Noncommutative Poisson Random Measure and its Applications in AQFT and Quantum Chaos

Abstract: One of the important problems in modern mathematical physics is to understand quantization - both quantizing classical systems to quantum systems and quantizing one-particle systems to many-body systems. Up to now, the mathematical second quantization relies on the Gaussian functor and produces generalized free theories. Interactions are generally modeled by perturbation theories (a la Glimm and Jaffe). In this talk, we take a new approach to second quantization by constructing a noncommutative analog of the classical Poisson random measure. This construction produces a functor which shall be called Poissonization. The analog of correlation functions in this construction cannot be factorized into the sum of products of two-point functions. After we outlined the main ideas of Poissonization, we will clearly state the construction and its functorial properties. Moreover, we will discuss the main properties of this construction, emphasizing on its versatility and physical relevance. Subsequently, we will focus on two main areas of physical applications. On one hand, we will discuss how Poissonization can be used to produce examples of algebraic quantum field theories starting from the data of unitary representations of real semisimple Lie groups (single particles a la Weinberg). On the other hand, we will discuss how Poissonization can be used to rigorously calculate quantum information quantities (e.g. relative entropy, tripartite information, out-of-time-order correlation etc.) in type III von Neumann algebras. In many toy examples, these quantities have the right quanlitative behaviors as conjectured in physics literature. Lastly, we will briefly discuss future directions, emphasizing on why we cannot yet claim to have constructed an interacting CFT in dimensions larger than 2 and how we can achieve this goal using Poissonization.

HEP - theorymathematical physicsquantum physics

Audience: researchers in the topic

( video )

Purdue HET

Series comments: The recorded talks will be available on YouTube here: www.youtube.com/playlist?list=PLxU3vHZccQj64m9zsQR74D5WP1z1g4t1J