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- Local Domains and Graph Diameter of Biquadric Fields in a Finite Geometry
- supervised by Klaus Mecke
- finished 2018-12
- [PDF]

In modern physics there are two main models describing very accurately different parts of physical phenomena. General relativity uses geometry and the properties of spacetime itself to describe the effects of gravitation where this spacetime is influenced by mass and energy. Quantum mechanics and quantum field theory, on the other hand, describe the physics of elementary and other small particles by the use of a wave function describing the probability amplitude and fields, respectively, of a quantum object. However, these two models describing the same nature, just on different scales, do differ quite a bit in their concepts. The quest of finding a theory that describes both parts, quantum effects and cosmic geometry, is still on.

In this and related work it is tested if such a unification of quantum field theory and general relativity might be based on finite projective geometry [1]. Therefore, the usual approach of using real (or complex) numbers as number field for coordinates (or wavefunctions) to perform calculations in is left behind in favor of a Finite Projective Geometry. The idea is to model spacetime similar to general relativity but based on a finite field such that quantization is not additionally imposed but emerges intrinsically from the finite geometry. Then, singularities and divergences cannot exist neither in a curved spacetime nor in a quantum world modeled over finite fields. However, the quite unusual properties of finite fields require additional care in defining physical quantities. Central to this approach is a `biquadric' that defines, similar to a metric, an idea of `closeness'. The long-time goal is to derive the properties of the standard model in a continuum limit for very large finite fields.

In the presented thesis, the local (world) domain, defining a subspace in the geometry where an Euclidean-like ordering of the points is possible, and the transformations between different local world domains are investigated. It is proven that the prior assumption that biquadric points (different from the ones selected as new basis vectors for the local coordinate system) are not part of the local world domain has to be dropped since it is indeed possible that these can be within the local world domain. For the transformation statistics investigating the possible transformations from the local world domain of one local coordinate system into the local world domain of other local coordinate systems some common traits for all considered cases are found. Most of the points within a local world domain can be successfully transformed into other local world domains about two times the prime order p of the Galois field, with a few about twice as much as this. The shape of the distribution resembles a Gaussian distribution and gets narrower for increasing prime numbers p.

Furthermore, the finite field and the neighbouring relations which are defined by the biquadrics are interpreted as a graph and its graph diameter is explored. It is proven that the diameter of a quadric field is diam = 2 in the general case of dimension d ≥ 3 and diam ≤ 3 for dimension d = 2.

[1] Mecke, K. (2017). Biquadrics configure finite projective geometry into a quantum spacetime. EPL (Europhysics Letters), 120(1).