“Models of noncommutative differential geometries”
Abstract: Quantum spacetime is the idea that spacetime coordinates are not classical variables but elements of a noncommutative ‘co-ordinate algebra’ much as in quantum theory. The first convincing models of (flat) quantum spacetime appeared in the 1990’s, typically on the basis of quantum symmetry using the then-new theory of quantum groups. The thinking is that such noncommutativity should arise from quantum gravity effects and allows us to model these in an effective description without full knowledge of quantum gravity itself (this not being known). Such flat quantum spacetimes are an area of existing strength for EU science. The project proposes a new generation of quantum spacetime models no longer tied to quantum symmetry developed within mathematical theory of ‘noncommutative Riemannian geometry’ including quantum differentials, quantum metrics, and quantum-Levi Civita connections and quantum curvature. The realisation of the project will validate and enrich the mathematical formalism and also promises a next generation of physical effects related to the conjunction of both gravity and quantum noncommutativy, which will stimulate original and creative approaches to quantum gravity across several EU institutes.
The outcomes of the NCDIFFGEO project:
with Shahn Majid
We study parallelisable bimodule noncommutative Riemannian geometries in small dimensions, working over the field F2 of 2 elements and with coordinate algebras up to dimension n≤3. We find a rich moduli of examples for n=3 and top form degree 2, many of them not flat. Their coordinate algebras are commutative but their differentials are not. We also study the quantum Laplacian Δ=( , )∇d on our models and characterise when it has a massive mode.
with Shahn Majid
It is known that connected translation invariant n-dimensional noncommutative differentials dxi on the algebra k[x1,⋯,xn] of polynomials in n-variables over a field k are classified by commutative algebras V on the vector space spanned by the coordinates. This data also applies to construct differentials on the Heisenberg algebra ‘spacetime’ with relations [xμ,xν]=λΘμν where Θ is an antisymmetric matrix as well as to Lie algebras with pre-Lie algebra structures. We specialise the general theory to the field k= F2 of two elements, in which case translation invariant metrics (i.e. with constant coefficients) are equivalent to making V a Frobenius algebras. We classify all of these and their quantum Levi-Civita bimodule connections for n=2,3, with partial results for n=4. For n=2 we find 3 inequivalent differential structures admitting 1,2 and 3 invariant metrics respectively. For n=3 we find 6 differential structures admitting 0,1,2,3,4,7 invariant metrics respectively. We give some examples for n=4 and general n. Surprisingly, not all our geometries for n≥2 have zero quantum Riemann curvature. Quantum gravity is normally seen as a weighted `sum’ over all possible metrics but our results are a step towards a deeper approach in which we must also `sum’ over differential structures. Over F2 we construct some of our algebras and associated structures by digital gates, opening up the possibility of ‘digital geometry’.
The following additional papers were supported by the NCDIFFGEO project:
- Observables and Dispersion Relations in kappa-Minkowski Spacetime. Its a result of collaboration with Paolo Aschieri and Andrzej Borowiec .Published in Journal of High Energy Physics October 2017, 2017:152.
- Remarks on simple interpolation between Jordanian twists. Its a result of collaboration with the group from Ruder Boskovic Institute in Zagreb (Croatia). Published in Journal of Physics A: Mathematical and Theoretical, Volume 50, Number 26 (2017).
- Twisted bialgebroids versus bialgebroids from Drinfeld twist. Its a result of collaboration with Andrzej Borowiec from University of Wroclaw (Poland). Published in Journal of Physics A: Mathematical and Theoretical, Volume 50, Number 5 (2017).