# München 2019 – wissenschaftliches Programm

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# GR: Fachverband Gravitation und Relativitätstheorie

## GR 17: Alternative Approaches

### GR 17.4: Vortrag

### Freitag, 22. März 2019, 12:15–12:30, HS 5

**Biquadric Fields on a Finite Geometry as a Quantum World** — •Judith Höfer, Alexander Laska, and Klaus Mecke — Institut für Theoretische Physik 1, FAU Erlangen-Nürnberg, Germany

A unification of quantum field theory and general relativity might be based on finite projective geometry [1]. To this end the standard approach of using real (or complex) numbers as number field for coordinates (or wavefunctions) is replaced by a Galois field. 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, distances and neighbourhoods. 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 work a local domain is defined as the subspace where an Euclidean-like ordering of the points is possible, and the coordinate transformations between different local domains are investigated. Furthermore, the finite field and the neighbouring relations defined by a homogeneous field of biquadrics are interpreted as a graph and its diameter is explored in order to elucidate non-local features in finite geometries.

[1] Klaus Mecke, Biquadrics configure Finite Projective Geometry into a Quantum Spacetime, EPL 120(1), 10007 (2017).