The polymer quantization in LQG: massless scalar field

Universe

We propose a new polymerization scheme for scalar fields coupled to gravity. It has the advantage of being a (non-bijective) canonical transformation of the fields, and therefore ensures the covariance of the theory. We study it in detail in spherically symmetric situations and compare to other approaches.

Classical and Quantum Gravity, 2007

We present a polymer(loop) quantization of a two dimensional theory of dilatonic gravity known as the CGHS model. We recast the theory as a parametrized free field theory on a flat 2-dimensional spacetime and quantize the resulting phase space using techniques of loop quantization. The resulting (kinematical) Hilbert space admits a unitary representation of the spacetime diffeomorphism group. We obtain the complete spectrum of the theory using a technique known as group averaging and perform quantization of Dirac observables on the resulting Hilbert space. We argue that the algebra of Dirac observables gets 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.

2006

We present a polymer(loop) quantization of a two dimensional theory of dilatonic gravity known as the CGHS model. We recast the theory as a parametrized free field theory on a flat 2-dimensional spacetime and quantize the resulting phase space using techniques of loop quantization. The resulting (kinematical) Hilbert space admits a unitary representation of the spacetime diffeomorphism group. We obtain the complete spectrum of the theory using a technique known as group averaging and perform quantization of Dirac observables on the resulting Hilbert space. We argue that the algebra of Dirac observables gets 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.

Classical and quantum gravity, 2007

Il Nuovo Cimento, 1962

In any quantum theory, in which the metric tensor of Einstein's gravitational theory is also quantized, it becomes meaningless to ask for an initial space-like surface on which to specify the conventional field commutators. The covariant quantum formalism, in which all fields either commute or fail to do so only when the field's points coincide, is proposed as being suitable to quantize gravity. The extension of the covariant quantum formalism to general boson fields that interact in an intrisically nonlinear way with external fields is analysed in some detail. This formalism is applied to the case of the free gravitational field. In a functional representation, the measure en metrics is found to be that proposed by Misner. A basic state of the quantized gravitational theory is proposed, which involves a summation over all permissible metrics in the entire space-time manifohl.

General Relativity and Gravitation, 2005

We present a first attempt to apply the approach of deformation quantization to linearized Einstein's equations. We use the analogy with Maxwell equations to derive the field equations of linearized gravity from a modified Maxwell Lagrangian which allows the construction of a Hamiltonian in the standard way. The deformation quantization procedure for free fields is applied to this Hamiltonian. As a result we obtain the complete set of quantum states and its discrete spectrum.

Journal of Physics: Conference Series, 2011

Physical Review D, 2013

The quantum-reduced loop-gravity technique has been introduced for dealing with cosmological models. We show that it can be applied rather generically: anytime the spatial metric can be gauge-fixed to a diagonal form. The technique selects states based on reduced graphs with Livine-Speziale coherent intertwiners and could simplify the analysis of the dynamics in the full theory.

Physical Review D, 2013

We introduce a new framework for loop quantum gravity: mimicking the spinfoam quantization procedure we propose to study the symmetric sectors of the theory imposing the reduction weakly on the full kinematical Hilbert space of the canonical theory. As a first application of Quantum-Reduced Loop Gravity we study the inhomogeneous Bianchi I model. The emerging quantum cosmological model represents a simplified arena on which the complete canonical quantization program can be tested. The achievements of this analysis could elucidate the relationship between Loop Quantum Cosmology and the full theory.

Arxiv preprint arXiv:1105.0636, 2011

We construct the smeared diffeomorphism constraint operator at finite triangulation from the basic holonomy-flux operators of Loop Quantum Gravity, evaluate its continuum limit on the Lewandowski-Marolf habitat and show that the action of the continuum operator provides an anomaly free representation of the Lie algebra of diffeomorphisms of the 3-manifold. Key features of our analysis include: (i) finite triangulation approximants to the curvature, F i ab of the Ashtekar-Barbero connection which involve not only small loop holonomies but also small surface fluxes as well as an explicit dependence on the edge labels of the spin network being acted on (ii) the dependence of the small loop underlying the holonomy on both the direction and magnitude of the shift vector field (iii) continuum constraint operators which do not have finite action on the kinematic Hilbert space, thus implementing a key lesson from recent studies of parameterised field theory by the authors.