From Planck to the Monster of Sporadic Groups

Could the Universe be a continuum with one light photon following another in a never-ending succession?

We experience this process of energy exchange as Time with a probabilistic future unfolding moment-by-moment, quanta by quanta. The continuous absorption and emission of light, along with the repetition of the quantum wave particle function collapsing and reforming would form a mechanism for generating a probabilistic uncertain future.

Photon spherical oscillations or vibrations precedes everything with the emerging spherical geometry forming greater degrees of freedom for statistical entropy and the spontaneous irreversible processes of classical physics. Such as heat energy always spontaneously flowing from hot to cold and friction always changing motion into heat.

In this theory the mathematics of Quantum Mechanics represent, the physics of time with Classical Physics represents processes over a ‘period of time’ as in Newton's differential equations.

The wave particle duality of light and matter in the form of electrons is forming a blank canvas that we, as atoms, can interact with forming a future relative to the energy and momentum of our actions. 

At the smallest scale of this process, we have the Planck constant as a constant of action. This constant is formed because when there is the absorption and emission of light the energy levels cannot drop below the centre of the electron sphere of probability that surrounds the nucleus of the atom. We have to square the radius because the process is relative to the centre of sphere, formed by the nucleus of the atom, and the spherical surface.

 

A repetition of this geometrical process will naturally form the Fibonacci Spiral and fractal self-similarities we see in nature. Because of this process of spherical symmetry forming and breaking the future is probabilistic, but not totally uncertain. There is a built in potential for greater symmetry to emerge, this formed the potential for the complexity of cell life and emerges of built in geometrical patterns to our mathematic.

In this theory sporadic group, represent exceptional symmetries describing the symmetries that remain after a high degree of initial symmetry as in spherical symmetry that has been broken through a series of quantized steps. This process of spherical symmetry forming and breaking could explain why many sporadic group are linked to sphere-packing lattices, including the Monster, are deeply connected to the symmetries of the Leech lattice, a key structure in the study of dense sphere packing.  Spherical lattice serves as a geometric foundation for the construction and understanding of many of these exceptional finite simple groups.

 The centre of each sphere within the lattice is a discrete set of points within a vector space.

In this theory, this is because of a geometrical process linked to Huygens Principle that says each point on the light wave front acts as a source of a secondary spherical wavelet. Each point on the wave front has the potential for a new photon contributing to the probability amplitude and thus the probability density squared at a future point. The weight of this modular form is half the dimension of the lattice because the process is based on the radius, that is squared, is half the diameter of the lattice. The spherical radiant energy, radiating out from the centre of the sphere could explain the potential for centralizers of an involution of rather smaller sporadic group. Now called the baby monster and the slight variation of it centre of an involution in the monster group.

A process of spherical symmetry forming and breaking can form a simpler uniform construction of all the finite simple groups. Spherical symmetry, a high degree of symmetry, is broken as part of a physical systems to give rise to directional properties and structures explaining why sporadic group are based on a few simple rules about how we are allowed to shuffle things, or why very simple axioms are hiding such complexity and beauty in them. We can think about this just in terms of pictures of a geometrical periodic pattern on the surface of a sphere. 

Something having positive curvature formed by the outer spherical surface and sometimes having constant negative curvature formed by the inner concaved spherical surface. 

If you add a point at infinity this fundamental domain is really a Riemann surface with a complex structure and the mapping that takes a point in tau and maps it to a point on the Riemann sphere is known as the modular j function.

Based on this dynamic geometry the modular J function, which shows up in all sorts of number theory, is linked to the monster group the largest sporadic group forming the idea that sounds so ridiculous that it was called moonshine. The fundamental domain is the Riemann sphere but if you take an arbitrary subgroup then the image of it will be some general Riemann surface, it won't be a sphere, it might be a toroidal doughnut-like shape or something more complicated formed out of broken spherical symmetry. 

In this theory, mathematics is discovered not invented.

The potential for this can be seen as Ford Circles emerging naturally in our mathematics. Pascal's triangle is a good example of this; it is constructed starting with a single '1' at the apex with increasing symmetry and complexity as it cascades down. 

The Fibonacci sequence is within Pascal's triangle, appears along the shallow diagonals of the triangle where each number is the sum of the two preceding ones. 

Everything is based on the same geometrical process forming the unity of mathematics and physics within three-dimensional space with one variable in the form of time.