Workshop on Non-Additive Measures and Generalized Probabilities in Physics

University of Michigan

July 7-8, 2026

All sessions will be in 335 West Hall. Zoom link for remote participation will be provided to registrants, and all presentations and discussions will also be livestreamed on YouTube.

Registration Form

Discussants and Presentation Descriptions

Gabriele Carcassi, University of Michigan, US
Title: Developing a general theory for statistical ensembles
Description: Interested in developing a theory of ensemble spaces that can serve as a base theory for all physical theories. The basic structures/axioms are motivated as follows: experimental verifiability -> topologies/sigma-algebras; statistical mixing -> convex structure; ensemble variability -> entropic structure; processes/equilibria -> Lie algebras/Poisson structures. Want to understand how to connect to and generalize results from different areas. Generalized notions of measure theory and probability theory seem to require non-additive measures. Some of these measures are defined on lattices that are connected to those of quantum logic. It is not yet clear how the Poisson/commutator algebras for classical/quantum mechanics can generalize.

Marcel Reginatto, German National Metrology Institute, Germany
Title: Ensemble of particles, observables, Lie algebras: Lessons from hybrid quantum-classical systems
Description: Hybrid theories describe interacting classical and quantum systems that may consist of discrete systems, particles, or fields. To formulate such a theory, you need a common mathematical framework for the classical and quantum sectors and you need to define explicitly the properties that you consider essential for distinguishing between classical and quantum systems. There are, in addition, consistency conditions that ought to be addressed. I consider two approaches to modeling hybrid systems, ensembles on configuration space and hybrid van Hove theory, and focus on the case of interacting classical and quantum particles. In these theories, the distinction between classical and quantum is fundamentally connected to the definitions of observables and the corresponding Lie algebras of observables. Thus, in addition to requiring a description in terms of statistical ensembles of particles, it turns out that algebraic considerations in these theories play a fundamental role, in contrast to other hybrid models (e.g., those based on ideas from quantum information theory or Sudarshan's Hilbert space formulation based on the Koopman-von Neumann approach). I discuss how algebraic aspects are naturally incorporated into the two approaches, how observables are constructed, and the relevance for the description of physical systems.

Zuzana Ontkovičová, Slovak University of Technology, Slovakia
Title: The world where 1 + 1 does not equal 2
Description: Fuzzy measures provide a powerful framework for describing real-world situations where agents dynamically interact with each other. While classical probability theory assumes strict independence — 1+1=2 always! — fuzzy measures successfully model interdependent connections. However, working within fuzzy measure theory introduces many challenges. How do we construct an integral using fuzzy measures, and is there just one suitable option? Is it possible to define a fuzzy analogue to the classical Radon-Nikodym derivatives for probability measures? These (and more) are still more or less open questions for researchers, with some partial answers that we will look into and discuss during the workshop.

Gert de Cooman, Ghent University, Belgium
Title: Can we get rid of the probability assumption?
Description: There’s a way of unifying classical and quantum probabilities by moving away from the probability point of view and moving towards the expectation point of view. Classical and quantum expectations both are linear functionals on a real linear space – of bounded functions in the classical case and Hermitian operators in the quantum case. Both express preferences between uncertain rewards. What distinguishes these functionals, then, is not the linear space, bur the background superlinear real functional that the expectation is supposed to dominate. This background represents the preferences under complete ignorance: those preferences You – a subject – will have regardless of any information You might have about the (classical or quantum) state. It turns out that this background functional can be determined by physical invariance properties, leading to the conclusion that it’s only the invariances/symmetries that the background embodies, that distinguish between classical and quantum probabilities. This point of view also allows for an immediate generalisation in terms of lower and upper expectations (imprecise probability models), and allows us to aim for a common uncertainty framework to describe both classical and quantum uncertainty. This approach still rests on the Hilbert space set-up for QM, and so an important question is: how to generalise it to a set-up with fewer initial structural assumptions?

Alessio Benavoli, Trinity College Dublin, Ireland
Title: What do we want to do?
Description: I am interested in understanding how physical theories can be derived from order structures (and topological structures) and generalised notions of probability theory and its more general mathematical dual, desirable gambles theory. More concretely, a question I am interested to address is the following. Suppose we are given an `ordered vector space’ with some additional properties and invariances, e.g., a (closed) convex set inside the positive cone and permutation invariance. Is it always possible to define the properties, invariances and an order which make this vector space `isomorphic’ to an operational definition (experimental verifiability) of a certain physical theory? This seems to work for quantum theory, but could it be extended to other physical theories, e.g., to understand spacetime? I am also interested in how physical systems can be used for computation and, vice versa, how computation can help us understanding physical systems.