K Giesel

T Thiemann
email: thiemann@aei.mpg.de

Algebraic Quantum Gravity (AQG) IV. Reduced Phase Space Quantisation of Loop Quantum Gravity

We perform a canonical, reduced phase space quantisation of General Relativity by Loop Quantum Gravity (LQG) methods.

The explicit construction of the reduced phase space is made possible by the combination of 1. the Brown -Kuchař mechanism in the presence of pressure free dust fields which allows to deparametrise the theory and 2. Rovelli's relational formalism in the extended version developed by Dittrich to construct the algebra of gauge invariant observables.

Since the resulting algebra of observables is very simple, one can quantise it using the methods of LQG. Basically, the kinematical Hilbert space of non reduced LQG now becomes a physical Hilbert space and the kinematical results of LQG such as discreteness of spectra of geometrical operators now have physical meaning. The constraints have disappeared, however, the dynamics of the observables is driven by a physical Hamiltonian which is related to the Hamiltonian of the standard model (without dust) and which we quantise in this paper.

Introduction

The objects of ultimate interest in a field theory with gauge symmetry are the gauge invariant observables. There are two major approaches to the canonical quantisation of such theories. In the so called Dirac approach one first constructs Hilbert space representations of gauge variant non observables and then imposes the vanishing of the quantised version of the classical gauge symmetry generators (constraints) as a selection principle for physical states. The associated physical Hilbert space then hopefully (if there are no anomalies) carries a representation of the observable algebra. In the so called reduced phase space approach one first constructs the classical observables and then directly looks for representations of that algebra.

The advantage of the Dirac approach is that the unreduced phase space of non observables is typically a smooth (Banach) manifold so that the algebra of non -observables is sufficiently simple and representations thereof are easy to construct. Its disadvantage is that one has to deal with spurious degrees of freedom which is the possible source of ambiguities and anomalies in the gauge symmetry algebra. The advantage of the reduced phase space approach is that one never has to care about kinematical Hilbert space representations. However, its disadvantage is that the reduced phase space typically no longer is a smooth manifold turning the induced algebra of observables so difficult that representations thereof are hard to find.

The reduced phase space of General Relativity with standard matter is hard to construct explicitly. However, on can combine two independent recent developments in order to make progress:

On the one hand, Brown & Kuchař have shown in a seminal paper [START_REF] Brown | Dust as a standard of space and time in canonical quantum gravity[END_REF] that there is hope to construct observables if one adds pressure free dust to the theory. This is because one can then write the constraints in deparametrised form 1 .

On the other hand, there is Rovelli's relational formalism [START_REF] Rovelli | What is observable in classical and quantum gravity?[END_REF] for constructing observables which we need in the extended form developed by Dittrich [START_REF] Dittrich | Partial and complete observables for Hamiltonian constrained systems[END_REF]. With this formalism one can write the observables as an infinite series F f ;T in terms of powers of so called clock variables T and with coefficients involving multiple Poisson brackets between constraints C and non observables f such that the series is (formally) gauge invariant 2 . Remarkably [START_REF] Dittrich | Partial and complete observables for Hamiltonian constrained systems[END_REF][START_REF] Thiemann | Reduced phase space quantization and Dirac observables[END_REF], the map F T : f → F f ;T is a Poisson homomorphism between the algebra of non observables f and the algebra of observables with respect to a certain Dirac bracket (which is uniquely determined by the constraints and the functions T ). Now usually Dirac brackets make the Poisson structure so complicated that one cannot find representations thereof. However, as observed in [START_REF] Thiemann | Reduced phase space quantization and Dirac observables[END_REF], if the system deparametrises, if one uses as clocks T the configuration variables conjugate to the momenta P in C = P + H and if one considers functions f which do not depend on T, P3 then F T becomes a Poisson isomorphism. Moreover, the functions H in C = P +H become physical, conserved Hamiltonian densities which drive the physical evolution of the observables. This implies that a reduced phase space quantisation strategy becomes available, since to find representations of the F f,T is as easy as for the f . The only non trivial problem left is to find representations which support the physical Hamiltonian 4 .

In [START_REF] Thiemann | Solving the problem of time in general relativity and cosmology with phantoms and k-essence[END_REF] these two independent observations were combined and the algebra of classical physical Observables was constructed explicitly by adding a general scalar field Lagrangian without potential to the Einstein -Hilbert and standard model Lagrangian. It turns out that among the, in principle, infinite number of physical Observables there is a unique, positive Hamiltonian selected.

In [START_REF] Giesel | Manifestly gauge-invariant general relativistic perturbation theory: I[END_REF][START_REF] Giesel | Manifestly gauge invariant general relativistic perturbation theory: II. FRW background and first order[END_REF] that framework was further improved by using as specific scalar field the pressure free dust of Brown & Kuchař. The corresponding Hamiltonian is positive, reduces to the ADM energy far away from the sources and to the standard model Hamiltonian on flat space. It generates equations of motion for the observables associated to the non dust variables that are in agreement with the Einstein equations for the system without dust, up to small corrections which originate from the presence of the dust. In particular one can develop a manifestly gauge invariant cosmological perturbation theory to all orders which was shown to reproduce the linear order as developed by Mukhanov, Feldmann and Brandenberger [START_REF] Mukhanov | Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations[END_REF]. The dust serves as a material reference system which we couple dynamically as fields rather than assuming the usual test observers in order to give the Einstein equations (modulo gauge freedom) the interpretation of evolution equations of observable quantities. This leads to in principle observable deviations from the standard formalism which however decay during the cosmological evolution.

In this paper we quantise the algebra of observables constructed in [START_REF] Giesel | Manifestly gauge-invariant general relativistic perturbation theory: I[END_REF]. Actually there is not much to do because that algebra is isomorphic to the Poisson algebra of General Relativity plus the standard model on R × S where S is the dust space manifold. Hence we can take over the kinematical Hilbert space representation that is used in Loop Quantum Gravity (LQG) [START_REF] Ashtekar | Representations of the holonomy algebras of gravity and non-Abelean gauge theories[END_REF][START_REF] Ashtekar | Representation theory of analytic holonomy C algebras[END_REF]. For recent reviews on LQG see [START_REF] Rovelli | Loop quantum gravity[END_REF], for books see [START_REF] Rovelli | Quantum Gravity[END_REF]. One may object that this representation is less natural here than in usual Dirac quantised LQG where it is uniquely selected on physical grounds [START_REF] Lewandowski | Uniqueness of diffeomorphism invariant states on holonomy -flux algebras[END_REF][START_REF] Fleischhack | Representations of the Weyl Algebra in Quantum Geometry[END_REF], namely one wants to have a unitary representation of the spatial diffeomorphism group of the coordinate manifold X which is a gauge group (passive diffeomorphisms) there. Since all our observables are gauge invariant, we have no diffeomorphism gauge group any longer, hence that physical selection criterion is absent. However, it is replaced by a different one: It turns out that the physical Hamiltonian has the diffeomorphism group of the dust label space as symmetry group. These diffeomorphisms change our observables, they are active diffeomorphisms since they map between physically distinguishable dust space labels. Thus we may apply the same selection criterion. Now the interesting remaining question is whether that representation allows us to define the quantised version of the physical Hamiltonian. Maybe not surprisingly, it turns out that the same techniques that allowed to construct the quantum Hamiltonian constraint [START_REF] Thiemann | Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity[END_REF] and the master constraint [START_REF] Thiemann | The Phoenix project: master constraint programme for loop quantum gravity[END_REF] in usual Dirac quantised LQG can be used to define the quantised physical Hamiltonian. This operator is positive, hence symmetric and upon taking its natural Friedrich extension, it becomes self -adjoint. In order to preserve its classical, active, spatial diffeomorphism symmetry it turns out that one has to define it in such a way that it preserves the graph of a spin network function that it acts on. The techniques developed in [START_REF] Giesel | Algebraic quantum gravity (AQG) I. Conceptual setup[END_REF] can now be applied to show, using the semiclassical states introduced in [START_REF] Thiemann | Gauge Field Theory Coherent States (GCS): I. General properties[END_REF], that the physical Hamiltonian has the correct semiclassical limit on sufficinently fine graphs. In fact, in order to get rid of the graph dependence one can use the generalisation of LQG to Algebraic Quantum Gravity [START_REF] Giesel | Algebraic quantum gravity (AQG) I. Conceptual setup[END_REF]. This casts quantum gravity completely into the framework of (Hamiltonian) lattice gauge theory [START_REF] Susskind | Hamiltonian formulation of Wilson's lattice gauge theories[END_REF][START_REF] Renteln | A Lattice Approach To Spinorial Quantum Gravity[END_REF] with one crucial difference: There is no continuum limit to be taken because we are in a background independent theory with active diffeomorphisms as symmetries.

The attractive feature of this reduced phase space approach is that we no longer need to deal with the constraints: No anomalies can arise, no master constraint needs to be constructed, no physical Hilbert space needs to be derived by complicated group averaging techniques. We map a conceptually complicated gauge system to the conceptually safe realm of an ordinary dynamical Hamiltonian system. The kinematical results of LQG such as discreteness of spectra of geometric operators now become physical predictions. This is a concrete implementation of the programme outlined for the full theory in [START_REF] Brunnemann | On (cosmological) singularity avoidance in loop quantum gravity[END_REF] and generalises the reduced phase space techniques recently adopted for the Loop Quantum Cosmology (LQC) truncation of LQG [START_REF] Bojowald | Absence of singularity in loop quantum cosmology[END_REF][START_REF] Ashtekar | Mathematical structure of loop quantum cosmology[END_REF][START_REF] Ashtekar | Black hole evaporation: A paradigm[END_REF][START_REF] Ashtekar | Quantum nature of the big bang[END_REF] which is a toy model for the cosmological sector of LQG, to the full theory.

It "remains" to analyse the physical Hamiltonian in detail since it encodes the complete dynamics of General Relativity coupled to the standard model. The following tasks should be addressed in the future:

1. Vacuum and spectral gap For a start we notice that the physical Hamiltonian does not depend explicitly on an external time parameter. Our Hamiltonian system which dynamically couples geometry and matter is a conservative system. This is in contrast to QFT on curved and in particular time dependent background spacetime metrics where one quantises matter propagating on an externally given background geometry. The Hamiltonian of that QFT is not preserved and thus even the notion of a ground state or vacuum as a lowest energy eigenstate becomes time dependent which leads to constant particle creation problems etc. [START_REF] Wald | Quantum field theory in curved space-time and black hole thermodynamics[END_REF]. In our approach the notion of a vacuum state would not suffer from those problems. This appears as a conceptual improvement although of course the lowest eigenvalue of the Hamiltonian could be vastly degenerate. Also, the minimum of the spectrum of the Hamiltonian might not lie in its discrete (more precisely, pure point) part so that the "ground state(s)" would not be normalisable.

Scattering theory

With a physical Hamiltonian H at our disposal we can in principle perform scattering theory, that is, we can compute matrix elements of the time evolution operator U (τ ) = exp(iτ H).

The analytical evaluation of those matrix elements is of course too difficult but as in ordinary QFT we may use Fermi's golden rule and expand, for short time intervals τ , the exponential as

U (τ ) = 1 H + iτ H +O(τ ) 2 .
The matrix elements of H seem hopeless to compute because it involves square roots of a positive self adjoint operator for whose precise evaluation we would need the associated projection valued measure which of course we do not have. However, since in scattering theory initial and final states are excitations over a ground state which we do not know exactly but presumably can approximate by kinematical coherent states, one can invoke the technique developed in [START_REF] Giesel | Algebraic quantum gravity (AQG) I. Conceptual setup[END_REF] to expand the square root of the operator around the square root of its expectation value. We will do this in a future project. Of course there are issues to be resolved such as those of the existence of asymptotic states [START_REF] Haag | Local Quantum Physics[END_REF] and how one implements them in our formalism, see e.g. [START_REF] Sahlmann | Towards the QFT on curved spacetime limit of QGR. 1. A general scheme[END_REF] for some basic ideas.

Anomalies

As already mentioned, the Hamiltonian H has a huge symmetry group of which Diff(S) is a subgroup and it is easy to implement this symmetry at the quantum level. However, there is another infinite classical, Abelian symmetry group N which is generated by the Hamiltonian density functions H(σ) and in terms of which the Hamiltonian reads H = S d 3 σ H(σ).

Classically one has {H(σ), H(σ )} = 0 which of course implies classically that {H(σ), H}=0.

The Lie algebra of the total classical symmetry group thus consists of infinitesimal active diffeomorphisms and infinitesimal transformations generated by the H(σ). The latter form an Abelian Poisson ideal and thus N is an Abelian invariant subgroup in the total symmetry group which hence is a semidirect product G = N Diff(σ). Presumably, in the naive quantisation of H that we consider as a preliminary proposal in this paper, the latter symmetry is explicitly broken, or anomalous although semiclassically it is preserved. In order to reinstall it, one can try to make use of renormalisation group techniques associated to so called improved or perfect actions [START_REF] Hasenfratz | The Theoretical Background and Properties of Perfect Actions[END_REF].

Lattice numerical methods

It transpires that within the framework proposed here many of the conceptual problems of canonical quantum gravity have been solved and the technical tasks have been simplified and reduced to a detailed analysis of the operator H, of course, at the price to have introduced additional, albeit unobservable, matter as a material reference system and a possibly only locally (in phase space) description. Since H is a complicated operator which is is formulated in terms of lattice like variables especially in the AQG version, it is natural to use Monte Carlo methods in order to study the operator numerically.

QFT on curved spacetimes and standard model

It is widely accepted that the framework of QFT on curved spacetimes [START_REF] Wald | Quantum field theory in curved space-time and black hole thermodynamics[END_REF] should be an excellent approximation to quantum gravity whenever the metric fluctuations are small. In particular, when the background spacetime is Minkowski, then the standard model must be reproduced. Besides that, one would like to see whether our background independent lattice theory which is manifestly UV finite and non perturbative can explore the non perturbative sector of the standard model such as QCD. Another interesting question is whether our explicitly geometry -mater coupled system can lead to an improved understanding of the Hawking effect due to the possibility to take care of backreaction effects.

Effective action, universality, ambiguities

Our framework presents a canonical quantisation of the field theory underlying the Einstein Hilbert action plus standard model action. Now computations within perturbative QFT and also string theory suggest that the effective action5 for gravity is an extension of the Einstein -Hilbert Lagrangian by higher derivative terms and an often asked question is whether one should not quantise these more general actions. There are several remarks in order:

A. The effective action is a complicated, often even non local, action which takes care of all higher loop diagrammes obtained from a simple bare action. It looks like a classical action but it actually encodes all quantum fluctuations. Therefore it is inappropriate to quantise that classical action anew, it would not produce the same quantum theory as the bare action.

B. Still one could just add all possible higher derivative terms from the outset. While one can canonically quantise such theories by the Ostrogradsky formalism, this leads in general to a drastic increase in the number of degrees of freedom [START_REF] Gitman | Quantization of Fields with Constraints[END_REF] due to the appearance of higher time derivatives.

C. In the Euclidean formulation of QFT on Minkowski as a path integral one entertains a related (Wilson) notion of effective action as the action that one obtains when integrating out degrees of freedom labelled by (in Fourier space) momenta above a certain energy scale 6 . This also produces various higher derivative terms at lower energies as compared to the bare action which is defined at infinite energy. Now the couplings of the bare action also are in principle unknown, however, for many theories that does not matter due to a phenomenon called universality: The couplings of the higher derivative terms depend on the energy scale and a coupling is called relevant, marginal or irrelevant respectively if it grows, remains constant or decreases in the low energy limit. A universal theory is such that all but a finite number of the couplings are irrelevant. One may ask whether one can see universality also in the canonical formalism, however, there are several obstacles in answering this question. First of all, the Euclidean formulation uses a Wick rotation which is only possible for background dependent theories where the background has a presentation with an analytic dependence on the time coordinate. In quantum gravity the metric becomes an operator, hence Wick rotation and therefore a Euclidean formulation is not possible. One should therefore define the Wilsonian effective action directly in the Hamiltonian (Lorentzian) formulation, however, that has not been done so far.

It seems to us that in order to make progress on this kind of questions one should first try to define a Hamiltonian notion of effective action, see [START_REF] Bojowald | Effective equations of motion for quantum systems[END_REF] for a possible direction. Then, if the symmetry arguments mentioned under [3.] are insufficient in order to fix the quantisation (discretisation) ambiguities in the definition of H, possibly universality studies may lead to further understanding.

Singularity avoidance

In quantum gravity we expect or want to resolve two types of singularities: First, QFT kind of short distance singularities which come from the fact that in interacting field theories one has to deal with products of operator valued distributions. Secondly, classical General Relativity kind of singularities which are simply a feature of the Einstein equations to predict that generically spacetimes are geodesically incomplete. An analytical measure for such spacetime singularities are typically divergences of curvature invariants. Now as shown in [START_REF] Thiemann | Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity[END_REF], UV type of singularities are absent at the non gauge invariant level, specifically, the quantum constraints are densely defined. In [START_REF] Brunnemann | On (cosmological) singularity avoidance in loop quantum gravity[END_REF] it was discovered, in the context of usual LQG that expectation values of non gauge invariant curvature operators with respect to non gauge invariant coherent states that are peaked on a classically singular (FRW) trajectory remain finite as one reaches the singularity, thus backing up the much more spectacular results of [START_REF] Bojowald | Absence of singularity in loop quantum cosmology[END_REF][START_REF] Ashtekar | Mathematical structure of loop quantum cosmology[END_REF][START_REF] Ashtekar | Black hole evaporation: A paradigm[END_REF][START_REF] Ashtekar | Quantum nature of the big bang[END_REF] which are at the level of the physical Hilbert space albeit for a toy model and not the full theory.

While these are encouraging results, they are at the kinematical level only and thus are inconclusive. However, with the technology developed in this paper we can transfer both results literally and with absolutely no changes to the physical Hilbert space. As far as the spacetime singularity resolution is concerned, this is still not enough because the coherent states that we are using, while being now physical coherent states, they are not adapted to the physical Hamiltonian and thus may spread out under the quantum dynamics generated by U (τ ). In other words, given gauge invariant initial data m(0) and a coherent state ψ 0 that we prepare at τ = 0 and which is peaked on m(0), it may be that after short time τ the state U (τ )ψ 0 is very different form the state ψ τ which is peaked on the classical trajectory τ → m(τ ). Therefore, in order to come to conclusions one should rather study expectation values with respect to the states U (τ )ψ 0 rather than ψ τ . In addition, one should try to construct dynamical coherent states for which such a spread does not happen. However, this is a difficult task already for the unharmonic oscillator.

The plan of the paper is as follows:

In section two we review the essentials of [START_REF] Giesel | Manifestly gauge-invariant general relativistic perturbation theory: I[END_REF][START_REF] Giesel | Manifestly gauge invariant general relativistic perturbation theory: II. FRW background and first order[END_REF] in order to make this article self -contained. This will lead to the reduced phase space and the classical physical Hamiltonian.

In section three we quantise the reduced phase space using methods from LQG and obtain the physical Hilbert space almost for free. Then we implement the physical Hamiltonian on that Hilbert space. We do this both for LQG and the AQG extension.

In section four we summarise and conclude.

2 Review of the Brown -Kuchař and relational framework

Brown -Kuchař Lagrangian

In [START_REF] Brown | Dust as a standard of space and time in canonical quantum gravity[END_REF] Brown and Kuchař add the following Lagrangian to the Einstein -Hilbert and standard model Lagrangian 7 on the spacetime manifold M

S D = - 1 2 M d 4 X | det(g)| ρ [g µν U µ U ν + 1] (2.1)
where the one form U is defined by U = -dT + W j dS j and the index j takes values 1, 2, 3 while µ, ν take values 0, 1, 2, 3. The action S D is a functional of the fields ρ, g µν , T, S j , W j . Here T, S j have dimension of length, W j is dimensionless and thus ρ has dimension cm -4 . As shown in [START_REF] Giesel | Manifestly gauge-invariant general relativistic perturbation theory: I[END_REF][START_REF] Giesel | Manifestly gauge invariant general relativistic perturbation theory: II. FRW background and first order[END_REF], in performing the Legendre transformation of (2.1) according to the 3+1 split of M ∼ = R × X into time and space one introduces momenta P, P j , I, I j conjugate to T, S j , ρ, W j respectively next to the momenta P ab , p, p a conjugate to q ab , n, n a respectively one encounters several primary constraints. Here one has introduced a foliation of M, that is, a one parameter family of embeddings t → X t : X → X t where X t are the leaves of the foliation and the coordinates on X are denoted by x a , a = 1, 2, 3. The vector field ∂ t X µ t = nn µ + n a X µ t,a can be decompsed in components normal and tangential to the laves where n µ is the future oriented normal. The functions n, n a are the usual lapse and shift functions and q ab = g µν X µ ,a X ν ,b defines the three metric intrinsic to X . The afore mentioned primary constraints are Z =: I = 0, Z j := I j = 0, Z j := P j + P W j = 0, z := p = 0, z a := p a = 0 (2.2)

The stability analysis of these constraints with respect to the corresponding primary Hamiltonian 7 Usually Lorentz symmetry breaking is discussed at the Lagrangian level before discussing the phase space and constraint structure of the system. In that sense, our system is perfectly Lorentz invariant (more precisely, spacetime diffeomorphism invariant). The Brown Kuchar Lagrangian which is our starting point is a bona fide spacetime diffeomorphism covariant Lagrangian scalar density. With respect to the reduced phase space, which we discuss below, it is physically completely obvious that different observers describe the same physics in different terms (rods and clocks) and one would therefore not expect the reduced action to be "label space" (in our case,dust space) diffeomorphism invariant, see our companian work [START_REF] Giesel | Manifestly gauge-invariant general relativistic perturbation theory: I[END_REF].

leads to the following secondary constraints

c tot = c + c D , c D = 1 2 [ P 2 ρ det(q) + ρ det(q)(1 + q ab U a U b )] c tot a = c a + c D a , c D a = P [T ,a -W j S j ,a ] c = n 2 [- P 2 ρ 2 det(q) + det(q)(1 + q ab U a U b )] (2.3)
and six more equations which can be solved for the Lagrange multipliers corresponding to constraints Z j , Z j and which we do not display here. Here U a = -T ,a + W j S j ,a = -c D a /P and c, c D a respectively are the contributions of geometry and standard matter to the usual Hamiltonian and spatial diffeomorphism constraint respectively.

The stability analysis of the secondary constraints with respect to the primary Hamiltonian which is a linear combination of the constraints (2.2) and the first two constraints in (2.3) reveals that there are no tertiary constraints. Moreover, the classification of the sets of constraints into first and second class shows that the constraints z, z a , c tot , c tot a are first class while, roughly speaking, the pairs (Z, c), (Z j , Z j ) form second class constraints with non degenerate matrix formed by their mutual Poisson brackets. Hence, to proceed, one passes to the corresponding Dirac bracket and solves the second class constraints explicitly by setting

I := 0, I J := 0, W j := - P j P , ρ 2 := P 2 det(q) [q ab U a U b + 1] (2.4)
Fortunately, the Dirac bracket reduced to the geometry variables q ab , p ab and the remaining matter variables is identical to the original Poisson bracket. After using (2.4) and solving z = z a by identifying lapse and shift as Lagrange multiplicator functions respectively we are left with the first class constraints

c tot = c + c D , c D = -P 2 + q ab c D a c D b c tot a = c a + c D a , c D a = P T ,a + P j S j ,a (2.5)
In principle we could have chosen the other sign to solve the quadratic equation for ρ in (2.4) but the detailed analysis in [START_REF] Giesel | Manifestly gauge-invariant general relativistic perturbation theory: I[END_REF] reveals that the other choice would produce the Einstein equations with the wrong sign in the limit of vanishing dust fields. In particular one must choose ρ, P < 0 so that the additional matter enters with negative sign into the Hamiltonian constraint. This has the important consequence that c > 0 thus enables close to flat space solutions 8 . As far as the physical interpretation of the additional matter is concerned we just mention that its Euler Lagrange equations imply that the vector field U µ = g µν U ν is a geodesic in affine parametrisation, that the fields W j , S j are constant along the geodesic and that the field T defines proper time 8 As explained in the text, the energy density of the observable matter is supposed to obey the usual energy conditions. This is not in contradiction with vanishing total Hamiltonian constraint and nowhere vanishing non positive dust energy conditions since the gravitational contribution to the energy density in indefinite but of course excludes a strictly flat spacetime in absence of observable matter ("vacuum") which in turn is not in contradiction to experiment (Minkowski space is only a local approximation to the metric in our universe). The presence of the dust leads to arbitrarily small, nowhere vanishing corrections where "small" means arbitrarily small in any metric topology. These do not exclude vacuum with respect to observable matter but rather that it is not possible to have both a flat spacetime and no observable matter as already said above. Addressing stability, we are only concerned with the stability of the reduced phase space with respect to the physical Hamiltonian which is manifestly positive and thus renders the system stable.

along each geodesic. It follows that S j = σ j =const. labels a geodesic while T = τ =const. is an affine parameter along the geodesic. Furthermore, its energy momentum tensor is that of a perfect fluid with vanishing pressure and negative energy density9 , hence it is pressure free phantom dust. It serves as a dynamical, material reference system which also plays the role of a phantom in the literal sense because it is not directly visible in the final picture while leaving its fingerprint on the dynamics.

Brown -Kuchař Mechanism

The observation of Brown and Kuchař was that the constraints (2.5) can be written in deparametrised form. This holds in more general circumstances, namely whenever we consider scalar fields without potential and mass terms as pointed out in [START_REF] Thiemann | Solving the problem of time in general relativity and cosmology with phantoms and k-essence[END_REF]. The observation consists in the fact that the only appearance of T, S j in c tot is in the form c D a . However, this means that using c tot a = 0 we may write (2.5) in the equivalent form

c tot = c + c D , c D = -P 2 + q ab c a c b c tot a = c a + c D a , c D a = P T ,a + P j S j ,a (2.6)
where equivalent means that in (2.5) and (2.6) the new constraints c can be expressed in terms of the old ones c as c I = M J I c J where M J I is a matrix non -singular on the constraint surface defined by the constraint c I . This means the constraints in equation (2.6) define the same constraint surface and the same gauge invariant functions.

We can now solve the first equation in (2.6) for P , remembering that P < 0 and the second equation for P j , making the assumption that the matrix S j ,a is everywhere non degenerate10 with inverse S a j . The result is ctot = P + h, h = + c 2 -q ab c a c b ctot

j = P j + h j , h j = S a j [c a -hT ,a ] ( 2 . 7 )
In solving (2.6) in terms of P we find at an intermediate step that P 2 = c 2 -q ab c a c b . Hence, while the argument of the square root in (2.7) is not manifestly positive, it is constrained to be positive.

Notice that the function h is independent of S j , T while h j still depends on both. Hence, we have achieved only partial deparametrisation. However, this will be sufficient for our purposes. An important consequence is that the constraints in the form (2.7) are mutually Poisson commuting. This follows immediately from an abstract argument11 [START_REF] Henneaux | Quantization of Gauge Systems[END_REF], although one can also verify this by direct computation [START_REF] Brown | Dust as a standard of space and time in canonical quantum gravity[END_REF]. This implies in particular that the h(x) are mutually Poisson commuting while the h(x) do not Poisson commute with the h j (y) and neither do the h j (y) among each other.

Relational framework

General theory

We first consider a general system with first class constraints C I with arbitrary index set I and later specialise to our situation.

Consider any set of functions T I on phase space such that the matrix defined by the Poisson bracket entries M J I := {C I , T J } is invertible. Consider the equivalent set of constraints

C I := J [M -1 ] J I C J (2.8)
such that {C I , T J } ≈ δ J I where ≈ means = modulo terms that vanish on the constraint surface. Let X I be the Hamiltonian vector field of C I and set for any set of real numbers β I X β :=

I β I X I (2.9)
For any function f on phase space we set

α β (f ) := exp(X β ) • f = ∞ n=0 1 n! X n β • f (2.10)
Now let τ I be another set of real numbers and define

O f (τ ) := [α β (f )] α β (T )=τ (2.11)
where α β (T ) = τ means α β (T I ) = τ I for all I. As one can check, α β (T I ) ≈ T I + β I so that (2.11) is weakly (i.e. on the constraint surface) equivalent to

O f (τ ) := [α β (f )] β=τ -T (2.12)
Notice that after equating β with τ -T , the previously phase space independent quantities β become phase space dependent, therefore it is important in (2.12) to first compute the action of X β with β treated as phase space independent and only then to set it equal to τ -T . The significance of (2.12) lies in the following facts:

1. The functions O f (τ ) are weak Dirac observables with respect to the C I , that is

{C I , O f (τ )} ≈ 0 ( 2 . 1 3 )
This remarkable property is due to the key observation that the X I weakly commute [START_REF] Dittrich | Partial and complete observables for Hamiltonian constrained systems[END_REF][START_REF] Thiemann | Reduced phase space quantization and Dirac observables[END_REF].

2. The multi parameter family of maps O τ : f → O f (τ ) is a homomorphism from the commutative algebra of functions on phase space to the commutative algebra of weak Dirac observables, both with pointwise multiplication, that is

O f (τ ) + O f (τ ) = O f +f (τ ), O f (τ ) O f (τ ) ≈ O ff (τ ) ( 2 . 1 4 )
The linear relation is obvious, the multiplicative one follows from the fact that

α β (f f ) = e X β • f f = e X β • f f e -X β • 1 = [e X β • f e -X β ][e X β f e -X β ] ( 2 . 1 5 )
where we used the identity

[e X β • f e -X β ] = ∞ n=0 1 n! [X β , f] (n) (2.16)
and where X β , f respectively are considered as derivation and multiplication operators respectively on the algebra of functions on phase space so that [X

β , f] = X β • f . Here [X, f] (0) = f, [X, f ] (n+1) = [X, [X, f ] (n) ].
3. The multi parameter family of maps O τ : f → O f (τ ) is in fact a Poisson homomorphism with respect to the Dirac bracket {., .} * defined by the second class system C I , T J , that is

{O f (τ ), O f (τ )} ≈ {O f (τ ), O f (τ )} * ≈ O {f,f } * (τ ) ( 2 . 1 7 )
where the Dirac bracket is explicitly given by

{f, f } * = {f, f } -{f, C I }[M -1 ] I J {T J , f } + {f , C I }[M -1 ] I J {T J , f} (2.18)
Here we have used in the first step that both O f (τ ), O f (τ ) have weakly vanishing brackets with the constraints. Relation (2.17) follows from the fact that the map α β is a Poisson automorphism on the algebra of functions on phase space and the Poisson bracket must be replaced by the Dirac bracket because in evaluating {O f (τ ), O f (τ )} we must take care of the fact that β = τ -T is phase space dependent. See [START_REF] Thiemann | Reduced phase space quantization and Dirac observables[END_REF] for the explicit proof.

The interpretation of O f (τ ) is that it is a relational observable, namely it is the value of f in the gauge β = T -τ .

Specialisation to deparametrised theories

For deparametrised theories it is possible to find canonical coordinates consisting of two sets of canonical pairs (P I , T I ) and (q a , p a ) respectively (where the Poisson brackets between elements of the first and second set set vanish) such that the constraints C I can be rewritten in the equivalent form

C I = P I + h I (q a , p a ) ( 2 . 1 9 )
that is, they no longer depend on the variables T I . This is a very special case and most gauge systems cannot be written in this form. Even with dust General Relativity is a priori not of that form, however, we will reduce it to that form with an additional manipulation below.

The simplifications that occur are now the following:

A. We obviously have

M J I = {C I , T J } = δ J I (2.20)
therefore C I = C I and we do not have to invert a complicated matrix.

B. By the same argument as in the footnote after (2.7) we have {C I , C J } = 0 identically on the full phase space, not only on the constraint surface which of course implies that [X I , X J ] = 0, the Hamiltonian vector fields of the constraints are mutually commuting. It also follows that {h I , h J } = 0 and thus {C I , h J } = 0 for all I, J which means that the h I are already Dirac observables.

These simplifications mean that all the previous weak equalities become strong ones, i.e. identities on the full phase space. The Dirac observable associated to

T I O T I (τ ) = [α β (T I )] α β (T )=τ = τ I (2.21)
is simply the constant (on phase space) function τ I . The momenta P I are already Dirac observables, however they can be expressed in terms of q a , p a via the constraints. Moreover, since O τ is a homomorphism we have on the constraint surface

P I = O P I (τ ) = -O h I (τ ) = -h I (O q a (τ ), O p a (τ )) =: -H I (2.22)
In fact we have h I = H I because h I is already a Dirac observable.

The reduced phase space (where the constraints hold and where the gauge transformations have been factored out) is therefore coordinatised by the functions

Q a (τ ) = O q a (τ ), P a (τ ) = O pa (τ ) ( 2 . 2 3 )
and in what follows we concentrate on functions f which only depend on q a , p a . On such functions the Dirac bracket reduces to the Poisson bracket since {T I , f} = 0 for all I. Therefore the reduced map O τ : f → O f (τ ) is now a multi -parameter Poisson automorphism with respect to the Poisson bracket. In particular we note

{P a (τ ), Q b (τ )} = {O pa (τ ), O q a (τ )} = O {pa,q b } (τ ) = O δ b a (τ ) = δ b a (2.24)
which means that the reduced phase space has a very simple symplectic structure in terms of the coordinates P a := P a (0), Q a := Q a (0) which in fact form a conjugate pair. It is this fact which makes reduced phase space quantisation feasible as observed in [START_REF] Thiemann | Reduced phase space quantization and Dirac observables[END_REF].

It seems that we have trivialised everything. However, this is not the case as we must interpret the τ dependence of our observables. We notice first of all that on functions f independent of T I , P I formula (2.12) reads explicitly

O f (τ ) = α τ (f ) = exp(X τ ) • f = ∞ n=0 1 n! X n τ • f (2.25)
where X τ is the Hamiltonian vector field of the function H τ = (τ I -T I )H I . Here we have used that the X I on f reduce to the Hamiltonian vector field of h I and since h I is independent of P J we may write O f (τ ) in the above compact form. It is now a simple exercise to verify that [START_REF] Thiemann | Reduced phase space quantization and Dirac observables[END_REF]

∂O f (τ )/∂τ I = {H I , O f (τ )} (2.26)
which means that the strongly Abelian group of Poisson automorphisms α τ is generated by the "Hamiltonians" H I . Thus, if we interpret the T I as clocks then we have a multi -fingered time evolution with Hamiltonians H I .

In quantum theory then one would like to select a suitable one parameter family by prescribing functions τ I (s) in terms of a single parameter such that the associated Hamiltonian is positive and has preferred physical properties.

The reduced phase space of General Relativity with dust

Now we specialise to our situation which is a special case of the general theory. This has been previously done in detail, including proofs, in [START_REF] Thiemann | Solving the problem of time in general relativity and cosmology with phantoms and k-essence[END_REF] and was also reviewed in [START_REF] Giesel | Manifestly gauge-invariant general relativistic perturbation theory: I[END_REF]. Here we summarise those results.

As previously mentioned, the Hamiltonian constraints in (2.7) are in deparametrised form, however, the spatial diffeomorphism constraints are not. However, the idea is to exploit the fact that the constraints (2.7) are mutually Poisson commuting so that one can perform the reduction of the phase space in two steps: First we reduce with respect to the spatial diffeomorphism constraint and then with respect to the Hamiltonian constraint. More precisely, consider arbitrary functions β 0 , β j on X and denote by X β the Hamiltonian vector field of the function

c tot β := X d 3 σ β µ (x) ctot µ (x) ( 2 . 2 7 )
where we have defined ctot 0 = ctot . Then for arbitrary functions τ 0 (x) = τ (x), τ j (x) := σ j (x) on X the general formula reads

O f (τ ) = [α β (f )] α β (T )=τ , α β (f ) = exp(X β ) • f (2.28)
where T 0 (x) = T (x), T j (x) = S j (x). We readily compute that

α β (T µ (x)) = T µ (x) + β µ (x) so that O f (τ ) = [α β (f )] β=τ -T (2.29)
Now since S j (x) Poisson commutes with ctot (y) we may rewrite (2.29) in the form

O f (τ ) = [α β 0 ([α β (f ))] β= σ-S )] β 0 =τ -T (2.30)
It turns out that one can compute the inner argument of (2.30) rather explicitly with an immediate physical interpretation for judicious choices of the functions σ j (x). Namely, for any scalar function f built from of T, P, q ab , p ab and the matter of the standard model one finds explicitly that for constant

functions σ j [α β (f (x))] β= σ-S = f (x) S(x)=σ (2.31)
In other words, whatever the value of x at which the function f is evaluated, (2.31) evaluates it at the point x a σ at which S j (x) assumes the value σ j . Since we have assumed that S j ,a is everywhere invertible and thus defines a diffeomorphism between X and the range of S j which is the dust space S, the value x σ is unique. Formula (2.31) is proved explicitly in [START_REF] Giesel | Manifestly gauge-invariant general relativistic perturbation theory: I[END_REF] and will not be repeated here. Thus, (2.31) takes a simple form if we choose as f one of the following functions on S T := T, P = P J , qjk := q ab S a j S b k , pjk := for connections a I b , electric fields e a I , fermions ψ αI , ψαI and Higgs fields φ I with conjugate momentum π I of the standard model where I labels a basis in the Lie algebra of the appropriate gauge group, see [START_REF] Thiemann | Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity[END_REF] for the canonical formulation of the standard model coupled to gravity including appropriate background independent Hilbert space representations.

p ab S j ,a S k ,b J (2.
It is clear that the evaluation of the functions (2.32) and (2.34) at x σ is nothing else than the pull back of the corresponding fields to S under the inverse of the diffeomorphism S j : X → S. We will denote the corresponding tensor fields on S as in (2.32) and (2.34). Notice that while these are scalars on X they are tensor densities of the same weight on S as they have 12 on X . In [START_REF] Brown | Dust as a standard of space and time in canonical quantum gravity[END_REF][START_REF] Giesel | Manifestly gauge-invariant general relativistic perturbation theory: I[END_REF] it is shown that one can arrive at the spatially diffeomorphism invariant functions (2.32) and (2.34) also by symplectic reduction with respect to the spatial diffeomorphism constraint which is an alternative proof of the fact that canonical pairs without tilde on X are mapped to canonical pairs on S. For instance {p jk (σ), qmn (σ

)} = κδ j (m δ k n) δ(σ, σ ) ( 2 . 3 5 )
where κ = 16πG Newton . This also shows that it is sufficient to consider constant σ j rather than arbitrary functions.

Returning to (2.30) we see that it remains to compute

O f (τ, σ) := [α β 0 (f (σ))] β 0 =τ -T (2.
c tot τ = X d 3 x (τ -T )(x) ctot (x) ( 2 . 3 7 )
which is expressed on the space X rather than dust space S. However, for constant τ (2.37) is the integral of a density of weight one and can then be written in the form

c tot τ = S d 3 σ (τ -T )(σ) [ P + h](σ) ( 2 . 3 8 )
where

h(σ) = c(σ) 2 -qjk (σ)c j (σ)c k )(σ) c(σ) = c J (x σ ) cj (σ) = c a S a j J (x σ ) ( 2 . 3 9 )
Notice that e.g. c is just the pull back of c and that one simply has to replace every tensor without tilde by their pulled back image with tilde. Thus constant τ is uniquely selected by the requirement that c tot τ is spatially diffeomorphism invariant.

12 This statement sounds contradictory because of the following subtlety: We have e.g. the three quantities P (x), P (x) = P (x)/J(x), P (σ) = P (x σ ). On X , P (x) is a scalar density while P (x) is a scalar. Pulling back P (x) to S = S(X ) by the diffeomorphism σ → S -1 (σ) results in P (σ). But pulling back P (x) back to S results in the same quantity P (σ). Since a diffeomorphism does not change the density weight, we would get the contradiction that P (σ) has both density weights zero and one on S. The resolution of the puzzle is that what determines the density weight of P (x) on X is its transformation behaviour under canonical transformations generated by the total spatial diffeomorphism constraint c tot a = c D a + c a where c D a , c a are the dust and non dust contributions respectively. After the reduction of c tot a , what determines the density weight of P (σ) on S is its transformation behaviour under ([c a + P T ,a ]S a j /J)(x σ ) = cj (σ) + P (σ) T,j (σ) and this shows that P (σ) has density weight one.

It follows now from section 2.3.2 that

O f (τ, σ) = ∞ n=0 1 n! {H τ , f(σ)} (n) , H τ = S d 3 σ [τ -T ](σ) h(σ) ( 2 . 4 0 )
and that

d dτ O f (σ, τ ) = {H, O f (σ, τ )}, H = S d 3 σ h(σ) ( 2 . 4 1 )
Since the h(x) are mutually Poisson commuting it follows that also the h(σ) are mutually Poisson commuting so that H(σ, τ

) := α β 0 ( h(σ)) β 0 =τ -T = h(σ) =: H(σ) ( 2 . 4 2 )
is independent of τ and already a Dirac observable.

Notice that the physical Hamiltonian H is positive. It enjoys the following symmetries: Since it is an integral over a density of weight one it is invariant under diffeomorphisms of S. Notice that S is a label space for geodesics and not a coordinate manifold, hence in contrast to the passive diffeomorphism group Diff(X ), the group Diff(S) are active diffeomorphisms. In particular, it follows that {H, cj (σ)} = 0 (2.43) which also is a consequence of having chosen constant τ , in which case the physical Hamiltonian has a maximal amount of symmetry. Had we not chosen constant τ then the physical Hamiltonian would not be a Dirac observable. This also implies that

{H, C j (σ)} = 0, C ( σ, τ ) := α β 0 (c j (σ)) β 0 =τ -T =: C j (σ) ( 2 . 4 4 )
is actually independent of τ , although cj = C j . Notice that

H(σ) = C(σ, τ ) 2 -Q jk (σ, τ )C j (σ)C k (σ), C(σ, τ ) := α β 0 (c(σ)) β 0 =τ -T (2.45)
The second symmetry of H is of course that

{H, H(σ)} = 0 (2.46)
Let us write for some scalar and vector test functions f, u j respectively

H(f ) := S d 3 σ f(σ) H(σ), C(u) := S d 3 σ u j (σ) C j (σ) ( 2 . 4 7 ) then {C(u), C(u )} = -κC([u, u ]) {C(u), H(f )} = -κH(u[f ]) {H(f ), H(f )} = 0 (2.48)
which shows that the symmetry generators generate an honest Lie algebra g in contrast to the Dirac algebra underlying GR as was pointed out already in [START_REF] Brown | Dust as a standard of space and time in canonical quantum gravity[END_REF] and further examined in [START_REF] Kuchař | Gravitational constraints which generate a lie algebra[END_REF]. That Lie algebra has a subalgebra generated by the C(u) and an Abelian ideal generated by the H(f ), hence it is not semisimple. The corresponding Lie group G = N Diff(S) is therefore the semidirect product of the Abelian invariant subgroup N to which the H(f ) exponentiate and the active diffeomorphism group of dust space.

Physical interpretation and comparison with unreduced formalism

The symmetry algebra g and the associated conservation laws play a crucial role in showing [START_REF] Giesel | Manifestly gauge-invariant general relativistic perturbation theory: I[END_REF] that the equations of motion for the canonical pairs of true degrees of freedom

(Q jk , P jk ); (A I j , E j I ); (Ψ αI , ΨαI ); (Φ I , Π I ) ( 2 . 4 9 )
which are the images of the canonical pairs (q jk , pjk ); (ã I j , ẽj I ); ( ψαI , ψαI ); ( φI , πI ) ( 2 . 5 0 ) under α β 0 (.) β 0 =τ -T at τ = 0 assume the standard form that they have in General Relativity without dust [START_REF] Wald | General Relativity[END_REF], with two important modifications: First, in usual General Relativity without dust the equations of motion generated by the canonical Hamiltonian h(n, n) = c(n) + c( n) which is a linear combination of the smeared Hamiltonian constraint c(n) = X d 3 xnc and spatial diffeomorphism constraint c( n) = X d 3 xn a c a , involve arbitrary lapse and shift functions n, n a on X which are independent of phase space. However, in our formalism lapse and shift functions become dynamical functions 13 on S, namely N = C/H and N j = -Q jk C k /H. Secondly, without dust we still have constraints c = c a = 0 while we have energy -momentum conservation laws H = , C j =j where , j are arbitrary functions on S independent of τ . This turns dynamical lapse and shift into a function of Q jk , j / . The functions , j express the influence of the dust on the other variables and are the price to pay for having a manifestly gauge invariant formalism rather than assuming non dynamical test observers that turn geometry and matter into observable quantities.

This concludes the classical analysis and the review of [START_REF] Giesel | Manifestly gauge-invariant general relativistic perturbation theory: I[END_REF].

3 Reduced phase space quantisation of General Relativity

Hilbert space representation

Let us summarise the result of the previous section: By using the relational formalism we can explicitly compute the reduced phase space of General Relativity with dust. It is identical to the unreduced phase space without dust with proper identification of X with S and of the gauge invariant canonical pairs (2.49) with the gauge variant canonical pairs (q ab , p ab ); (a I b , e b I ); (ψ αI , ψαI ); (φ I , π I ) (

of geometry and standard matter. The constraints have disappeared, they have been solved and reduced. Instead of a linear combination of constraints on the gauge variant phase space coordinatised by (3.1) which generates gauge transformations, there is a physical Hamiltonian (2.41) which generates physical time evolution on the gauge invariant phase coordinatised by (2.49). From the classical point of view one should now simply solve those equations in physically interesting situations. In [START_REF] Giesel | Manifestly gauge-invariant general relativistic perturbation theory: I[END_REF][START_REF] Giesel | Manifestly gauge invariant general relativistic perturbation theory: II. FRW background and first order[END_REF] we have done this in the context of cosmological perturbation theory [START_REF] Giesel | Manifestly gauge-invariant general relativistic perturbation theory: I[END_REF][START_REF] Giesel | Manifestly gauge invariant general relativistic perturbation theory: II. FRW background and first order[END_REF] which is written in manifestly gauge invariant form. This not only reproduces the standard results [START_REF] Mukhanov | Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations[END_REF] but also will allow us to investigate higher order perturbation theory without running into problems with gauge invariance.

In the quantum theory we are looking for representations of the Poisson * -algebra generated by (2.49) which supports a quantised version of the Hamiltonian H. The selection of appropriate representations will be guided by the symmetry group G unveiled in the previous section. First of all, since we consider fermionic matter we are forced to work with tetrads rather four metrics. We use the second order formalism as displayed in [START_REF] Thiemann | Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity[END_REF] (that is, we write the Einstein Hilbert Lagrangian in terms of the spin connection of the tetrad which involves second order derivatives rather than using the first order Palatini formalism) in order to avoid torsion. This means that we formulate the geometry phase space in terms of su(2) connections and canonically conjugate fields (A I j , E j I ) rather than in terms of the ADM variables Q jk , P jk where I is an su(2) index. This casts the geometry sector of the phase space into a SU(2) Yang -Mills theory description. Th price to pay is that there is an additional Gauss constraint on the phase space (which has been reduced only with respect to the Hamiltonian and spatial diffeomorphism constraint) given by

G I := ∂ j E j I + IJK A J j E j K + fermion terms (3.2)
just as for the matter Yang -Mills variables (we assume that the Cartan Killing metric is always δ JK by appropriate normalisation of the Lie algebra basis).

The gauge field language suggests to formulate the theory in terms of holonomies along one dimensional paths and electric fluxes through two dimensional surfaces, just as in unreduced LQG. There one has a uniqueness result [START_REF] Lewandowski | Uniqueness of diffeomorphism invariant states on holonomy -flux algebras[END_REF][START_REF] Fleischhack | Representations of the Weyl Algebra in Quantum Geometry[END_REF] which says that cyclic representations of the holonomy -flux algebra which implement a unitary representation of the spatial diffeomorphism gauge group Diff(X ) are unique and are unitarily equivalent to the Ashtekar -Isham -Lewandowski representation [START_REF] Ashtekar | Representations of the holonomy algebras of gravity and non-Abelean gauge theories[END_REF][START_REF] Ashtekar | Representation theory of analytic holonomy C algebras[END_REF]. In our case we do not have a diffeomorphism gauge group but rather a diffeomorphism symmetry group Diff(X ) of the physical Hamiltonian H. This is physical input enough to also insist on cyclic Diff(S) covariant representations and correspondingly we can copy the uniqueness result.

Thus we simply choose the background independent and active diffeomorphism covariant Hilbert space representation of LQG used extensively in [START_REF] Thiemann | Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity[END_REF] and we ask whether that representation supports a quantum operator corresponding to H.

Subtleties with the Gauss constraints

Before we analyse this question in detail, we should mention a subtlety: When one rewrites the geometry and standard matter contributions c, c a to the total Hamiltonian and and spatial diffeomorphism constraint in terms of the gauge theory variables, one can do this is G invariant form (where G is the compact gauge group underlying the corresponding Yang Mills theory) only by introducing terms proportional to the Gauss constraint, see e.g. [START_REF] Rovelli | Quantum Gravity[END_REF]. For instance, the contribution to the spatial diffeomorphism constraint of a Yang Mills field on the unreduced phase space is given by

c Y M a = f I ab e b I -a I a g Y M I = cY M a -a I a g Y M I (3.3)
where Likewise, the geometry contribution c geo to c contains a term proportional to g geo I [START_REF] Rovelli | Quantum Gravity[END_REF] (however, c Y M does not). As far as the definition of the complete constraint surface is concerned, one can drop the various Gauss law contributions to c, c a since we impose the Gauss laws independently anyway. This gives an equivalent set of constraints which is such that c, c a are manifestly invariant under Yang -Mills type of Gauss transformations. However, now the algebra of the c tot , c tot a only closes up to a term proportional to the various Gauss laws.

f I ab = 2∂ [a a I b] + IJK a J a a K b is
The question is now whether this spoils the argument that the constraints in the form (2.7) are mutually Poisson commuting. In fact, we only can conclude that their Poisson brackets are proportional to ctot , ctot a and the various g Y M I while they must not depend on the dust momenta P, P j . This means that their Poisson brackets are proportional to a Yang Mills gauge invariant linear combination of Gauss constraints. Hence, indeed the constraints ctot , ctot a are Abelian only on the constraint surface of the Gauss constraints.

This poses the question which consequences this has for the formalism developed in the previous section. First of all, all relations that we have written there remain valid modulo terms proportional to the Gauss constraints. Secondly, the physical Hamiltonian is manifestly Yang -Mills gauge invariant, manifestly Diff(X ) invariant and invariant modulo the Gauss constraints under N .

The strategy that we adopt is the following. In the presence of gauge fields we actually work with the non Gauss invariant contributions to the spatial diffeomorphism constraints as in (3.3) and with the non Gauss invariant contribution to c geo such that algebra of Hamiltonian and spatial diffeomorphism constraints closes without involvement of the Gauss constraints. This makes the analysis of the previous section go through without modifications at the price that the physical Hamiltonian is not Gauss invariant. When we quantise it turns out that one can actually solve the various Gauss constraints explicitly by Dirac constraint quantisation. That is, the Hilbert space can be projected to the Gauss invariant subspace which has an explicitly known orthonormal basis given by the Gauss invariant spin network functions (and their analog for the gauge group of the standard model). Therefore, on the Gauss invariant Hilbert space one can actually replace the C Y M j by CY M j because the correction term proportional to the Gauss constraint vanishes on the Gauss invariant Hilbert space (upon appropriate ordering of the Gauss constraint operator to the right so that no commutator terms arise). Thus C j is replaced by its Gauss invariant analog and similarly one can replace C by its Gauss invariant analog so that H and H become manifestly Gauss invariant operators and H should have the symmetry group G as well.

An alternative route would be to also reduce the phase space with respect to the Gauss constraints, possibly using the framework of [START_REF] Leigh | Solving pure QCD in 2+1 dimensions[END_REF] and references therein.

Quantum Hamiltonian

Sign issues and strategy

Before we go into details we must worry about yet another issue: As we have seen in the classical analysis, the expression

H 2 = C 2 -Q JK C J C K is
constrained to be non negative. Actually we have seen this only for c 2 -q ab c a c b but as we showed

(C 2 -Q jk C j C k )(σ) = ([c 2 -q ab c a c b ]/J)(x σ ) ( 3 . 5 )
and J > 0 by assumption (we have imposed J = 0 everywhere, hence either J > 0 everywhere or J < 0 everywhere by continuity and we choose the first option). However, on the full, reduced phase space C 2 -Q jk C J C K maybe indefinite. In the quantum theory we therefore should derive, roughly speaking, a self adjoint operator (valued distribution) for H 2 (σ) and restrict the spectral resolution of the Hilbert space to the positive spectrum part. This has to be done for every σ. This maybe impossible because the corresponding spectral conditions could be incompatible. However, as already pointed out by Brown and Kuchař [START_REF] Brown | Dust as a standard of space and time in canonical quantum gravity[END_REF], if we indeed manage to quantise H 2 (σ) in such a way that they are mutually commuting 14 then the corresponding spectral projections commute and the above requirement is consistent. Unfortunately, not only may it be hard to achieve commutativity of the operators corresponding to the various H 2 (σ), moreover it will be hard to compute the corresponding projection valued measures. Therefore, as a first step, in this article we adopt the following strategy: Classically, in the interesting part of the phase space we have C 2 -Q jk C j C k ≥ 0. Therefore on this part of the phase space we have trivially

C 2 -Q jk C j C k = |C 2 -Q jk C j C k |.
Hence on that part of the phase space we have the identity

H = |C 2 -Q jk C j C k | = 1 2 ([C 2 -Q jk C j C k ] + |C 2 -Q jk C j C k |) ( 3 . 6 )
The virtue of this rewriting is that both expressions, which are identical on the physically interesting piece of the phase space, can be extended to the full phase space without becoming imaginary. In the second version, the function actually vanishes on the unphysical part of the phase space. In either form, the square root now makes sense in the quantum theory because its argument is now a non negative expression. We remark that a discussion of similar sign issues and whether one should allow states in the quantum theory which violate the classical positivity of H 2 (σ) which is enforced by a constraint of the form P 2 -H 2 = 0 and where H 2 is not manifestly positive while P 2 surely is, can be found for instance in [START_REF] Ashtekar | On the canonical approach to quantum gravity[END_REF]. There the authors argue that one should allow negative energy states because otherwise one would exclude the tunnelling effects into the classically not allows regions which, as we know from quantum mechanical experiments, do happen. What happens mathematically is that in the operator constraint method (Dirac approach) one quantises both P and H 2 as self -adjoint operators on the kinematical Hilbert space and then solves the quantum constraint. The elements of the corresponding physical Hilbert space may have support in the classically not allowed region of the configuration space (where they typically decay rather than oscillate) so that the expectation value of H 2 = P 2 becomes negative. This is possible only because the operator corresponding to P , while being a quantum Dirac observable, does not descend to a self adjoint operator on the physical Hilbert space. In a strict reduced phase space quantisation one would have to restrict the physical Hilbert space to states which have support only in the classically allowed region of the phase space and this may well be the physically correct procedure. However, for the moment, as we do not yet have sufficient control over the spectrum of H 2 , we comply with the conclusion of [START_REF] Ashtekar | On the canonical approach to quantum gravity[END_REF] and do not make any restriction on the physical Hilbert space.

Thus, in this article we therefore propose to quantise the first version of (3.6) which is a classically valid starting point 15 . We then adopt a naive quantisation strategy and are able to construct a well defined Hamiltonian operator. That quantisation not necessarily has the property that the quantised versions of the H 2 (σ) are mutually commuting and therefore the operator constructed in this paper should only be considered as a preliminary step. However, that operator has the following three properties: It is manifestly Gauss invariant, manifestly Diff(S) covariant and has the correct classical limit in the sense of expectation values and fluctuations with respect to coherent states. However, it maybe anomalous with respect to the group N . In fact, the absence of that anomaly would be 14 More precisely, one has to demand that the projection valued measures E σ for the H 2 (σ) mutually commute in order to avoid domain questions. Notice that the Poisson commutativity of the H(σ) implies the Poisson commutativity of the H 2 (σ) and vice versa. 15 A similar strategy was adopted for the quantisation of the volume in LQG: Classically we have det(q) = det(E) > 0 but in order to give meaning to det(q) in the quantum theory we must start from | det(E)|.

mathematically equivalent to showing that the Dirac algebra of General Relativity is implemented non anomalously. We stress, however, that the gauge symmetries of General Relativity have been exactly taken care of in the reduced phase space approach. We are talking here about a symmetry group and not a gauge group. To break a local gauge group is usually physically unacceptable especially in renormalisable theories where the corresponding Ward identities find their way into the renormalisation theorems. However, it may or may not be acceptable that a physical symmetry is (spontaneously, explicitly ...) broken. For instance, the explicit breaking of the axial vector current Ward identity in QED, also called the ABJ anomaly, is experimentally verified.

In lack of a physical justification for why the N symmetry should be broken, we view that potential anomaly as an indication that the quantisation of the present paper has to be improved. In fact, since we are effectively working with a background independent lattice gauge theory, it is useful to adopt strategies from lattice gauge theory in order to restore symmetries on the lattice that are broken in a naive quantisation. It turns out that in fortunate cases one can restore the symmetry by making the operator quasi non local. That is, in addition to next neighbour interactions one has to consider next to next neighbour interactions etc. which makes the action non local, however the coefficients of those additional interactions decay exponentially with the lattice distance. See e.g. [START_REF] Hasenfratz | The Theoretical Background and Properties of Perfect Actions[END_REF] and references therein.

We consider the completion of this step as a future research programme. In the course of that analysis we might even be able to fix the quantisation (discretisation) ambiguities, i.e. the coefficients in front of the various n-th neighbour contributions.

With this cautionary remarks out of the way, we can now consider a naive quantisation of the Hamiltonian which is strongly guided by analogous techniques developed for the Hamiltonian and Master constraint of unreduced LQG [START_REF] Thiemann | Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity[END_REF][START_REF] Thiemann | The Phoenix project: master constraint programme for loop quantum gravity[END_REF] so that these constructions are also helpful in the present reduced phase space approach.

Naive Quantisation

Classical regularisation

We begin with some classical considerations and we focus on the gravitational contributions to C, C j and for C only on the Euclidean piece. For the matter contributions and the Lorentzian piece the necessary, completely analogous manipulations can be found in [START_REF] Giesel | Algebraic quantum gravity (AQG) I. Conceptual setup[END_REF]. Consider a partition P of S into cubes 2 so that

H = 2∈P 2 d 3 σ |C 2 -Q jk C j C k |(σ) ( 3 . 7 )
Let V 0 (2) be the coordinate volume of 2 in any coordinate system and let σ(2) be some coordinate point inside 2 with respect to the same coordinate system. Then we can write (3.7) as limit, in which the partition becomes the continuum, of the following Riemann sum approximation of the above integral

H = lim P→S 2∈P V 0 (2) |C 2 -Q jk C j C k |(σ(2)) (3.8)
Using the classical identities

Q jk = E j I E k J δ IJ det(Q) , E j I = det(Q)e j I (3.9)
where I, J, .. = 1, 2, 3 label a basis τ I = -iσ I (where σ I denote the Pauli matrices) in su( 2) and e j I denotes th triad it is not difficult to verify that

C 2 = [Tr(B)] 2 , Q jk C j C k = [Tr(Bτ I )] 2 /4 =: C 2 I (3.10)
Here we have introduced the magnetic field B j I = 1 2 jkl F I kl and have set B j = B j I τ I , e j = e I j τ I , B = B j e j where e I j denotes the cotriad. We may further write

H = lim P→S 2∈S |C(2) 2 -δ IJ C I (2)C J (2)| (3.11)
where

C(2) := 2 d 3 σ C(σ), C I (2) := 2 d 3 σ C I (σ) ( 3 . 1 2 )
The strategy is now to quantise the objects (3.12) and to define

H := lim P→S 2∈S | Ĉ(2) † Ĉ(2) -δ IJ ĈI (2) † ĈJ (2)| (3.13)
provided the limit exists. For C(2) this has been done in the literature [START_REF] Thiemann | Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity[END_REF][START_REF] Thiemann | The Phoenix project: master constraint programme for loop quantum gravity[END_REF] and we follow the same strategy here. In fact we can treat both C, C I in a unified way. We have with τ 0 := 1 2

2 d 3 σ Tr(Bτ µ ) = 2 Tr(F ∧ e τ µ ) = 1 κ 2 Tr(F ∧ {V (2), A} τ µ ) ( 3 . 1 4 )
where

V (2) = 2 d 3 σ det(Q) ( 3 . 1 5 )
is the physical volume of 2. Actually there is a sign of det(e) involved in (3.15) but this is cancelled in the squares that appear in (3.11).

The virtue of writing (3.14) in this form is that (3.15) can be quantised on the LQG Hilbert space, hence one replaces the Poisson bracket by the commutator divided by i . Thus one is left with the quantisation of the connection A and its curvature F . This is the source of many ambiguities already in unreduced LQG because A, F do not exist as operators, what exists are holonomies along paths and loops respectively which can be used in order to approximate A, F respectively. However, while classically there are infinitely many ways to do this with the same continuum limit, in the quantum theory each choice leads to a different regularised operator in unreduced LQG, see [START_REF] Thiemann | Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity[END_REF]. In unreduced LQG one can still argue that most of the uncountably infinite number of choices are gauge related under the spatial diffeomorphism group and in fact spatial diffeomorphism invariance is used in order to carry out the limit P → X in a specific operator topology [START_REF] Rovelli | Loop quantum gravity[END_REF][START_REF] Thiemann | Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity[END_REF]. However, in reduced LQG the spatial diffeomorphism group is no longer a gauge group, it is a symmetry group of the dynamics. Therefore these two arguments are no longer available and therefore the ambiguity issue appears to be much worse in reduced LQG. This is the first indication that calls for the AQG generalisation.

In the next paragraph we will discuss to what extent those ambiguities persist in reduced LQG, in the paragraph after that we use the AQG reformulation.

Reduced LQG: Embedded graphs

We want to define the Hamiltonian operator H on the Gauss invariant Hilbert space of LQG which we will denote by H. This Hilbert space has an orthonormal basis consisting of spin network functions T γ,j,I where γ is a (semianalytic) graph embedded into S, j = {j e } e∈E(γ) is a collection of non vanishing spin quantum numbers (one for each edge) and I = {I v } v∈V (γ) is a collection of Gauss invariant intertwiners (one for each vertex). There is a unitary action of the active diffeomorphisms on this Hilbert space defined by U (ϕ)T γ,j,I = T ϕ(γ),j,I (3.16) In unreduced LQG the diffeomorphisms are considered as gauge transformations and therefore the states (3.16) are all gauge related. In the reduced formalism of this paper the states of the form are physically distinguishable. Therefore it does not make physical sense to construct diffeomorphism invariant distributions which sometimes are used in the construction of Hamiltonian or master constraint operators as already pointed out. This last point has crucial bearing on the quantisation strategy: If we want to preserve the classical symmetry of the Hamiltonian operator under diffeomorphisms, then this operator must be quantised in a graph non changing way [START_REF] Ashtekar | Quantization of diffeomorphism invariant theories of connections with local degrees of freedom[END_REF] on H. By this is meant the following: Let H γ be the closed linear span of spin network states over γ. Then H is the direct sum of the H γ , that is

H = ⊕ γ H γ (3.17)
which shows that the physical Hilbert space H is non separable. This is an important difference with non reduced LQG where the physical Hilbert space can be made separable if one extends the spatial passive diffeomorphism group beyond the differentiable category [START_REF] Fairbairn | Separable Hilbert space in loop quantum gravity[END_REF]. This is a second indication that one should possibly leave the strict realm of (reduced) LQG and pass to another framework where non separable Hilbert spaces can be avoided. This calls for the AQG extension [START_REF] Giesel | Algebraic quantum gravity (AQG) I. Conceptual setup[END_REF] which we discuss in the subsequent paragraph.

In any case, graph non changing in the sense of [START_REF] Ashtekar | Quantization of diffeomorphism invariant theories of connections with local degrees of freedom[END_REF] now means that the operator H should preserve each H γ separately! This appears as if we had to assume an infinite number of conservation laws that the classical theory did not have which is a second point to worry about and presents a third motivation to switch to the AQG extension of LQG. However, let us see how far we can get within the usual formalism. To that end, we use the notion of a minimal loop originally introduced in [START_REF] Sahlmann | Towards the QFT on curved spacetime limit of QGR. 1. A general scheme[END_REF] and also used to some extent in [START_REF] Thiemann | The Phoenix project: master constraint programme for loop quantum gravity[END_REF]. Definition 3.1. Given a graph γ, consider a vertex v ∈ V (γ) and a pair e, e ∈ E(γ) of distinct edges incident at v and with outgoing orientation. A loop α γ,v,e,e in γ starting at v along e and ending at v along (e ) -1 is said to be minimal provided that there exists no other loop in γ with the same properties and fewer edges traversed. The set of minimal loops in γ with data v, e, e will be denoted by L γ,v,e,e .

Notice that the definition is background independent and diffeomorphism covariant. Given a graph γ and a vertex v ∈ V (γ) we define for µ = 0, 1, 2, 3 Ĉµ,γ,v := 1 is the projection of the volume operator [START_REF] Ashtekar | Quantum theory of geometry II: volume operators[END_REF] to 16 H γ for an infinitesimal neighbourhood of v. Here σ v (e 1 , e 2 , e 3 ) is the sign of the determinant of the matrix formed by the tangents of those three edges at v and X e denotes the right invariant vector field on SU(2) associated with the copy of SU(2) coordinatised by A(e).

Finally we set

H γ := v∈V (γ) |P γ [ Ĉ † γ,v Ĉγ,v - 1 4 Ĉ † I,γ,v ĈI,γ,v ]P γ | (3.20)
where P γ : H → H γ denotes the orthogonal projection and makes sure that H is not graph changing, i.e. preserves H γ . The dagger operation is that on H for the operator defined in (3.18) using that entries of holonomies matrices are just multiplication operators and that Vγ,v is self adjoint.

The operator H is now simply

H = ⊕ γ H γ (3.21)
It is easy to check that it is diffeomorphism invariant

U (ϕ) H U (ϕ) -1 = H (3.22)
for all ϕ. Moreover, it is manifestly Gauss invariant. One may ask what happened to the limit P → S. The answer is that we define the operator H as in (3.22) and just check that its expectation values with respect to suitable semiclassical states reproduces the classical function H. Such states in particular must use sufficiently large and fine graphs in order to fill out S. What the operator does on small graphs is irrelevant from the point of view of the classical limit.

With the methods of [START_REF] Giesel | Algebraic quantum gravity (AQG) I. Conceptual setup[END_REF] one should be able to verify that on such graphs the semiclassical limit of the operator is correct. However, that calculation is of course graph dependent.

Reduced AQG: Abstract (algebraic) graphs

One of the motivations for the AQG extension of LQG is the graph dependence of the semiclassical calculations. The other is the necessarily graph preserving feature of diffeomorphism invariant operators which appears to say that there is an uncountably infinite number of conservation laws that the classical theory does not have. Finally, the non separability of the Hilbert space H even if S is compact without boundary is disturbing. In a sense, to use all graphs is a vast over-counting of degrees of freedom, at least from the classical perspective. To see this, suppose for simplicity that S is topologically R 3 (or an open neighbourhood thereof) and thus can be covered by a single coordinate system. Consider piecewise analytic paths which consist of segments along the coordinate axes. Likewise, consider piecewise analytic surfaces which are composed out of segments of coordinate planes. It is clear that the holonomies along those kind of paths and fluxes through that kind of surfaces separates the points of the reduced phase space.

It is true that also in canonical QFT the quantum configuration space is always a distributional enlargement of the classical configuration space. However, there it is never the case that the label set of those fields is uncountable. For instance, in free scalar field theory on Minkowski space the quantum configuration space consists of Schwartz distributions rather than smooth functions. The label set of the fields consists of test functions of rapid decrease which are dense in the Hilbert space of square integrable functions on R 3 and there exists a countable orthonormal basis of that Hilbert space consisting of Schwarz functions (e.g. Hermite functions times a Gaussian). Thus, the quantum fields are tested by a countable set of test functions and an orthonormal basis in the QFT Hilbert space is labelled by that countable set. In LQG on the other hand the quantum connections are tested by all graphs which is an uncountable set and states over different graphs are orthogonal. So the situation is completely different which seems to be the price of having a diffeomorphism covariant theory [START_REF] Lewandowski | Uniqueness of diffeomorphism invariant states on holonomy -flux algebras[END_REF][START_REF] Fleischhack | Representations of the Weyl Algebra in Quantum Geometry[END_REF].

One could of course restrict the labels to those mentioned above but these would not be preserved by diffeomorphisms. It is true that the diffeomorphic image of a coordinate segment can be approximated by coordinate segments, however, the length of say a rotated segment when approximated by a staircase will differ largely from the original length. The same happens for areas of surfaces. The only chance that this does not happen is for observables that are integrals over three dimensional regions as pointed out in [START_REF] Thiemann | Complexifier coherent states for canonical quantum general relativity[END_REF].

To make progress on those issues we therefore will restrict attention to operators that come from integrals over regions of S such as the volume operator or the Hamiltonian operator. This does not mean that one cannot construct length and volume operators, one just has to define them in an indirect way, see [START_REF] Giesel | Algebraic Quantum Gravity (AQG) V. Geometric Operators[END_REF]. In fact, we will only consider quantising functions which are Diff(S) invariant. The motivation for doing this is that in physics we do not specify spatial regions by considering a 3D subset R of S and define, say, a Diff(X ) invariant volume functions (i.e. a function invariant under passive diffeomorphism starting from the unreduced formalism) by

V (R) := X d 3 x χ R (S(x)) det(q)(x) ( 3 . 2 3 )
where χ R denotes the characteristic function of the set R. Rather we use observable matter for doing this. To be sure, (3.23) is Diff(X ) invariant, being the integral of a scalar density over all of X . In fact, we can pull back this expression to S and obtain

V (R) = S d 3 σ χ R (σ) det(q)(σ) = R d 3 σ det(q)(σ) ( 3 . 2 4 )
where q = (S -1 ) * q which would be a mathematically natural object to consider in the reduced theory (after further applying α β 0 (.) β 0 =τ -T ). It is, however, not Diff(S) invariant. However, from the point of view of observation one would rather like to consider an object of the form

V (I) := X d 3 x χ I (φ(x)) det(q)(x) ( 3 . 2 5 )
where I is a subset of the real axis and φ is a scalar field. Notice that (3.25) is Diff(X ) invariant but it is not a Dirac observable yet. It measures the volume of the subset of X in which φ has range in I. Now we apply the map O τ and obtain immediately Having motivated to consider only Diff(S) invariant observables we are now ready to consider the AQG framework. Since for such observables the coordinate system plays no role we generalise from embedded to non embedded graphs and the above argument shows that infinite cubic algebraic graphs should be sufficient although a generalisation to arbitrary countable algebraic graphs as sketched in [START_REF] Giesel | Algebraic quantum gravity (AQG) I. Conceptual setup[END_REF] would be desirable 17 . In this paper we will just consider the cubic graph for simplicity.

O V (I) (τ, σ) := S d 3 σ χ I (Φ(τ, σ)) det(Q)(τ, σ) ( 3 . 2
At the algebraic level the notion of Diff(S) and even of S itself is meaningless. Notice that in AQG the infinite algebraic graph is a fundamental object. This fundamental graph does not change. What does change under the dynamics are subgraphs of the algebraic graph. In other words, subgraphs of the fundamental algebraic graph are not preserved under the quantum dynamics 18 . The definition of H in AQG is much simpler and no longer involves the projection operators P γ , so we do not have the awkward conservation laws any longer. In fact, there is no dependence on any algebraic subgraph whatsoever. In complete analogy 19 to [START_REF] Giesel | Algebraic quantum gravity (AQG) I. Conceptual setup[END_REF] it is given by the following list of formulae

Ĉµ,v := 1 24 2 P s 1 ,s 2 ,s 3 =±1 s 1 s 2 s 3 I 1 I 2 I 3 ×Tr(τ µ A(α v;I 1 s 1 ,I 2 s 2 )A(e v;I 3 s 3 )[A(e v;I 3 s 3 ) -1 , Vv ]) (3.27)
where e v;Is is the edge beginning at v in positive (s = 1) or negative (s = -1) I direction and α v;Is,Js is the unique minimal loop in the cubic algebraic graph with data v, e v;Is , e v;Js . Formula (3.27) is actually the specialisation of (3.18). The operator Vv is the algebraic volume operator

Vv = 3 P | 1 48 s 1 ,s 2 ,s 3 =±1 s 1 s 2 s 3 IJK LM N X L e v;Is 1 X M e v;Js 2 X N e v;Ks 3 | (3.28)
Finally

H := v | Ĉ † 0,v Ĉ0,v - 1 4 Ĉ † I,v ĈI,v ]| (3.29)
where the sum is over all of the infinite number of vertices of the algebraic graph. The operator (3.29) is manifestly Gauss invariant. The Hilbert space of AQG is the infinite tensor product (ITP) of Hilbert spaces L 2 (SU(2), dµ H ), one for each edge of the graph (this can be generalised to defining different ITP's that come into play when constructing Gauss invariant states). This Hilbert space is not separable but it is a direct sum of separable Hilbert spaces which assume a Fock like structure and which are preserved 20 by H. As far as the symmetry group G is concerned, at the algebraic level for instance we no longer 17 The idea would be to consider the most general such graph which is the maximal algebraic graph. This is an algebraic graph with a countably infinite number of vertices and with a countably infinite number of edges between each pair of vertices including loops. This generalises the notion of a complete graph which is a graph in which a single edge connects each pair of vertices. 18 This bears some resemblance with the models for emergent gravity considered in [START_REF] Konopka | Quantum Graphity[END_REF] although the dynamics of those models not obviously models the dynamics of H. 19 In [START_REF] Giesel | Algebraic quantum gravity (AQG) I. Conceptual setup[END_REF] we considered the extended Master constraint which involves, in the language of this paper, [C 2 + Q jk C j C k ]/ det(Q) rather than |C 2 -Q jk C j C k |. Hence apart from the sign in front of Q jk C j C k we only need to change the power of the volume operator from V 1/2 v in [START_REF] Giesel | Algebraic quantum gravity (AQG) I. Conceptual setup[END_REF] to V v here. 20 We do not know whether H is densely defined on all of the ITP. However, if it is defined on a single vector in a given separable sector then it is densely defined on the entire sector. Now each separable sector of the ITP is labelled by a cyclic vector Ω which is explicitly known. Now H is defined on a given Ω if and only if it is densely defined on the corresponding sector. Hence, for each Ω we just have to perform this test and we simply remove the sectors from the ITP on which H it is not densely defined, if any, since they are unphysical. H is certainly densely defined on the sectors built from semiclassical Ω, hence the surviving part of the ITP certainly includes all the semiclassical states. where b > a > 0 are any real numbers. Then a classical function F is invariant with respect to transformations generated by H, C j respectively if and only if {F, {F, M}} M =0 = 0. The functional (3.30) can be quantised on the AQG Hilbert space by literally the same techniques as in [START_REF] Giesel | Algebraic quantum gravity (AQG) I. Conceptual setup[END_REF]. Thus, we have the possibility to analyse the anomaly issue with respect to G at the algebraic level as well. Finally, we can compute the expectation value of H with respect to semiclassical states as in [START_REF] Giesel | Algebraic quantum gravity (AQG) I. Conceptual setup[END_REF] and to zeroth order in we should find that the classical value is reproduced with small fluctuations. As for the master constraint, the really astonishing fact is that H is a finite operator without renormalisation thanks to our manifestly background independent formulation. Namely, at the fundamental quantum level the operator algebra is labelled by a single, countably infinite abstract, that is non embedded, graph Γ. There is no such thing as a lattice distance which would need a background metric. However, the semiclassical states depend on a differential manifold X , an embedding Y of the algebraic graph Γ into X , a cell complex Y (Γ) * dual to Y (Γ) as well as a point (A 0 , E 0 ) in the classical reduced phase space. Thus, the semiclassical states make contact to the usual (reduced) LQG formulation which in particular uses an at least topological manifold X . Hence AQG describes all topologies simultaneously. The point is now that, since Γ is an infinite graph, the embedding of Γ can be as fine as we wish, with respect to the spatial geometry described by E 0 even if X is not compact 21 . The expectation values of our operators such as H will now give, to zeroth order in , a Riemann sum approximation of the desired continuum integral H as in (3.14) in terms of holonomies along edges of the embedded graph and the volume of the cubes in the dual cell complex. That Riemann sum will approximate the integral the better, the finer the embedding. It is in this sense that in (3.29) no continuum limit has to be performed.

Summary and Outlook

As compared to the Master Constraint Programme [START_REF] Thiemann | The Phoenix project: master constraint programme for loop quantum gravity[END_REF] the present framework has the advantage that the Master constraint and its solutions are not needed. We directly consider a representation of the gauge invariant phase space and its Hamiltonian. Celebrated results of unreduced LQG such as the discreteness of the spectrum of kinematical geometric operators [START_REF] Ashtekar | Quantum theory of geometry II: volume operators[END_REF][START_REF] Rovelli | Discreteness of volume and area in quantum gravity[END_REF][START_REF] Ashtekar | Quantum theory of geometry I: Area Operators[END_REF][START_REF] Thiemann | A length operator for canonical quantum gravity[END_REF] which is not granted to survive when passing to the physical Hilbert space [START_REF] Dittrich | Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?[END_REF][START_REF] Rovelli | Comment on 'Are the spectra of geometrical operators in Loop Quantum Gravity really discrete[END_REF] in the usual Dirac constraint quantisation now becomes a physical prediction if the curves, surfaces and regions that one measures length, area and volume of are labelled by dust space. The Gauss invariant [START_REF] Bahr | Gauge-invariant coherent states for Loop Quantum Gravity I: Abelian gauge groups[END_REF] kinematical coherent states [START_REF] Thiemann | Gauge Field Theory Coherent States (GCS): I. General properties[END_REF] of unreduced LQG now become physical coherent states.

However, the physical Hilbert space of reduced LQG is non separable which appears to be a vast over-counting of quantum degrees of freedom. Passing to AQG means to switch from embedded to non embedded graphs and thus removes the over-counting. Since for spatially diffeomorphism invariant operators (on dust space) such as the Hamiltonian H or any other operationally interesting observable (which does not refer directly to the dust label space) the embedding of a graph is immaterial, we can consider the AQG reformulation as an economic description of reduced LQG in the sense that diffeomorphism related embeddings would lead to isomorphic sectors superselected by this kind of observables. The additional advantage of AQG is that it does not require a topological manifold and that it is free from complications that have to do with graph preservation.

The challenge of the present framework is to implement the (algebraic version of) the symmetry group G in the definition of H which will require tools from lattice gauge theory. The final AQG version of the reduced phase space is in any case very similar to Hamiltonian lattice gauge theory with the important difference that no continuum limit has to be taken which is why the theory is UV finite. Another important question is how one can understand from the complicated, non perturbative Hamiltonian H the significance of the standard model Hamiltonian on Minkowski space. The answer to this question must lie in the construction of a minimum energy eigenstate of H which is simultaneously a minimal uncertainty state Ω for all the observables and which is peaked around flat vacuum (no excitations of observable matter) spacetime. Presumably, if one studies matter excitations of Ω and considers matrix elements of H in such states then the resulting matrix elements can be considered as the matrix elements of an effective matter Hamiltonian on Minkowski space which should be close the Hamiltonian of the standard model on Minkowski space. This expectation is supported by the analysis of [START_REF] Giesel | Manifestly gauge-invariant general relativistic perturbation theory: I[END_REF][START_REF] Giesel | Manifestly gauge invariant general relativistic perturbation theory: II. FRW background and first order[END_REF] which shows that the equations of motion of the gauge invariant geometry and matter degrees of freedom perturbed around a homogeneous and isotropic (FRW) solution is described effectively by the usual Hamiltonian on a FRW background expanded to second order in the perturbations. Of course, this is only a classical argument. See [START_REF] Sahlmann | Towards the QFT on curved spacetime limit of QGR. 1. A general scheme[END_REF] for more details about the quantum aspects of this idea. We leave this and the research projects mentioned in the introduction for future analysis.

  = ψ αI , ψαI := ψαI , φI := φ I , πI := π I J (2.34)

  the curvature of the connection a I a and IJK are the structure constants of the corresponding Lie algebra. The function (3.3) really generates Yang Mills gauge transformations, however, it is itself of course not Yang -Mills gauge invariant due to the term proportional to the Gauss constraint g Y M I = ∂ a e a I + IJK a J a

2 P 3 P | 1 48 e 1

 23481 |T v (γ)| (e 1 ,e 2 ,e 3 )∈Tv (γ) IJK 1 |L γ,v,e I ,e J | α∈Lγ,v,e I ,e J ×Tr(τ µ A(α)A(e K )[A(e K ) -1 , Vγ,v ]) (3.18) where T v (γ) is the set of ordered triples (i.e. order matters) of distinct edges of γ incident at v taken with outgoing orientation, A(p) denotes the holonomy of the connection A along a path p and Vγ,v = ,e 2 ,e 3 ∈Tv(γ) σ(e 1 , e 2 , e 3 ) LM N X L e 1 X M e 2 X N e 3 | (3.19)

6 )

 6 where Φ(τ, σ) = O φ(x) (τ, σ) (for any x) is the Dirac observable associated to φ. Curiously, (3.26) is a Dirac observable and it is Diff(S) invariant. It measures the physical volume of the region in S where the physical scalar field Φ ranges in I. The argument shows that Diff(S) invariant observables naturally arise from the point of the unreduced theory and from operational considerations.

S d 3 σ aH 2 +

 2 have spatial diffeomorphisms. However, we have its algebraic version which consists in the following: Consider the master constraint like functionalM := bQ jk C j C k det(Q) (3.30)

  [START_REF] Wald | General Relativity[END_REF] where f is now an arbitrary function of the spatially diffeomorphism invariant functions (2.32) and (2.34). Now we can use the simplified theory of section 2.3.2 because ctot is written in deparametrised form, i.e. it does not involve T, S j any longer. Actually, formula (2.36) would be awkward for non constant functions τ because it depends on

Given a system of constraints C I on a phase space, deparametrisation means that one can find local coordinates in the form of two mutually commuting sets of canonical pairs (q a , p a ), (T I , π I ) such that the constraints can be written in the locally equivalent form C I = π I + H I where the H I only depend on the (q a , p a ).

It is manifestly gauge invariant in an open neighbourhood of the phase space if the series converges with non zero convergence radius which has to be checked.

This is no loss of generality because P can be eliminated in terms of the other degrees of freedom via the constraints and T is pure gauge.

The caveat is that the deparametrisation and thus the reduced phase space quantisation is generically only locally valid in phase space. Thus, the globally valid Dirac quantisation programme should be developed further in parallel.

There are several loosely equivalent definitions for the effective action. The notion we mean here is the following: Consider first a renormalisable theory. Given a defining action with a finite number of finite but unknown couplings and masses (parameters) one can perform perturbation theory and discovers, within a given regularisation scheme, that the parameters are to be altered by functions of the distance cutoff which diverge in the limit of vanishing cutoff in order to avoid singularities in loop diagrammes. If one does this order by order then one ends up with the so called bare action which produces finite higher loop diagrammes to all orders. The effective action is a vehicle that produces the same scattering amplitudes or n-point functions as the bare action but of which one only needs to compute tree diagrammes (no loops). The definition for a non renormalisable theory such as gravity is the same, just that then number of parameters is infinite. In renormalisable theories a finite number of experiments is sufficient to fix the unknown parameters while non renormalisable theories have no predictive power.

That energy scale has nothing to do with a perturbative cutoff, we are talking here about an already well defined theory.

We are not violating any energy conditions because we still require that the energy momentum tensor of observable (standard) matter plus dust satisfies the energy conditions. In fact, it would be sufficient if the energy conditions are satisfied by the standard matter alone because in the final analysis the dust completely disappears while the equations of motion for observable matter and geometry assume their standard form plus small corrections, see[START_REF] Giesel | Manifestly gauge-invariant general relativistic perturbation theory: I[END_REF][START_REF] Giesel | Manifestly gauge invariant general relativistic perturbation theory: II. FRW background and first order[END_REF].

This is a classical restriction of the same kind as det(q) > 0. As far as the dynamical stability of the condition is concerned, the expression det(S j ,a ) Poisson commutes with the function h whose observable is the Hamiltonian density of the physical Hamiltonian. Therefore if the condition det(S j ,a ) > 0 is satisfied on the inital hypersurface it remains stable under the dynamics. It is also obviously spatially diffeomorphism invariant. Notice that we use the equivalent Hamiltonian constraint c = P + h which is a linear combination of the original constraints with phase space dependent non singular coefficients.

The constraints (2.7) are first class. Hence their Poisson brackets are linear combinations of constraints. Since the constraints are linear in the momenta P, P j , their Poisson brackets are independent of P, P j . Therefore we can evaluate the linear combination of the constraints that appear in the Poisson bracket computation in particular at P = -h, P j = -h j .

This is similar in spirit to[START_REF] Ashtekar | Phase space of general relativity revisited: A canonical choice of time and simplification of the Hamiltonian[END_REF] where one replaces lapse and shift test fields by hand by phase space dependent functions, carefully chosen (via Witten spinor techniques that enter the proof of the gravitational positive energy theorem) so that the resulting Hamiltonian is positive, at least on shell.

The alternative volume operator[START_REF] Rovelli | Discreteness of volume and area in quantum gravity[END_REF] was ruled out in[START_REF] Giesel | Consistency check on volume and triad operator quantisation in loop quantum gravity[END_REF] as inconsistent with the classical Poisson bracket identity(3.14). In unreduced LQG one could still say that the volume operator and the Poisson bracket identity are relations among non observable objects but this is no longer true in reduced LQG as considered here and hence the objection[START_REF] Giesel | Consistency check on volume and triad operator quantisation in loop quantum gravity[END_REF] must be taken seriously.

In the compact case the embedding necessarily has accumulation points but we can choose our states not to be excited on edges that are mapped under the embedding into a suitably small neighbourhood of every accumulation point.

Acknowledgements

The authors thank the KITP in Santa Barbara for hospitality during the workshop "The Quantum Nature of Spacetime Singularities" held in January 2007 during which parts of this project were completed. T.T. was supported there in part by the National Science Foundation under Grant No. PHY99-07949. We thank Abhay Ashtekar, Martin Bojowald, Steven Giddings, Jim Hartle, Gary Horowitz, Ted Jacobson, Jurek Lewandowski, Don Marolf, Rob Myers, Hermann Nicolai, Joe Polchinski, Stephen Shenker, Eva Silverstein, Lukasz Szulc and Erik Verlinde for inspiring discussions. K.G. is grateful to the Perimeter Institute for Theoretical Physics for hospitality and financial support where parts of the present work were carried out. Research performed at the Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI.