Since its creation at the beginning of the 20th century, Quantum Mechanics (QM) has posed challenging mathematical problems, for example, the determination of the spectra of operators or finding representations of operator algebras. Especially after the introduction of Quantum Field Theory (QFT), the mathematical problems became more difficult; one of the main tools to help with their solution in the case of the Standard Model being the idea of a path integral and the idea of a renormalizible QFT. However, these new mathematical ideas turned out to be inadequate for the problem of quantizing gravity, leading to the development of various new mathematical approaches such as supersymmetry, string theory, noncommutative geometry and state-sum models.

At GFM we study quantum gravity using the path-integral approach where the smooth-manifold spacetime is replaced with a piecewise linear (PL) manifold, which represents a triangulation of the smooth manifold. Defining a smooth-manifold limit of a PL path integral is still a difficult problem; however, if one assumes that the short-distance spacetime structure is a PL manifold, then the smooth spacetime can be considered as an approximation when the average edge length in a spacetime region is sufficiently small and there is a large number of simplices. In this case the corresponding effective action can be approximated by a QFT effective action for the smooth spacetime, such that the corresponding QFT has a cutoff determined by the average edge length.

We are also interested in Higher Gauge Theory, as the use of higher categorical groups is an elegant way to generalize gauge symmetry. One can then obtain new topological QFTs, as well as new ways to formulate the symmetries for elementary particles and gauge fields.

Standard quantum mechanics is also an area of interest for us; the developement of quantum computers has stimulated interest in quantum systems defined in finite-dimensional Hilbert spaces. We investigate the general problem of constructing simple and computable criteria to characterize classical and quantum states, as well as pure, mixed, separable and entangled quantum states for systems defined in infinite-dimensional Hilbert spaces. We also study the problem of characterising the general form of the mappings between these sets of states.

These are longstanding problems whose (partial) solutions are important for applications in quantum information theory and quantum computing. We work mainly in the context of the phase space formulation of quantum mechanics which, in our view, provides a superior framework to address these problems, unifying the formulation of both classical and quantum mechanics, and of the different types of quantum states. The central objects of this formulation are quantum distributions, particularly the Wigner function. Our work is especially focussed on studying the general properties of these objects, and in exploring the uncertainty principle in order to construct the necessary conditions for a quantum behaviour.

A further area of interest at GFM is exactly integrable classical and quantum systems, in particular, the exactly integrable models of General Relativity and Gaudin models.

Concretely, we are currently working on the following topics:

- Higher gauge theory
- State-sum models of quantum gravity and TQFTs
- Piecewise flat quantum gravity
- Criteria for quantumness
- Phase space quantum mechanics
- Algebraic Bethe ansatz

Members working in this area: Nuno Costa Dias, Nenad Manojlović, Aleksandar Miković, João Nuno Prata

Recent book:

- A. Mikovic and M. Vojinovic, State-sum models of PL quantum gravity, World Scientific, 2023

URL: https://doi.org/10.1142/13233

Selected other recent publications:

- N. C. Dias, M. de Gosson and J. N. Prata, Partial traces and the geometry of entanglement; sufficient conditions for the separability of Gaussian states,
*Rev. Math. Phys.*34 (2022), 2250005

URL: https://doi.org/10.1142/S0129055X22500052 - N. C. Dias, F. Luef and J. N. Prata, Uncertainty Principle Via Variational Calculus on Modulation Spaces,
*J. Funct. Anal.*283 (2022), 109605

URL: https://doi.org/10.1016/j.jfa.2022.109605 - N. C. Dias and J. N. Prata, Quantum mappings acting by coordinate transformations on Wigner distributions,
*Rev. Mat. Iberoamericana*35 (2019), 317-337

URL: https://doi.org/10.4171/rmi/1056 - N. Manojlovic and I. Salom, Rational so(3) Gaudin model with general boundary terms,
*Nucl. Phys. B*978 (2022), 115747.

URL: https://doi.org/10.1016/j.nuclphysb.2022.115747 - A. Mikovic and N. Manojlovic, Self-dual cosmological Bianchi IX and VIII metrics,
*Adv. Theor. Math. Phys.*26 (2022), 2841-2857

URL: https://dx.doi.org/10.4310/ATMP.2022.v26.n8.a10