Parameterized field theories and Loop Quantization[HBNI Th10]

Abstract

This thesis shows how two dimensional parametrized field theories constitute perfect toy models for Loop Quantum Gravity. Two dimensional massless scalar field theories on a Minkowskian plane and show how various aspects Loop quantization - like construction of quantum observables, determination of physical Hilbert Space and emergence of discrete spacetime can be explicitly illustrated within these models. Also loop quantized parametrized field theories are demonstrated for quantum theories capturing non-perturbative aspects of two dimensional quantum black holes. This thesis comprises of two parts, first part presents a polymer quantization of a parametrized scalar field theory on 2 dimensional flat cylinder. Both the matter fields as well as the embedding variables are quantized in LQG type polymer representations. The quantum constraints are solved via group averaging techniques and analogous to the case of spatial geometry in LQG, the smooth(flat) space time geometry is replaced by a discrete quantum structure. It is shown that physical weaves necessarily underly such states and that such states display semiclassicality with respect to at most a countable subset of the set of observable type. The second part presents a polymer(loop) quantization of a two dimensional theory of dilatonic gravity known as the CGHS model. The theory is recasted as a parametrized free field theory on a flat 2-dimensional space time and quantize the resulting phase space using techniques of loop quantization. The complete spectrum of the theory is obtained using a technique known as group averaging and perform quantization of Dirac observables on the resulting Hilbert Space. It is argued that the algebra of Dirac observables get deformed in the quantum theory. Combining the ideas from parametrized field theory, with certain relational observables evolution is defined in the quantum theory in the Heisenberg Picture. Finally the dilaton field is quantized on the Physical Hilbert Space, which carries information about quantum geometry.