Geometric properties of the Livine-Speziale coherent intertwiner - Sahlmann, Hanno; Seeger, Robert; Seeger, Robert (2015)

Revisiting loop quantum gravity with selfdual variables: classical theory - Sahlmann, Hanno; Seeger, Robert (Classical and Quantum Gravity, 2024)
We review the classical formulation of general relativity as an S L ( 2 , C ) gauge theory in terms of Ashtekar’s self-dual variables and reality conditions for the spatial metric (first reality condition) and its evolution (second reality condition), and we add some new observations and results. We first explain in detail how a connection taking values in the Lie algebra of the complex Lorentz group yields two spin(3,1) connections, one self-dual and one anti-self-dual, without the need for a spin structure. We then demonstrate that the self-dual part of the complexified Palatini action in Ashtekar variables requires a holomorphic phase space description in order to obtain a non-degenerate symplectic structure. The holomorphic phase space does not allow for the implementation of the reality conditions as additional constraints, so they have to be taken care of ‘by hand’ during the quantisation. We also observe that, due to the action being complex, there is an overall complex phase that can be chosen at will. We then review the canonical formulation and the consequences of the implementation of the reality conditions. We pay close attention to the transformation behaviour of the various fields under (complex) basis changes, as well as to complex analytic properties of the relevant functions on phase space.
CRIS-Team Scopus Importer:2024-03-22
Keywords: Ashtekar variables; loop quantum gravity; self-dual variables

Revisiting loop quantum gravity with selfdual variables: Hilbert space and first reality condition - Sahlmann, Hanno; Seeger, Robert (Classical and Quantum Gravity, 2024)
We consider the quantization of gravity as an S L ( 2 , C ) gauge theory in terms of Ashtekar’s selfdual variables and reality conditions for the spatial metric (RCI) and its evolution (RCII). We start from a holomorphic phase space formulation. It is then natural to push for a quantization in terms of holomorphic wave functions. Thus we consider holomorphic cylindrical wave functions over S L ( 2 , C ) connections. We use an overall phase ambiguity of the complex selfdual action to obtain Poisson brackets that mirror those of the real theory. We then show that there is a representation of the corresponding canonical commutation relations in the space of holomorphic cylindrical functions. We describe a class of cylindrically consistent measures that implements RCI. We show that spin networks with S U ( 2 ) intertwiners form a basis for gauge invariant states. They are still mutually orthogonal, but the normalisation is different than for the Ashtekar-Lewandowski measure for S U ( 2 ) . We do not consider RCII in the present article. Work on RCII is ongoing and will be presented elsewhere.
CRIS-Team Scopus Importer:2024-03-22
Keywords: Ashtekar variables; loop quantum gravity; measures on spaces of connections; selfdual variables

Towards Gaussian states for loop quantum gravity - Sahlmann, Hanno; Seeger, Robert (Physical Review D, 2020)
An important challenge in loop quantum gravity is to find semiclassical states-states that are as close to classical as quantum theory allows. This is difficult because the states in the Hilbert space used in loop quantum gravity are excitations over a vacuum in which geometry is highly degenerate. Additionally, fluctuations are distributed very unevenly between configuration and momentum variables. Coherent states that have been proposed to balance the uncertainties more evenly can, up to now, only do this for finitely many degrees of freedom. Our work is motivated by the desire to obtain Gaussian states that encompass all degrees of freedom. We reformulate the U(1) holonomy-flux algebra in any dimension as a Weyl algebra. We then define and investigate a new class of states on this algebra which behave as quasifree states on the momentum variables. Using a general result on representations of the holonomy-flux algebra, we define analogous representations also in the case of non-Abelian compact structure groups. For the case of SU(2), we study an explicit example of such a representation and the consequences for quantum geometry. This kind of state, with Gaussian fluctuations in the spatial geometry, seems well suited to investigate problems related to structure formation in cosmology.
CRIS-Team WoS Importer:2020-05-29

Towards Gaussian States for the Holonomy-Flux Weyl Algebra - Seeger, Robert; Sahlmann, Hanno (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2018)
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.<br