Operator-Algebraic Construction of Gauge Theories and Jones’ Actions of Thompson’s Groups

Abstract

Using ideas from Jones, lattice gauge theory and loop quantum gravity, we construct 1+1-dimensional gauge theories on a spacetime cylinder. Given a separable compact group G, we construct localized time-zero fields on the spatial torus as a net of C*-algebras together with an action of the gauge group that is an infinite product of G over the dyadic rationals and, using a recent machinery of Jones, an action of Thompson's group T as a replacement of the spatial diffeomorphism group. Adding a family of probability measures on the unitary dual of G we construct a state and obtain a net of von Neumann algebras carrying a state-preserving gauge group action. For abelian G, we provide a very explicit description of our algebras. For a single measure on the dual of G, we have a state-preserving action of Thompson's group and semi-finite von Neumann algebras. For G " S the circle group together with a certain family of heat-kernel states providing the measures, we obtain hyperfinite type III factors with a normal faithful state providing a nontrivial time evolution via Tomita-Takesaki theory (KMS condition). In the latter case, we additionally have a non-singular action of the group of rotations with dyadic angles, as a subgroup of Thompson's group T , for geometrically motivated choices of families of heat-kernel states.

Scaling Limits of Lattice Quantum Fields by Wavelets

“…Natural generalizations of the latter arising in the context of lattice gauge theory are [13,14,59]:…”

“…4, where the goal is to obtain an infrared scaling limit. From this we can make three basic observations, see also [13,14]:…”

“…In the present paper, we start from a lattice approximation to quantum field theories closer in spirit to the earlier constructive works by Friedrichs, Glimm-Jaffe and others [36,40] and the setting of quantum statistical mechanics. We explicitly implement an operator-algebraic approach [13,14] to the Wilson-Kadanoff renormalization group for scalar lattice quantum fields in any dimension (see also [76] for a short overview). Specifically, using results of wavelet theory [23,58], we build a real-space (as opposed to momentum-space) renormalization group by identifying the lattice fields with the continuum fields smeared with suitable functions which generate a wavelet basis.…”

Hamiltonian Renormalization V: Free Vector Bosons

“…In a recent series of paper (Lang et al, 2018a;Lang et al, 2018b;Lang et al, 2018c;Lang et al, 2018d) we introduced a Hamiltonian formulation of the renormalization group which is rather close in methodology to density matrix renormalization (Brothier and Stottmeister, 2019;Brothier, 2019;Stottmeister et al, 2020) and projective renormalization (Okołow, 2013;Kijowski and Okołow, 2017;Lanéry and Thiemann, 2017a;Lanéry and Thiemann, 2017b;Lanéry and Thiemann, 2018;Lanéry, 2018;Lanéry, 2016;Yamasaki, 1985) which in turn are based on the seminal ideas of Wilson, Kadanov and Fisher (Fisher, 1974;Wilson, 1975;Kadanoff, 1977). The proposal is motivated by formulations of the renormalization group in the covariant setting (Fisher, 1974;Wilson, 1975;Kadanoff, 1977;Wilson and Kogut, 1974;Peter, 1998) which can be reformulated in Hamiltonian terms using Osterwalder-Schrader reconstruction and in fact gives rise to a natural flow of inductive structures and Hamiltonian quadratic forms (Lang et al, 2018a;Lang et al, 2018b).…”

“…Of course, this does not guarantee universality under any changes of coarse graining map-a property which cannot be true in general. However, it is possible to show that for the Hamiltonian RG formulation all coarse graining maps are unitary equivalent, albeit the initial discretisations may change under said unitary map, see(Bahr and Liegener, 2020) for all details.Frontiers in Astronomy and Space Sciences | www.frontiersin.org January 2021 | Volume 7 | Article 547550…”