Gravity and Quantum Mechanics
This course demonstrates how gravity and quantum mechanics work in Knot Physics. Topics covered include the double slit experiment, virtual particles, and spacetime curvature.
The spacetime manifold is a 4-dimensional manifold embedded in a 5+1 Minkowski space.
Knots on Manifolds
A n-manifold in a n+2-dimensional space can be knotted.
Those knots are elementary particles in Knot Physics.
The knots of Knot Physics are topological defects in the
spacetime manifold. Here we show creation of pairs of topological defects on a
2-dimensional manifold. This is analogous to the production of a particle/antiparticle
A Topological Defect
We show a closer view of a topological defect on a
2-dimensional manifold. This is a simplified model of the topological defects of Knot
Topologically Flat Spacetime
If spacetime is topologically flat, then the
curvature constraints require that the only geometric feature of the manifold is
Geometric Degrees of Freedom
A knot (topological defect) on the spacetime
manifold adds geometric degrees of freedom.
In Knot Physics, the spacetime manifold is a branched
manifold. Here we show examples of branches that split and recombine. (Geometric
separation between branches is exaggerated for clarity.)
Branches and Knots
When branches split and recombine, the knots on the
branches also split and recombine.
Branch Weight and Branching
There is a branch weight w defined at every
point on the spacetime manifold. That weight is conserved at branching.
Branch Weight and Geometry
The branch weight w is conserved by stretching
of the manifold.
Branch Weight, Branching, and Geometry
The branch weight w is conserved by
both branching and stretching. This implies a relationship between branching and
Entropy of Virtual Particles
The spacetime manifold can spontaneously
produce virtual particles. These virtual particles contribute to the entropy. (Here we
show the production of virtual photons. For more on electromagnetism, please see the
Entropy of Branching
Recombination of branches of the manifold is random.
This recombination contributes to the entropy.
Entropy of branch recombination is reduced when branches
are far from each other. For this reason, branches tend to stay close to each other. We
call this branch cohesion.
Knots on the spacetime manifold can rotate and change size.
We describe this with a single complex number, which we call the knot amplitude.
When knots recombine, their knot amplitudes recombine to
the weighted average amplitude.
We can define a quantum amplitude that is based on the knot
amplitude. The quantum amplitude is additive at recombination. (For derivation of the
Born rule of quantum probability, please see the presentations.)
Quantum Path Integral
Using the formula for addition of quantum amplitudes,
we derive a formula for summing over the contributions of all knots. In the limit, the
formula converges to a path integral.
Single Particle Distribution
Using the notion of quantum interference from
the path integral, we can demonstrate the behavior of a single real particle. A single
real particle has many knots, with one knot on each branch of the manifold.
Double Slit Experiment
The double slit experiment displays quantum
interference. Here we show an example of the double slit experiment as it works in Knot
Physics. (For a description of quantum measurement, please see the paper.)
Geometry of Spacetime
We show how an embedded spacetime manifold has
geometric characteristics that qualitatively match general relativity.
Lagrangian of Curvature
Branch recombination has entropy. Stretching
reduces branch recombination. We give a formula relating stretching and entropy. The
formula has the form of a Lagrangian.
Lagrangian of Matter
Matter reduces the entropy of the spacetime manifold.
We give a formula relating matter and entropy. The formula has the form of a
We combine spacetime geometry with entropy to produce a
Lagrangian that has the form of the Einstein-Hilbert action of general relativity.