In our companion papers [1, 2, 3, 4, 5], we studied the canonical formulation of General Relativity (gravitons) coupled to standard matter in terms of connection variables for a compact gauge group without second class constraints in order that Loop Quantum Gravity (LQG) quantisation methods, so far formulated only in three and four spacetime dimensions [6, 7], apply. The actual motivation for doing this comes from Supergravity and String theory [8, 9]: String theory is considered as a candidate for a UV completion of General Relativity, which in its present formulation requires extra dimensions and supersymmetry. Supergravity is considered as the low energy effective field theory limit of String theory. One may therefore call String theory a top – bottom approach. In this series of papers we take first steps towards a bottom – top approach in that we try to canonically quantise the Supergravity theories by LQG methods. While String theory in its present form needs a background dependent and perturbative quantum formulation, the LQG quantum formulation is by design background independent and non perturbative. On the other hand, quantum String theory is much richer above the low energy field theory limit, containing an infinite tower of higher excitation modes of the string, which come into play only when approaching the Planck scale and which are necessary in order to find a theory which is finite at least order by order in perturbation theory. The quantisation of Supergravity is therefore the ideal arena in which to compare these two complementary approaches to quantum gravity, which was not possible so far. At least at low energies, that is, in the semiclassical limit, the two theories should agree with each other, as otherwise they would quantise two different classical theories. Evidently, this opens the very exciting possibility of cross fertilisation between the two approaches, which we are going to address in future publications. The new field content of Supergravity theories as compared to standard matter Lagrangians are 1. Majorana (or Majorana-Weyl) spinor fields of spin 1/2, 3/2 including the Rarita-Schwinger field (gravitino) and 2. additional bosonic fields that appear in order to obtain a complete supersymmetry multiplet in the dimension and the amount N of supersymmetry charges under consideration. The treatment of the Rarita- Schwinger sector and its embedding in the framework of [1, 2, 3, 4, 5] was accomplished in [10]. In this paper, we complete the quantisation of the extra matter content of many Supergravity theories by considering the quantisation of the additional bosonic fields, in particular, p-form fields. Specifically, for reasons of concreteness, we quantise the 3-index photon of 11d Supergravity but it will transpire that the methods employed generalise to arbitrary p. What makes the quantisation possible is that the Gauß constraints of the 3-index photon form an Abelian ideal in the constraint algebra. If this ideal (or subalgebra) would be non – Abelian, then our methods would be insufficient and we most probably would have to use methods from higher gauge theory [11, 12, 13, 14, 15] such as p-groups, p-holonomies etc., a subject which at the moment is not yet sufficiently developed from the mathematical perspective (see [16] for the state of the art of the subject). Despite the Abelian character of this additional Gauß constraint, the quantisation of the theory is not straightforward and cannot be performed in complete analogy to the treatment of the Abelian Gauß constraint of standard 1-form matter [17]. This is due to a Chern-Simons term in the Supergravity action, whose presence is dictated by supersymmetry and which makes the theory in fact self-interacting, that is, the Hamiltonian is a fourth order polynomial in the 3-connection and its conjugate momentum just like in Yang-Mills theory. In particular, while one can define a holonomy flux algebra as for Abelian Maxwell theory, the Ashtekar-Isham-Lewandowski representation [18, 19] is inadequate because the Abelian gauge group does not preserve the holonomy flux algebra. A solution to the problem lies in performing a reduced phase space quantisation in terms of a twisted holonomy flux algebra, which is in fact Gauß invariant. We were not able to find a background independent representation of the corresponding Heisenberg algebra, which also differs by a twist from the usual one, however, one succeeds when formulating the quantum theory in terms of the corresponding Weyl elements. The resulting Weyl algebra is not of standard form and to the best of our knowledge it has not been quantised before. We show that it admits a state of the Narnhofer-Thirring type [20] whence the Hilbert space representation follows by the GNS construction. The Hamiltonian (constraint) can be straightforwardly expressed in terms of the Weyl elements, in fact it is quadratic in terms of the classical observables, that is, the generators of the Heisenberg algebra. inspirehep.net/record/900476
Posted on: Sat, 02 Aug 2014 13:58:44 +0000
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