Quantum Gravity

My field is the foundations of quantum mechanics, in particular Bohmian mechanics, a non-relativistic theory that is empirically equivalent to standard quantum mechanics while solving all of its paradoxes in an elegant and simple way, essentially by assuming that particles have trajectories. Bohmian mechanics possesses a straightforward generalization to relativistic space-time, be it flat or curved, if one assumption against the spirit of relativity is granted: the existence of a "time foliation", i.e., a physical object mathematically represented by a slicing of space-time into spacelike 3-surfaces, which evolves according to a Lorentz-invariant law. On the basis of this kind of theory, describing particles in a background 4-geometry, I propose an extension in which the space-time geometry is dynamically generated, as in general relativity. Whether my model is empirically equivalent to any known type of quantum gravity I don't know. In this model, there is a Lorentzian metric on configuration-space-time, evolving according to the higher-dimensional analog of the Einstein field equation. The 4-metric is obtained from the configuration-space-time metric and the actual particle configuration. Thus, this Bohm-like model generates (up to diffeomorphisms) a 4-metric and particle world lines from a given wave function.

We introduce a new top down approach to canonical quantum gravity, called Algebraic Quantum Gravity (AQG): The quantum kinematics of AQG is determined by an abstract $*-$algebra generated by a countable set of elementary operators labelled by an algebraic graph. The quantum dynamics of AQG is governed by a single Master Constraint operator. While AQG is inspired by Loop Quantum Gravity LQG), it differs drastically from it because in AQG there is fundamentally no topology or differential structure. The missing information about the topology and differential structure of the spacetime manifold as well as about the background metric to be approximated is supplied by coherent states and is therefore only available in the semiclassical sector of the theory. Given such data, the corresponding coherent state defines a sector in the Hilbert space of AQG which can be identified with a usual QFT on the given manifold and background. Thus, AQG contains QFT on all curved spacetimes at once, possibly has something to say about topology change and provides the contact with the familiar low energy physics. We will show that AQG admits a semiclassical limit whose infinitesimal gauge symmetry generators agree with the ones of General Relativity.

I will first argue that the notion of black hole entropy extends universally to causal horizons. Then I will deduce the causal dynamics of spacetime from the equilibrium thermodynamics of causal horizons. Specifically, it will be shown how the Clausius relation dS = dQ/T between entropy change, energy flux, and acceleration temperature for all local causal horizons implies the Einstein equation, with Newton's constant determined by the universal horizon entropy density. Implications, non-equilibrium processes, and relations to AdS/CFT duality will also be discussed.

Not only general relativity but also quantum theory plays important roles in current cosmology. Quantum fluctuations of matter fields are supposed to have provided the initial seeds of all the structure of the current universe, and quantum gravity is assumed to have been essential in the earliest stages. Both issues are not fully understood, although several heuristic effects have been discussed. In this talk, implications of an effective framework taking into account the coupling of matter and gravity are discussed. This touches on interpretational issues of quantum mechanics, cosmological observations and properties of quantum gravity.

At low energy and small curvature, general relativity has the form of an effective field theory. I will describe the structure of the effective field theory, and show how it can be used to calculate low energy quantum effects.

Category theory is a general language for describing things and processes - called "objects" and "morphisms". In this language, many counterintuitive features of quantum theory turn out to be properties shared by the category of Hilbert spaces and the category of cobordisms, in which objects are choices of "space" and emorphisms are choices of "spacetime". This striking fact suggests that "n-categories with duals" are a promising language for a quantum theory of spacetime. We sketch the historical development of these ideas from Feynman diagrams to string theory, topological quantum field theory, spin networks and spin foams, and especially recent work on open-closed string theory, 3d quantum gravity coupled to point particles, and 4d BF theory coupled to strings.

While modern theories lavishly invoke several spatial dimensions within models that seek to unify relativity theory and quantum mechanics, none seems to consider the possibility that a yet-unfamiliar aspect of time may do the work. I introduce the notion of Becoming and then consider its consequences for physical theory. Becoming portrays a possible aspect of time that is "curled" very much like the extra spatial dimensions in superstring theories. Within the resulting picture of spacetime, some fundamental aspects of quantum mechanics, special and general relativity, thermodynamics and modern cosmology fit in very naturally. The proposed model is not yet a scientific theory as it still lacks a rigorous formalism and experimental predictions, yet it points out an entire family of possible theories that merit serious consideration.

Space-time measurements and gravitational experiments are made by the mutual relations between objects, fields, particles etc... Any operationally meaningful assertion about spacetime is therefore intrinsic to the degrees of freedom of the matter (i.e. non-gravitational) fields and concepts such as ``locality'' and ``proximity'' should, at least in principle, be definible entirely within the dynamics of the matter fields. We propose to consider the regions of space just as general ``subsystems''. By writing the Hilbert space of the matter fields as a generic tensor product of subsystems we analyse the evolution of a state vector on an information theoretical basis and discuss general principles to recover a posteriori the usual space-time relations. We apply such principles to generic interacting second quantized models with a finite number of fermionic degrees of freedom. Finally, we discuss the possible role of gravity in this framework.

We calculate analytically the highly damped quasinormal mode spectra of generic single-horizon black holes using the rigorous WKB techniques of Andersson and Howls. We thereby provide a firm foundation for previous analysis, and point out some of their possible limitations. The numerical coefficient in the real part of the highly damped frequency is generically determined by the behavior of coupling of the perturbation to the gravitational field near the origin, as expressed in tortoise coordinates. This fact makes it difficult to understand how the (in)famous ln(3) could be related to the quantum gravitational microstates near the horizon.

The phenomenology of quantum gravity can be examined even though the underlying theory is not yet fully understood. Effective extensions of the standard model allow us to study specific features, such as the existence of extra dimensions or a minimal length scale. I will talk about some applications of this approach which can be used to make predictions for particle- and astrophysics, and fill in some blanks in the puzzle of quantum gravity. A central point of this investigations is the physics of black holes. I will comment on possible ways to proceed and on the missing pieces I find most important to look for.