This series consists of talks in the area of Foundations of Quantum Theory. Seminar and group meetings will alternate.
In this talk I would like to put forward Wasserstein-geometry as a natural language for Quantum hydrodynamics. Wasserstein-geometry is a formal, infinite dimensional, Riemannian manifold structure on the space of probability measures on a given Riemannian manifold. The basic equations of Quantum hydrodynamics on the other hand are given by the Madelung equations. In terms of Wasserstein-geometry, Madelung equations appear in the shape of Newton's second law of motion, in which the geodesics are disturbed by the influence of a quantum potential. This was pointed out in 2008 by Max. K.
In quantum theory every state can be diagonalised, i.e. decomposed as a convex combination of perfectly distinguishable pure states. This fact is crucial in quantum statistical mechanics, as it provides the foundation for the notions of majorisation and entropy. A natural question then arises: can we give an operational characterisation of them? We address this question in the framework of general probabilistic theories, presenting a set of axioms that guarantee that every state can be diagonalised: Causality, Purity Preservation, Purification, and Pure Sharpness.
Recent findings on quantitative growth patterns have revealed striking generalities across the tree of life, and recurring over distinct levels of organization. Growth-mass relationships in 1) individual growth to maturity, 2) population reproduction, 3) insect colony enlargement and 4) community production across wholeecosystems of very different types, often follow highly robust near ¾ scaling laws. These patterns represent some of the most general relations in biology, but the reasons they are so strangely similar across levels of organization remains a mystery.
As discussed in last week’s colloquium, the use of the p-adic metric in state space provides a route to resolving the Bell Theorem in favour of realism and local causality, without fine tuning. Here the p-adic integers provide a natural way to describe the fractal geometry of Invariant Set Theory’s state space. In this talk I first explore the role of complex numbers in Invariant Set Theory (arXiv:1605.01051), and describe a novel realistic perspective on quantum interferometry.
In this talk I will: 1) review the results of my work on a geometric approach to foundations for a postquantum information theory; 2) discuss how it is related to other foundational approaches, including some resource theories of knowledge and quantum histories; 3) present some of my research on a category theoretic framework for a multi-agent information relativity. More details on part 1: this approach does not rely on probability theory, spectral theory, or Hilbert spaces. Normalisation of states, convexity, and tensor products are allowed but not assumed foundationally.
In the study of closed quantum system, the simple harmonic oscillator is ubiquitous because all smooth potentials look quadratic locally, and exhaustively understanding it is very valuable because it is exactly solvable. Although not widely appreciated, Markovian quantum Brownian motion (QBM) plays almost exactly the same role in the study of open quantum systems. QBM is ubiquitous because it arises from only the Markov assumption and linear Lindblad operators, and it likewise has an elegant and transparent exact solution.
In the de Broglie-Bohm pilot-wave formulation of quantum theory, standard quantum probabilities arise spontaneously through a process of dynamical relaxation that is broadly similar to thermal relaxation in classical physics. If we are to regard this process as the cause of the quantum probabilities we observe today, then we must infer a primordial ‘quantum nonequilibrium’ in the remote past.
Our physical theories often admit multiple formulations or variants. Although these variants are generally empirically indistinguishable, they nonetheless appear to represent the world as having different structures. In this talk, I will discuss several criteria for comparing empirically equivalent theories that may be used to identify (1) when one variant has more structure than another (i.e., when a formulation of a theory has “excess structure”) and (2) when two variants are theoretically equivalent, even though they appear to represent the world differently.
Bell inequalities bound the strength of classical correlations arising between outcomes of measurements performed on subsystems of a shared physical system. The ability of quantum theory to violate Bell inequalities has been intensively studied for several decades. Recently, there has been an increased interest in studying physical correlations beyond the scenario of Bell inequalities, to more general network structures involving many sources of physical states and observers that may be measuring on subsystems of independent states.
Quantum theory can be understood as a theory of information processing in the circuit framework for operational probabilistic theories. This approach presupposes a definite casual structure as well as a preferred time direction. But in general relativity, the causal structure of space-time is dynamical and not predefined, which indicates that a quantum theory that could incorporate gravity requires a more general operational paradigm. In this talk, I will describe recent progress in this direction.