This series consists of talks in the area of Foundations of Quantum Theory. Seminar and group meetings will alternate.
We present a quantization method based on theEhrenfest theorem embedded in an extended algebraic structure capableof consistently describing hybrid quantum-classical systems, where thestandard quantum and classical mechanics are two limiting cases. TheWigner phase space formulation and the Schordinger equation are foundto be two alternative representations of the quantum case while theKoopman-von Neumann equation is the corresponding classicalcounterpart.
Categorical quantum mechanics (CQM) uses symmetric monoidal categories to formalize quantum theory, in order to extract the key structures that yield protocols such as teleportation in an abstract way. This formalism admits a purely graphical calculus, but the causal structure of these diagrams, and the formalism in general, is unclear. We begin by considering the signaling abilities of probabilistic devices with inputs and outputs and we show how a non-signaling device can become a perfect signaling device under time-reversal.
I will present a new approach to information theoretic foundations of quantum theory, developed in order to encompass quantum field theory and curved space-times. Its kinematics is based on the geometry of spaces of integrals on W*-algebras, and is independent of probability theory and Hilbert spaces. It allows to recover ordinary quantum mechanical kinematics as well as emergent curved space-times.
We prove an uncertainty relation for energy and arrival time, where the arrival of a particle at a detector is modeled by an absorbing term added to the Hamiltonian. In this well-known scheme the probability for the particle's arrival at the counter is identified with the loss of normalization for an initial wave packet. The result is obtained under the sole assumption that the absorbing term vanishes on the initial wave function. Nearly minimal uncertainty can be achieved in a two-level system.
The experimental violation of Bell inequalities using spacelike separated measurements precludes the explanation of quantum correlations through causal influences propagating at subluminal speed. Yet, it is always possible, in principle, to explain such experimental violations through models based on hidden influences propagating at a finite speed v>c, provided v is large enough. Here, we show that for any finite speed v>c, such models predict correlations that can be exploited for faster-than-light communication.
It is my contention that non-commutative geometry is really "ordinary geometry" carried out in a non-commutative logic. I will sketch a specific project, relating groupoid C*-algebras to toposes, by means of which I hope to detect the nature of this non-commutative logic.
Quantum theory can be thought of as a noncommutative generalization of Bayesian probability theory, but for the analogy to be convincing, it should be possible to describe inferences among quantum systems in a manner that is independent of the causal relationship between those systems.
We use the mathematical language of sheaf theory to give a unified treatment of non-locality and contextuality, which generalizes the familiar probability tables used in non-locality theory to cover Kochen-Specker configurations and more. We show that contextuality, and non-locality as a special case, correspond exactly to *obstructions to the existence of global sections*.