Year: 2015
Type: dissertation
Abstract: In this work, we extend loop quantum gravity (LQG) both, to higher dimensions and supersymmetry (i.e. supergravity theories), thus overcoming the current limitation to 3+1 dimensions with standard model matter fields. On the one hand, this gives a proof of principle that LQG is in accordance with these two theoretical concepts, and on the other hand hopefully allows contact with superstring/M - theory, which necessarily is supersymmetric and formulated in ten or eleven spacetime dimensions. Symmetry arguments suggest that supergravity theories in the corresponding dimensions constitute the low energy effective field theory limit of superstring/M - theory. This makes a study of the loop quantisation thereof, which we start here, a promising endeavour at the border between the two approaches.
In more detail, our findings are the following: firstly, a new canonical formulation for general relativity in D + 1 spacetime dimensions (D ≥ 2) on a Yang Mills theory phase space is presented for the first time, with the core properties that 1. the canonical variables encoding the metric information are a real connection and its real conjugate momentum, in particular satisfying the standard canonical Poisson bracket relations, 2. the gauge group can be chosen to be a compact group (namely SO(D + 1)) for both, Lorentzian and Euclidean signature spacetimes, and 3. the system of constraints is first class (in Dirac’s terminology). Up to now, such a formulation was only known for D = 3 (and D = 2), corresponding to Ashtekar Barbero variables, constituting the classical foundation of the loop quantisation programme.
The quantisation procedure itself is formulated almost independently of the number of spacetime dimensions and the choice of compact gauge group, and therefore the lack of higher dimensional analogues of LQG only was caused by the missing classical canonical formulation satisfying 1. - 3. Thus it is not surprising and we show explicitly that the new formulation we present can be quantised using the methods developed in the loop community straightforwardly to obtain LQG theories in higher dimensions.
The formulation which we present is genuinely new in that it does not reduce to the Ashtekar Barbero formulation for D = 3, and furthermore for D > 2 comes with an additional set of constraints, the so called simplicity constraints, which pose the only conceptually new challenge when quantising. Interestingly, these constraints are not at all unknown in (quantum) gravity research, and in particular are a standard ingredient in the covariant approach to LQG called spin foam models. The formulation in this sense builds a novel bridge between the covariant and canonical approaches to LQG. The quantum anomalies known for this constraint from spin foams are recovered, which lead to problems when implementing it at the quantum level. We present some new proposals of how to deal with these problems.
In the second part of this work, we give an extension of the above framework to the loop quantisation of a large class of Lorentzian signature supergravities, including in particular the D + 1 = 4 N = 8, D + 1 = 11 N = 1 and D + 1 = 10 N = 1 theories. Concretely, we incorporate standard and also non-standard matter fields, which appear in supergravity theories due to the requirement of supersymmetry, into the afore developed framework of higher dimensional LQG.
Coupling to standard model matter fields has already been achieved in usual LQG and the results obtained there carry over to the case at hand. The only exception is the treatment of Dirac fermions, which needs slight adjustment: coming from an action principle, the Dirac field transforms in the spinor representation of the gauge group SO(1, D) for the physically relevant Lorentzian theory, but due to the strong similarity of the Lorentzian and the Euclidean Clifford algebras, the gauge group can be exchanged for SO(D + 1) to fit in with the gravitational degrees of freedom.
Typical non-standard fields appearing in supergravity theories are the spin 3/2 Rarita Schwinger field (“gravitino”) on the fermionic side, and (Abelian) higher p-form fields as novel bosonic fields (i.e. generalisations of the Maxwell field to higher form degree).
The former usually is a Majorana fermion (i.e. it is its own antiparticle) and therefore belongs to a real representation space of SO(1, D). In order to formulate supergravities in terms of SO(D + 1) gauge theories, we again have to exchange the gauge group SO(1, D) with SO(D + 1), but there is no action of SO(D + 1) on these real representation spaces, which hugely complicates the passage when compared to the case of Dirac fermions. We present a solution to this problem and for the first time, to the best of the author’s knowledge, provide a background independent Hilbert space representation for the gravitino field.
Concerning novel bosonic fields, we exemplarily treat the three-form field (“three index photon”) of D + 1 = 11 N = 1 supergravity. Due to an additional Chern Simons term in the action, this field is not a simple generalisation of the Maxwell field to three-forms, but actually becomes self interacting and the equivalent of the electric field is not gauge invariant. We propose a reduced phase space quantisation with respect to the equivalent of the Gaus constraint, and the background independent representation we use is given by a state of Narnhofer-Thirring type, which already has been used in the loop literature in Thiemann’s treatment of the closed bosonic string.
In the third part of this work, as a first application of the new variables, we extend the isolated horizon treatment (a quasi-local notion of black holes) in LQG to higher dimensions. In D = 3, the use of Ashtekar Barbero variables induces a Chern Simons theory on the horizon and the quantisation thereof and subsequent state counting led to the derivation of the famous Bekenstein Hawking entropy formula for black holes from LQG. Here, we study (non-distorted) isolated horizons in 2(n + 1) dimensional spacetimes and find that using the new variables induces an SO(2(n+1)) Chern Simons theory thereon. Since this theory, unlike its D = 3 counterpart, has local degrees of freedom, the quantisation and finally rederivation of the entropy formula become significantly more intricate and are left for further research.
We want to stress that several aspects of both, the higher dimensional as well as the supersymmetric extension, definitely deserve further study to actually catch up with the current status of usual canonical LQG. In the non-supersymmetric case, this concerns mainly the implementation of the simplicity constraint and its interplay with the dynamics. In the supersymmetric case, of course the supersymmetry constraint needs intensive study, in particular its role in the quantum super Dirac algebra. We hope that the generalisation of LQG to higher dimensions and supersymmetry achieved in this work will spark further development to clarify the mentioned open problems and finally lead to new interrelations between LQG and superstring/M - theory.
Author: Andreas Thurn
Language: English
Subfield: Nuclear and High Energy Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2338105899