Three lectures on coherent quantization and field theory

Coherent Quantization I: Coherent spaces for linear fields

Coherent Quantization II: Causal groups and quantum fields

Coherent Quantization III: Infra Foock spaces and nonlinear fields

The first two lectures can be followed independently. The third lecture is based on the first two.

Coherent Quantization I: Coherent spaces for linear fields

Monday, December 11, 2023, 14:15-15:45,
AG Lie-Gruppen,
Übungsraum Ü2, 01.251, Cauerstr. 11, Erlangen

The notion of a coherent space is a nonlinear version of the notion of a complex Hilbert space: The vector space axioms are dropped while the notion of inner product, now called a coherent product, is kept.

Every coherent space can be uniquely embedded into a Hilbert space, its completed quantum space, by suitably extending the coherent product to an inner product. In the interesting examples, the coherent space is an extended classical phase space, and there is a quantization functor that turns the symmetries of the coherent space into unitary operators in the corresponding quantum space. Thus the quantum space is a representation space for quantum dynamics.

This provides a universal framework for quantization, extending the traditional geometric quantization of finite-dimensional symplectic manifolds to more general situations, and in particular to the quantization of certain classical field theories.

Fields defined by linear field equations can be quantized by means of Klauder spaces, a class of coherent spaces discussed by Neumaier and Ghani Farashahi in Anal. Math. Phys. 12 (2022), 1-47.

We describe another quantization of linear field equations in terms of symplectic and orthogonal Hua spaces (for bosons and fermions), a new class of coherent spaces based on the geometric analysis by Loo-Keng Hua in Trans. Amer. Math. Soc. 57 (1945), 441-481. Their symmetry groups are infinite-dimensional metaplectic or metagonal (spin) groups. They allow one to describe the full quantum scattering behavior of linear field equations in terms of classical scattering and Maslov corrections for the phase of the S-matrix.

Slides of the lecture

Coherent Quantization II: Causal groups and quantum fields

Thursday, December 14, 2023, 16:15-18:00,
AG Mathematische Physik,
Übungsraum Ü1, 01.250, Cauerstr. 11, Erlangen

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 Buchholz and Fredenhagen (who obtain their dynamical C^*-algebras from an analysis of causal perturbation theory), causal groups are motivated in a fully nonperturbative way through 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.

It is shown 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.

Slides of the lecture

Preprint: A. Neumaier, Causal groups and local fields, Manuscript (December 2023)

Coherent Quantization III: Infra Fock spaces and nonlinear fields

Monday, December 18, 2023, 14:15-15:45,
AG Lie-Gruppen,
Übungsraum Ü2, 01.251, Cauerstr. 11, Erlangen

This is the last of three lectures on coherent quantization and field theory to be given 11.-18.12.2023 in Erlangen (Germany).

Infra Fock spaces organize the infrastructure available in quantum field theories. They lead naturally to quantum fluids, which form a framework for diffeomorphism invariant quantum field theories. Quantum fluid dynamics interpolates between the superselection sectors of a quantum field theory.

We introduce coherent field quantization, an infinite-dimensional generalization of geometric quantization, and discuss associated conjectures for the interface between classical and quantum observables and for the construction of QED, Yang-Mills, QCD, and quantum gravity.

This lecture depends on the concepts and results of the first two lectures. 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.

This lecture depends on the concepts and results of the first two lectures.

Slides of the lecture