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
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
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)
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
where one can also find background material (abstracts and some
references) for all lectures.