12 research outputs found
Towards Loop Quantum Supergravity (LQSG) II. p-Form Sector
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.Comment: 12 pages. v2: Journal version. Minor clarifications and correction
New Variables for Classical and Quantum Gravity in all Dimensions II. Lagrangian Analysis
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.Comment: 42 pages. v2: Journal version. Some nonessential sign errors in
section 2 corrected. Minor clarification
New Variables for Classical and Quantum Gravity in all Dimensions III. Quantum Theory
We quantise the new connection formulation of D+1 dimensional 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 more elaborate
treatment.Comment: 36 pages. v2: Journal version. Discussion on simplicity constraints
extended. Conclusion and outlook extended. Minor clarification
On a partially reduced phase space quantisation of general relativity conformally coupled to a scalar field
The purpose of this paper is twofold: On the one hand, after a thorough
review of the matter free case, we supplement the derivations in our companion
paper on 'loop quantum gravity without the Hamiltonian constraint' with
calculational details and extend the results to standard model matter, a
cosmological constant, and non-compact spatial slices. On the other hand, we
provide a discussion on the role of observables, focussed on the situation of a
symmetry exchange, which is key to our derivation. Furthermore, we comment on
the relation of our model to reduced phase space quantisations based on
deparametrisation.Comment: 51 pages, 5 figures. v2: Gauge condition used shown to coincide with
CMC gauge. Minor clarifications and correction
Towards Loop Quantum Supergravity (LQSG)
Should nature be supersymmetric, then it will be described by Quantum
Supergravity at least in some energy regimes. The currently most advanced
description of Quantum Supergravity and beyond is Superstring Theory/M-Theory
in 10/11 dimensions. String Theory is a top-to-bottom approach to Quantum
Supergravity in that it postulates a new object, the string, from which
classical Supergravity emerges as a low energy limit. On the other hand, one
may try more traditional bottom-to-top routes and apply the techniques of
Quantum Field Theory. Loop Quantum Gravity (LQG) is a manifestly background
independent and non-perturbative approach to the quantisation of classical
General Relativity, however, so far mostly without supersymmetry. The main
obstacle to the extension of the techniques of LQG to the quantisation of
higher dimensional Supergravity is that LQG rests on a specific connection
formulation of General Relativity which exists only in D+1 = 4 dimensions. In
this Letter we introduce a new connection formulation of General Relativity
which exists in all space-time dimensions. We show that all LQG techniques
developed in D+1 = 4 can be transferred to the new variables in all dimensions
and describe how they can be generalised to the new types of fields that appear
in Supergravity theories as compared to standard matter, specifically
Rarita-Schwinger and p-form gauge fields.Comment: 9 pages. v2: minor improvements in presentation, virtually identical
to published versio
New Variables for Classical and Quantum Gravity in all Dimensions I. Hamiltonian Analysis
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.Comment: 28 pages. v2: Journal version. Minor clarification
Towards Loop Quantum Supergravity (LQSG) I. Rarita-Schwinger Sector
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.Comment: 43 pages. v2: Journal version. Some nonessential sign errors in
sections 2 and 3 corrected. Minor clarifications and correction
Loop quantum gravity without the Hamiltonian constraint
We show that under certain technical assumptions, including the existence of
a constant mean curvature (CMC) slice and strict positivity of the scalar
field, general relativity conformally coupled to a scalar field can be
quantised on a partially reduced phase space, meaning reduced only with respect
to the Hamiltonian constraint and a proper gauge fixing. More precisely, we
introduce, in close analogy to shape dynamics, the generator of a local
conformal transformation acting on both, the metric and the scalar field, which
coincides with the CMC gauge condition. A new metric, which is invariant under
this transformation, is constructed and used to define connection variables
which can be quantised by standard loop quantum gravity methods. While it is
hard to address dynamical problems in this framework (due to the complicated
'time' function), it seems, due to good accessibility properties of the CMC
gauge, to be well suited for problems such as the computation of black hole
entropy, where actual physical states can be counted and the dynamics is only
of indirect importance. The corresponding calculation yields the surprising
result that the usual prescription of fixing the Barbero-Immirzi parameter beta
to a constant value in order to obtain the well-known formula S = a(Phi) A/(4G)
does not work for the black holes under consideration, while a recently
proposed prescription involving an analytic continuation of beta to the case of
a self-dual space-time connection yields the correct result. Also, the
interpretation of the geometric operators gets an interesting twist, which
exemplifies the deep relationship between observables and the choice of a time
function and has consequences for loop quantum cosmology.Comment: 8 pages. v2: Journal version. Black hole state counting based on
physical states added. Applications to loop quantum cosmology discussed.
Gauge condition used shown to coincide with CMC gauge. Minor clarifications.
v3: Erroneous topology dependence of the entropy in journal version
corrected, conclusions fixed accordingly. Main results unaffecte
On the Implementation of the Canonical Quantum Simplicity Constraint
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 map of the spin networks in the kernel of these simplicity
constraint operators 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. Also, we comment on
the relation of our proposals to existing work in the spin foam literature and
how these works could be used in the canonical 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.Comment: 30 pages, 2 figures. v2: Journal version. Comparison to existing
approaches added. Discussion extended. References added. Sign error in
equation (2.15) corrected. Minor clarifications and correction
Quantization of Lorentzian 3d Gravity by Partial Gauge Fixing
D = 2+1 gravity with a cosmological constant has been shown by Bonzom and
Livine to present a Barbero-Immirzi like ambiguity depending on a parameter. We
make use of this fact to show that, for positive cosmological constant, the
Lorentzian theory can be partially gauge fixed and reduced to an SU(2)
Chern-Simons theory. We then review the already known quantization of the
latter in the framework of Loop Quantization for the case of space being
topogically a cylinder. We finally construct, in the same setting, a quantum
observable which, although non-trivial at the quantum level, corresponds to a
null classical quantity.Comment: Notation defect fixed on pages 5 (bottom) and 6 (around Eqs. 3.1)--
19 pages, Late