fields. Therefore we have an explanation of particles and fields that only assumes the spacetime manifold.
Has anyone else tried this?
There is one result that discouraged further research into knots as an explanation of particles: a Lorentzian manifold cannot change topology. If we take constant time slices of a manifold that is everywhere Lorentzian, then those slices must always have the same topology. If the manifold cannot change topology then there can be no knots and therefore no particles in this description. People typically assume that the spacetime manifold is everywhere Lorentzian.
So... that's the end of the story, right?
In knot physics the spacetime manifold is embedded in a Minkowski 6-space. An embedded manifold is Lorentzian if it is moving at less than light speed. If it is moving at light speed then the metric is degenerate. The manifold must have finite energy. By relativity, if the manifold is moving then the energy density is proportional to γ. But γ is infinite at light speed. That means that the manifold must be Lorentzian everywhere except a set of measure zero. The metric is degenerate on the measure zero set. A degenerate metric means the manifold can be kinked. This allows topology change.
An example.
Let's try this with an example in lower dimensions. This is a cylinder.
To describe a spacetime manifold we use a n-manifold embedded in a Minkowski space. In this example our spacetime is a 3-manifold with boundary. This picture is a spacelike slice of the spacetime. It is a 2-manifold with boundary; specifically, it is a cylinder.
Now rotate the cylinder in the direction of the red arrow. Every point is in motion and the blue line segment is moving at light speed.
We can describe the manifold as "moving". We can also describe the manifold as having a shape in space and time. The descriptions are equivalent.
This induces a relativistic length contraction along the blue line segment. The metric is degenerate at the point where the cylinder meets the blue line.
For any speed less than light speed, the manifold is Lorentzian: we can make a change of reference frame that puts the manifold velocity equal to zero. At light speed, there is no such frame change. At that speed the metric is degenerate.
We can twist around that point of degeneracy. This produces a twist in the cylinder.
At the metric degeneracy the manifold tapers to a point and it can rotate freely around that point.
If we stop moving the manifold it returns to being Lorentzian. The metric degeneracy disappears. However, the twist in the cylinder remains. It is now a Mobius strip.
The Mobius strip has different topology than the cylinder. This topology change is a convenient example because it is easy to visualize. However, the Mobius strip is a manifold with boundary, unlike the spacetime manifold. Performing an analogous change on a 2-manifold without boundary makes a P2 (a projective plane). If the manifold has a field (like spacetime) then the P2 must be produced in pairs, like particle/anti-particle pairs.
What happened? Relativity uses a Minkowski metric. If the manifold is always moving at less than light speed then no topology change is possible because the Minkowksi metric makes the manifold everywhere Lorentzian. However, that same metric allows a topology change if we allow some part of the manifold to move at light speed.
We can use this same technique to allow topology change on the spacetime manifold. For a cylinder, the topology change can make a Mobius strip. The degenerate metric allows the manifold to "kink". Then we can twist around that kink. For spacetime, a degenerate metric also produces a kink. When we twist around that kink, the topology change can produce a pair of twists. Each twist is a topology called S1×P2. A S1×P2 is a fermion in knot physics. In the papers we show how S1×P2 can produce fractional spin statistics, charge, and spin angular momentum. The correspondence between S1×P2 and the fermions is precise. There are three generations with particles corresponding to charged leptons, neutrinos, and quarks.
For hadrons we link multiple copies of S1×P2. This is similar to the way that circles can be linked, but in higher dimensions. Each S1×P2 is a quark and the linked quarks are a hadron. Linked S1×P2 cannot be separated, in the same way that quarks cannot be removed from their hadron.
Linked circles embedded in 3 dimensions cannot be separated. Similarly, linked P2 in 4 dimensions cannot be separated. Let L be a link of multiple P2. Then S1×L is a link of multiple S1×P2 in 5 dimensions. Each S1×P2 is a quark and the quarks cannot be separated. This corresponds to quark confinement. At close distances the links exert no force on each other. This corresponds to asymptotic freedom. Extending this idea produces a field equation that is quite similar to QCD.
This is only a brief introduction. The papers cover this material in greater detail as well as many other aspects of the theory. In particular, the papers demonstrate the fields and forces, quantum field theory, and a variety of other topics.
If you find this interesting, feel free to contact me. (Contact info is available in the papers.)
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