# AG Mathematische Physik, Arnold Neumaier (Wien): Coherent Quantization II: Causal groups and quantum fields

**Arnold Neumaier (Wien)**

**Coherent Quantization II: Causal groups and quantum fields**

Abstract:

This is the second of three lectures on coherent quantization and field

theory to be given 11.-18.12.2023 in Erlangen (Germany).

Causal groups are a new class of mathematical objects abstracted from

the concept of dynamical C^*-algebras for quantum field theories

introduced by Buchholz and Fredenhagen in their paper

Comm. Math. Phys. 377 (2020), 947-969.

Unlike these authors (who obtain their dynamical C^*-algebras from an

analysis of causal perturbatiion theory) we motivate causal groups in

a fully nonperturbative way from the consideration of classical

discrete-time dynamical systems. This gives an intuitive understanding

of the properties later assumed axiomatically for causal groups over

causal spaces (generalizing Minkowski spacetime).

Each causal group over Minkowski space gives rise to a particular

dynamical C^*-algebra. The dynamical C^*-algebras of Buchholz and

Fredenhagen arise from abstract causal groups defined by generators and

relations.

We show how to construct causal groups over causal spaces having a

Tomonaga-Schwinger structure associated with an appropriate classical

many-fingered time dynamics. This gives a clear geometric meaning to

the new concept.

Their unitary representations give nonperturbative constructions of

nonrelativistic and relativistic quantum field theories. Conditions

are given under which the Haag-Kastler axioms or the Wightman axioms

can be established. This reduces the rigorous construction of realistic

quantum field theories such as QED or QCD to the (still unsettled)

construction of unitary representations of causal groups with the

properties defining QED or QCD. A constructed QFT can be identified

with a particular perturbatively defined one (such as QED or QCD)

by performing aposteriori causal perturbation theory to lowest (1 loop)

order.

This lecture is essentially independent of the first lecture. The slides

of the lecture will probably become available – at least three hours

before the lecture – at

https://arnold-neumaier.at/cohErlangen2023.html

where one can also find background material (abstracts and some

references) for all lectures.