Mathematical Physics

One of the main challenges that we face both as individual persons and as a species concerns the distribution and use of resources, such as water, time, capital, computing power or negatively valued
resources like nuclear waste. Also within theoretical physics, one frequently deals with resources like free energy or quantum entanglement. I will describe a mathematical theory of resources which makes quantitative predictions about how many resources are required for producing a certain commodity and outline some applications to information theory.
Joint work with Bob Coecke and Rob Spekkens.

Modelling continuum dynamics on discrete space time
We will discuss perfect discretizations which aim at mirroring exactly continuum physics on a given lattice. Such discretizations avoid typical artifacts like Lorentz violation, energy dissipation, particle doubling and in particular breaking of diffeomorphism symmetry. Thus the question arises how to distinguish such lattice dynamics from continuum physics.

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Turning Weyl’s tile argument into a mathematically rigorous no-go theorem
Weyl's tile argument notes that if space was fundamentally discrete then the set of allowed velocities of a classical particle would not be isotropic. I will generalize Weyl's heuristic argument to a no-go theorem applying to any discrete periodic structure. Since this theorem does not take quantum mechanics into account it should only be regarded as the first step of a program of understanding the phenomenology of discrete spacetimes in a mathematically rigorous way.See arXiv:1109.1963

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On the Observability of Discrete Spatial Geometry
 If quantum geometry is an accurate model of microscopic spatial geometry then two related questions arise, one observational and one theoretical: How and at what scale is the discreteness manifest? And, how is the general relativistic limit achieved? These questions will be discussed in the context of studies of a single atom of geometry. It will be shown that the effective scale of the discreteness could be much larger than the Planck scale. Before this scale can be predicted, the relations between discrete geometry, coherent states, and the semiclassical limit need to be clarified. Work towards this goal, using coherent states in spin foams and the spin geometry theorem of Penrose and Moussouris will be described.

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Asymptotic safety and minimal length
Since asymptotic safety - if true - would make a quantum field theory of gavity consistent "up to arbitrarily high energy", it would seem that this notion is incompatible with the existence of a minimal length.  I will argue that this is not necessarily the case, due to ambiguity in the notion of minimal length.