Viqar Husain

2+1 quantum gravity as a toy model for the 3+1 theory

2+1 Einstein gravity is used as a toy model for testing a program for nonperturbative canonical quantisation of the 3+1 theory. The program can be successfully implemented in the model and leads to a surprisingly rich quantum theory.

Time and a physical Hamiltonian for quantum gravity.

We present a nonperturbative quantization of general relativity coupled to dust and other matter fields. The dust provides a natural time variable, leading to a physical Hamiltonian with spatial diffeomorphism symmetry. The surprising feature is that the Hamiltonian is not a square root. This property, together with the kinematical structure of loop quantum gravity, provides a complete theory of quantum gravity, and puts applications to cosmology, quantum gravitational collapse, and Hawking radiation within technical reach.

Background-independent quantization and the uncertainty principle

It is shown that polymer quantization leads to a modified uncertainty principle similar to that motivated by string theory and non-commutative geometry. When applied to quantum field theory on general background spacetimes, corrections to the uncertainty principle acquire a metric dependence. For Friedmann–Robertson–Walker cosmology this translates to a scale factor dependence which gives a large effect in the early Universe.

Quantum resolution of black hole singularities

We study the classical and quantum theory of spherically symmetric spacetimes with scalar field coupling in general relativity. We utilize the canonical formalism of geometrodynamics adapted to the Painleve–Gullstrand coordinates, and present a new quantization of the resulting field theory. We give an explicit construction of operators that capture curvature properties of the spacetime and use these to show that the black hole curvature singularity is avoided in the quantum theory.

Self-dual gravity as a two-dimensional theory and conservation laws

Starting from the Ashtekar Hamiltonian variables for general relativity, the self-dual Einstein equations (SDE) may be rewritten as evolution equations for three divergence-free vector fields given on a three-dimensional surface with a fixed volume element. From this general form of the SDE, it is shown how they may be interpreted as the field equations for a two-dimensional field theory. It is further shown that these equations imply an infinite number of non-local conserved currents. A specific way of writing the vector fields allows an identification of the full SDE with those of the two-dimensional chiral model, with the gauge group being the group of area-preserving diffeomorphisms of a two-dimensional surface. This gives a natural Hamiltonian formulation of the SDE in terms of that of the chiral model.

Exactly solvable quantum cosmologies from two killing field reductions of general relativity

Abstract An exact and, possibly, general solution to the quantum constraints is given for the sector of general relativity containing cosmological solutions with two space-like, commuting, Killing fields. The dynamics of these model space-times, which are known as Gowdy space-times, is formulated in terms of Ashtekars new variables. The quantization is done by using the recently introduced self-dual and loop representations. On the classical phase space we find four explicit physical observables, or constants of motion, which generate a GL(2) symmetry group on the space of solutions. In the loop representations we find that a complete description of the physical state space, consisting of the simultaneous solutions to all of the constraints, is given in terms of the equivalence classes, under Diff(S1), of a pair of densities on the circle. These play the same role that the link classes play in the loop representation solution to the full 3+1 theory. An infinite dimensional algebra of physical observables is found on the physical state space, which is a GL(2) loop algebra. In addition, by freezing the local degrees of freedom of the model, we find a finite dimensional quantum system which describes a set of degenerate quantum cosmologies on T3 in which the length of one of the S1s has gone to zero, while the area of the remaining S1×S1 is quantized in units of the Planck area. The quantum kinematics of this sector of the model is identical to that of a one-plaquette SU(2) lattice gauge theory.

Propagator in polymer quantum field theory

We study free scalar field theory on flat spacetime using a background independent (polymer) quantization procedure. Specifically we compute the propagator using a method that takes the energy spectrum and position matrix elements of the harmonic oscillator as inputs. We obtain closed form results in the infrared and ultraviolet regimes that give Lorentz invariance violating dispersion relations, and show suppression of propagation at sufficiently high energy.

Scalar field collapse in three-dimensional AdS spacetime

We describe the results of a numerical calculation of circularly symmetric scalar field collapse in three spacetime dimensions with a negative cosmological constant. The procedure uses a double null formulation of the Einstein-scalar equations. We see evidence of black hole formation on first implosion of a scalar pulse if the initial pulse amplitude A is greater than a critical value A*. Sufficiently near criticality the apparent horizon radius rAH grows with pulse amplitude according to the formula rAH~(A-A*)0.81.

Quantum black holes from null expansion operators

Using a recently developed quantization of spherically symmetric gravity coupled to a scalar field, we give a construction of null expansion operators that allow a definition of general, fully dynamical quantum black holes. These operators capture the intuitive idea that classical black holes are defined by the presence of trapped surfaces, that is, surfaces from which light cannot escape outward. They thus provide a mechanism for classifying quantum states of the system into those that describe quantum black holes and those that do not. We find that quantum horizons fluctuate, confirming long-held heuristic expectations. We also give explicit examples of quantum black hole states. The work sets a framework for addressing the puzzles of black hole physics in a fully quantized dynamical setting.

Exact solution for scalar field collapse.

We give a class of exact spherically symmetric solutions for the Einstein-scalar field system. The solutions may be interpreted as inhomogeneous dynamical scalar field cosmologies. The spacetimes have a timelike conformal Killing vector field and are asymptotically conformally flat. They also have black- or white-hole-like regions containing trapped surfaces. The properties of the apparent horizons are described in detail.