Quantum Geons and Noncommutative Spacetimes

TL;DR

This paper explores noncommutative spacetime models for quantum geons, generalizing twist operations to complex groups, leading to non-associative algebras and potential violations of fundamental principles like the Pauli exclusion principle.

Contribution

It introduces a generalized Drinfel'd twist for complex groups, enabling the construction of noncommutative and non-associative spacetime algebras for geons with rich topological structures.

Findings

Noncommutative geon spacetimes support twisted diffeomorphism actions.

Non-associative algebras naturally arise in this framework.

Potential physical effects include violations of the Pauli principle.

Abstract

Physical considerations strongly indicate that spacetime at Planck scales is noncommutative. A popular model for such a spacetime is the Moyal plane. The Poincar\`e group algebra acts on it with a Drinfel'd-twisted coproduct. But the latter is not appropriate for more complicated spacetimes such as those containing the Friedman-Sorkin (topological) geons. They have rich diffeomorphism groups and in particular mapping class groups, so that the statistics groups for N identical geons is strikingly different from the permutation group $S_N$. We generalise the Drinfel'd twist to (essentially) generic groups including to finite and discrete ones and use it to modify the commutative spacetime algebras of geons as well to noncommutative algebras. The latter support twisted actions of diffeos of geon spacetimes and associated twisted statistics. The notion of covariant fields for geons is formulated and their twisted versions are constructed from their untwisted versions. Non-associative spacetime algebras arise naturally in our analysis. Physical consequences, such as the violation of Pauli principle, seem to be the outcomes of such nonassociativity. The richness of the statistics groups of identical geons comes from the nontrivial fundamental groups of their spatial slices. As discussed long ago, extended objects like rings and D-branes also have similar rich fundamental groups. This work is recalled and its relevance to the present quantum geon context is pointed out.