QSD III : Quantum Constraint Algebra and Physical Scalar Product in Quantum General Relativity

This paper deals with several technical issues of non-perturbative four-dimensional Lorentzian canonical quantum gravity in the continuum that arose in connection with the recently constructed Wheeler-DeWitt quantum constraint operator. 1) The Wheeler-DeWitt constraint mixes the previously discussed diffeomorphism superselection sectors which thus become spurious, 2) Thus, the inner product for diffeomorphism invariant states can be fixed by requiring that diffeomorphism group averaging is a partial isometry, 3) The established non-anomalous constraint algebra is clarified by computing commutators of duals of constraint operators, 4) The full classical constraint algebra is faithfully implemented on the diffeomorphism invariant Hilbert space in an appropriate sense, 5) The Hilbert space of diffeomorphism invariant states can be made separable if a natural new superselection principle is satisfied, 6) We propose a natural physical scalar product for quantum general relativity by extending the group average approach to the case of non-self-adjoint constraint operators like the Wheeler-DeWitt constraint and 7) Equipped with this inner product, the construction of physical observables is straightforward.Comment: 37p, LATE

Lectures on Loop Quantum Gravity

Quantum General Relativity (QGR), sometimes called Loop Quantum Gravity, has matured over the past fifteen years to a mathematically rigorous candidate quantum field theory of the gravitational field. The features that distinguish it from other quantum gravity theories are 1) background independence and 2) minimality of structures. Background independence means that this is a non-perturbative approach in which one does not perturb around a given, distinguished, classical background metric, rather arbitrary fluctuations are allowed, thus precisely encoding the quantum version of Einstein's radical perception that gravity is geometry. Minimality here means that one explores the logical consequences of bringing together the two fundamental principles of modern physics, namely general covariance and quantum theory, without adding any experimentally unverified additional structures. The approach is purposely conservative in order to systematically derive which basic principles of physics have to be given up and must be replaced by more fundamental ones. QGR unifies all presently known interactions in a new sense by quantum mechanically implementing their common symmetry group, the four-dimensional diffeomorphism group, which is almost completely broken in perturbative approaches. These lectures offer a problem -- supported introduction to the subject.Comment: 90 pages, Latex, 18 figures, uses graphicx and pstricks for coloured text and graphics, based on lectures given at the 271st WE Heraeus Seminar ``Aspects of Quantum Gravity: From Theory to Experimental Search'', Bad Honnef, Germany, February 25th -- March 1st, to appear in Lecture Notes in Physic

QSD V : Quantum Gravity as the Natural Regulator of Matter Quantum Field Theories

It is an old speculation in physics that, once the gravitational field is successfully quantized, it should serve as the natural regulator of infrared and ultraviolet singularities that plague quantum field theories in a background metric. We demonstrate that this idea is implemented in a precise sense within the framework of four-dimensional canonical Lorentzian quantum gravity in the continuum. Specifically, we show that the Hamiltonian of the standard model supports a representation in which finite linear combinations of Wilson loop functionals around closed loops, as well as along open lines with fermionic and Higgs field insertions at the end points are densely defined operators. This Hamiltonian, surprisingly, does not suffer from any singularities, it is completely finite without renormalization. This property is shared by string theory. In contrast to string theory, however, we are dealing with a particular phase of the standard model coupled to gravity which is entirely non-perturbatively defined and second quantized.Comment: 34p, LATE

Projective Loop Quantum Gravity I. State Space

Instead of formulating the state space of a quantum field theory over one big Hilbert space, it has been proposed by Kijowski to describe quantum states as projective families of density matrices over a collection of smaller, simpler Hilbert spaces. Beside the physical motivations for this approach, it could help designing a quantum state space holding the states we need. In [Oko{\l}\'ow 2013, arXiv:1304.6330] the description of a theory of Abelian connections within this framework was developed, an important insight being to use building blocks labeled by combinations of edges and surfaces. The present work generalizes this construction to an arbitrary gauge group G (in particular, G is neither assumed to be Abelian nor compact). This involves refining the definition of the label set, as well as deriving explicit formulas to relate the Hilbert spaces attached to different labels. If the gauge group happens to be compact, we also have at our disposal the well-established Ashtekar-Lewandowski Hilbert space, which is defined as an inductive limit using building blocks labeled by edges only. We then show that the quantum state space presented here can be thought as a natural extension of the space of density matrices over this Hilbert space. In addition, it is manifest from the classical counterparts of both formalisms that the projective approach allows for a more balanced treatment of the holonomy and flux variables, so it might pave the way for the development of more satisfactory coherent states.Comment: 81 pages, many figure