Canonical Quantum Gravity, Constructive QFT and Renormalisation

The canonical approach to quantum gravity has been put on a firm mathematical foundation in the recent decades. Even the quantum dynamics can be rigorously defined, however, due to the tremendously non-polynomial character of the gravitational interaction, the corresponding Wheeler-DeWitt operator valued distribution suffers from quantisation ambiguities that need to be fixed. In a very recent series of works we have employed methods from constructive quantum field theory in order to address those ambiguities. Constructive QFT trades quantum fields for random variables and measures thereby phrasing the theory in the language of quantum statistical physics. The connection to the canonical formulation is made via Osterwalder-Schrader reconstruction. It is well known in quantum statistics that the corresponding ambiguities in measures can be fixed using renormalisation. The associated renormalisation flow can thus be used to define a canonical renormalisation programme. The purpose of this article is to review and further develop these ideas and to put them into context with closely related earlier and parallel programmes.


Introduction
The canonical approach to quantum gravity has been initialised long time ago [1]. However, the mathematical foundations of the theory remained veiled due to the tremendous non-linearity of the gravitational interaction. This has much changed with the reformulation of General Relativity as a Yang-Mills type gauge theory in terms of connection rather than metric variables [2] and has culminated in a research programme now known as Loop Quantum Gravity (LQG) (see e.g. [3] for monographs and recent reviews on the subject). The qualifier "loop" stems from the fact that for gauge theories of Yang-Mills type it has proved useful to formulate the theory in terms of holonomies of the connection along closed paths (loops) in order to maintain manifest gauge invariance. Such so-called (Wilson) loop variables are widely used for instance in (lattice) QCD [4].
LQG has succeeded in providing a rigorous mathematical framework: The representation theory of the canonical commutation relations and the * relations has been studied and a unique representation has been singled out [5] that allows for a unitary representation of the spatial diffeomorphism group. Moreover, the generators of temporal diffeomorphisms, sometimes referred to as Wheeler-DWitt operators, could be rigorously quantised on the corresponding Hilbert space [6] and in contrast to the perturbative approach to quantum gravity [7], no ultraviolet divergences were found. It should be emphasised, that this was achieved 1. in the continuum, rather than on a lattice, i.e. there is no artificial cut-off left over, 2. for the physical Lorentzian signature rather than unphysical Euclidian one, and 3. non-perturbatively and background independently, that is, one does not perturb around a classical background metric and then quantises the fluctuations which thus manifestly preserves the diffeomorphism covariance of all constructions. However, the theory is not yet completed: Due to the tremendously non-polynomial nature of the gravitational interaction, the ususal factor ordering ambiguity in the quantisation of operator valued distributions which are nonlinear in the fields is much more severe. Thus, the operators defined in [6] suffer from those ambiguities. Moreover, the following problem arises: In the classical theory, the canonical generators of spacetime diffeomorphisms (that is, their Hamiltonian vector fields) form a Lie algebroid (that is, a Lie algebra except that the structure constants are replaced by structure functions on the phase space) known as the hypersurface algebroid [8]. The structure functions are themselves promoted to operator valued distributions upon quantisation, thus it becomes even harder to find a quantisation of those generators such that the algebroid is represented without anomalies than it would be for an honest Lie algebra. Specifically, the commutator between two temporal diffeomorphism generators is supposed to 1. be proportional to a linear combination of spatial diffeomorphism generators with operator valued distributions as coefficients and 2. in an ordering, such that the following holds: The image of any such commutator of a dense domain of vectors in the Hilbert space must be in the kernel of the space of spatially diffeomorphism invariant distributions on that domain. In [10] it was shown that both conditions 1. and 2. hold, however, the coefficients in that linear combination do not qualify as quantisations of their classical counterpart. Thus while the quantisation of the hypersurface algebroid closes, it does so with the wrong operator valued distributions as coefficients.
Thus, the status of LQG can be summarised as follows: As compared to [1] it is now possible to ask and answer precise questions about the mathematical consistency of the whole framework. As compared to the perturbative approach [7] the framework does not suffer from ultraviolet divergences and one does not have to worry about the convergence of a perturbation series due to the manifestly non-perturbative definition of LQG. However, just as in the perturbative approach, one needs further input in order to draw predictions from the theory, although of a different kind: In the perturbative approach, there are an infinite number of counter terms necessary due to non-perturbative non-renormalisability all of which come with coefficients that have to be measured but one can argue that only a finite number of them is of interest for processes involving energies not exceeding a certain threshhold (effective field theory point of view). In LQG there are in principle infinitely many quantisation ordering prescriptions possible, each of which comes with definite coefficients in order to yield the correct naive continuum limit but it is not clear which ordering to choose so that presently one resorts to the principle of least technical complexity.
Various proposals have been made in order to improve the situation. In [11] one exploits the fact that classically one can always trade a set of first class constraints by a single weighted sum of their squares (called the master constraint). Since a single constraint always closes with itself and the weights can be chosen such that the master constraint commutes with spatial diffeomorphisms, one can now focus on the quantisation ambiguities involved in the master constraint without having to worry about anomalies. In [12] the case of General Relativity coupled to perfect fluid matter was considered which allows to solve the constraints before quantisation so that the remaining quantisation ambiguity now only rests in the corresponding physical Hamiltonian that drives the time evolution of the physical (that is, spacetime diffeomorpohism invariant) observables. In [13] the constraints are quantised on a suitable space of distributions with respect to a dense domain of the Hilbert space rather than the Hilbert space itself in order to find a representation of the the hypersurface algebroid directly on that space of distributions which would at least partially fix the afore mentioned ordering ambiguity.
It transpires that additional input is necessary in order to fix the quantisation ambiguity in the dynamics of LQG and thus to complete the definition of the theory. This would also put additional faith into applications of LQG for instance to quantum cosmology [14] (where the amount of ambiguity is drastically reduced) which are believed to be approximations of LQG by enabling to make the connection between LQG and those approximations precise including an error control, see [15] for recent progress in that respect. In the recent proposal [16] which we intend to review in this article, the authors were inspired by Wilson's observation [16] that renormalisation methods help to identify among the principally infinitely many interaction terms in Hamiltonians relevant for condensed matter physics the finitely many relevant ones that need to be measured. This insight implies that a theory maybe perturbatively non-renomalisable but non-perturbatively renormalisable, also known as asymptotically safe [17]. The asymptotic safety approach to quantum gravity for Euclidian [18] and Lorentzian signature [19] precisely rests on that idea and has received much attention recently. In fact, there is much in common between our proposal and asymptotically safe quantum gravity (especially for Lorentzian signature) and we will have the opportunity to spell out more precisely points of contact in the course of this article. Also, there is a large body of work on renormalisation [20] in the so-called spin foam approach [23] and the related group field theory [21] and tensor model 1 [22] approach to quantum gravity. The spin foam approach is loosely connected to LQG in the following sense: The states of the Hilbert space underlying LQG are labelled by collections of loops, that is, 3D graphs. A spin foam is an operator that maps such states excited on a graph to states excited on another graph. The operator depends on a specific class of 4D cell complex (foam) such that its boundary 3D complex is dual to the union of the two graphs corresponding to the incoming and outgoing Hilbert spaces. The operator is supposed to form the rigging map [24] of LQG, i.e. a generalised projector onto the joint kernel of the Wheeler-Dewitt constraints. We say that the connection is loose because the rigging nature of current spin foams in 4D is not confirmed yet. In any case, a spin foam operator can be formulated as a state sum model and thus renormalisation ideas apply. For applications of renormalisation group ideas in the cosmological sector of LQG see [25].
Most of the work on renormalisation is either within classical statistical physics (e.g. [26]) or the Euclidian (also called constructive) approach to quantum field theory [27]. In the Euclidian approach, the quantum field, which is an operator valued distribution on Minkowski space, is replaced by a distribution valued random variable on Euclidian space. While the dynamics in the Minkowski theory is given by Heisenberg equations, in the Euclidian theory it is encoded in a measure on the space of random variables. We are then back in the realm of statistical physics because losely speaking the measure can be considered as a Gibbs factor for a Hamiltonian (sometimes called Euclidian action) in 4 spatial dimensions. How then should one use renormalisation ideas for quantum gravity? Quantum gravity is not a quantum field theory on Minkowski space (unless one works in the perturbative regime, but then it is non renormalisable). Also, while the Minkowski and Euclidian signature of metrics are related by simple analytic rotation in time from the real to the imaginary axis, this does not even work for classical metrics with curvature, not to mention the quantum nature of the metric (in ordinary QFT, the metric is just a non-dynamic background structure). One can of course start with Euclidian signature GR and try to build a measure theoretic framework, but then the relation to the Lorentzian signature theory is unclear. Moreover, while as an Ansatz for the Euclidian signature measure we can take the exponential of the Euclidian Einstein-Hilbert action, that action is not bounded from below and thus the measure cannot be a probability measure which is one of the assumptions of constructive QFT. Finally, in contrast to constructive QFT, in quantum gravity expectation values (operator language) or means (measure language) of basic operators (or random variables) such as the metric tensor have no direct physical meaning because coordinate transformations are considerd as gauge transformations, hence none of the basic fields correspond to observables.
In our approach [16] we will use the framework [12], that is, we do not consider vacuum GR but GR coupled to matter which acts as a dynamical reference field. This enables us 1. to solve the spatial diffeomorphism and Hamiltonian constraints classically, 2. to work directly on the physical Hilbert space (i.e. the generalised kernel of all constraints equipped with the inner product induced by the rigged Hilbert space structure), 3. to have at our disposal immediately the gauge invariant degrees of freedom such that the physical Hilbert space is the representation space of a * representation of those observables and 4. to be equipped with a physical Hamiltonian that drives the physical time evolution of those observables. Concretely and out of mathematical convenience we use the perfect fluid matter suggested in [28] but for what follows these details are not important. Important is only that it is possible to rephrase GR coupled to matter as a conservative Hamiltonian system and that all the machinery that was developed for LQG can be imported. Now the quantisation ambiguity rests of course in the physical Hamiltonian and it is that object and its renormalisation on which we focus our attention.
As we just explained, we can bring GR coupled to matter somewhat closer to the usual setting of ordinary QFT or statistical physics but still we cannot apply the usual path integral renormalisation scheme because we work in the canonical (or Hamiltonian) framework. The idea is then to make use of Feynman-Kac-Trotter-Wiener like ideas in order to generate a Wiener measure theoretic framework from the Hamiltonian setting and vice versa to use Osterwalder-Schrader reconstruction to map the measure theoretic (or path integral) frameork to the Hamiltonian one. This way we can map between the two frameworks and thus import path integral renormalisation techniques into the Hamiltonian framework which are strictly equivalent to those employed in path integral renormalisation.
In order that this works one needs to check of course that the Wiener measure constructed obeys at least a minimal subset [29] of Osterwalder-Schrader axioms [30] in order for the reconstruction to be applicable, most importantly reflection positivity.
This was one of the goals of [16], namely to define a renormalisation group flow directly within the Hamiltonian setting with strict equivalence to the path integral flow. Specifically the flow is a flow of Osterwalder-Schrader triples (H, H, Ω) consisting of a Hilbert space H a self-adjoint Hamiltonian H thereon bounded from below and a vacuum vector Ω ∈ H annihilated by H. While physically well motivated, of course, one does not need to do this. Indeed, renormalisation techniques for Hamiltonians and vacua directly within the Hamiltonian setting were invented before and we devote the next section for putting our framework into context with schemes closely related to ours. The fact that we have a precise relation between Hamiltonian and path integral renormalisation makes it possible to bring Hamiltonian formulations of Quantum Gravity such as LQG and path integral formulations such as asymptotically safe Quantum Gravity, into closer contact.
The architecture of this article is as follows: In the second section we give an incomplete overview over and sketch Hamiltonian renormalisation frameworks closely related to ours and point out differences and similarities.
In the third section we review how classical General Relativity coupled to suitable matter can be brought into the form of a conservative Hamiltonian system as well as the LQG quantisation thereof. The necessity to remove quantisation ambiguities will be highlighted.
In the fourth section we recall some background material on constructive QFT, the Feynman-Kac-Trotter-Wiener construction as well as Osterwalder-Schrader reconstruction.
In the fifth section we derive the natural relation between families of cylindrically defined measures, coarse graining, renormalisation group flows and their fixed points. We then use Osterwalder-Schrader reconstruction to map the flow into the Hamiltonian framework. This section contains new material as compared to [16] in the sense that we 1. develop some systematics in the choice of coarse graining maps that are motivated by naturally available structures in the classical theory, 2. clarify the importance of the choice of random variable or stochastic process when performing OS reconstruction and 3. improve the derivation of the Hamiltonian renormalisation flow by adding the uniqueness of the vacuum as an additional assumption (also made in the OS framework of Euclidian QFT [27]) as well as some machinery concerning degenerate contraction semigroups and associated Kato-Trotter formulae.
In the sixth section we mention concrete points of contact between the scheme developed here and others in the conext of density matrix, entanglement and projective renormalisation.
In the seventh section we sketch a relation between Hamiltonian renormalisation via Osterwalder Schrader reconstruction and the functional renormalisatiin group which is the underlying technique of the asymptotic safety programme.
In the eighth section we summarise, spell out implications of the renormalisation programme for the anomaly free implementation of the hypersurface algebroid and outline the next steps when trying to apply the framework to interacting QFT and finally canonical quantum gravity such as LQG.
In appendix A we prove some properties for a coarse graining scheme appropriate for non Abelian gauge theories, in appendix B we prove a lemma on the existence of certain Abelian C * −algebras needed for the construction of stochastic processes during OS reconstruction, in appendix C we collect some renormalisation terminology for readers more familiar with actions rather than measures, in appendix D we give a proof for the Kato-Trotter product formula for semi-groups and projections in the simple case that the semi-group has a bounded generator and in appendix E we prove a strong limit identity between projections needed in section 5.3.

Overview over related Hamiltonian renormalisation schemes
Purpose of this section is not to give a complete scan of the vast literature on the subject of Hamiltonian renormalisation but just to give an overview over those programmes that we believe are closest to ours. Also we leave out many finer details as we just want to sketch their relation to our framework in broad terms. In sections 6, 7 we will give a few more details on the connection of our approach with the density matrix and functional renormalisation group.
The starting point are of course the seminal papers by Kadanoff [31] and Wilson [32]. Kadanoff introduced the concept of a block spin transformation in statistical physics, i.e. a coarse graining transformation in real space (namely on the location of the spin degrees of freedom on the lattice) rather than in some more abstract space (e.g. momentum space blocking/suppressing as used e.g. in the asymptotically safe quantum gravity approach). This kind of real space coarse graining map is widely used not only in statistical physics but also in the path integral approach to QFT as for instance in lattice QCD [33]. On the other hand, Wilson introduced the concept of Hamiltonian diagonalisation to solve the Kondo problem (the low temperature behaviour of the electrical resistance in metals with impurities). This defines a renormalisation group flow directly on the space of Hamiltonians and its lowest lying energy eigenstates. More precisely, one considers a family of Hamiltonians labelled by an integer valued cut-off on the momentum mode label of the electron anhilation and creation operators. The renormalisation group flow is defined by diagonalising the Hamiltonian given by a certain cut-off label, and to use the eigenstates so computed to construct the matrix elements of the Hamiltonian at the next cut-off label. To make this practical, Wilson considered a truncation, at each renormalisation step, of the full energy spectrum to the 10 3 lowest lying energy levels which was sufficient for the low temperature Kondo problem. This is in fact nothing but the concrete application ot the Rayleigh-Ritz method. The concept of truncation plays an important role also in most other renormalisation schemes, as otherwise the calculations become unmanagable.
The next step was done by Wegner [34] as well as Glazek and Wilson [35] which can be considered as a generalisation of the Hamiltonian methods of [32]. It could be called perturbative Hamiltonian block diagonalisation and was applied in QFT already (e.g. [36] and references therein). Roughly speaking, one introduces a momentum cut-off on the modes of the annihilation and creation operators involved in the free part of the Hamiltonian, then perturbatively (with respect to the coupling constant) constructs unitarities which at least block diagonalise that Hamiltonian with respect to a basis defined by modes that lie below half the cut-off and and those that lie between half and the full cut-off, and then projects the Hamiltonian onto the Hilbert space defined by the modes below half of the cut-off to define a new Hamiltonian at half the cut-off. This can be done for each value of the cut-off and thus defines a flow of Hamiltonians (and vacua defined as their ground states). Another branch of work closely related to this is the projective programme due to Kijowski [37]. Here a flow of Hamiltonians on Hilbert spaces for different resolutions is given by the partial traces of the corresponding density matrices given by minus their exponential (Gibbs factors -assuming that hese are trace class). See also [38,39,40] for more recent work on renormalisation building on this programme.
In these developments the spectrum of the Hamiltonian was directly used to define the flow. Another proposal was made by White [41] who defined the density matrix renormalisation group. This is a real space renormalisation group flow which considers the reduced density matrix corresponding to the tensor product split of a vector (e.g. the ground state of a Hamiltonian) of the total Hilbert space into two factors corresponding to a block and the rest (or at least a much larger "superblock"). This density matrix is diagonalised and then the Hilbert space is truncated by keeping only a certain fixed number of highest lying eigenvalues of the reduced density matrix. Finally the Hamiltonian corresponding to the block is projected and then the resulting structure is considered as the new structure on the coarser lattice resulting from collapsing the blocks to new vertices (we are skipping here some finer details). This method thus makes use of entanglement ideas since the reduced density matrix defines the degree of entanglement via its von Neumann entropy.
A variant of this is the tensor renormalisation group approach due to Levin and Nave [42]. It is based on the fact that each vector in a finite tensor product of finite dimensional Hilbert spaces can be written as a matrix product state, i.e. the coefficients of the vector with respect to the tensor product base can be written as a trace of a product of matrices of which there are in general as many as the dimensionality of the Hilbert space. One now performs a real space renormalisation scheme directly in terms of those matrices which are considered to be located on a lattice with as many vertices as tensor product factors. Importantly, this work connects renormalisation to the powerful numerical machinery of tensor networks [43].
Finally, as observed by Vidal [44] and Evenbly and Vidal [45], one can improve [41,42] by building in an additional unitary disentanglement step into the tensor network renormalisation scheme. This is quite natural because a tensor network can also be considered as a quantum circuit with the truncation steps involved considered as isometries but a quantum circuit in quantum computing [46] consists of a network of unitary gates some of which have a disentangling nature depending on the state that they act upon. The resulting scheme is called multi-scale entanglement renormalisation ansatz (MERA).
As this brief and incomplete discussion reveals, there are numerous proposals in the literature for how to renormalise quantum systems. They crucially differ from each other in the choice of the coarse graining map. There are various aspects that discriminate between these maps, such as: 1. Real space versus other labels The degrees of freedom to be coarse grained are labelled by points in spacetime or else (momentum, energy, ...). 2. Kinematic versus dynamical Real space block spin transformations are an example of a kinematic coarse graining, i.e. the form of the action, a Hamiltonian, its vacuum vector, its associated reduced density matrix and the corresponding degree of entanglement do not play any role. By contrast, Hamiltonian block diagonalisation, density matrix and entanglement renormalisation take such dynamical information into account. 3. Truncated versus exact In principle any renormalisation scheme can be performed exactly, e.g. in real space path integral renormalisation one can just integrate the excess degrees of freedom that live on the finer lattice but not on the coarser thus obtaining the measure (or effective action) on the coarser lattice from that of the finer one. The same is true e.g. for the procedure followed in asymptotically safe quantum gravity. However, in practice this may quickly become unmanagable and thus one resorts to approximation methods e.g. by truncation in the space of coupling constants, energy eigenstates or reduced density matrix eigenstates.
For the newcomer to the subject, this plethora of suggestions may appear confusing. Which choice of coarse graining is preferred? Do different choices lead to equivalent physics? What can be said about the convergence of various schemes and what is the meaning of the fixed point(s) if it (they) exist(s)? The physical intuition is that different schemes should give equivalent results if 1. the corresponding fixed point conditions capture necessary and sufficient properties that the theory should have in order to qualify as a continuum theory and 2. when performed exactly. The first condition is obvious, we start from what we believe to be an initial guess for how the theory looks at different resolutions and then formulate a coarse graining flow whose fixed points are such that they qualify to define a continuum theory. The seond condition entails that the coarse graining maps just differ in the separation of the total set of degrees of freedom into subsets corresponding to coarse and fine resolution, hence corresponds to choices of coordinate systems which of course can be translated into each other. However, when truncations come into play, this equivalence is lost because different schemes truncate different sets of degrees of freedom which are generically no longer in bijection. It is conceivable therefore that dynamically driven truncation schemes perform better at identifying the correct fixed point structure of the theory in the sense that they may converge faster and are less vulnerable to truncation errors or automatically pick the truncation of irrelevant couplings. This seems to be confirmed in spin system examples but we are not aware of a general proof. Recently, the importance of the kinematic versus dynamic issue has also been emphasised for the LQG and spin foam approach [47].
In our work we currently are not concerned with issues of computationability, that is, we consider an exact scheme. Next, as far as the coarse graining map is concerned, we currently favour a kinematic scheme. The reason for doing this is that kinematic schemes are naturally suggested by measure theoretic questions. Namely, measures on spaces of infinitely many degrees of freedom are never of the type of the exponential of some action times a normalisation constant times Lebesgue measure. Neither of these three ingredients is well defined. What is well defined are integrals of certain probe functions of the field with respect to that measure. These probe functions in turn are naturally chosen to depend on test functions that one integrates the field against. Thus these test functions provide a natural notion of resolution, discretisation and coarse graining. By integrating the measure against probe functions one obtains a family of measures labelled by the test functions involved. The relation between test functions at different resolution induces a corresponding relation between members of the family of measures which must hold exactly for a true measure of the continuum QFT. In turn, such consistency relations called cylindrical consistency can be used to define a measure on a space of infinitely many degrees of freedom [48], called a projective limit. The idea is then to formulate measure renormalisation in such a way that its fixed points solve the consistency relations. This approach has been advocated in [49] for Euclidian Yang-Mills theory and in [50] for spin foams. Note that spin foams strictly speaking do not construct measures but rather are supposed to construct a rigging map so that Hamiltonian methods come also into play. Indeed, in [47] it was shown that the cylindrically consistent coarse graining of the rigging map and its underlying spacetime lattice, thought of as an anti-linear functional on the kinematical Hilbert space, induces a coarse graining of the spatial lattice on its boundary and thus the Hilbert space thereon, equipping it with a system of consistent embeddings, a structure similar to inductive limits of Hilbert spaces (an inductive structure requires in addition the injections to be isometric). That latter structure underlies the kinematical Hilbert space of LQG and a renormalisation procedure based on inductive limits was already proposed in [51] due to the similarity of LQG to lattice gauge theory.
Another reason for why picking real space coarse graining schemes as compared to say, momentum space based ones is their background independence which especially important for quantum gravity. In our work, as we consider the version of LQG in which the constraints already have been solved, we will work with probability measures. As we will see, the connection between inductive limits of Hilbert spaces and projective limits of path integral measures can be made crystal clear in this case. The price we pay by using an exact, kinematical scheme is that the fixed point (or renormalised) Hamiltonian becomes spatially non-local at finite resolution. However, in the free QFT examples studied [16], which are spatially local in the continuum, by blocking the known fixed point theory from the continuum one can see that this is natural and must happen for such schemes, hence it is not a reason for concern but in fact physical reality. The degree of spatial non-locality in fact decreases as we increase the resolution scale.
When applying the framework to interacting QFT one will have to resort to some kind of approximation scheme and possibly tools from entanglement renormalisation combined with tensor network techniques may prove useful. However, note that QFT of bosonic fields (gravity is an example) deals with infinite dimensional Hilbert spaces even when the theory depends only on a finite number of degrees of freedom say by discretising it on a lattice and confining it to finite volume. Thus, to apply tensor network techniques which, to the best of our knowledge, require the factors in the tensor product to be finite dimensional Hilbert spaces, one would have to cut off the dimensions of those Hilbert spaces right from the beginning, i.e. one would have to work with three cut-offs rather than two (see e.g. [52] where quantum group representations are used in gauge theories rather than classical group representations and performs real space renormalisation or [53] where one combines both the UV and the dimension cut-off into one by turning the dimension of tensor spaces in tensor models into a finite coarse graining parameter and otherwise performs the asymptotic safety programme which is often formulated in the presence of a cut-off anyway). Some sort of truncation or approximation has to be made in practice when treating complex systems numerically. The physical insight behind the tensor network and density matrix/entanglement renormalisation developments, namely that the dynamically interesting vectors in a Hilbert space appear to lie in a "tiny" subspace thereof is presumably a profound one and the truncation of the Hilbert space to the corresponding subspaces appears to be well motivated by the model (spin) systems studied so far. Still, what one would like to have is some sort of error control or convergence criteria on those truncations. We appreciate that this is a hard task for the future. For the time being we phrase our framework without incorporating a cut-off on the dimension of Hilbert spaces as we are not yet concerned with numerical investigations, however, we may have to use some of these ideas in the future.

Canonical Quantum Gravity coupled to reference matter
The physical idea is quite simple and goes back to [54]: General Relativity is a gauge theory, the gauge group being the spacetime diffeomorphism group. Thus the basic tensor and spinor fields in terms of which one writes the Einstein-Hilbert action and the action of the standard model coupled to the metric (or its tetrad) are not observable. However, the value of, say a scalar field Φ at that spacetime point X y , at which four reference scalar fields φ 0 , ..., φ 3 take values y 0 , .., y 3 , that is, Φ(X y ); φ µ (X y ) = y µ is spacetime diffeomorphism invariant. For this to work, the relation φ µ (X) = y µ must of course be invertible, in particular the reference scalar fields must not vanish anywhere or anytime. This seems to be a property of dark matter [55].
These kind of relational observables have been further developed by various authors, see in particular [56,57]. When one couples General Relativity and such reference matter preserving general covariance, it becomes possible to formulate the theory in a manifestly gauge invariant way. The form of that gauge invariant formulation of course strongly depends on the type of reference matter used and its Lagrangian. In what follows we use the concrete model [12] out of mathematical convenience, but we emphasise that the same technique works in a fairly general context. In the next subsection, that model will be introduced and the classical gauge invariant formulation will be derived. After that we quantise it using LQG methods which will be introduced in tandem.

Gaussian dust model
The Lagrangian of the theory takes the form where L EH is the Einstein-Hilbert Lagrangian, L SM the standard model Lagrangian coupled to GR via the metric, its tetrad or its spin connection and L D is the Gaussian dust Lagrangian [28] where g is the Lorentzian signature metric tensor, ∇ µ its Levi-Civita covariant differential, φ 0 := T, φ j := S j ; j = 1, 2, 3 are the reference scalar fields introduced above and ρ, W j are additional four scalar fields. The latter four fields appear without derivatives and thus give rise to primary constraints in addition to those present even in vacuum GR. One easily show that the contribution of L D to the energy momentum tensor is of perfect fluid type. Further physical properties and motivations are discussed in [28]. For what follows it suffices to know that the equations of motion for T, S j say that ∇ µ T is a timelike geodesic co-tangent and that S j is constant along the geodesic spray. Thus, those geodesics can be interpreted as worldlines of dynamically coupled test observers.
The full constraint analysis of (3.1) is carried out in [12]. There are secondary constraints and the full set of constraints contains those of first and second class (see [57] for a modern treatment of Dirac's algorithm [58]). One has to introduce a Dirac bracket and solves the second class constraints in the course of which the variables ρ, W j are eliminated. The remaining constraints are then of first class and read Here C is the Wheeler-DeWitt constraint function (including standard matter), C a , a = 1, 2, 3 are the spatial diffeomorphism functions (including standard matter). The Dirac bracket reduces to the Poisson bracket on all the variables involved in (3.3) and P, P j are the momenta conjugate to T, S j , e.g. {P (x), T (y)} = δ(x, y). Here a = 1, 2, 3 are tensorial indices on the spatial hypersurface σ of the Arnowitt-Deser-Misner foliation underlying the Hamiltonian formulation of GR [59] with intrinsic metric tensor q ab . For the moment it is just necessary to know that C, C a do not involve the variables T, P, S j , P j . The constraints (3.3) encode the spacetime diffeomorphism gauge symmetry in Hamiltonian form, in particular they represent the hypersurface deformation algebra [8]. It is possible to solve these remaining constraints, to determine the complete set of gauge invariant (so called Dirac) observables and to determine the physical Hamiltonian H that drives their physical time evolution [12]. Equivalently, we may gauge fix (3.3). The above interpretation of T, S j suggest to use the gauge conditions G = T − t, G a = δ a j S j − x a . The stabilisation of these gauge conditions fixes the Lagrange multipliers λ, λ a in the gauge generator which when evaluated at G = G a = 0 yields the unique solution λ = 1, λ a = 0. Likewise, in this gauge the constraints can be uniqely solved for P = −C, P j = −δ a j C a while T, S j are pure gauge. This shows that the physical degrees of freedom are those not involving T, P, S j , P j .
For any function F independent of these variables the reduced or physical Hamiltonian is that function on the phase space coordinatised by the physical degrees of freedom which generates the same time evolution as K when the constraints, gauge conditions and stabilising Lagrange multipliers are installed Thus the final picture is remarkably simple: The physical phase space is simply coordinatised by all metric and standard matter degrees of freedom (and their conjugate momenta) while the physical Hamiltonian is just the integral of the usual Wheeler-DeWitt constraint. The influence of the reference matter now only reveals itself in the fact that H is not constrained to vanish as it only involves the geometry and standard matter contribution C of C tot and that the number of physical degrees of freedom has increased by four as compared to the system without reference matter. This phenomenon is of course well known from the electroweak interaction: One can solve the three isospin SU(2) Gaus constraints for three of the four degrees of freedom sitting in the complex valued Higgs isodublett, leaving a single scalar Higgs field and three massive rather than massless vector bosons. See [60] for further discussion.
We close this subsection with three remarks: First, a complete discussion requires to show that the gauge cut G = G a = 0 on the constraint surface of the phase space be reachable from anywhere on the constraint surface. As (3.5) shows, this requires that S j ,a be invertible. We thus impose this as an anholonomic constraint on the total phase space. One easily verifies from (3.5) that this condition is gauge invariant, i.e. compatible with the dynamics. Second, the simplicity of the final picture is due to the particular choice of reference matter. Other reference matter most likely will increase the complexity, see e.g. [61] which produces a square root Hamiltonian! One may argue that the dust is a form of cold dark matter [55], but it is unclear whether this is physically viable. Nevertheless, the present model serves as a proof of principle, namely that GR coupled to standard matter and reference can be cast into the form of a conservative Hamiltonian system. Third, it should be appreciated that the reference matter helps us to accomplish a huge step in the quantum gravity programme: It frees us from quantising and solving the constraints, construct the physical inner product, the gauge invariant observables and their physical time evolution. All of these steps are of tremendous technical difficulty [3]. All we are left to do is to quantise the physical degrees of freedom and the physical Hamiltonian.

Loop Quantum Gravity quantisation of the reduced physical system
In order to keep the technical complexity to a minimum, we consider just the contribution to H coming from the gravitational degrees of freedom, see [6,3] for more details on standard matter coupling. The Hamiltonian directly written in terms of SU (2) gauge theory variables reads (we drop some numerical coefficients that are not important for our discussion) Here A is an SU(2) connection and E an SU(2) non-Abelian electric field that one would encounter also in a SU (2) Yang-Mills theory. However, the geometric interpretation of A, E is different: Namely, e a j := E a j / | det(E)| is a triad, that is, q ab = δ jk e a j e b k is the inverse spatial metric. Here, as before, a, b, c, .. = 1, 2, 3 denote spatial tensor indices while now j, k, l, .. = 1, 2, 3 denote su(2) Lie algebra indices. Further, let Γ j a be the spin connection of e a j . Then K ab := (A j a − Γ j a )e k b δ jk has the meaning of the extrinsic curvature of the ADM slices [59] on the kernel of the SU(2) Gauss constraint The important quantity V is recognised as the total volume of the hypersurface σ and H E , H L are known as the Euclidian and Lorentzian contributions to H. See [6] for further details. The Poisson brackets displayed are with respect to the standard symplectic structure where κ = ℓ 2 P is the Planck area. The definition of the phase space is completed by the statement that the elementary fields A, E are real valued The traces involved in 3.8 are carried out by introducing the Lie algebra valued 1-forms A = A j a τ j dx a where 2iτ j are the Pauli matrices and F = 2(dA + A ∧ A) is the curvature of A. The non polynomiality of GR is hidden in the Poisson brackets that appear in (3.8). The reason for why we use these particular Poisson bracket structure will become clear only later.
To quantise the theory we start from functions on the phase space that are usually employed in lattice gauge theory, see e.g. [62], namely non-Abelian magnetic holonomy and electric flux variables where P denotes path ordering, c is a piecewise analytic real curve, S a piecewise real analytic surface, f an su(2) valued function and * E = ǫ abc E a dx b ∧ dx c /2 the pseudo 2-form corresponding to the su(2) valued vector where c −1 is the same curve as c but with the opposite orientation. The simplest non-trivial Poisson brackets are in case that S∩c is a single point in the interior of both c, S, see [5] for a complete discussion. The relations (3.13), (3.14) are the defining relations of a non-commutative abstract * −algebra A generated by fluxes and complex valued smooth functions F of a finite number of holonomy variables [5]. It is the free algebra generated by them and divided by the two sided ideal generated by the canonical commutation relations , F } and the adjointness relations (3.13). See [5] for more details. Interestingly, the physical Hamiltonian H has a large symmetry group. Namely, it is invariant under the group G = SU (2) loc ⋊ Dif f (σ) where SU (2) loc denotes the group of local SU(2) valued gauge transformations and Dif f (σ) denotes the group of (piecewise real analytic) diffeomorphisms of σ. An element of G is given by a pair g = (g, ϕ) which acts on the basic variables as where ϕ * denotes the pull-back action of diffeomorphisms on differential forms. This action lifts to the algebra A, specifically where b(c), f (c) denote beginning and final point of c and this simple covariant transformation behaviour was part of the reason why the particular "smearing" of A along curves involved in holonomies is used. Note also the different character of the two groups: While we still have to find the gauge invariant observables with respect to the Gauss constraint the diffeomorphism constraint is already solved. The diffeomorphisms in G are thus to be considered as active diffeomorphisms rather than passive ones. The mathematical problem in quantising the theory consists in constructing a * representation of A, that is, a representation (π, H) of elements a ∈ A as operators π(a) densely defined on a common, invariant domain D of a Hilbert space H such that the * relations are implemented as adjointness relations and such that the canonical commutation relations are implemented as commutators between them. Thus we want in particular that In QFT this problem is known to typically have an uncountably infinite number of unitarily inequivalent solutions, there is no Stone-von Neumann uniqueness theorem when the number of degrees of freedom in infinite. Hence, to make progress, we must use additional physical input. That input can only come from the Hamiltonian. Thus, we require in addition that the representation supports H as a selfadjoint operator (H is real valued) also densely defined on D and such that H carries a unitary representation U of G (such that its generators are self-adjoint by Stone's theorem). Using the powerful machinery of the Gel'fand-Naimark-Segal construction [63], the representation property and the unitarity property can be granted if we find a positive linear and G invariant functional ω : In [5] it was found that there is a unique ω satisfying (3.18). While the derivation is somewhat involved, the final result can be described in a compact form. The dense domain D consists of functions of the form i.e. ψ γ is a complex valued, smooth functions of a finite number of holonomy variables. The union of the curves of these holonomies forms a finite graph γ where E(γ) denotes the set of its edges. Note that the elements of A that just depend on the connection are themselves of the form (3.19) and thus their action by multiplication is densely defined. The fluxes are densely defined when acting by derivation which also solves the canonical commutation relations. To see that the adjointness conditions hold we need the inner product. To define it, we note that graphs defined by finiteley many piecewiese analytic curves are partially ordered by set theoretic inclusion and they are directed in the sense that for any two graphs γ 1 , γ 2 there exists Then we can decompose all edges of γ 1 , γ 2 with respect to the edges of γ 3 , use the algebraic relations of the holonomy in order to write ψ 1 , ψ 2 excited over γ 1 , γ 2 respectively as functions excited over γ 3 . Thus it is sufficient to know the inner product of functions excited over the same graph γ which is given by where µ H is the Haar measure on SU(2). One can check that the adjointness relations are indeed satisfied, in fact π(E f (S)) is an unbounded but essentially self-adjoint operator (i.e. a symmetric operator with unique self-adjoint extension). In fact (3.22) defines a cylindrical family of measures µ γ , one for every graph γ. One has to check that (3.22) is well defined because a function excited on γ can be written also as a function excited over any finer graph γ ′ by extending it trivially to the additional edges. This is in fact the case [5]. Then the Kolmogorov type extension theorems grant that the family extends to an honest continuum measure µ on the quantum configuration space A of distributional connections. We will not go into the details here which can be found in [5] but just mention for the interested reader that this space coincides with the so-called Gel'fand spectrum of the Abelian C * algebra that one obtains by completing the space of functions (3.19) in the sup norm. It follows that the Hilbert space is given by H = L 2 (A, dµ).
By construction, the Hilbert space H carries a unitary representation U of G given by To check this, one uses the properties of the Haar measure (translation invariance) and the diffeomorphism invariance of (3.22) which does not care about the location and shape of the curves involved. The Hilbert space comes equipped with an explicitly known orthonormal basis called spin network functions (SNWF). This makes use of harmonic analysis on compact groups G [64], in particular the Peter & Weyl theorem which states that the matrix element functions of the irreducible representations of G, which are all finite dimensional and unitary without loss of generality, are mutually orthogonal, unless equivalent, with respect to the inner product defined by the Haar measure on G, moreover, they span the whole Hilbert space. As the irreducible representations of SU(2) are labelled by spin quantum numbers, the name SNWF comes at no surprise. More in detail a SNWF T γ,j,ι is labelled by a graph γ, a tuple j = {j c } c∈E(γ) of spin quantum numbers decorating the edges and a tuple ι = {ι v } v∈V (γ) of intertwiners decorating the vertices v in the vertex set V (γ) of γ. Here an intertwiner ι v projects the tensor product of irreducible representations corresponding to the edges incident at v onto one of the irreducible representations appearing in its decomposition into irreducibles (Clebsch-Gordan theory). Besides providing an ONB convenient for concrete calculations, SNF make it easy to solve the Gauss constraint: A detailed analysis [3] shows that (3.9) can be quantised in the given representation and just imposes that the space of intertwiners be restricted to those projecting on the trivial (spin zero) representation. We call such intertwiners gauge invariant. Hence the joint kernel of the Gauss constraints is a closed subspace of H which is explicitly known. We will abuse the notation and will not distinguish between that subspace and H and henceforth consider the Gauss constraint as solved. All operators considered in what follows are manifestly gauge invariant and preserve that subspace.
As a historical remark, solutions of the Gauss constraint are excited on closed graphs since there is no nontrivial intertwiner between the trivial representation and a single irreducible one, hence open ends are forbidden. For closed graphs, one can alternatively label SNWF by homotopically independent closed paths (loops) with a common starting point (vertex) on that graph. Originally one used loops as labels, hence the name Loop Quantum Gravity (LQG).
One of the many unfamiliar features of H is that it is not separable which easily follows from the uncountable cardinality of the set of graphs. This is a direct consequence of the diffeomorphism invariance of the inner product: Two graphs that are arbitrarily close but disjoint are simultaneously also arbitrarily far apart under the inner product. Thus if the measure clusters for far apart support of the smearing functions (here the graphs) then the orthogonality of the corresponding spn network functions comes at no surprise. A direct consequence of this is that the diffeomorphism operators U (ϕ) do not act (strongly) continuously, hence a generator of of infinitesimal diffeomorphisms generated by the integral curves of vector fields cannot exist. Yet another direct consequence is that the connection operator A itself does not exist, only its holonomies do.
The remaining task is to quantise the Hamiltonian and it is at this point where the afore mentioned quantisation ambiguities arise. The strategy followed in [6] is as follows: It turns out that the volume operator appearing in (3.8) can be quantised on H as an essentially self-adjoint operator whose spectrum is pure point (discrete) [65]. It is densely defined on the span of the SNWF and it acts vertex wise, with no contribution from gauge (in)variant vertices that are not at least three (four) valent or from vertices whose incident edges have tangents in a common two dimensional or one dimensional space. Next the holonomy along an open curve c can be expanded as A(c) = 1 2 + c A+... and along a closed curve α as A(α) = 1 2 + S,∂S=α F so that the functions A, F that appear in (3.8) can be approximated by suitable holonomies where the approximation is in terms of the "length" of the curves involved which are matched with the coordinate volume assigned by the Lebesgue measure d 3 x appearing in (3.8) approximating the integral by a Riemann sum (this is a regularisation step). Suppose then that somehow a well defined operator H E can be defined by replacing the classical functions by operators and the Poisson brackets by commutators times . Then the same argument can be applied to the Lorentzian piece. As a final piece of information, one uses the observation that a spatially diffeomorphism invariant operator, densely defined on the span of SNWF cannot have non-trivial matrix elements between SNWF excited over different graphs [9]. This has the following consequence: Let H γ be the closed linear span of SNWF excited precisely over γ. Then, if H is supposed to preserve its classical diffeomorphism invariance upon quantisation we necessarily have where each H γ is self-adjoint on H γ , in particular it preserves this space. Let now P γ : H → H γ be the orthogonal projection. Then the following concrete expression for H can be given [12] (again we drop some numerical coefficients and set = 1) The sum is over vertices of γ and triples of edges incident at them (taken with outgoing orientation). For each vertex v and pairs of edges c, c ′ outgoing from v one defines α γ,v,c,c ′ as that loop within γ starting at v along c and ending at v along (c ′ ) −1 with the minimal number of elements of E(γ) used (if that loop is not unique, we average over them). It has been shown, that the concrete expression (3.25) has the correct semiclassical limit in terms of expectation values with respect to semiclassical coherent states [66] on sufficiently fine graphs of cubic topology [67].
Remarkably, (3.25) defines an essentially self-adjoint, diffeomorphism invariant, continuum Hamiltonian operator for Lorentzian quantum gravity in four spacetime dimensions, densely defined on the physical continuum Hilbert space H which is manifestly free ov ultraviolet divergences. That is, while for each given graph γ the theory looks like a lattice gauge theory on γ, the theory is defined on all lattices simultaneously which makes it a continuum theory. Moreover, note that the vector Ω = 1 has norm unity and that HΩ = 1.
Yet, one cannot be satisfied with (3.25) for the following reasons: 1.
While it is true that one can give a better motivated derivation than we could sketch here for reasons of space, there are some ad hoc steps involved.

2.
There are several ordering ambiguities involved in (3.25): Not only could we have written the factors in different orders but instead of using the fundamental representation to approximate connections in terms of holonomies we could have used higher spin representions [68] or an avarage over several of them and in each case we would have different coefficients appearing in front of these terms.

3.
Of particular concern is definition of the minimal loop. While this gives good semiclassical results on sufficiently fine lattices, the theory lives on all lattices also those which are very coarse and on those the expression (3.25) is doubtful, because the Riemann approximation mentioned above would suggest to use a much finer loop. In fact, one is supposed to take the regulator (i.e. the coordinate volume ǫ of the Riemann approximants) away and in that limit the loop would shrink to zero. One can justify that this does not happen by using a sufficiently weak operator topoloy [6]. Namely, there exist diffeomorphism invariant distributions (linear functionals) l on the dense span of SNWF ψ [9] and we define an operator O ǫ to converge to an operator O in that topology if l([O ǫ − O]ψ) → 0 for all l, ψ. Now due to diffeomorphism invariance we can deform for any ǫ the small loop to any diffeomorphic one as long as we do not cross other edges of the graph, in particular we can deform is as close as we want to the minimal one. Then the result mentioned above about the matrix elements of diffeomorphism invariant operators in fact forces us to choose that loop precisely, not only approximately. Of course, while the diffeomorphism symmetry of H makes the space of diffeomorphism invariant distributions a natural space to consider, it is still not perfectly justified to use it in order to define a topology. 4. The naive dequantisation of (3.25) will perform poorly on very coarse graphs and will be far from the continuum expression (3.8) but one could argue that that vectors supported on coarse graphs simply do not qualify as good semiclassical states. 5.
Using the same argument as in 3. there is nothing sacred about the minimal loop and one could take again other loops and/or average of over them with certain weights. However, then the locality of (3.25) is lost. 6. The block diagonal or superselection structure 3.24) which is forced on us by the non-separability of the Hilbert space and its spatial diffeomorphism covariance appears unphysical, one would expect that the Hamiltonian creates also new excitations.
It transpires that we must improve (3.25) and the discussion has indicated a possible solution: Blocking free QFT from the continuum (i.e. restricting the Hilbert space to vectors of finite spatial resolution) with respect to a kinematic real space coarse graining scheme exactly produces such a high degree of non-locality at finite resolution even if the continuum measure or the continuum Hamiltonian is local [16,20,33]. This bears the chance that what we see in (3.25) is nothing but a naive guess of a continuum Hamiltonian which is blocked from the continuum but whose off-block diagonal form we cannot determine with the technology used so far. Accordingly this calls for shifting our strategy which was already started in [67] in the sense that the block diagonal structure was dropped but only one infinite graph was kept: We take the above speculation serious and consider the operators H γ as projections onto the subspaces H γ of H of a continuum Hamiltonian H but we will drop the unphysical block diagonal structure 3.24 which arises from the non-separability of H. Rather the relation between the H γ is to be imposed by a renormalisation scheme induced by the path integral renormalisation scheme adopted in quantum statistical physics. To do this, we must first derive a path integral measure µ γ from the OS data H γ , H γ , Ω γ where Ω γ is the vacuum of H γ by the usual Feynman-Kac-Trotter-Wiener formalism. Then we can compute the flow of the µ γ in the ususal way and then translate into a flow of OS data by OS reconstructing them from the measures. The fixed points of the flow will then define the possible continuum theories and these may be "phases" quite different from (3.25). The details of this programme will be the subject of the following sections.

Constructive QFT, Feynman-Kac-Trotter-Wiener construction and Osterwalder-Schrader reconstruction
The purpose of this section is to provide some background information on constructive QFT and related topics such as the Feynman-Kac-Trotter-Wiener construction of measures (path integrals) from a Hamiltonian formulation (operator formulation) and vice versa the Osterwalder-Schrader reconstruction of a Hamiltonain framework from a measure. Our description will be minimal. The prime textbook references are [70,27].

Measure theoretic glossary
The measure µ is called a probability measure if µ(S) = 1. One uses the notation where χ s (p) = 1 if p ∈ s and χ s (p) = 0 else is called the characteristic function of s ∈ B.
Consider now a second measurable space (S,B). A function X : S →S is called measurable or a random variable if the pre-images X −1 (s) = {p ∈ S; f (p) ∈s} of measurable setss ⊂S are measurable in S. Let F be the set of random variables X : S →s then for X ∈ F the set functioñ defines also a probability measure called the distribution of Φ. We consider real valued functions f : where the sum is over at most finitely many terms and define their integral as One can show that this identity extends from simple functions to Borel functions i.e. measurable functions f :S → R where R is equipped with the Borel σ−algebra (the smallest σ algebra containing all open intervals). We can then also extend it to those complex functions whose real and imaginary parts are Borel by linearity. In what follows we assume that for each N ∈ N 0 there exists a distinguished system W N of complex valued, bounded elementary functions W on N copies ofS such that the corresponding cylinder functions enjoy the following properties: 1. They generate an Abelian * algebra, that is, for all I, I ′ ∈ I the product W I W ′ I ′ is a finite, complex linear combination of suitable W ′′ I ′′ , I ′′ ∈ I, W ′′ ∈ W |I ′′ | and also W I is of that form. 2. W N contains the constant function. 3. For each I ∈ I, the moments µ(W I ), W ∈ W |I| determineμ I uniquely. 4. These properties show that the W I are L 2 (dμ I ,S |I| ) functions. We require their span to be dense.
We saw that a probability measure µ together with a stochastic process gives rise to a family of cylindrical probabilty measures (μ I ) I∈I onS |I| . The converse question is under which circumstances a cylindrical family of cylinder probability measures determines a measure µ. A necessary criterion is as follows: The set I is partially ordered and directed by inclusion, that is, for each I, J ∈ I we find K ∈ I such that I, J ⊂ K (for instance K = I ∪ J). Suppose that I ⊂ J. Then where X I = {X i } i∈I ,s I ⊂B |I| . Furthermore for any permutation π on N = |I| elements set π · I = {i π(1) , .., i π(N ) } and π ·s I = {(p π(1) , ..,p π(N ) ); (p 1 , ..,p N ) ∈s I }. Then µ(X −1 π·I (π ·s I ) =μ π·I (π ·s I ) = µ(X −1 I (s I ) =μ I (s I ) (4.9) Even more generally, a partial order on the set I of finite subsets I of I is a transitive, reflexive and antisymmetric relation, that is, I < J ∧ J < K ⇒ I < K and I < I and I < J ∧ J < I ⇒ I = J for all I, J, K ∈ I. The set I is called directed with respect to < provided that for all I, J ∈ I we find K ∈ I such that I, J < K.
For I < J we may have surjective maps P JI :S |J| →S |I| such that X I (p) = P JI (X J (p)) and such that for I < J < K whe have P JI • P KJ = P KI . Then similar as in (4.8) we necessarily must have for I < J µ I (s I ) =μ J (P −1 JI (s I )) (4.10) It turns out that these two conditions, either (4.8), (4.9) or (4.10) are also sufficient in fortunate cases (for instance ifS = R, which is the classical Kolmogorov theorem, see [48]). That is, we can then reconstruct the measure space (S, B, µ) and a stochastic process {X i } i∈I such that theμ I are the cylinder measures of µ. It follows that the W I ∈ W |I| , I ∈ I lie dense in L 2 (S, dµ).

Physical meaning:
We consider the elements p ∈ S to be spacetime fields Φ or spatial fields φ respectively. The index set I will have the meaning of a set of test functions or more generally distributions whose elements i label the random variables X i . These map the fields smeared with test functions to a finite dimensional manifold (usually copies of R or more generally of a Lie group). For instance for a scalar field Φ we may consider the random variable . It is also customary to consider the field p = Φ itself as a random variable indexed by the same index set or to simply write X i (p) = p(i) as an abbreviation.

Constructive QFT
The application of interest of the previous subsection is a stochastic process indexed by either R × L or just by L where the label set L is a certain set of distributions on the spatial manifold. We distinguish between random variables Φ indexed by a pair (t, f ) ∈ R × L and random variables φ indexed by f ∈ L. Some examples are: 1. Real quantum scalar fields with smooth smearing: Consider L = S(R 3 ), the space of smooth test functions of rapid decrease andS = R equipped with the Borel σ−algebra. Then The space W N of elementary functions on N copies of R can be chosen to be generated by the exponentials with r 1 , .., r N ∈ R labelling the (necessarily one dimensional) unitary irreducible representations of U (1).
In fact, since in this case the space L is a vector space, it is sufficient to consdider the functions w(φ(f )) = exp(iφ(f )), f ∈ L. Analogously the space of elementary functions for the time dependent fields can be chosen which of course reduces to exp(iΦ(t T , f ′ T )).. exp(iΦ(t 1 , f ′ 1 )), f ′ k ∈ L (4.13) for certain f ′ k ∈ L. Obviously the Abelian * −algebra and boundedness conditions are satisfied. That these elementary functions suffice to determine the cylindrical measures requires a more involved argument (Bochner's theorem, [48]).
2. Real quantum scalar fields with distributional smearing: Consider a subset L =⊂ S ′ (R 3 ) of the tempered distributions andS = U (1) equipped with the Borel σ−algebra. In applications to scalar fields coupled to General Relativity elements f ∈ L are typically δ−distributions supported at a single point. Then The space W N of elementary functions on N copies of U (1) can be chosen to be generated by the exponentials with r 1 , .., r N ∈ R labelling the (necessarily one dimensional) unitary irreducible representations of U (1). Analogously the space of elementary functions for the time dependent fields can be chosen as (w k ∈ W N k ) In this case we could still equip L with the structure of a real vector space if we extend L to the finite real linear combinationsL of its generating set L. Since this is no longer possible for the non-Abelian gauge theory example below, we will refrain from doing this, in order to highlight the structural similarity between the examples.
3. Non-Abelian gauge fields for compact gauge groups G: where c is a one dimensional path in iσ. We takeS = G equipped with the natural Borel σ − algebra and where we have identified φ as a G connection and P denotes path ordering. Thus (4.17) is the direct analog of the scalar field construction (note that the Lie generators are anti-self adjoint since G is compact so that (4.17) is unitary) and φ(f c ) is simply the holonomy of φ along c. Likewise Note that the form factors do not form a vector space, in general they cannot be added (unless two curves share a boundary point) and they can never be multiplied by a non-integer real scalar (there is a certain groupoid structure behind this [3]). Accordingly, our space of generating set of elementary functions W N on N copies of G need to be more sophisticated. We consider the space L of form factors and for each F = (f c 1 , .., f c N ) ∈ L N the pairing φ(F ) = (φ(c 1 ), .., φ(c N )) ∈ G N . Then a possible choice of generating set W N of elementary functions is with δ := {(j 1 , m 1 , n 1 ), .., (j N , m N , n N )}. In fact it is sufficient to consider mutually disjoint (up to end points), piecewise real analytic curves c k . Here j labels an irreducible representation π j of G of dimension d j and [π j (g)] m,n ; m, n = 1, .., d j its matrix element functions. By the Peter& Weyl theorem these functions suffice to determine the cylindrical measures uniquely at least if they are absolutely continuous with respect to the product Haar measure. Likewise we consider the elementary functions The fact that these functions satisfy all requirements is the statement of Clebsch-Goradan decomposition theory together with the properties of the holonomy to factorise along segments of a curve (note the piecewise analyticity condition).
This ends our list of examples. We will denote the measure related to the stochastic process {Φ(t, f )} by µ and the measure related to the stochastic process {φ(f )} by ν. As the notation suggests, Φ is a field defined on spacetime M = R × σ while φ is a field defined on space σ. Note that M = R × σ with σ any 3D manifold is a consequence of the requirement of global hyperbolicity [71].
The measures µ underlying a relativistic QFT are not only probability measures. In addition, they need to satisfy a set of axioms [27,30] called Osterwalder-Schrader axioms which, however, are tailored to M = R 4 , stochastic processes with L being a vector space and with an Euclidean background metric at one's disposal.
In quantum gravity and more generally in non-Abelian gauge theories one typically must or may want to drop some of these structures. As a consequence we will only keep those axioms that can also be applied in this more general context. Some of them generalise to stochastic processes not indexed by a vector space, some do not. Some generalise from the manifold R 4 to the general spacetime manifold R × σ allowed by global hyperbolicity, some do not. Fortunately, those that do generalise are sufficient for the reconstruction process [29]. We call them the minimal OS axioms and we call a probability measure that satisfy them an OS measure. They are can be phrased as follows: Let θ(t, x) := (−t, x) and T s (t, x) := (t + s, x) denote time reflection and time translation respectively. Let w k ∈ W N k , k = 1, .., T, F k ∈ L N k , t k ∈ R and Then we have the following conditions on the generating functional µ(W (t 1 , F 1 ), .., (t T , F T )) (4.22)
It is often argued that this index set provides an insufficient "smearing" in the time direction and fails to cover interacting QFT at least in 3+1 spacetime dimensions (in 1+1 and 2+1 dimensions, there are examples for which this works [72]). However, this argument rests on perturbative results as on 3+1 dimensioonal Minkowski space so far no interacting QFT (obeying the Wightman axioms) has been rigorously constructed. It is still conceivable [73] that in a non-perturbative construction of the theory, for which constructive QFT is designed, one can deal with fields at sharp time. One could of course be more general and consider stochastic processes indexed by some L which now also includes smearing in the time direction and the formulation of reflection positivity will then constrain to elements of L with positive time support, however, then the Wiener measure construction sketched below will not work. Our viewpoint is that this more general situation can be obtained from the sharp time construction because integrals of smearing functions with respect to time can be approximated by Riemann sums which in turn are nothing but integrals with respect to sharp time smearing functions.
At the moment it is rather unclear how and why µ, Φ define a relativistic QFT. This will become clear in the next subsection.

Osterwalder-Schrader (OS) Reconstruction
The following abstract argument is standard [27]. See [16] for a proof adapted to the notation in this article.
Due to reflection positivity (4.26) defines a postive semi-definite sesqui-linear form on V . We compute its null space N and complete the quotient of equivalence classes V /N in the inner product (4.24) to a Hilbert space. Given Ψ ∈ V we denote its equivalence class Ψ + N by The constraint s ≥ 0 is due to the time support condition in the definition of V . One must show that this is well defined (independent of the representative) [27]. By virtue of their definition (4.21) the U (s) form a one parameter Abelian group of operators U (s)U (s ′ ) = U (s + s ′ ) on L 2 (S, dµ). This implies that the K(s) form a one parameter Abelian semi-group due to the constraint s ≥ 0 (again one must show that the definition is well-defined). Time translation continuity (3.23) translates into weak continuity of the semi-group. Furthermore, by time translation invariance 4.24 the U (s) define unitary, in particular bounded operators on L 2 (S, dµ) which translates into the statement that the K(s) form a contraction semi-group. Thus [27] there exists a positive self-adjoint operator H, called "Hamiltonian" on H such that K(s) = e −sH . Obviously K(s)Ω = Ω, thus Ω is a ground state for H which justifies the name "vacuum".
This elegant argument is deceivingly simple. To actually compute the Osterwalder-Schrader triple (H, Ω, H) from µ and to relate it to the fields and Hamiltonian in terms of which one would construct the quantum theory using canonical quantisation is not clear yet. However, one can again use the following abstract argument [16]. Suppose that there is an Abelian C * −algebra B of bounded operators on H such that BΩ is dense (the C * − norm is inherited from the uniform operator topology). It is not difficult to show that this is always the case when H is separable which is the only case that we will consider in our application to renormalisation, but it it also holds in many non-separable situations, see appendix B for a proof. Let ∆(B) be its Gel'fand spectrum [74] (which is a compact space) i.e. the space of all * homomorphisms φ : B → C. Then by Gel'fands theorem, B can be thought of as the space C(∆(B)) i.e. the continuous functions on the spectrum which is an Abelian C * −algebra with respect to the sup-norm. The correspondence (Gel'fand isomorphism) is given byb(φ) = φ(b) for all φ ∈ ∆(B) and in fact this is an isometric isomorphism of C * −algebras. Consider now the linear functional which by construction is positive ν(|b| 2 ) = ||bΩ|| 2 . By the Riesz-Markov theorem [70] there exists a (regular Borel) probability measure on S ′ := ∆(B) which by abuse of notation we also denote by ν such that That is to say, the Hilbert space H obtained from OS reconstruction can be thought of as L 2 (∆(B), dν) under the isomorphism bΩ →b, in particular Ω corresponds to the constant function equal to 1. We thus have managed to cast H into the language of measure theory on the set S ′ = ∆(B). The fields φ that come out of this construction are random variables indexed by some index set L ′ , that is, we have shown that we can always construct such a measure and a corresponding stochastic process. We think of the field φ as the spatial configuration fields underlying a canonical quantisation approach. A priori, however, it is not clear what L ′ is altough it must be related in some way to R + ×L. In the case of free fields one can show that in fact one can choose B in such a way that L ′ = L due to the quotient construction involved in H but even then it is a priori not clear how Φ(t, f ) and φ(f ), f ∈ L are related. Again in the case of free fields one shows that φ(f ) can be thought of as Φ(0, f ), the spacetime field at sharp time zero. However, in general the relation between the stochastic processes underlying Φ and φ may be more complex. In any case, the operator H translates in this language into the operator Hb := HbΩ (4.30)

Feynman-Kac-Trotter-Wiener (FKTW) construction
Given an OS triple (H, Ω, H) we saw at the end of the previous subsection that without loss of generality we can assume that H = L 2 (S ′ , dν) where ν is a probability measure on S equipped with a Borel σ−algebra and that we are given a stochastic propcess φ(f ), f ∈ L indexed by some index set L, at least when H is separable (which will be the case in our applications). Moreover, Ω = 1 in this presentation of H is cyclic for some C * −algebra of functions on S ′ . We pick some set W N , N ∈ N 0 of elementary functions w ∈ W N subject to the conditions 1.-4. spelled out just after (4.7) and for Let now T ∈ N 0 , t 1 < t 2 < .. < t T and F k ∈ L N k , w k ∈ W N k . We consider the expectation value functional Consider now a stochastic process Φ(s, f ) indexed by (s, f ) ∈ R × L and the elementary functions Then the Wiener measure µ, if it exists, evaluated on (4.33) is supposed to equal (4.31). The non-trivial question is why this should be the case, under which circumstances and how to construct µ. For this we consider the integral kernel K β of the operator e −βH , β > 0, that is, Note the semigroup property Define S := t∈R S ′ . For each T ∈ N 0 consider t 1 < .. < t T and measurable sets s ′ t k ⊂ S ′ and define the set function It is not clear that this is a positive set function, but when it is, it is called the Wiener measure generated by the OS triple. For sufficient criteria for this property called Nelson-Symanzik positivity in the case of scalar fields, see [75]. Basically one needs to show that matrix elements of e −βH between positive functions is positive. Note that for s ′ k = S ′ for all k we get µ(S) =< Ω, e −(s T −s 1 )H Ω >= 1 (4.38) This shows that µ is a probability measure on S. For quantum mechanical Schrödinger Hamiltonians one can use the Trotter product formula and the Wienner measure of the heat kernel to prove positivity [76] (Feynman-Kac formula).
One can now show the following [16]: Theorem. i. Suppose that OS data (H, H, Ω) are given and that the corresponding Wiener measure µ exists. Then µ is an OS measure and its OS reconstruction reproduce the given OS data up to unitary equivalence. ii. Suppose that an OS measure µ is given thus producing OS data (H, Ω, H). Then the corresponding Wiener measure exists and reproduces µ up to equivalence of measure spaces.
Here measure spaces (S j , B j , µ j ); j = 1, 2 are called equivalent if there exists a bijection F : S 1 → S 2 such that both F, F −1 are measurable and such that µ 1 = µ 2 • F . The reason for why we generically only reproduce an equivalent and not an identical starting point lies in the large freedom in the choice of the stochastic process φ when performing the OS reconstruction step.

Motivation
Our motivation for renormalisation comes from the current state of affairs with respect to the definition of the quantum dynamics in LQG as outlined in section 3. In that case the Hilbert space H = L 2 (S ′ , dν) is precisely of the form we envisage here. Moreover we have a vacuum Ω for a candidate Hamiltonian H that, however, we are not sure whether all steps of the quantisation process that led to H are justified. Namely we have defined H as H γ on certain mutually orthogonal subspaces H γ preserving it using a choice of discretisation of the classical continuum expression which has naively the correct dequantisation if the graph γ fills the spatial manifold σ sufficiently densely. The definition of elementary functions in (4.19) precisely reproduces the SNWF and thus the spatial connection defines a stochastic process indexed by graphs.
As already mentioned at the emnd of section 3 we would like to take a fresh look at the problem. As usual in constructive QFT, if σ is not already compact, we replace it with a compact manifold σ R where R is an infrared cut-off (IR) which we remove in the end R → ∞ (thermodynamic limit). In order not to clutter the notation, the dependence on R of all considerations that follow will be suppressed. Next we do not consider all finite graphs γ (taking all finite graphs leads to a non-separable Hilbert space) but only a controllable countable family M therein which however is such that the discretised classical variables (configuration and momentum fields) in terms of which we perform the quantisation separate the points of the classical phase space when all the graphs in M are at our disposal. The set M is supposed to partially ordered and directed. The motivation for doing so stems from the spatial diffeomorphism invariance of the classical LQG Hamiltonian: The algebraic form of the Hamiltonian discretised on diffeomorphic graphs is identical. This is precisely the starting point of the algebraic quantum gravity proposal [67] where it was emphasised that one can quantise gravity in terms of abstract graphs which gain their physical meaning only after choosing an embedding supplied, for anstance, by a semiclassical state.
To have some intuitive picture in mind consider σ = R 3 with toroidal compactification σ R = T 3 (where each direction has length R with respect to the Euclidian background metric on R 3 and with periodic boundary conditions installed) and Γ the set of all finite graphs σ R of cubic topology. This is still an uncountable set which we now restrict to a countable one as follows. Each element of Γ is uniquely labelled by M ∈ N where M 3 is the number of vertices of the graph (one could generalise this and have different numbers of vertices in each direction). We pick once and for all a coordinate system and locate the vertices of γ M at the points where the edges of the graph are straight lines into the coordinate directions between the vertices. We equip M := N with the following partial order: with m ′ ∈ Z 3 M ′ and because the edges of the graphs are straight lines into the coordinate directions. This is certainly not a linear order because not all natural numbers are in relation but still equips Γ with a direction: Given M, M ′ take for instance M ′′ = M M ′ then M, M ′ < M ′′ (more efficiently take M ′′ as the least common multiple). It is clear that for M sufficiently large discretised phase space variables obtained by integrating continuum variables over 0− or 1− dimensional subsets of γ M (vertices or edges) or by integrating momentum variables over 3− or 2− dimensional subsets of the cell complex corresponding to γ M (faces and cubes) will separate the points of the continuum phase space. Instead of γ M one could also use the cubic cell complex γ * M dual to γ M defined by saying that the barycentres of the cubes of γ * M coincide with the vertices of γ M . However, in the spirit of economy we will not use the additional structure γ * M in what follows.

Discretisation of phase space
In canonical quantisation we start with a continuum phase space coordinatised by configuration fields φ J and canonically conjugate momentum fields π J in terms of which the classical continuum Hamiltonian H is formulated.
Here the index J corresponds to an internal symmetry and is typically Lie algebra valued. We now would like to consider a discretisation of both the phase space and the Hamiltonian, one for each lattice M , while keeping track of how these fields φ J M , π M J are related to the continuum fields φ J , π J . The idea for how to do this stems from the observation that by construction of generally covariant field theories the fields φ J , π J are dual in the sense that there is a natural bilinear form < π, φ > ′ I ′ ×K ′ := J < π J , φ J > I×K on the phase space (usually a cotangent bundle T * K ′ ) I ′ × K ′ of momentum and configuration fields respectively where < ., . > is spatially diffeomorphism invariant. Note that < ., . > ′ , < ., . > just differ by tracing over the internal directions in field space i.e. I ′ = I d , K ′ = K d where d is the number of internal directions in field space.
For instance, the momentum of a scalar field is geometrically a scalar density of weight one, so that < π, φ > ′ =< π, φ >:= σ d 3 x π(x) φ(x). The momentum of a G connection is geometrically a Lie algebra valued vector field density so that < π, φ > ′ = J σ d 3 x π a J (x)φ J a (x) This also holds for higher p−forms as they occur in some supergravity theories as well as for (standard model or Rarita-Schwinger) fermions. Note that the bilinear form is in general not invariant under the internal symmetry group but this will not be important for what follows. The fact that π, φ are conjugate is the statement that their canonical brackets are The fact that the bilinear form < ., . > is naturally at our disposal motivates a natural choice for the index set L, L * of the stochastic process φ, π. Namely we choose L to be a certain distributional extension of I and likewise L * as a certain distributional extension of K. These extensions should be such that < i, k > remains well defined for i ∈ L, k ∈ L * . For instance, for a scalar field we may choose L as the set of δ distributions f p (x) = δ p (x) with support at single points p ∈ σ and L * as the set of characterisic functions g R (x) = χ R (x) of connected D−dimensional submanifolds R of σ. For a compact G−connection we can choose L as the set of form factors f a c (x) := c dy a δ(x, y) with support on (piecewise analytic) curves c. For L * we would consider the set of dual form factors of the form g S a (x) : with support on (piecewise analytic) D − 1 submanifolds S. We may also have opportunity to consider their Lie algebra valued versions f aJ x) ∈ (L ′ ) * were τ J , τ J are dual bases in the defining representation of the Lie algebra of G such that Tr(τ J τ K ) = δ J K . Note that we deliberatively do not make use of the fact that these distributions are elements of vector spaces. This is because we would like to have a uniform description of both linear and non-linear theories. In the case of linear theories the description can be significantly simplified as we have done in [16].
The connection to section 4.2 is then as follows: For each f ∈ L, φ ∈ K we consider a map (f, φ) → φ(f ) ∈S. For linear theories one usually takesS = U (1) and for a G gauge theory one takesS = G. The object φ(f ) exploits the existence of the natural bilinear form < ., . >. For instance for a scalar field one considers φ(f p ) = exp(i < f p , φ >) while for a G connection we consider the holonomy φ(f c ) = P exp(< f c , φ J > τ J ).
ii. For any M < M ′ define the injection maps where the first condition (5.6) was used. Thus, (5.6) makes sure that the discretisations (5.10) enjoy canonical brackets, so we call (5.6) the symplectomorphism property. The motivation for the second condition (5.9) will become clear only later, however we note that it implies that for all M < M ′ < M ′′ the Hamiltoinian depends only quadratically on the fields then one may try (including discretisations of spatial derivatives and some spatial averages) For interacting Hamiltonians, more sophisticated approximations must be used. Certainly the expression for H M is in general plagued by a large amount of discretisation ambiguity beyond the choice of discretised variables. On the other hand, the fact that π M = K ′ M π, φ M = I ′ M φ are conjugate will be convenient when constructing H M and it is efficient to construct them motivated by the naturally available bilinear form < ., . > ′ on the phase space.
To see that there are non-trivial examples for such maps, consider a scalar field in D spatial dimensions compactified on a torus with Euclidian coordinate length R in all directions. Then (recall ǫ M = R/M ) where where the latter denotes the characteristic functions of left closed -right open intervals. This clopen interval structure is very important in order that (5.6) and (5.9) are satisfied [16]. Similar constructions work for gauge fields, see appendix A. Note that we changed here the notation as compared to [16]: The maps I M , E M used there are called here K M , I ′ M respectively. The motivation for this change of notation is to make it manifest how much of the structure is in fact already canonically provided by the structure of the classical theory. Here j m,a labels an irreducible representation π jm,a of G (one from each equivalence class), d jm,a is its dimension, and [π jm,a (.)] nm,a,ñm,a denote its matrix elements with n m,a ,ñ m,a ∈ {1, .., d jm,a }. To see how (5.16) interacts with the map I M M ′ in the case of non-Abelian gauge theorie we note that the cylindrical consistency property of where the notation is as follows (see where the sum over α involves a finite, unique set of generating functions w M ′ ∈ W M ′ and z α are certain, definite complex numbers. Similar statements then of course hold for the stochastic process labelled by R × L and for the functions

Hamiltonian Renormalisation
Abstracting from the concrete lattice implementation and field content above we are in the following situation: There

.20)
Using I M = I M ′ I M M ′ for M < M ′ we would find the identity called the condition of cylindrical consistency. As reviewed in section 3, condition (5.21) grants the existence of µ under rather generic conditions. The strategy (see also [50]) is therefore to construct an iterative sequence of measure families N 0 ∋ n → {µ (n) M } M ∈M called renormalisation (group) flow with initial family as above such that the fixed point family satisfies (5.21). We refer to section C for the reader interested in more notions of the renormalisation group in the language of measure theory.
The scheme that we will employ in fact does not make use of (5.21) for all M < M ′ but only M ′ = p n M where p is a prime. The simplest choice is p = 2 but we have tested the formalism also for p = 3, 5 [16] and mixtures thereof in the case of free scalar QFT. This in fact does cover all possible M because any natural number can be written as kp l ; k, p relative prime but the fixed point family could depend on k. Of course one assumes that the fixed point family is independent of the choices of p, k as an expression of universality as confirmed again for simple systems [16]. Thus we define as renormalisation flow for M ′ := 2M . Having then obtained µ * from cylindrically consistent projections µ * M we want to construct the OS triple (H * , Ω * , H * ) using OS reconstruction. However, while we are sure that µ M is an OS measure, i.e. that the flow preserves the OS measure class. This in fact shown in [16].

Theorem.
The renormalisation flow (5.22) preserves the OS measure class and its fixed points define OS measures µ * .
Responsible for this result is the fact that the time operations that define an OS measure commute with the spatial coarse graining operation.
Thus in principle we can perform renormalisation in the measure (or path integral) language and then carry out OS reconstruction in order to find the continuum Hamiltonian theory that we are interested in. On the other hand, the fact that FKTW construction and OS reconstruction are inverses of each other (theorem 4.4) allows for the possibility to map the renormalistion flow of measures directly into a renormalisation flow of OS triples. In detail: Step 1: Identifying the stochastic processes We need to work out the null space of the reflection positive sesquilinear form determined by the measure µ  . However, the abstract description in terms of equivalence classes is not very useful in practice, rather we would like to describe them concretely in terms of stochastic processes and measures ν (n) M as outlined in section 4.3. As the Hilbert spaces we deal with are separable, this is always possible, see appendix B, however, that construction does not directly refer to the spacetime stochastic process Φ we started from. The reason for why this happens is because of the appearing equivalence classes: To perform concrete calculations one will work with representatives which makes the construction non-canonical because the choice of such representatives is largely a matter of taste. In our setting, if µ can be formulated in terms of the fields φ M (t, l M ) = Φ M ′ (t, I M M ′ l M ) for a minimal number of distinguished times t ∈ τ where the set τ is determined by the quotienting process, see [16] for an example. Alternatively one can view the fields φ M (t, l M ), t ∈ τ as fields at time zeroφ M (l M ) but in a larger space of fields, i.e. a stochastic processφ M with a larger index setB M = τ × B M that still lives on the lattice labelled by M [16]. It follows that without further input, which will be provided below, φ M (.) = Φ M ′ (0, I M M ′ .) is in general a compound field, i.e. a collective degree of freedom composed out of theφ M which together with its momentum π M is insufficient to define the Hamiltonian H (n+1) M which will generically depend on the larger set of variablesφ M and its conjugate momentuñ π M .
Step 2: Working out the flow of OS triples Using the correspondence between the Wiener measures µ (n) and the corresponding operator expressions we M M ′ an arbitrary number of times and at arbitary places on the right hand side of (5.24). In particular this means that we must replace on the right hand side of (5.24) the operator e −βH To see this, we write in (5.24) for each k = 2, .., T and for any N ∈ N on the lhs e −(t k −t k−1 )H (n+1) by the approximants (5.38) or more precisely the P (n, k, β) of appendix E for P = 1 H (n+1) M . Using multi-linearity of (5.24) we can rewrite the resulting expression in terms of (5.24) again, just that now we have not T insertions of w operators but T ′ = 2 N T insertions at times for k = 1, .., T − 1; l = 1, .., N . By the correspondence (5.24) this translates into the corresponding expressions on the rhs with approximants (5.36) or more precisely the P (n, k, β) of appendix E for P = P (n) M M ′ . Then one takes strong limits in the appropriate order, see appendix E, in particular β → ∞, keeping t k − t k−1 fixed. As this is to hold for all N , we take N → ∞.
Formula (5.39) is known in the mathematics literature [80] as as a degenerate case of a Kato-Trotter product [79] of which there are many versions. One of them states that for contraction semigroups generated by selfadjoint operators A, B such that A + B is essentially self-adjoint on the dense domain D M M ′ . In [80] sufficient criteria for the existence of a degenerate semigroup K(β), K(0) an invariant projection, rather than the identity are studied, such that in, say the strong operator topology K(β) = lim N →∞ (e −β/N A P ) N . Assuming that existence K(β) of the limit (5.39) is secured, we deduce We stress that H is not the inductive limit of the H M since that would require H M ′ J M M ′ = J M M ′ H M . This inductive limit condition is much stronger than the quadratic form condition H M = J † M M ′ H M ′ J M M ′ which can be seen by multiplying the inductive limit condition from the left with J † M M ′ and using isometry. It is not possible to derive the inductive limit condition from the quadratic form condition because J † M M ′ has no left inverse. We emphasise that this Hamiltonian renormalisation scheme can be seen as an independent, real space, kinematical renormalization flow different from the OS measure (or path integral) scheme even if the assumptions that were made during its derivation from the measure theoretic one are violated. Note that both schemes are exact, i.e. make no truncation error. This is possible because we do not need to compute the spectra of the Hamiltonians (which is practically impossible to do analytically without error even at finite resolution) but only matrix elements which is computationally much easier and can often performed anaytically, even if the Hilbert spaces involved are infinite dimensional as is the case in bosonic QFT even at finite resolution.
As a final remark, recall that the reduction of (5.24) to (5.29) and (5.29) rests crucially on the assumption that the vacuum vectors Ω M ′ with respect to the decomposition (5.34). When deleting those terms by hand then (5.24) indeed becomes equivalent to (5.29) and (5.32). This is reminescent of the Raleigh-Ritz procedure of diagonalising a self-adjoint operator [79]: There the statement is that for any self-adjoint operator H bounded from below (which is precisely our situation) and any finite dimensional projection P the dim(P ) eigenvalues of P HP ordered by size are upper bounds of the the dim(P ) eigenvalues, ordered by size, in the discrete part of the spectrum (i.e. isolated eigenvalues of finite multiplicity) of H. Here we deal with an infinite projection instead of a finite one, but the general setting is the same. The idea is that as we increase M we approach the continuum Hamiltonian for which eventually there are no off-diagonal elements.

Connection with density matrix and entanglement renormalisation
In this brief section we display natural points of contact between density matrix renormalisation [32,41], entanglement renormalisation [44,45] and projective renormalisation [37,38,39,40]. In particular the notions of ascending and descending superoperators, isometries and disentanglers from the multi scale entanglement renormalisation Ansatz (MERA) can be applied.
Recall that our Hamiltonian renormalisation scheme produced the following structures: At each resolution M we are given an OS triple (H M , Ω M , H M ) and in addition we have the C * −algebra A M of bounded operators acting on H M . The w(φ M (l M )) form a commutative subalgebra B M of A M for which Ω M is cyclic and separating. The full algebra A M contains in addition operators that depend on the momenta π M conjugate to φ M . Furthermore, the renormalisation scheme has naturally produced an inductive system of isometric injections In section 5 we established that the Hamiltonian renormalisation derived from OS reconstruction, under the assumption of vacuum uniqueness at each resolution, has given rise to a flow of contraction semigroups and density matrix renormalisation amounts to taking partial traces with respect to the factor H M ′ |M [32]. The map (6.14) can be described in terms of isometric embeddings which explains why just one isometry is not sufficient to make α M M ′ unital. Such a tensor product situation is available if we consider as label set the mutually commuting modes of a Fock representation. The latter can be obtained by semaring creation and annihilation operators with respect an orthonormal basis of the 1-particle Hilbert space and if the corresponding smearing is supposed to be smooth and spatially local as one would like for a real space coarse graining scheme then one can use a (e.g. Meyer) wavelet basis [83]. However in view of Haag's theorem [73] it is unclear whether Fock space techniques can be applied in the interacting case, which is why the scheme adopted in this paper keeps track of the interacting rather than free vacuum during the flow. Even for free theories note that in case one is dealing with an infinite number of degrees of freedom (continuum limit) a Fock representation must be carefully adapted to the Hamiltonian as otherwise it is not even densely defined. Thus our Hamiltonian renormalisation scheme also suggests as a variant an independent density matrix renormalisation scheme corresponding to the Gibbs operator at fixed temperature that simultaneously keeps track of the vacuum. Various combinations and generalisations of these ideas are conceivable. For instance, one could use as density matrices the Gibbs operator corresponding to a Hamiltonian ρ (n) M = e −βH (n) M and any fixed inductive limit Hilbert space with inductive structure J M M ′ or a fixed consistent system of unital homomorphisms and then use the flow defined by the scheme considered in this section to derive a flow of Hamiltonians as an independent technique, see also [32,39,40]. This will likely give rise to a (finite) temperature state rather than a vacuum state as a fixed point of the renormalisation flow. Or one could keep the density matrices independent of the Hamiltonians and study the time evolution of the reduced density matrices ρ M ′ (.) e −itH (n) M ′ whose infinitesimal form gives rise to a Lindblad equation [78] in the unital case (with tivial non-linear corrections in the non-unital case that can be computed from the linear flow that one obtains by dropping the normalisation by the partition function). Its non-dissipative component can be used to define a flow of Hamiltonians. Yet more generally we could in fact consider a system of quadruples (ω M , A M , H M , α M M ′ ) where the Gibbs states have been replaced by general algebraic states ω M and we could study their analogous flow.

Connections with the functional renormalisation group
The functional renormalisation group (FRG) (see the first two references in [18]) is the technique underlying the asymptotic safety aproach to quantum gravity. We will sketch it here in measure theoretic terms to display the rather obvious point of contact with the Hamiltonian renormalisation programme which arises from OS reconstruction. We will restrict to scalar field theories for simplicity. The discussion will rely on several assumptions which we spell out in some detail and will be only qualitative in nature.
Suppose that µ is a probability measure with stochastic process Φ labelled by smearing functions F and that R + ∋ k → µ k is a one parameter family of such probability measures defined in terms of its generating functional as Here ρ k is a µ measurable, positive L 1 (dµ) function with positive µ measure. This means that dµ k = ρ k µ(ρ k ) dµ, i.e. the measures µ k are mutually absolutely continuous and ρ k is essentially the Radon-Nikodym derivative of µ k with respect to µ [70].
The family of measures enjoy the obvious cylindrical consistency identity Suppose now that in fact the continuum measure µ is not known but that we guess a family of measures µ k and use (7.2) to define a renormalisation group flow where the ρ k are fixed and supposed to be measurable functions for all the µ k in the process of the flow. This is, in most general terms, the idea of the FRG, that is, the rhs of (7.2) is the renormalisation of µ k ′ and serves as the new definition of µ k . If we take the derivative of the rhs of (7.2) with respect to k at k ′ = k we obtain what is known in the literature as the Polchinsky equation. If we take the Legendre transformation with respect to F of obe obtains the effective average action Γ k [ϕ]. One finds a closed equation for Γ k for the case that ln(ρ k )(ϕ) = − 1 2 < ϕ, R k ϕ > is quadratic in ϕ known as the Wetterich equation where R k = R k (x, y) and the second functional derivative Γ (2) k (x, y) are symmetric integral kernels. As (7.4) is a consequence of having a consistent family of measures derived from µ, it can be used as a flow equation as well which is strictly equivalent to the Polchinsky equation.
To translate (7.2) into Hamiltonian renormalisation we must be more specific on the choice of ρ k which we take to be of the form cited above in order that it yields the Wetterich equation together with the assumptions usually made: We consider a translation invariant kernel R k whose Fourier transformR k (p) enjoys the following properties: 1.R k (p) ≥ 0, 2.R k (p) is monotonously growing in k at fixed p and 3. lim p→∞Rk (p) = 0. One usually adds 4.R k=0 (p) = 0,R k (p = 0) > 0, lim k→ΛRk (p) = ∞ (where Λ is a cut-off which for a a continuum theory can be taken to ∞). We will only need 1.-3. in what follows. They imply that R k is an infrared cut-off i.e. momentum modes p lower than k (with respect to the Euclidian metric) are suppressed, high momentum modes are unaffected. Thus as we increase k we suppress more and more modes, hence the number of degrees of freedom becomes smaller. Accordingly, the coarse graining here happens by increasing k rather than lowering, i.e. the flow is towards the UV rather than towards the IR in contrast to real space block spin transformations from fine to coarse lattices.
Accordingly, we update a measure at k from a measure at k ′ < k. Let us further assume that the measure µ k ′ (.) and µ k ′ (ρ k /ρ k ′ .)/µ k ′ (ρ k /ρ k ′ ) are in fact Wiener mesures with respect to the same stochastic process φ and Hilbert space H k ′ but with underlying Hamiltonians and vacua given by H k ′ , Ω k ′ and H ′ k := H k ′ + h k − h k ′ , Ω kk ′ respectively. This puts some restriction on R k as far as the dependence on the time component of the momentum is concerned if we wish h k to be an at most quadartic expression in the momenta π conjugate to φ. Thus we consider e.g.R k (p) = [(p 0 ) 2 κ k (|| p||) + σ k (|| p||)] where κ k , σ k enjoy the properties 1.-3. above but only with respect to the spatial momenta and λ > 0. Similar restrictions have been made in [19] which is concerned with making contact to Lorentzian signature GR. This way temporal momentum is not suppressed which is in accordance with our Hamiltonian renormalisation scheme which does not renormalise in the time direction.
Then the rhs of (7.2) can be written for To get rid of the dependence on k in Ω kk ′ we use as usually that assuming that the strong limit exists and thus the rhs of (7.2) can be written Using the Trotter product formula we can replace every exponential of the form e −sH ′ k by [e −sH k ′ /N e −s[h k −h k ′ ]/N ] N as N turns to ∞. By construction h k − h k ′ is non-vanishing for momentum modes k ′ ≤ || p|| ≤ k and thus its Gibbs exponential suppresses those. As emphasised in the first two references of [18] one obtains an actual coarse graining renormalisation group only when R k (p) takes the more special form Z( p)[(p 0 ) 2 + p 2 ]I k ( p) 2 /[1 − I k ( p) 2 ] where i. I k ( p)I k ′ ( p) = Ik (k,k ′ ) ( p) for somek(k, k ′ ) which ensures the group property and moreover ii. I k ( p) approaches a step function θ(k 2 − p 2 ). In this limit e −s(h k −h k ′ )/N → P kk ′ becomes a projection operator which projects on a subspace H k = P kk ′ H k ′ independent of the excess modes k ′ < || p|| < k. To see this note that the form of R k − R k ′ allows us to write h k − h k ′ in terms of suitable annihilation and creation operators on a suitable Fock space which we assume to be unitarily related to H k ′ (this will be the case in presence of both IR and UV cut-offs). On that Fock space in the step function limit the above is a projector onto the subspace with no excitations from the excess modes and a unitary transformation does not change the projection property. Accordingly the Trotter product becomes a degenerate one and e −sH ′ k becomes P kk ′ e −sP kk ′ H k ′ P kk ′ see appendix D. Taking β → ∞ in (7.7) the rhs of (7.2) can be written If we now compare to the end of section 5 we find qualitatively an exact match, just that the projections now involve momentum modes with respect to a Fock representation determined by R k .

Conclusion
In this contribution we have reviewed, extended and clarified the proposal [16]. The extension consisted in i. an improved derivation of the renormalisation scheme (5.29), (5.32) from OS reconstruction using an extended minimal set of OS axioms that also includes the uniqueness of the vacuum (which is in fact always assumed in QFT on Minkowski space) as well as ii. a much more systematic approach to the choice of coarse graining maps for a general QFT which are motivated by structures naturally provided already by the classical theory. The clarification consisted in separating off the null space quotienting process imposed by OS reconstruction as an independent part of the renormalisation flow whose formulation naturally uses the language of stochastic processes.
We also had the opportunity to make several points of contact with other renormalisation programmes that are currently being further developed. For instance, the reduced density matrix approach on which entanglement renormalisation schmemes rest occurs naturally in our scheme as well when looking at the flow of the vacuum and Hilbert space. Next, since we consider a real space renormalisation scheme, when translated in terms of the flow of Wiener measures that we obtain from the flow of OS data we are rather close to the asymptotic safety programme because our spatial lattices can of course be translated into momentum lattices by Fourier transformation that are used in the asymptotically safe quantum gravity programme. Finally, our proposal is obviously very close in language and methods to all other Hamiltonian renormalisation schemes and while we currently focus on a kinematical coarse graining scheme our approach also contains dynamical components such as the flow of the vacuuum.
In [16] and [81] we have successfully applied our scheme to free QFT (scalar fields and Abelian gauge theories) exploiting their linear structure. Obviously one should construct further solvable examples of interacting theories, e.g. interacting 2D scalar QFT [72] or free Abelian gauge theories but artificially discretised in terms of non-linear holonomies in order to simulate the situation in Loop Quantum Gravity, see [81] for further remarks.
Of course the ultimate goal is to use Hamiltonian renormalisation to find a continuum theory for canonical Quantum Gravity. Here we can use the LQG candidate as a starting point because it is rather far developed, but of course the flow scheme developed can be applied to any other canonical programme. However, using LQG and the concrete scheme that employs a fixed subset of graphs γ M labelled by M ∈ M of cubical topology is, at each resolution M , precisely the AQG version of LQG [67]. Hence we can already speculate on what can be expected from the renormalisation flow: The Hamiltonian H Haar measure) could be, but not needs to be, ordered in such a way as to annihilate the vacuum Ω M but not each subspace defined by sublattices of γ M and is thus not super local in contrast to the definition [6], for instance it will use volume operators local to a vertex and holonomies along plaquettes incident at that vertex (next neighbour interaction). When running the renormalisation scheme, next to next neighbour interactions will be switched on (this is exactly what happens in the examples [16], [81]) etc. and upon reaching the fixed point the Hamiltonian H M will involve all possible interactions with precise coefficients and thus be spatially non local but hopefully quasi local (i.e. the interactions die off exponentially with the distance between vertices defined by the 3D taxi driver metric on the graph (each edge counting one unit)). Note that this quasi locality at finite resolution can be straightforwardly computed in the examples [16], [81] by using the spatially local continuum Hamiltonian and projecting it with J M , J † M (blocking from the continuum) and is thus physically correct. In other words spatial locality in the continuuum is not in conflict with spatial non locality at finite resolution. In fact we even expect a high degree of spatial non-locality for very small M for which the naive dequantisation of H (0) M at any phase space point p will be far off the classical value H(p) which matches with the remarks made at the end of section 3.
Several questions arise from this picture should the flow display any fixed points: First, for compact σ and if indeed we use a countable set of lattices γ M as above, the resulting inductive limit Hilbert space could be separable (since there is a countable basis defined by vectors at finite resolution), thus would not be the standard LQG representation space H LQG = L 2 (A, dµ AL ) of square integrable functions with respect to the Ashtekar-Lewandowski measure µ AL on a space A of distributional connections 2 . In view of the uniqueness theorem [5] one of its assumptions will then be violated. The most likely possibility is that the corresponding vacuum expectation value functional is not spatially diffeomorphism invariant since the diffeomorphism symmetry was explicitly broken in the renormalisation process. If the continuum Hamiltonian is still spatially diffeomorphism invariant we would be in the situation of spontaneous symmetry breakdown and could view this as a phase transition from the symmetric H LQG phase to this broken phase. Note that in our gauge fixed situation the diffeomorphism group is considered as a continuous symmetry group and not as a gauge group. Next, precisely due to this separability the resulting theory may not suffer from the discontinuity of holonomy operators which otherwise gives rise to what has been called the "staircase problem" in the literature [82]: The cubical graphs γ M contain paths only along the coordinate axes. Since all M ∈ M are allowed, these paths separate the points of the classical configuration space but not of the distributional space A. In particular, any path that is not a "staircase" path cannot be accomodated at any finite resolution. Yet, the continuum Hamiltonian in the example [81] does not care about the fact that it was defined as a fixed point of a flow of its finite resolution projections of cubical lattices only, it also knows how to act on states which are excited on non-"staircase" paths. The reason for why this happens is as follows: Consider any path c and some staircase approximantc with the same end points as c which has zero winding number with respect to c so that c •c −1 = ∂S bounds a surface. Then for an Abelian connection A we have c A − c A = S dA and in the classical theory the surface integral converges to zero. In the quantum theory a similar calculation can be made because the Hilbert space measure is supported on a different kind of distributional connections than A. Finally, although the scheme strictly speaking was derived for theories with gauge fixed spacetime diffeomorphism constraints and a true physical Hamiltonian bounded from below, we may of course "abuse" it and also consider constraint operators C(f ) as Hamiltonians, define their finite resolution expressions C(f ) (0) M and let them flow (here f is a test function on the spatial manifold σ) 3 . This will involve as a new ingredient also a discretisation of the smearing function f which could be done using the maps I M , K M for scalar fields, see [81].
the C(f ) form an inductive family which is not expected. Of course the correction term in (8.1) is expected to become "small" in the limit M → ∞ in which P M → 1 H and thus an appropriate criterion for closure of the continuum algebra using only finite resolution projections can be formulated (see [16] for the simpler case of rotational invariance). Note that the quantisation performed for spatially diffeomorphism invariant Hamiltonian operators on the Hilbert space H LQG displayed in section 3 was forced to have the unphysical property [H, P M ] = 0, see the statement just before (3.24). But the underlying theorem exploits in a crucial way the non-separability of H LQG and thus fortunately does not hold on separable Hilbert spaces.
Before closing, note that even if this approach of taking the UV limit can be completed and unless the manifold σ is compact, we still must take the thermodynamic or infrared limit and remove the IR cut-off R (compactification scale). As is well known from statistical quantum field theory [63] interesting phenomena related to phase transitions can happen here. Moreover, constructible eamples of low dimensional interacting QFT show that the thermodynamic limit requires techniques that go beyond what was displayeed here [72]. However, we consider this momentarily as a "higher order" problem and reserve it for future research.
from which we read off which is non vanishing iff for all b = a we have m ′′b /n ′ ∈ Z M ′ and m ′′b /(n ′ n) ∈ Z M . Since the latter condition implies the first, the first factor in the last line of (A.7) can be dropped. Next let m ′′a = n ′ m ′a + k ′ with m ′a ∈ Z M ′ , k ′ = 0, .., n ′ − 1 and m ′a = nm a + k with m a ∈ Z M , k = 0, .., n − 1. Then m ′′a = n ′ nm a + (n ′ k + k ′ ) ≤ n ′ nm a + n ′ n − 1 whence l M (m, a) = 1 means that the edge with starting point mǫ M that extends one lattice unit into direction a is excited, otherwise not. The reason why this information is sufficient is that the information about the strength of the excitation sits in the group representation label of the corresponding holonomy. For Abelian groups we could take group representations labelled by Z or R and thus absorb the representation label into l M which then becomes Z D or R D valued and which has the structure of a module over the ring Z or a vector space over the field R. That absorption of the representation label into l M is no longer possible for non-Abelian gauge groups. Thus, in order to develop a unified language for all physical theories we use the bit valued space B M in the main text. Note that in the bit valued case, B M is formally a vector space over the field F D 2 = {0, 1} D . However, these module or vector space structures are irrelevant for the purposes of renormalisation.

B Cyclic Abelian C * −algebras
Recall that a (unital) C * algebra A is a * −algebra with a norm topology with respect to which it is complete (i.e. A is a (unital) Banach algebra) such that ||a * a|| = ||a|| 2 for all a ∈ A. One of the best known examples is the algebra B(H) of bounded operators on a Hilbert space H. Given a Hilbert space H and some sub-C * algebra B ⊂ B(H) a vector Ω ∈ H is called cyclic for B if BΩ is dense in H. It is called separating if bΩ = 0 implies b = 0 for all b ∈ B.
In the OS reconstruction algorithm we construct H, Ω in terms of equivalence classes of the vector space V of vectors which are linear combinations of vectors of the form Ψ[Φ] = w T (Φ(t T , l T ))..w(Φ(t 1 , l 1 ) where Φ is a stochastic process labelled by R × L for the measure µ with t T > t T −1 > .. > t 1 and the reflection positive inner product is < ψ, ψ ′ >= µ(ψ * Rψ ′ ) where R denotes the time reflection. Here the w k belong to some space of bounded functions W on the range of the Φ(t, l), l ∈ L which form an Abelian * -algebra with respect to point wise operations. The equivalence class [ψ] = ψ + N is with respect to the null space N of < ., . >, H is the completion of V /N and Ω = [1]. Consider the operators [Q(w, l)], w ∈ W, l ∈ L defined by where Q(w, l) * R = RQ(w, l) * and the Cauchy-Schwarz inequality was used. Consequently it is not clear that Ω is cyclic for the Q(w, l) and in general this will not be the case.
Nevertheless, in the main text, we quoted the following result:

Theorem.
Let H be a separable Hilbert space and Ω ∈ H a unit vector. Then there exists an Abelian, unital sub-C * −algebra B ⊂ B(H) of bounded operators on H such that Ω is cyclic for B.
We could not find a proof of this statement in the literature, hence we supply an elementary one below.
Proof. Since H is separable, it has a countable orthonormal basis b I , I ∈ N 0 and we may assume b 0 = Ω using the Gram-Schmidt algorithm. The idea is to construct inductively a basis e I , I ∈ N 0 such that e 0 = b 0 = Ω, < Ω, e I >> 0 for all I ≥ 0 and < e I , e J >= δ IJ for all I, J > 0. Then the bounded operators given by the self-adjoint projections P 0 =: 1 H , P I := e I < e I , . > generate the searched for unital Abelian C * −algebra since P 0 P I = P I P 0 = P I , P I P J = δ IJ P J , I, J > 0 and P * I = P I , I ≥ 0 and P I Ω =< e I , Ω > e I , I ≥ 0 thus the finite linear combinations of the P I applied to Ω have dense range.
< Ω, e 1 >= I! (I + 2)! = 1 (I + 1)(I + 2) (B.8) which extends also to I = 0. To see that the span of the e I is dense suppose that < ψ, e I >= 0 for all I ≥ 0. This means < ψ, b 0 >= 0 and thus because e I is in the span of the b 0 , .., b I succesively < ψ, b I >= 0 for all b I , I ≥ 0 whence ψ = 0 as the b I provide an orthonormal basis. It is instructive to see that even the e I , I > 0 already lie dense which is demonstrated by showing that Bessel's inequality is saturated for Ω. Indeed In the non-separable case a non-constructive argument can be provided.

Theorem.
Suppose that H is not separable but that its dimension has the cardinality of R.
Let Ω ∈ H, ||Ω|| = 1. Then there exists an Abelian, unital sub-C * −algebra B ⊂ B(H) of bounded operators on H such that Ω is cyclic for B.
Proof. Let b I be an orthonormal basis of H with b 0 = Ω (this needs the axiom of choice) and consider the Abelian one-parameter group of unitary operators densely defined by U s b I = b I+s , s ∈ R. Since U * s = U −s these generate an unital, Abelian C * −algebra B of bounded operators and obviously Ω is cyclic for B as U I Ω = b I .
In the case of LQG we may label the basis b I by spin net work functions. These have discrete labels (adjacency matrix (which edge connects which vertices), irreducuible representations on the edges and intertwiners at the vertices) and continuous labels (germs of piecewise analytic edges). These data have the cardinality of finite unions of Z and R respectively and thus have the cardinality of R again [77] (even the space of real valued sequences R N has the same cardinality as R), hence the above construction can be applied to LQG in principle.
In the separable case we may alternatively consider the index set Z and the unitary one parameter group U s b I = b I+s , s ∈ Z to apply the same construction. As elucidated by the proofs, the algebra B while always existent under the assumptions made has in general not a very direct relation to the operators that have an immediate physical interpretation.

C Brief Account of Renormalisation Group Terminology
The purpose of this brief appendix is to relate notions of the path integral renormalisation group usually formulated in terms of actions [33] now in terms of measures.
Recall from the main text that condition (5.21) grants the existence of µ under rather generic conditions and that the strategy is therefore to construct an iterative sequence of measure families N 0 ∋ n → {µ (n) M } M ∈M called renormalisation (group) flow with initial family as above such that either i. it converges or ii. a subsequence of it converges or iii. it has a fixed point. The limit or the fixed point family then satisfy (5.21). As there is no natural choice for such a sequence, there are many possibilities which are guided mostly by the principle of least technical complexity. In that sense, case ii. is really case i. because we might have simply defined the sequence by passing to the subsequence right from the beginning. On the other hand, a fixed point maybe unstable (i.e. it has non-convergent directions) so that iii. is qualitatively different from i. In practice, one is generically confronted with case iii. and what one finds is the following: After performing a sufficient number n of iterations, one observes that the flow stabilises in the sense that the family member µ strongly converges to a strongly continuous degenerate contraction semi-group, that is, K(0) = Q ≤ P is a subprojection, QH is an invariant subspace of K(s) and K(s)(1 − Q) = 0. A general proof of this result generalising the ususal case that P in (D.1) is replaced by another contraction semigroup S(s) appears not be available yet in the literature.
In what follows we consider therefore the simpler case that T (s) = e −sA has a bounded generator A.

Theorem.
Let T (s), P be as above with T (s) = e −sA , ||A|| < ∞ bounded. Then K(s) = P e −s P AP (D.2) that is Q = P and K(s) is a strongly continuous semigroup with generator P AP on P H.
Proof. Consider the identity (no domain questions arise) Using ||P || ≤ 1 and the sub-multiplicativity of the operator norm we estimate hence the convergence is even uniform in this case.
Note that the sign of s played no role in the proof, but it will in the unbounded case.

E Projections and Limits
In the main text we need the following result: Theorem. Let H be a separable Hilbert space with projection P . Let Ω be cyclic for a C * −subalgebra B ⊂ B(H). Suppose that a self-adjoint operator H has spectrum bounded from below by zero and that zero is an isolated eigenvalue with eigenvector Ω. Then there exist elements w I,k ∈ B; I, k ∈ N such that thus s − lim β→∞ e −βH ψ = Ω < Ω, ψ >. Finally, P H is the closure of the finite linear span of the b I , thus s − lim n→∞ P n ψ = P ψ where P n ψ = n I=1 b I < b I , ψ > It remains to write P (n, k, β) − P as a telescopic sum (P −P (n, k, β))ψ = (P −P n )ψ+ n I=1 [b I < b I , ψ > −w I,k Ω < w I,k Ω, ψ >]+ n I=1 w I,k [Ω < Ω, w † I,k ψ > −e −βH w † I,k ψ] (E.4) and to take the strong limits in the indicated order, i.e. first β → ∞ at finite n, k removes the third term, then k → ∞ at finite n removes the second term and finally n → ∞ removes the first term.