http://arxiv.org/abs/1105.3480
Towards a Spin-foam unification of gravity, Yang-Mills interactions and matter fields
Stephon Alexander, Antonino Marciano, Ruggero Altair Tacchi
(Submitted on 17 May 2011)
"We propose a new method of unifying gravity and the Standard Model by introducing a spin-foam model. We realize a unification between an SU(2) Yang-Mills interaction and 3D general relativity by considering a Spin(4) Plebanski action. The theory is quantized a la spin-foam by implementing the analogue of the simplicial constraints for the broken phase of the Spin(4) symmetry. A natural 4D extension of the theory is shown. We also present a way to recover 2-point correlation functions between the connections as a first way to implement scattering amplitudes between particle states, aiming to connect Loop Quantum Gravity to new physical predictions."
5 pages, 2 figures
http://arxiv.org/abs/1105.3703
New Variables for Classical and Quantum Gravity in all Dimensions I. Hamiltonian Analysis
Norbert Bodendorfer, Thomas Thiemann, Andreas Thurn
(Submitted on 18 May 2011)
"Loop Quantum Gravity heavily relies on a connection formulation of General Relativity such that 1. the connection Poisson commutes with itself and 2. the corresponding gauge group is compact. This can be achieved starting from the Palatini or Holst action when imposing the time gauge. Unfortunately, this method is restricted to D+1 = 4 spacetime dimensions. However, interesting String theories and Supergravity theories require higher dimensions and it would therefore be desirable to have higher dimensional Supergravity loop quantisations at one's disposal in order to compare these approaches. In this series of papers, we take first steps towards this goal. The present first paper develops a classical canonical platform for a higher dimensional connection formulation of the purely gravitational sector. The new ingredient is a different extension of the ADM phase space than the one used in LQG, which does not require the time gauge and which generalises to any dimension D > 1. The result is a Yang-Mills theory phase space subject to Gauss, spatial diffeomorphism and Hamiltonian constraint as well as one additional constraint, called the simplicity constraint. The structure group can be chosen to be SO(1,D) or SO(D+1) and the latter choice is preferred for purposes of quantisation."
28 pages
http://arxiv.org/abs/1105.3704
New Variables for Classical and Quantum Gravity in all Dimensions II. Lagrangian Analysis
Norbert Bodendorfer, Thomas Thiemann, Andreas Thurn
(Submitted on 18 May 2011)
"We rederive the results of our companion paper, for matching spacetime and internal signature, by applying in detail the Dirac algorithm to the Palatini action. While the constraint set of the Palatini action contains second class constraints, by an appeal to the method of gauge unfixing, we map the second class system to an equivalent first class system which turns out to be identical to the first class constraint system obtained via the extension of the ADM phase space performed in our companion paper. Central to our analysis is again the appropriate treatment of the simplicity constraint. Remarkably, the simplicity constraint invariant extension of the Hamiltonian constraint, that is a necessary step in the gauge unfixing procedure, involves a correction term which is precisely the one found in the companion paper and which makes sure that the Hamiltonian constraint derived from the Palatini Lagrangian coincides with the ADM Hamiltonian constraint when Gauss and simplicity constraints are satisfied. We therefore have rederived our new connection formulation of General Relativity from an independent starting point, thus confirming the consistency of this framework."
43 pages
http://arxiv.org/abs/1105.3705
New Variables for Classical and Quantum Gravity in all Dimensions III. Quantum Theory
Norbert Bodendorfer, Thomas Thiemann, Andreas Thurn
(Submitted on 18 May 2011)
"We quantise the new connection formulation of D+1 General Relativity developed in our companion papers by Loop Quantum Gravity (LQG) methods. It turns out that all the tools prepared for LQG straightforwardly generalise to the new connection formulation in higher dimensions. The only new challenge is the simplicity constraint. While its 'diagonal' components acting at edges of spin network functions are easily solved, its 'off-diagonal' components acting at vertices are non trivial and require a Master constraint treatment."
34 pages
http://arxiv.org/abs/1105.3706
New Variables for Classical and Quantum Gravity in all Dimensions IV. Matter Coupling
Norbert Bodendorfer, Thomas Thiemann, Andreas Thurn
(Submitted on 18 May 2011)
"We employ the techniques introduced in the companion papers to derive a connection formulation of Lorentzian General Relativity coupled to Dirac fermions in dimensions D+1 > 2 with compact gauge group. The technique that accomplishes that is similar to the one that has been introduced in 3+1 dimensions already: First one performs a canonical analysis of Lorentzian General Relativity using the time gauge and then introduces an extension of the phase space analogous to the one employed in the first paper of this series to obtain a connection theory with SO(D+1) as the internal gauge group subject to additional constraints. The success of this method rests heavily on the strong similarity of the Lorentzian and Euclidean Clifford algebras. A quantisation of the Hamiltonian constraint is provided."
13 pages
http://arxiv.org/abs/1105.3708
On the Implementation of the Canonical Quantum Simplicity Constraint
Norbert Bodendorfer, Thomas Thiemann, Andreas Thurn
(Submitted on 18 May 2011)
"In this paper, we are going to discuss several approaches to solve the quadratic and linear simplicity constraints in the context of the canonical formulations of higher dimensional General Relativity and Supergravity developed in our companion papers. Since the canonical quadratic simplicity constraint operators have been shown to be anomalous in any dimension D > 2, non-standard methods have to be employed to avoid inconsistencies in the quantum theory. We show that one can choose a subset of quadratic simplicity constraint operators which are non-anomalous among themselves and allow for a natural unitary 1-1 map to the SU(2)-based Ashtekar-Lewandowski Hilbert space in D = 3. The linear constraint operators on the other hand are non-anomalous by themselves, however their solution space will be shown to differ in D = 3 from the expected Ashtekar-Lewandowski Hilbert space. We comment on possible strategies to make a connection to the quadratic theory. We emphasise that many ideas developed in this paper are certainly incomplete and should be considered as suggestions for possible starting points for more satisfactory treatments in the future."
22 pages, 2 figures
http://arxiv.org/abs/1105.3709
Towards Loop Quantum Supergravity (LQSG) I. Rarita-Schwinger Sector
Norbert Bodendorfer, Thomas Thiemann, Andreas Thurn
(Submitted on 18 May 2011)
"In our companion papers, we managed to derive a connection formulation of Lorentzian General Relativity in D+1 dimensions with compact gauge group SO(D+1) such that the connection is Poisson commuting, which implies that Loop Quantum Gravity quantisation methods apply. We also provided the coupling to standard matter. In this paper, we extend our methods to derive a connection formulation of a large class of Lorentzian signature Supergravity theories, in particular 11d SUGRA and 4d, N = 8 SUGRA, which was in fact the motivation to consider higher dimensions. Starting from a Hamiltonian formulation in the time gauge which yields a Spin(D) theory, a major challenge is to extend the internal gauge group to Spin(D+1) in presence of the Rarita-Schwinger field. This is non trivial because SUSY typically requires the Rarita-Schwinger field to be a Majorana fermion for the Lorentzian Clifford algebra and Majorana representations of the Clifford algebra are not available in the same spacetime dimension for both Lorentzian and Euclidean signature. We resolve the arising tension and provide a background independent representation of the non trivial Dirac antibracket *-algebra for the Majorana field which significantly differs from the analogous construction for Dirac fields already available in the literature."
43 pages
http://arxiv.org/abs/1105.3710
Towards Loop Quantum Supergravity (LQSG) II. p-Form Sector
Norbert Bodendorfer, Thomas Thiemann, Andreas Thurn
(Submitted on 18 May 2011)
"In our companion paper, we focussed on the quantisation of the Rarita-Schwinger sector of Supergravity theories in various dimensions by using an extension of Loop Quantum Gravity to all spacetime dimensions. In this paper, we extend this analysis by considering the quantisation of additional bosonic fields necessary to obtain a complete SUSY multiplet next to graviton and gravitino in various dimensions. As a generic example, we study concretely the quantisation of the 3-index photon of 11d SUGRA, but our methods easily extend to more general p-form fields. Due to the presence of a Chern-Simons term for the 3-index photon, which is due to local SUSY, the theory is self-interacting and its quantisation far from straightforward. Nevertheless, we show that a reduced phase space quantisation with respect to the 3-index photon Gauss constraint is possible. Specifically, the Weyl algebra of observables, which deviates from the usual CCR Weyl algebras by an interesting twist contribution proportional to the level of the Chern-Simons theory, admits a background independent state of the Narnhofer-Thirring type."
12 pages
http://arxiv.org/abs/1105.3724
Loop quantum cosmology of k=1 FRW: A tale of two bounces
Alejandro Corichi, Asieh Karami
(Submitted on 18 May 2011)
"We consider the k=1 Friedman-Robertson-Walker (FRW) model within loop quantum cosmology, paying special attention to the existence of an ambiguity in the quantization process. In spatially non-flat anisotropic models such as Bianchi II and IX, the standard method of defining the curvature through closed holonomies is not admissible. Instead, one has to implement the quantum constraints by approximating the connection via open holonomies. In the case of flat k=0 FRW and Bianchi I models, these two quantization methods coincide, but in the case of the closed k=1 FRW model they might yield different quantum theories. In this manuscript we explore these two quantizations and the different effective descriptions they provide of the bouncing cyclic universe. In particular, as we show in detail, the most dramatic difference is that in the theory defined by the new quantization method, there is not one, but two different bounces through which the cyclic universe alternates. We show that for a 'large' universe, these two bounces are very similar and, therefore, practically indistinguishable, approaching the dynamics of the holonomy based quantum theory."
18 pages, 3 figures
Brief mention:
http://arxiv.org/abs/1105.3504
Beyond Einstein-Cartan gravity: Quadratic torsion and curvature invariants with even and odd parity including all boundary terms
Peter Baekler (Duesseldorf), Friedrich W. Hehl (Cologne & Columbia, Missouri)
(Submitted on 17 May 2011)
Recently, gravitational gauge theories with torsion have been discussed by an increasing number of authors from a classical as well as from a quantum field theoretical point of view. The Einstein-Cartan(-Sciama-Kibble) Lagrangian has been enriched by the parity odd pseudoscalar curvature (Hojman, Mukku, and Sayed) and by torsion square and curvature square pieces, likewise of even and odd parity. (i) We show that the inverse of the so-called Barbero-Immirzi parameter multiplying the pseudoscalar curvature, because of the topological Nieh-Yan form, can only be appropriately discussed if torsion square pieces are included. (ii) The quadratic gauge Lagrangian with both parities, proposed by Obukhov et al. and Baekler et al., emerges also in the framework of Diakonov et al.(2011). We establish the exact relations between both approaches by applying the topological Euler and Pontryagin forms in a Riemann-Cartan space expressed for the first time in terms of irreducible pieces of the curvature tensor. (iii) Only in a Riemann-Cartan spacetime, that is, in a spacetime with torsion, parity violating terms can be brought into the gravitational Lagrangian in a straightforward and natural way. Accordingly, Riemann-Cartan spacetime is a natural habitat for chiral fermionic matter fields.
12 page
http://arxiv.org/abs/1105.3612
Braided Tensor Products and the Covariance of Quantum Noncommutative Free Fields
Jerzy Lukierski, Mariusz Woronowicz (IFT, Wroclaw Univ.)
(Submitted on 18 May 2011)