- Institute for Quantum Gravity, FAU Erlangen–Nürnberg, Erlangen, Germany
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 the 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 was to review and further develop these ideas and to put them into context with closely related earlier and parallel programmes.
1. Introduction
The canonical approach to quantum gravity has been initialised long time ago [1–14]. 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 [15,16], and has culminated in a research programme now known as loop quantum gravity (LQG) (see e.g., Refs. 17–21 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 [22].
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 [23–27] that allows for a unitary representation of the spatial diffeomorphism group. Moreover, the generators of temporal diffeomorphisms, sometimes referred to as Wheeler–DeWitt operators, could be rigorously quantised on the corresponding Hilbert space [28–32], and in contrast to the perturbative approach to quantum gravity [33, 34], no ultraviolet divergences were found. It should be emphasised that this was achieved 1) in the continuum, rather than on a lattice, that is, 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 usual factor ordering ambiguity in the quantisation of operator-valued distributions which are non-linear in the fields is much more severe. Thus, the operators defined in Refs. 28–32 suffer from those ambiguities. Moreover, the following problem arises: In the classical theory, the canonical generators of space-time diffeomorphisms (i.e., their Hamiltonian vector fields) form a Lie algebroid (i.e., a Lie algebra except that the structure constants are replaced by structure functions on the phase space) known as the hypersurface algebroid [35]. The structure functions are themselves promoted to operator-valued distributions upon quantization; thus, it becomes even harder to find quantization 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 Ref. 37, it is 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 Refs. 1–14, it is now possible to ask and answer precise questions about the mathematical consistency of the whole framework. As compared to the perturbative approach, 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 threshold (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 Ref. 38, 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 Ref. 39, the case of general relativity coupled to perfect fluid matter was considered, which allows solving 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 (i.e., space-time diffeomorphism-invariant) observables. In Refs. 40–42, 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 hypersurface algebroid directly on that space of distributions which would at least partially fix the aforementioned 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 in applications of LQG, for instance to quantum cosmology [43–46] (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 Refs. 47–53 for recent progress in that respect). In the recent proposal [54–57] which we intend to review in this article, the authors were inspired by Wilson’s observation [54–57] that renormalisation methods help 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 may be perturbatively non-renomalisable but non-perturbatively renormalisable, also known as asymptotically safe [58]. The asymptotic safety approach to quantum gravity for Euclidian [59–68] and Lorentzian signature [69, 70] 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 longer version of this article [196].
Also, there is a large body of work on renormalisation [71–75] in the so-called spin foam approach [85–92] and the related group field theory [76–81] and tensor model1 [82–84] 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 [93] of LQG, that is, 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 Refs. 94–96.)
Most of the work on renormalisation is either within classical statistical physics (e.g., Ref. 97) or the Euclidian (also called constructive) approach to the quantum field theory [98–100]. 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 loosely speaking, the measure can be considered as a Gibbs factor for a Hamiltonian (sometimes called Euclidian action) in four 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 considered as gauge transformations; hence, none of the basic fields correspond to observables.
In our approach [54–57], we will use the framework [39], 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 Refs. 101 and 102, 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) framework 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 [103] of Osterwalder–Schrader axioms [104] in order for the reconstruction to be applicable, most importantly reflection positivity.
This was one of the goals of [54–57], 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
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 and 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, and 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 [54–57] 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 [98–100]) as well as some machinery concerning degenerate contraction semi-groups and associated Kato–Trotter formulae.
In the sixth 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.
The paper is supplemented by the following appendices:
In Supplementary Appendix A, we prove some properties for a coarse graining scheme appropriate for non-Abelian gauge theories; in Supplementary Appendix B, we prove a lemma on the existence of certain Abelian
In Supplementary Appendix F, we mention concrete points of contact between the scheme developed here and others in the context of density matrix, entanglement, and projective renormalisation.
In Supplementary Appendix G, we sketch a relation between Hamiltonian renormalisation via Osterwalder–Schrader reconstruction and the functional renormalisation group which is the underlying technique of the asymptotic safety programme. This article is the journal version of Ref. 196 which is organised slightly differently in the sense that Appendices F, G of this article are part of the main text of Ref. 196.
2. Overview Over Related Hamiltonian Renormalisation Schemes
The 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 and 7 of Ref. 196, we will give a few more details on the connection between our approach and the density matrix and functional renormalisation group.
The starting point is, of course, the seminal works by Kadanoff [105] and Wilson [106, 107]. Kadanoff introduced the concept of a block spin transformation in statistical physics, that is, 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 [108]. 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 annihilation 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
The next step was done by Wegner [109, 110] as well as Glazek and Wilson [111] which can be considered as a generalisation of the Hamiltonian methods of Refs. 106 and 107. It could be called perturbative Hamiltonian block diagonalisation and was applied in QFT already (e.g., Refs. 112 and 113 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 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 [114, 115]. 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 these are trace class). (See also Refs. 116–123 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 [124] 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 [125]. 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, that is, 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 [126].
Finally, as observed by Vidal [127] and Evenbly and Vidal [128, 129], one can improve [124, 125] 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 [130] 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 the following:
(1) Real space vs. other labels
The degrees of freedom to be coarse grained are labelled by points in space-time or else (momentum, energy, etc.).
(2) Kinematic vs. dynamical
Real-space block spin transformations are an example of a kinematic coarse graining, that is, 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 vs. exact
In principle, any renormalisation scheme can be performed exactly, for example, 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, for example, for the procedure followed in asymptotically safe quantum gravity. However, in practice, this may quickly become unmanageable, and thus, one resorts to approximation methods, for example, 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 second 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 vs. dynamic issue has also been emphasised for the LQG and spin foam approach [131–133].
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 [134], 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 Refs. 135 and 136 for Euclidian Yang–Mills theory and in Refs. 137 and 138 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 Refs. 131–133, it was shown that the cylindrically consistent coarse graining of the rigging map and its underlying space-time lattice, thought of as an anti-linear functional on the kinematical Hilbert space, induce 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 Refs. 139–141 due to the similarity of LQG to the 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 is 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 [54–57], 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, that is, one would have to work with three cut-offs, rather than two (see, e.g., Refs. 142 and 143 where quantum group representations are used in gauge theories, rather than classical group representations, and perform real-space renormalisation or [144] 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, 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.
3. Canonical Quantum Gravity Coupled to Reference Matter
The physical idea is quite simple and goes back to Ref. 145: General relativity is a gauge theory, the gauge group being the space-time 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
These kinds of relational observables have been further developed by various authors, in particular [147–153]. 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 [39] 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.
3.1. Gaussian Dust Model
The Lagrangian of the theory takes the form
where
where g is the Lorentzian signature metric tensor,
The full constraint analysis of Eq. 3.1 is carried out in Ref. 39. There are secondary constraints, and the full set of constraints contains those of the first and second classes (see Ref. 153 for a modern treatment of Dirac’s algorithm [154]). One has to introduce a Dirac bracket and solve the second-class constraints in the course of which the variables
Here, C is the Wheeler–DeWitt constraint function (including standard matter) and
The constraints (Eq. 3.3) encode the space-time diffeomorphism gauge symmetry in Hamiltonian form, in particular they represent the hypersurface deformation algebra [111]. It is possible to solve these remaining constraints to determine the complete set of gauge-invariant (the so-called Dirac) observables and to determine the physical Hamiltonian H that drives their physical time evolution [39]. Equivalently, we may gauge fix Eq. 3.3. The above interpretation of
namely,
which when evaluated at
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
which shows that
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
We close this subsection with three remarks: First, a complete discussion requires to show that the gauge cut
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., Ref. 158), which produces a square root Hamiltonian! One may argue that the dust is a form of cold dark matter [146], 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 accomplish a huge step in the quantum gravity programme: It frees us from quantising and solving the constraints and constructing the physical inner product, the gauge-invariant observables, and their physical time evolution. All of these steps are of tremendous technical difficulty [17–21]. All we are left to do is to quantise the physical degrees of freedom and the physical Hamiltonian.
3.2. 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 [17–21, 28–32] for more detail on standard matter coupling). The Hamiltonian directly written in terms of
Here, A is an SU(2) connection and E an SU(2) non-Abelian electric field that one would encounter also in an SU(2) Yang–Mills theory. However, the geometric interpretation of
The important quantity V is recognised as the total volume of the hypersurface σ, and
where
The traces involved in 3.2 are carried out by introducing the Lie algebra-valued 1-forms
To quantise the theory, we start from functions on the phase space that are usually employed in the lattice gauge theory (see, e.g., Ref. 159), namely, non-Abelian magnetic holonomy and electric flux variables
where
where
in case that
Interestingly, the physical Hamiltonian H has a large symmetry group, namely, it is invariant under the group
where
where
The mathematical problem in quantising the theory consists in constructing a
for all
In Refs. 23–27, 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
that is,
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 finitely many piecewise analytic curves are partially ordered by set theoretic inclusion, and they are directed in the sense that for any two graphs
where
In fact, Eq. 3.22 defines a cylindrical family of measures
By construction, the Hilbert space
To check this, one uses the properties of the Haar measure (translation invariance) and the diffeomorphism invariance of Eq. 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 (SNWFs). This makes use of harmonic analysis on compact groups G [161], in particular the Peter and 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, an SNWF
As a historical remark, solutions of the Gauss constraint are excited on closed graphs since there is no non-trivial intertwiner between the trivial representation and a single irreducible one; hence, open ends are forbidden. For closed graphs, one can alternatively label SNWFs 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
The remaining task is to quantise the Hamiltonian, and it is at this point where the aforementioned quantisation ambiguities arise. The strategy followed in Refs. 28–32 is as follows: It turns out that the volume operator appearing in Eq. 3.2 can be quantised on
where each
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
Remarkably, Eq. 3.25 defines an essentially self-adjoint, diffeomorphism-invariant, continuum Hamiltonian operator for Lorentzian quantum gravity in four space-time dimensions, densely defined on the physical continuum Hilbert space
Yet, one cannot be satisfied with Eq. 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 Eq. 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 representations [173] or an average 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 semi-classical results on sufficiently fine lattices, the theory lives on all lattices, also those which are very coarse, and on those, expression Eq. 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 ϵ 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 topology [28–32], namely, there exist diffeomorphism-invariant distributions (linear functionals) l on the dense span of SNWF ψ [36], and we define an operator
4. The naive dequantisation of Eq. 3.25 will perform poorly on very coarse graphs and will be far from the continuum expression Eq. 3.8, but one could argue that that vectors supported on coarse graphs simply do not qualify as good semi-classical 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 Eq. 3.25 is lost.
6. The block diagonal or superselection structure (Eq. 3.24) which is forced on us by the non-separability of the Hilbert space and its spatial diffeomorphism covariance appears unphysical, and one would expect that the Hamiltonian creates also new excitations.
It transpires that we must improve Eq. 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 [54–57, 71–75, 108]. This bears the chance that what we see in Eq. 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 Refs. 169–172, 174 (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
4. 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 Hamiltonian framework from a measure. Our description will be minimal. The prime textbook references are [98–100, 175].
4.1. Measure Theoretic Glossary
Let S be a set. A collection B of the so-called measurable subsets of S is called a
The measure μ is called a probability measure if
where
Consider now a second measurable space
defines also a probability measure called the distribution of X. We consider real-valued functions
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 that is, measurable functions
A stochastic process indexed by an index set
The probability measures
Functions on S of the form
In what follows, we assume that for each
(1) They generate an Abelian
(2)
(3) For each
(4) These properties show that
(5) We saw that a probability measure μ together with a stochastic process gives rise to a family of cylindrical probability measures
where
Even more generally, a partial order on the set
It turns out that these two conditions, Eqs 4.8, 4.9, or 4.10 is also sufficient in fortunate cases (for instance, if
Physical meaning: We consider the elements
4.2. Constructive QFT
The application of interest of the previous subsection is a stochastic process indexed by either
Real quantum scalar fields with smooth smearing:
Consider
with
In fact, since in this case, the space L is a vector space, it is sufficient to consider the functions
which, of course, reduces to
for certain
(2) Real quantum scalar fields with distributional smearing:
Consider a subset
Then,
with
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 combinations
(3) Non-Abelian gauge fields for compact gauge groups G:
A form factor is a distribution
where c is a one-dimensional path in σ. We take
where we have identified ϕ as a G connection and
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 [17–21]). Accordingly, our space of generating set of elementary functions
with
The fact that these functions satisfy all requirements is the statement of Clebsch–Gordan 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
The measures μ underlying a relativistic QFT are not only probability measures. In addition, they need to satisfy a set of axioms [98–100, 104] called Osterwalder–Schrader axioms which, however, are tailored to
Some of them generalise to stochastic processes not indexed by a vector space, and some do not. Some generalise from the manifold
An important remark is that the measures for gauge theories (such as general relativity) are to be formulated in terms of observable (gauge-invariant) fields which are typically composites of the elementary fields. That is why we work in a manifestly gauge (diffeomorphism)-invariant (equivalently, gauge-fixed) context as outlined in Section 3. In fact, in Ref. 39, we find an explicit formula that relates the observable composite fields to the elementary ones. The crucial condition is that the algebra of those observable fields is under sufficient mathematical control in order that Hilbert space representations can be found. This is the case for the construction sketched in Section 3.
The minimal set of OS axioms can be phrased as follows:
Let
Then, we have the following conditions on the generating functional
I. Time reflection invariance:
II. Time translation invariance
III. Time translation continuity
IV. Reflection positivity
Consider the vector space V of the complex span of functions of the form
Note that the stochastic process indexed by
At the moment, it is rather unclear how and why
4.3. Osterwalder–Schrader (OS) Reconstruction
The following abstract argument is standard [98–100]. (See Refs. 54–57 for a proof adapted to the notation in this article.) Due to reflection positivity, Eq. 4.26 defines a positive semi-definite sesquilinear form on V. We compute its null space N and complete the quotient of equivalence classes
The constraint
This elegant argument is deceivingly simple. To actually compute the Osterwalder–Schrader triple
which by construction is positive
that is, to say, the Hilbert space
4.4. Feynman–Kac–Trotter–Wiener (FKTW) Construction
Given an OS triple
Let now
Consider now a stochastic process
Then, the Wiener measure μ, if it exists, evaluated on Eq. 4.33
is supposed to equal Eq. 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
Note the semi-group property
Define
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 Refs. 183 and 184). Basically, one needs to show that matrix elements of
This shows that μ is a probability measure on S. For quantum mechanical Schrödinger Hamiltonians, one can use the Trotter product formula and the Wiener measure of the heat kernel to prove positivity [185] (Feynman–Kac formula).
One can now show the following [54–57]:
Theorem.
Suppose that OS data
Suppose that an OS measure μ is given thus producing OS data
Here, measure spaces
5. Renormalisation
5.1. 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
As already mentioned at the end 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
To have some intuitive picture in mind, consider
where the edges of the graph are straight lines in the coordinate directions between the vertices. We equip
with
5.2. Discretisation of Phase Space
In canonical quantisation, we start with a continuum phase space coordinatised by configuration fields
For instance, the momentum of a scalar field is geometrically a scalar density of weight one, so that
for all
The fact that the bilinear form
The connection to Section 4.2 is then as follows: For each
For each
for any
Definition.
A discretisation of the continuum phase space
with the following properties:
For any
That is, to say
ii. For any
Then, we require
To see how this gives rise to discretised configuration and momentum variables let
enjoy canonical brackets
where the first condition (Eq. 5.6) was used. Thus, Eq. 5.6 makes sure that the discretisations Eq. 5.10 enjoy canonical brackets, so we call Eq. 5.6 the symplectomorphism property. The motivation for the second condition Eq. 5.9 will become clear only later; however, we note that it implies that for all
which we thus call cylindrical consistency property. Likewise,
Finally, we will impose a further restriction on the maps
We note that Eq. 5.9 defines elements
For interacting Hamiltonians, more sophisticated approximations must be used. Certainly, the expression for
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
where
where the latter denotes the characteristic functions of left closed—right open—intervals. This clopen interval structure is very important in order that Eqs 5.6 and 5.9 are satisfied [54–57]. Similar constructions work for gauge fields (see appendix A or [196]). Note that we changed here the notation as compared to [54–57]: The maps
Given the lattice in D spatial dimensions labelled by
The space of elementary functions
Here,
To see how Eq. 5.16 interacts with the map
where the notation is as follows (see appendix A or [196]):
In general, therefore we see that for any generating function
where the sum over α involves a finite, unique set of generating functions
5.3. Hamiltonian Renormalisation
Abstracting from the concrete lattice implementation and field content above, we are in the following situation: There is a partially ordered and directed label set
Suppose that for each
Using the Feynman–Kac–Trotter–Wiener (FKTW) construction, we obtain a family of OS measures
Using
called the condition of cylindrical consistency.
As reviewed in Section 3, condition Eq. 5.21 grants the existence of μ under rather generic conditions. The strategy (see also Refs. 137 and 138) is therefore to construct an iterative sequence of measure families
The scheme that we will employ in fact does not make use of Eq. 5.21 for all
for
Theorem.
The renormalisation flow (Eq. 5.22) preserves the OS measure class, and its fixed points define OS measures
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 renormalisation 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
with
The Hilbert space
However, we are in a better situation than in the generic case because it is clear that
Step 2: Working out the flow of OS triples
Using the correspondence between the Wiener measures
for all choices of
We consider Eq. 5.24 as the master equation from which everything must be deduced. To avoid the compound field phenomenon mentioned above, we use that Eq. 5.24 i) is supposed to hold for an arbitrary number of time steps and ii) we add as further input one more OS axiom, namely, uniqueness of the vacuum which is, in fact, a standard axiom to impose in QFT on Minkowski space [98–100]. In terms of measures, it can be stated as ergodicity of time translations
We separate this axiom from the minimal ones because it enters in a crucial way only at this last stage of the renormalisation process. The subsequent discussion considerably extends the arguments of Refs. 54–57.
First of all, going back to Eq. 5.24 and picking
Using the fact that
which is a subspace of
is an isometry, that is,
which implies that
is a projection.
Next for
and using again the
Note that (choose
hence, the new vacuum is automatically annihilated by the new Hamiltonian.
We notice that for finite β, Eq. 5.31 is not implied by Eq. 5.32, unless
since they together with their adjoints leave
Let
becomes the projection on the ground state for
Let
On the other hand, if
Since the identity operator
To see this, we write in Eq. 5.24 for each
for
Equation 5.39 is known in the mathematics literature [189–192] as a degenerate case of a Kato–Trotter product [188], of which there are many versions. One of them states that for contraction semi-groups generated by self-adjoint operators
In our case, the second contraction semi-group,
In particular, if the solution of Eq. 5.42 is given by
we recover Eq. 5.32, since
In what follows, we will assume this to hold also when
To conclude this step, under the assumption that uniqueness of the vacuum is preserved under the renormalisation flow and that the degenerate Kato–Trotter product formula applies to general contraction semi-groups, we can strictly derive Eq. 5.29 and Eq. 5.32 as equivalent to Eq. 5.24. Unfortunately, it is not possible to show that the uniqueness property is automatically preserved under the flow: Suppose that
Step 3: Constructing the continuum theory from the fixed point data
Once we found a fixed point family
We stress that H is not the inductive limit of
We emphasise that this Hamiltonian renormalisation scheme can be seen as an independent, real-space, kinematical renormalisation 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, that is, 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 analytically, 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 Eqs 5.24–5.29 and Eq. 5.29 rests crucially on the assumption that the vacuum vectors
6. Conclusion
In this contribution, we have reviewed, extended, and clarified the proposal [54–57]. The extension consisted in i. an improved derivation of the renormalisation scheme (5.29) and (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) and 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 quotient 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 schemes 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 vacuum.
In [54–57, 193], 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, for example, interacting 2D scalar QFT [178–180] 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 [193] 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
The Hamiltonian
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
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 [194]: The cubical graphs
Finally, although the scheme, strictly speaking, was derived for theories with gauge-fixed space-time diffeomorphism constraints and a true physical Hamiltonian bounded from below, we may, of course, ‘abuse’ it and also consider constraint operators
Thus, even if the continuum algebra closes, one does not see this at any finite resolution, unless
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 [160], interesting phenomena related to phase transitions can happen here. Moreover, constructible examples of low dimensional interacting QFT show that the thermodynamic limit requires techniques that go beyond what was displayed here [178–180]. However, we consider this momentarily as a ‘higher order’ problem and reserve it for future research.
Author Contributions
The author confirms being the sole contributor of this work and has approved it for publication.
Conflict of Interest
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Acknowledgments
The author thanks Thorsten Lang for in depth discussions about reduced density matrices, decoherence, and the Lindblad equation in the context of renormalisation; Klaus Liegener for clarifying conversations about renormalisation of constraints; and Alexander Stottmeister for very fruitful exchanges about renormalisation in terms of algebraic states.
Supplementary Material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2020.548232/full#supplementary-material.
Footnotes
1In principle, any field theory with a polynomial Lagrangian can be written as a (coloured) tensor model as follows: Pick any orthonormal basis with respect to the measure appearing in the action, expand the field in that basis, call the expansion coefficients a coloured (by the space-time or internal indices) tensor in an infinite-dimensional
2In the non-compact case, one may need to take the infinite tensor product extension [69] which is also non-separable but in a different sense, and there one regains separability by passing to irreducible representations of the observable algebra.
3In fact, the physical Hamiltonian of Section 3 is not manifestly bounded from below, hence we to abuse the formalism in the sense that we assumed the semi-boundedness.
References
1. Dirac PAM. Quantum theory of localizable dynamical systems. Phys Rev (1948). 73:1092. doi:10.1103/physrev.73.1092.
2. Dirac PAM. Forms of relativistic dynamics. Rev Mod Phys (1949). 21:392.doi:10.1103/revmodphys.21.392.
4. DeWitt BS. Quantum theory of gravity. I. the canonical theory. Phys Rev (1967). 160:1113.doi:10.1103/physrev.160.1113.
5. DeWitt BS. Quantum theory of gravity. II. the manifestly covariant theory. Phys Rev (1967). 162:1195.doi:10.1103/physrev.162.1195.
6. DeWitt BS. Quantum theory of gravity. III. applications of the covariant theory. Phys Rev (1967). 162:1239.doi:10.1103/physrev.162.1239.
7. Komar A. General-relativistic observables via Hamilton-Jacobi functionals. Phys Rev D (1971). 4:923–7. doi:10.1103/physrevd.4.923.
8. Komar A. Commutator algebra of general-relativistic observables. Phys Rev D (1974). 9:885–8. doi:10.1103/physrevd.9.885.
9. Komar A. Generalized constraint structure for gravitation theory. Phys Rev D (1983). 27:2277–81. doi:10.1103/physrevd.27.2277.
10. Komar A. Consistent factor ordering of general-relativistic constraints. Phys Rev D (1979). 20:830–3. doi:10.1103/physrevd.20.830.
11. Bergmann PG, Komar A. The coordinate group symmetries of general relativity. Int J Theor Phys (1972). 5:15–28. doi:10.1007/bf00671650.
12. Bergmann PG, Komar A. The phase space formulation of general relativity and approaches towards its canonical quantization. Gen Rel Grav (1981). 1:227–54.
13. Kuchař K. Quantum gravity II: a second Oxford Symposium. CJ, Isham, R, Penrose, and DW, Sciama, editors. Oxford, UK: Clarendon Press (1981).
14. Isham CJ, Kuchar KV. Representations of spacetime diffeomorphisms. I. canonical parametrized field theories. Ann Phys (1985). 164:288–315. doi:10.1016/0003-4916(85)90018-1.
15. Isham CJ, Kuchar KV. Representations of spacetime diffeomorphisms. II. canonical geometrodynamics. Ann Phys (1985). 164:316. doi:10.1016/0003-4916(85)90019-3.
16. Ashtekar A. New variables for classical and quantum gravity. Phys Rev Lett (1986). 57:2244–7. doi:10.1103/physrevlett.57.2244.
17. Barbero JFG. A real polynomial formulation of general relativity in terms of connections. Phys Rev (1994). D49:6935–8.
19. Thiemann T. Modern canonical quantum general relativity. Cambridge, UK: Cambridge University Press (2007).
20. Pullin J, Gambini R. A first course in loop quantum gravity. New York, NY: Oxford University Press (2011).
21. Rovelli C, Vidotto F. Covariant loop quantum gravity. Cambridge, NY: Cambridge University Press (2015).
22. Giesel K, Sahlmann H. From classical to quantum gravity: introduction to loop quantum gravity, 002. PoS QGQGS2011 (2011). [arXiv:1203.2733]
24. Ashtekar A, Isham CJ. Representations of the holonomy algebras of gravity and nonAbelian gauge theories. Classical Quant Grav (1992). 9:1433. doi:10.1088/0264-9381/9/6/004.
25. Ashtekar A, Lewandowski J. Representation theory of analytic Holonomy algebras. In: J, Baez, editor Knots and quantum gravity. Oxford, UK: Oxford University Press (1994).
26. Ashtekar A, Lewandowski J. Projective techniques and functional integration for gauge theories. J Math Phys (1995). 36:2170–91. doi:10.1063/1.531037.
27. Fleischhack C. Representations of the Weyl algebra in quantum geometry. Commun Math Phys (2009). 285:67–140. doi:10.1007/s00220-008-0593-3.
28. Lewandowski J, Okołów A, Sahlmann H, Thiemann T. Uniqueness of diffeomorphism invariant states on holonomy-flux algebras. Commun Math Phys (2006). 267:703–33. doi:10.1007/s00220-006-0100-7.
29. Thiemann T. Anomaly-free formulation of non-perturbative, four-dimensional lorentzian quantum gravity. Phys Lett B (1996). 380:257–64. doi:10.1016/0370-2693(96)00532-1.
30. Thiemann T. Quantum spin dynamics (QSD). Class Quantum Grav (1998). 15:839–73. doi:10.1088/0264-9381/15/4/011
31. Thiemann T. Quantum spin dynamics (QSD): II. the kernel of the Wheeler-DeWitt constraint operator. Classical Quant Grav (1998). 15:875–905. doi:10.1088/0264-9381/15/4/012.
32. Thiemann T. Quantum spin dynamics (QSD): IV. Euclidean quantum gravity as a model to test Lorentzian quantum gravity. Classical Quant Grav (1998). 15:1249–80. doi:10.1088/0264-9381/15/5/011.
33. Thiemann T. Quantum spin dynamics (QSD): V. quantum gravity as the natural regulator of the Hamiltonian constraint of matter quantum field theories. Classical Quant Grav (1998). 15:1281–314. doi:10.1088/0264-9381/15/5/012.
34. Goroff MH, Sagnotti A, Sagnotti A. Quantum gravity at two loops. Phys Lett B (1985). 160:81–6. doi:10.1016/0370-2693(85)91470-4.
35. Hojman SA, Kuchař K, Teitelboim C. Geometrodynamics regained. Ann Phys (1976). 96:88–135. doi:10.1016/0003-4916(76)90112-3.
36. Ashtekar A, Lewandowski J, Marolf D, Mourão J, Thiemann T. Quantization of diffeomorphism invariant theories of connections with local degrees of freedom. J Math Phys (1995). 36:6456–93. doi:10.1063/1.531252.
37. Thiemann T. Quantum spin dynamics (QSD): III. quantum constraint algebra and physical scalar product in quantum general relativity. Classical Quant Grav (1998). 15:1207–47. doi:10.1088/0264-9381/15/5/010.
38. Thiemann T. Quantum spin dynamics: VIII. the master constraint. Classical Quant Grav (2006). 23:2249–65. doi:10.1088/0264-9381/23/7/003.
39. Giesel K, Thiemann T. Scalar material reference systems and loop quantum gravity. Classical Quant Grav (2015). 32:135015. doi:10.1088/0264-9381/32/13/135015.
40. Laddha A, Varadarajan M. The diffeomorphism constraint operator in loop quantum gravity. Classical Quant Grav (2011). 28:195010. doi:10.1088/0264-9381/28/19/195010.
41. Tomlin C, Varadarajan M. Towards an anomaly-free quantum dynamics for a weak coupling limit of euclidean gravity. Phys Rev D (2013). 87:044039. doi:10.1103/physrevd.87.044039
42. Tomlin C, Varadarajan M. Towards an anomaly-free quantum dynamics for a weak coupling limit of euclidean gravity: diffeomorphism covariance. Phys Rev D (2013). 87:044040. doi:10.1103/physrevd.87.044040
43. Ashtekar A, Singh P. Loop quantum cosmology: a status report. Classical Quant Grav (2011). 28:213001. doi:10.1088/0264-9381/28/21/213001.
45. Elizaga Navascues B, Martín-Benito M, Mena Marugan GA. Hybrid models in loop quantum cosmology. Int J Mod Phys D (2016). 25:1642007. doi:10.1142/S0218271816420074
46. Castello Gomar L, Mena Marugan GA, Martin De Blas D, Olmedo J. Hybrid loop quantum cosmology and predictions for the cosmic microwave background. Phys Rev D (2017). 96:103528. doi:10.1103/physrevd.96.103528
47. Han M, Liu H. Effective dynamics from coherent state path integral of full loop quantum gravity. Phys Rev D (2020). 101:046003. doi:10.1103/physrevd.101.046003
48. Han M, Liu H, Improved (
49. Dapor A, Liegener K. Cosmological effective Hamiltonian from full loop quantum gravity dynamics. Phys Lett B (2018). 785:506–10. doi:10.1016/j.physletb.2018.09.005.
50. Dapor A, Liegener K. Cosmological coherent state expectation values in loop quantum gravity I. isotropic kinematics. Classical Quant Grav (2018). 35:135011. doi:10.1088/1361-6382/aac4ba.
51. JEngle , IVilensky . Uniqueness of minimal loop quantum cosmology dynamics. Phys Rev D (2019). 100:121901. doi:10.1103/physrevd.100.121901
52. Engle J, Vilensky I. Deriving loop quantum cosmology dynamics from diffeomorphism invariance. Phys Rev D (2018). 98:023505. doi:10.1103/physrevd.98.023505
53. Fleischhack C. Continuity of states on non-unital differential algebras in loop quantum cosmology. Commun Math Phys (2019). 370:531–8. doi:10.1007/s00220-019-03486-5.
54. Lang T, Liegener K, Thiemann T. Hamiltonian renormalisation I: derivation from Osterwalder-Schrader reconstruction. Classical Quant Grav (2018). 35:245011. doi:10.1088/1361-6382/aaec56 | .
55. Lang T, Liegener K, Thiemann T. Hamiltonian renormalisation II. Renormalisation flow of 1+1 dimensional free scalar fields: derivation. Classical Quant Grav (2018). 35:245012. doi:10.1088/1361-6382/aaec54.
56. Lang T, Liegener K, Thiemann T. Hamiltonian renormalization III. Renormalisation flow of 1 + 1 dimensional free scalar fields: properties. Classical Quant Grav (2018). 35:245013. doi:10.1088/1361-6382/aaec3a.
57. Lang T, Liegener K, Thiemann T. Hamiltonian renormalisation IV. Renormalisation flow of D + 1 dimensional free scalar fields and rotation invariance. Classical Quant Grav (2018). 35:245014. doi:10.1088/1361-6382/aaec43.
58. Weinberg S. Ultraviolet divergences in quantum theories of gravitation. In: SW, Hawking, and W, Israel, editors General relativity: an Einstein centenary survey, Cambridge, UK: Cambridge University Press (1979). p. 790–831.
59. Wetterich C. Average action and the renormalization group equations. Nucl Phys B (1991). 352:529–84. doi:10.1016/0550-3213(91)90099-j.
60. Wetterich C. Exact evolution equation for the effective potential. Phys Lett B (1993). 301:90–4. doi:10.1016/0370-2693(93)90726-X
61. Reuter M, Wetterich C. Effective average action for gauge theories and exact evolution equations. Nucl Phys B (1994). 417:181214. doi:10.1016/0550-3213(94)90543-6.
62. Reuter M. Nonperturbative evolution equation for quantum gravity. Phys Rev D (1998). 57:971985. doi:10.1103/physrevd.57.971.
63. Lauscher O, Reuter M. Ultraviolet fixed point and generalized flow equation of quantum gravity. Phys Rev D (2001). 65:025013. doi:10.1103/physrevd.65.025013.
64. Reuter M, Saueressig F. Renormalization group flow of quantum gravity in the Einstein-Hilbert truncation. Phys Rev D (2002). 65:065016. doi:10.1103/physrevd.65.065016.
65. Niedermaier M, Reuter M. The asymptotic safety scenario in quantum gravity. Living Rev Relativ (2006). 9:5. doi:10.12942/lrr-2006-5.
66. Litim D. Renormalisation group and the Planck scale. Philos Trans A Math Phys Eng Sci (2006). 69:27592778. doi:10.1098/rsta.2011.0103
67. Percacci R. Asymptotic safety. In: D, Oriti, editor Approaches to quantum gravity: towards a new understanding of space, time and matter. Cambridge, UK: Cambridge University Press (2009).
68. Eichhorn A. An asymptotically safe guide to quantum gravity and matter. Front Astron Space Sci (2019). 5:47. doi:10.3389/fspas.2018.00047.
69. Manrique E, Rechenberger S, Saueressig F. Asymptotically safe lorentzian gravity. Phys Rev Lett (2011). 106:251302. doi:10.1103/physrevlett.106.251302.
70. Bonanno A, Gabriele Gionti SJ, Platania A. Bouncing and emergent cosmologies from Arnowitt-Deser-Misner RG flows. Classical Quant Grav (2018). 35:065004. doi:10.1088/1361-6382/aaa535.
71. Bahr B, Dittrich B. Improved and perfect actions in discrete gravity. Phys Rev D (2009). 80:124030. doi:10.1103/physrevd.80.124030
72. Bahr B, Dittrich B, Steinhaus S. Perfect discretization of reparametrization invariant path integrals. Phys Rev (2011). D83:105026. doi:10.1103/physrevd.83.105026
73. Bahr B, Dittrich B, He S. Coarse-graining free theories with gauge symmetries: the linearized case. New J Phys (2011). 13:045009. doi:10.1088/1367-2630/13/4/045009.
74. Bahr B. On background-independent renormalization of spin foam models. Classical Quant Grav (2017). 34:075001. doi:10.1088/1361-6382/aa5e13.
75. Bahr B, Dittrich B, Hellmann F, Kaminski W. Holonomy spin foam models: definition and coarse graining. Phys Rev D (2013). 87:044048. doi:10.1103/physrevd.87.044048
76. Livine ER, Oriti D. Coupling of spacetime atoms and spin foam renormalisation from group field theory. J. High Energy Phys (2007). 092:0702. doi:10.1088/1126-6708/2007/02/092
77. Freidel L, Gurau R, Oriti D. Group field theory renormalization-the 3d case: Power counting of divergences. Phys Rev D (2009). D80:044007. doi:10.1103/physrevd.80.044007
78. Carrozza S, Oriti D, Rivasseau V. Renormalization of tensorial group field theories: abelian U(1) models in four dimensions. Commun Math Phys (2014). 327:603–641. doi:10.1007/s00220-014-1954-8.
79. Carrozza S, Oriti D, Rivasseau V. Renormalization of a SU(2) tensorial group field theory in three dimensions. Commun Math Phys (2014). 330:581–637. doi:10.1007/s00220-014-1928-x.
80. Geloun JB, Koslowski TA, Oriti D, Pereira AD. Functional renormalization group analysis of rank 3 tensorial group field theory: the full quartic invariant truncation. Phys Rev (2018). D97:126018. doi:10.1103/physrevd.97.126018
81. Gurau R. Colored group field theory. Commun Math Phys (2011). 304:69–93.doi:10.1007/s00220-011-1226-9.
82. Gurau R. The complete 1/N expansion of colored tensor models in arbitrary dimension. Ann. Henri Poincaré (2012). 13:399–423. doi:10.1007/s00023-011-0118-z.
83. Gurau R, Ryan JP. Colored tensor models-a review. Sigma (2012). 8:020. doi:10.3842/SIGMA.2012.020
84. Bonzom V, Gurau R, Riello A, Rivasseau V. Critical behavior of colored tensor models in the large N limit. Nucl Phys B (2011). 853:174–95. doi:10.1016/j.nuclphysb.2011.07.022.
85. Reisenberger MP, Rovelli C. Sum over surfaces form of loop quantum gravity. Phys Rev D (1997). 56:3490–508. doi:10.1103/physrevd.56.3490.
86. Freidel L, Krasnov K. Spin foam models and the classical action principle. Adv Theor Math Phys (1999). 2:1183–247.
87. Barrett JW, Crane L. Relativistic spin networks and quantum gravity. J Math Phys (1998). 39:3296–302. doi:10.1063/1.532254.
88. Barrett JW, Crane L. A lorentzian signature model for quantum general relativity. Classical Quant Grav (2000). 17:3101–18. doi:10.1088/0264-9381/17/16/302.
89. Freidel L, Krasnov K. A new spin foam model for 4d gravity. Classical Quant Grav (2008). 25:125018. doi:10.1088/0264-9381/25/12/125018.
90. Engle J, Livine E, Pereira R, Rovelli C. LQG vertex with finite Immirzi parameter. Nucl Phys B (2008). 799:136–49. doi:10.1016/j.nuclphysb.2008.02.018.
91. Engle J. A spin-foam vertex amplitude with the correct semiclassical limit. Phys Lett B (2013). 724:333–7. doi:10.1016/j.physletb.2013.06.024.
92. Kaminski W, Kisielowski M, Lewandowski J. Spin-foams for all loop quantum gravity. Class. Quant Grav (2010).27:095006. Erratum: Class Quant Grav (2012) 29:049502. [arXiv:0909.0939]
93. Gel’fand IM, Vilenkin YN. Generalised functions. In: Applications of Harmonic Analysis, vol. 4. New York, NY and London, UK: Academic Press (1964).
94. Bojowald M. The BKL scenario, infrared renormalization, and quantum cosmology. JCAP (2019). [arXiv:1810.00238]
95. Bodendorfer N, Haneder F. Coarse graining as a representation change. Phys Lett B (2019). 792:69–73. doi:10.1016/j.physletb.2019.03.020.
96. Bodendorfer N, Wuhrer D. Renormalisation with SU(1, 1) coherent states on the LQC Hilbert space. [arXiv:1904.13269]
97. Itzykson C, Drouffe J-M. Statistical field theory. Cambridge, UK: Cambridge University Press (1989).
98. Froehlich J. An introduction to some topics in constructive QFT. New York, NY: Springer-Verlag (1978).
100. Rivasseau V. Constructive field theory and applications: Perspectives and open problems. J Math Phys (2000). 41:3764–5. doi:10.1063/1.533326.
101. Brown JD, Kuchar KV. Dust as a standard of space and time in canonical quantum gravity. Phys Rev (1995). D51:5600–29. doi:10.1103/physrevd.51.5600
102. Kucha KV, Torre CG. Gaussian reference fluid and interpretation of quantum geometrodynamics. Phys Rev D (1991). 43:419–41. doi:10.1103/physrevd.43.419.
103. Ashtekar A, Marolf D, Mourão J, Thiemann T. Constructing Hamiltonian quantum theories from path integrals in a diffeomorphism-invariant context. Classical Quant Grav (2000). 17:4919–40. doi:10.1088/0264-9381/17/23/310.
104. Osterwalder K, Schrader R. Axioms for euclidean greens functions. Commun Math Phys (1973). 31:83. doi:10.1007/bf01645738.
105. Kadanoff LP. Scaling laws for ising models near Tc. Physics (1966). 2:263–72. doi:10.1103/physicsphysiquefizika.2.263.
106. Wilson KG. The renormalization group: critical phenomena and the Kondo problem. Rev Mod Phys (1975). 47:773. doi:10.1103/revmodphys.47.773.
107. Fisher ME. The renormalization group in the theory of critical behavior. Rev Mod Phys (1974). 46:597. doi:10.1103/revmodphys.46.597. Erratum Rev. Mod. Phys. 47 (1975) 543.
108. Hasenfratz P. Prospects for perfect actions. Nucl Phys Proc Suppl (1998). 63:53–8. doi:10.1016/s0920-5632(97)00696-8.
110. Wegner FJ. Flow equations for Hamiltonians. Nucl Phys B Proc Suppl (2000). 90:141–6. doi:10.1016/s0920-5632(00)00911-7.
111. Glazek SD, Wilson KG. Renormalization of Hamiltonians. Phys Rev (1993). D48:5863–72. doi:10.1103/physrevd.48.5863
112. Weinstein M. Hamiltonians, path integrals, and a new renormalization group. Phys Rev D (1993). 47:5499.doi:10.1103/physrevd.47.5499.
113. Elias-Miro J, Rychkov S, Vitale LG, NLO renormalization in the Hamiltonian truncation. Phys Rev (2017). D96:065024. doi:10.1103/physrevd.96.065024
114. Kijowski J, Szczyrba W. A canonical structure for classical field theories. Commun Math Phys (1976). 46:183–206. doi:10.1007/bf01608496.
115. Kijowski J. Geometric structure of quantization. Rep Math Phys (1977). 11:97–109. doi:10.1016/0034-4877(77)90022-2.
116. Okolow A. Construction of spaces of kinematic quantum states for field theories via projective techniques. Classical Quant Grav (2013). 30:195003. doi:10.1088/0264-9381/30/19/195003
117. Kijowski J, Okołów A. A modification of the projective construction of quantum states for field theories. J Math Phys (2017). 58:062303. doi:10.1063/1.4989550.
118. Lanéry S, Thiemann T. Projective limits of state spaces I. Classical formalism. J Geom Phys (2017). 111:6–39. doi:10.1016/j.geomphys.2016.10.010.
119. Lanéry S, Thiemann T. Projective limits of state spaces II. Quantum formalism. J Geom Phys (2017). 116:10–51. doi:10.1016/j.geomphys.2017.01.011.
120. Stottmeister A, Thiemann T. Coherent states, quantum gravity, and the Born-Oppenheimer approximation. III: applications to loop quantum gravity. J Math Phys (2016). 57:083509. doi:10.1063/1.4960823.
121. Brothier A, Stottmeister A. Operator-algebraic construction of gauge theories and Jones’ actions of Thompson’s groups. Commun Math Phys (2020). 376:841–891. doi:10.1007/s00220-019-03603-4.
122. Brothier A, Stottmeister A. Canonical quantization of 1+1-dimensional Yang-Mills theory: an operator-algebraic approach. arXiv:1907.05549.
123. Stottmeister A, Morinelli V, Morsella G, Tanimoto Y. Operator-algebraic renormalization and wavelets. arXiv:2002.01442.
124. White SR. Density-matrix algorithms for quantum renormalisation groups. Phys Rev (1993). B48:10345–10356. doi:10.1103/physrevb.48.10345.
125. Levin M, Nave C. Tensor renormalization group approach to 2D classical lattice models. Phys Rev Lett (2207). 99:120601. doi:10.1103/physrevlett.99.120601.
126. Orús R. A practical introduction to tensor networks: matrix product states and projected entangled pair states. Ann Phys (2014). 349:117–158. doi:10.1016/j.aop.2014.06.013.
127. Vidal G. Entanglement renormalization. Phys Rev Lett (2007). 99:220405. doi:10.1103/physrevlett.99.220405.
128. Evenbly G, Vidal G, Algorithms for entanglement renormalization, Phys Rev B (2009). 79:144108. doi:10.1103/physrevb.79.149903
129. Evenbly G, Vidal G. Entanglement renormalization in free bosonic systems: real-space versus momentum-space renormalization group transforms. New J Phys (2010). 12:025007. doi:10.1088/1367-2630/12/2/025007.
130. Kaye P, Laflamme R, Mosca M. An introduction to quantum computing. Oxford, UK: Oxford University Press (2007).
131. Dittrich B. From the discrete to the continuous: towards a cylindrically consistent dynamics. New J Phys (2012). 14:123004. doi:10.1088/1367-2630/14/12/123004.
132. Dittrich B, Steinhaus S. Time evolution as refining, coarse graining and entangling. New J Phys (2014). 16:123041. doi:10.1088/1367-2630/16/12/123041.
133. Dittrich Y. The continuum limit of loop quantum gravity-a framework for solving the theory. [arXiv:1409.1450]
134. Yamasaki Y. Measures on infinite dimensional spaces. Singapore, Singapore: World Scientific (1985).
135. Balaban T, Imbrie J, Jaffe A. Exact renormalisation group for gauge theories. In: Proceedings of the 1983 Cargèse summer school.
136. Balaban T, Jaffe A. Constructive gauge theory. In: Proceedings of the 1986 Erichi summer school.
137. Bahr B. Operator Spin Foams: holonomy formulation and coarse graining. J Phys: Conf Ser (2012). 360:012042. doi:10.1088/1742-6596/360/1/012042.
138. Bahr B, Hellmann F, Kamiński W, Kisielowski M, Lewandowski J. Operator spin foam models. Classical Quant Grav (2011). 28:105003. doi:10.1088/0264-9381/28/10/105003.
139. Zapata JA. Loop quantization from a lattice gauge theory perspective. Class Quant Grav (2004). 21:L115–22. doi:10.1088/0264-9381/21/17/L01
140. Corichi A, Zapata J. On diffeomorphism invariance for lattice theories. Nucl Phys B (1997). 493:475–90. doi:10.1016/s0550-3213(97)00141-7.
141. Kijowski J, Okolow A. A modification of the projective construction of quantum states for field theories. J Math Phys (2016). 58:062303. doi:10.1063/1.4989550.
142. Dittrich B, Martin-Benito M, Steinhaus S. Quantum group spin nets: refinement limit and relation to spin foams. Phys Rev (2014). D90:024058. doi:10.1103/physrevd.90.024058
143. Dittrich B, Schnetter E, Cameron JS, Steinhaus S. Coarse graining flow of spin foam intertwiners. Phys Rev (2016). D94:124050. doi:10.1103/physrevd.94.124050
144. Eichhorn A, Koslowski T, Pereira AD. Status of background-independent coarse-graining in tensor models for quantum gravity. Universe (2019). 5:53. doi:10.3390/universe5.020053.
145. Komar A. General-relativistic observables via Hamilton-Jacobi functionals. Phys Rev D (1971). 4:923–7.doi:10.1103/physrevd.4.923.
146. Popolo AD, Le Delliou M, “Small scale problems of the
147. Rovelli C. Time in quantum gravity: an hypothesis. Phys Rev D (1991). 43:442–456.doi:10.1103/physrevd.43.442.
148. Vytheeswaran AS. Gauge unfixing in second-class constrained systems. Ann Phys (1994). 236:297–324.doi:10.1006/aphy.1994.1114.
149. Mitra P, Rajaraman R. Gauge-invariant reformulation of an anomalous gauge theory. Phys Lett B (1989). 225:267–271.doi:10.1016/0370-2693(89)90816-2.
150. Anishetty R, Vytheeswaran AS. Gauge invariance in second-class constrained systems. J Phys Math Gen (1993). 26:5613–5619. doi:10.1088/0305-4470/26/20/040.
151. Dittrich B. Partial and complete observables for canonical general relativity. Classical Quant Grav (2006). 23:6155–84. doi:10.1088/0264-9381/23/22/006.
152. Thiemann T. Reduced phase space quantization and Dirac observables. Classical Quant Grav (2006). 23:1163–80. doi:10.1088/0264-9381/23/4/006.
153. Henneaux M, Teitelboim C. Quantisation of gauge systems. Princeton, NJ: Princeton University Press (1992).
154. Dirac PAM. Lectures on quantum mechanics. New York, NY: Belfer Graduate School of Science, Yeshiva University Press (1964).
156. Giesel K, Hofmann S, Thiemann T, Winkler O. Manifestly gauge-invariant general relativistic perturbation theory: I. foundations. Classical Quant Grav (2010). 27:055005. doi:10.1088/0264-9381/27/5/055005.
157. Giesel K, Hofmann S, Thiemann T, Winkler O. Manifestly gauge-invariant general relativistic perturbation theory: II. FRW background and first order. Classical Quant Grav (2010). 27:055006. doi:10.1088/0264-9381/27/5/055006.
158. Domagala M, Giesel K, Kaminski W, Lewandowski J. Gravity quantized: loop quantum gravity with a scalar field. Phys Rev (2010). D82:104038. doi:10.1103/PhysRevD.82.104038
159. Kogut J, Susskind L. Hamiltonian formulation of Wilson's lattice gauge theories. Phys Rev D (1975). 11:395–408. doi:10.1103/physrevd.11.395.
160. Bratteli O, Robinson DW. Operator algebras and quantum statistical mechanics. Vols. 1 & 2. Berlin, Germany: Springer Verlag (1997).
161. Hall BC. Lie groups, lie algebras, and representations. Berlin, Germany: Springer Verlag (2015).
162. Rovelli C, Smolin L. Discreteness of area and volume in quantum gravity. Nucl Phys B (1995). 442:593–622. doi:10.1016/0550-3213(95)00150-q; Erratum: Nucl. Phys. B456 (1995) 753.
163. Ashtekar A, Lewandowski J. Quantum theory of geometry I: area operators. Class Quant Grav (1997). 14:A55–82. doi:10.1088/0264-9381/14/1a/006.
164. Ashtekar A, Lewandowski J. Quantum theory of geometry II: volume operators. Adv Theor Math Phys (1997). 1:388–429. doi:10.4310/atmp.1997.v1.n2.a8.
165. Thiemann T. Complexifier coherent states for quantum general relativity. Class Quant Grav (2006). 23:2063–117. doi:10.1088/0264-9381/23/6/013.
166. Thiemann T. Gauge field theory coherent states (GCS): I. General properties. Class Quant Grav (2001). 18:2025–64. doi:10.1088/0264-9381/18/11/304.
167. Thiemann T, Winkler O. Gauge field theory coherent states (GCS): II. peakedness properties. Class Quant Grav (2001). 18:2561–636. doi:10.1088/0264-9381/18/14/301.
168. Thiemann T, Winkler O. Gauge field theory coherent states (GCS): III. Ehrenfest theorems. Class Quant Grav (2001). 18:4629–81. doi:10.1088/0264-9381/18/21/315.
169. Giesel K, Thiemann T. Algebraic quantum gravity (AQG): I. conceptual setup. Classical Quant Grav (2007). 24:2465–97. doi:10.1088/0264-9381/24/10/003.
170. Giesel K, Thiemann T. Algebraic quantum gravity (AQG): II. Semiclassical analysis. Class Quant Grav (2007). 24:2499–564. doi:10.1088/0264-9381/24/10/004.
171. Giesel K, Thiemann T. Algebraic quantum gravity (AQG): III. semiclassical perturbation theory. Classical Quant Grav (2007). 24:2565–588. doi:10.1088/0264-9381/24/10/005.
172. Giesel K, Thiemann T. Algebraic quantum gravity (AQG): IV. reduced phase space quantization of loop quantum gravity. Classical Quant Grav (2010). 27:175009. doi:10.1088/0264-9381/27/17/175009.
173. Gaul M, Rovelli C. A generalized Hamiltonian constraint operator in loop quantum gravity and its simplest Euclidean matrix elements. Classical Quant Grav (2001). 18:1593–624. doi:10.1088/0264-9381/18/9/301.
174. Thiemann T, Winkler O. Gauge field theory coherent states (GCS): IV. Infinite tensor product and thermodynamical limit. Classical Quant Grav (2001). 18:4997–5053. doi:10.1088/0264-9381/18/23/302.
177. Bernal AN, Sánchez M. On smooth cauchy hypersurfaces and geroch's splitting theorem. Commun Math Phys (2003). 243:461–470. doi:10.1007/s00220-003-0982-6.
178. Simon B. The P(ϕ)2 Euclidean (quantum) field theory. Princeton, NJ:Princeton Unviersity Press (1974).
179. Glimm J. Boson fields with the :Φ4: interaction in three dimensions. Commun Math Phys (1968). 10:1–47. doi:10.1007/bf01654131.
180. Glimm J, Jaffe A. Positivity of the Hamiltonian. Fortschr Phys (1973). 21:327376. doi:10.1002/prop.19730210702.
182. Kadison RV, Ringrose JR. Fundamentals of the theory of operator algebras. Vols. 1 & 2. London, UK: Academic Press (1983).
183. Simon B. Positivity of the Hamiltonian semigroup and the construction of euclidean region fields. Helvetica Physics Acta (1973). 46:686–96.
184. Simon B, Høegh-Krohn R. Hypercontractive semigroups and two dimensional self-coupled Bose fields. J Funct Anal (1972). 9:121–80. doi:10.1016/0022-1236(72)90008-0.
185. Roeppstorff G. Path integral approach to quantum physics. Berlin, Germany: Springer-Verlag (1996).
187. Zurek WH. Decoherence, einselection, and the quantum origins of the classical. Rev Mod Phys (2003). 75:715. doi:10.1103/revmodphys.75.715.
188. Reed M, Simon B. Methods of modern mathematical physics. vol. I-IV. London, UK: Academic Press (1980).
189. Zagrebnov VA. The Trotter-Lie product formula for Gibbs semigroups. J Math Phys (1988). 29:888–91. doi:10.1063/1.527985.
190. Neidhardt H, Zagrebnov VA. The Trotter-Kato product formula for Gibbs semigroups. Commun Math Phys (1990). 131:333–46. doi:10.1007/bf02161418.
191. Matolcsi M, Shvidkoy R. Trotter’s product formula for projections. Arch Math (2003). 81:309–17. doi:10.1007/s00013-003-0537-2.
192. Arendt W, Batty CJK. Absorption semigroups and Dirichlet boundary conditions. Math Ann (1993). 295:427–48. doi:10.1007/bf01444895.
194. Sahlmann H, Thiemann T, Winkler O. Coherent states for canonical quantum general relativity and the infinite tensor product extension. Nucl Phys B (2001). 606:401–40. doi:10.1016/s0550-3213(01)00226-7.
195. Federbush P. A new formulation and regularization of gauge theories using a non-linear wavelet expansion. Prog Theor Phys (1995). 94:1135–46. doi:10.1143/ptp.94.1135.
Keywords: Canonical quantum gravity, lattice gauge field theory, constructive quantum field theory, renormalisation, Euclidian formulation
Citation: Thiemann T (2020) Canonical Quantum Gravity, Constructive QFT, and Renormalisation. Front. Phys. 8:548232. doi: 10.3389/fphy.2020.548232
Received: 01 April 2020; Accepted: 14 September 2020;
Published: 18 December 2020.
Edited by:
Astrid Eichhorn, University of Southern Denmark, FranceReviewed by:
Reinhard Alkofer, University of Graz, GrazKasia Rejzner, University of York, United Kingdom
Copyright © 2020 Thiemann. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Thomas Thiemann, thomas.thiemann@gravity.fau.de