Coupling any interacting quantum mechanical system to gravity in one (time) dimension requires the cosmological constant to belong to the matter energy spectrum and thus to be quantised, even though the gravity sector is free of any quantum dynamics. Furthermore, physical states are also confined to the subspace of the matter quantum states for which the energy coincides with the value of the c...

This Festschrift in honour of J. A. de Azcárraga gives an introduction to the concept of duality, i.e., to the relativity of the notion of a quantum, in the context of the quantum mechanics of a finite number of degrees of freedom. Although the concept of duality arises in string and M–theory, Vafa has argued that it should also have a counterpart in quantum mechanics, before moving on to secon...

We reviewed the canonical quantisation of geometry spacetime in cases a simply and non-simply connected manifold. In former, we analysed information contained solutions Wheeler–DeWitt equation showed their interpretation terms customary boundary conditions that are typically imposed on semiclassical wave functions. particular, three different paradigms for quantum creation homogeneous isotropic...

These are short notes from a series of lectures given at the University of Rennes in June 2013, at the University of Bonn in July 2013, at the XVIIth Brazilian School of Probability in Mambucaba in August 2013, and at ETH Zurich in September 2013. They give a concise overview of the theory of regularity structures as exposed in the article [Hai14]. In order to allow to focus on the conceptual a...

The Hilbert space of a free massless particle moving on a group manifold is studied in details using canonical quantisation. While the simplest model is invariant under a global symmetry, G × G, there is a very natural way to “factorise” the theory so that only one copy of the global symmetry is preserved. In the case of G = SU(2), a simple deformation of the quantised theory is proposed to giv...

Motivated by the success of classical random walks and chaotic dynamical systems, we study the quantisation of the random walk over Z and its relationships with a classical chaotic system x → 2xmod1, x ∈ [0,1]. In the physics literature, quantum random walks have been studied, for instance, by Ambainis et al. [1] and Konno et al. [8, 9]. In [8], Konno shows that the quantum random walk over Z c...

In quantum physics, the operators associated with the position and the momentum of a particle are unbounded operators and C∗-algebraic quantisation does therefore not deal with such operators. In the present article, I propose a quantisation of the Lie-Poisson structure of the dual of a Lie algebroid which deals with a big enough class of functions to include the above mentioned example. As an ...

In three-dimensional QED, a quantum Hall effect occurs in the absence of any magnetic field. We give a simple and transparent explanation. In solid-state physics, the Hall conductivity for rational magnetic flux is expected to be given by a Chern number. In our field-free situation, however, the conductivity is ±1/2 in natural units. We explain why the integrality of the conductivity breaks dow...

The classical mechanics of a finite number of degrees of freedom requires a symplectic structure on phase space C, but it is independent of any complex structure. On the contrary, the quantum theory is intimately linked with the choice of a complex structure on C. When the latter is a complex–analytic manifold admitting just one complex structure, there is a unique quantisation whose classical ...