Electroweak

Spacetime has an electromagnetic field.

In Knot Physics, the spacetime manifold is embedded in a larger space. There is an electromagnetic field that is defined on every point of spacetime but not elsewhere in the larger space. Changes to the electromagnetic field propagate as photons.

More detail: The spacetime manifold is a 4-dimensional manifold embedded in a 6-dimensional Minkowski space. Both Knot Physics and the Standard Model have an electromagnetic potential \( A^\nu \) and a corresponding electromagnetic field tensor \( F^{\mu\nu} = A^{\nu,\mu} - A^{\mu,\nu} \) . In both cases the action is proportional to \( F^{\mu\nu}F_{\mu\nu} .\) For more information, see Theory Summary: Electromagnetism.

Fermions are knots in spacetime and can have charge.

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

Knots have a topology that can be a source of the electromagnetic field. Knots that are a source of the field are charged fermions.

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. This topology can be a source of the electromagnetic field such that the field cusp is distributed across a torus. For that reason, the electromagnetic field has finite energy. For more information on fermion topology and field energy on fermions see the papers Physics on a Branched Knotted Spacetime Manifold and Deriving the Fine Structure Constant.

On knots, the electromagnetic field has both energy and mass.

The mass of the electromagnetic field is determined by the geometry of spacetime. On flat space, the electromagnetic field has energy but no mass. The geometry of a knot is not parallel to flat spacetime; here, the field has both energy and mass.

More detail: The stress-energy tensor \(T^{\mu\nu}\) is defined at every point on the spacetime manifold, including points that are on a knot. If the knot is in motion, the velocity vector of the knot will be parallel to flat spacetime. Because the knot geometry is not parallel to flat spacetime, the velocity vector of the knot motion may not be in the tangent space of the manifold on the knot. This causes the stress-energy tensor to Lorentz boost with rest mass. For that reason, field energy on a knot has mass. For more information on fermion topology and field energy on fermions see the paper Physics on a Branched Knotted Spacetime Manifold.

Oscillations of the electromagnetic field on a knot propagate as Z bosons.

The Z boson is an oscillation of the electromagnetic field on a knot that leaves the charge unchanged. On a knot, the electromagnetic field has mass; therefore, the Z boson has mass.

Knots can transfer charge with a W boson.

Muon decay is an example of a W boson carrying charge on a knot. The charge of the muon transfers onto the electron via a W boson. The charge moves down the side of the muon knot, which is not parallel to flat spacetime; therefore, the W boson must have mass.