Towards Gaussian States for the Holonomy-Flux Weyl Algebra

(Master thesis)

Publication Details

Author(s): Seeger R, Sahlmann H
Publication year: 2018
Language: English


The objective of this Master’s thesis is to consider the well-known framework of Weyl algebras and
quasifree states in order to find out if it is possible to apply the general ideas, coming from algebraic
quantum theory, to the theory of loop quantum gravity.

Starting from the U(1) toy-model of the canonical commutation relation of the holonomy-flux
algebra, underlying loop quantum gravity, we construct a Weyl C*-algebra generated by so-called
Weyl elements that arise from combining holonomies and exponentiated electric fluxes, which are
the canonically conjugated variables of the theory. Quasifree states are a certain notion of Gaussian
states, directly emerging from Weyl algebras. Because it seems to be impossible to establish such
states on the algebra we found, we develop a different notion states that is only Gaussian in one of
the variables and hence is referred to as almost-quasifree states. For such a state, which is Gaussian
in the fluxes, we find a representation on a Hilbert space that combines the Hilbert space of loop
quantum gravity with the Fock space of a scalar field.

For the canonical commutation relation of the actual theory, which involves SU(2) Yang-Mills
holonomies and the corresponding fluxes, we try to generalize our results. It is possible to define
Weyl-like elements for holonomies along a single path and a set of exponentiated fluxes. We work
toward a notion of elements that take care of more distinct edges or even graphs. It is, however,
not clear if these also generate a C*-algebra. Without an underlying Weyl algebra we successfully
generalize the almost-quasifree representation, found for the toy-model, and analyze its properties
by re-deriving the area operator of loop quantum gravity in this new representation.

FAU Authors / FAU Editors

Sahlmann, Hanno Prof. Dr.
Professur für Theoretische Physik

How to cite

Seeger, R., & Sahlmann, H. (2018). Towards Gaussian States for the Holonomy-Flux Weyl Algebra (Master thesis).

Seeger, Robert, and Hanno Sahlmann. Towards Gaussian States for the Holonomy-Flux Weyl Algebra. Master thesis, 2018.


Last updated on 2018-10-08 at 23:42