New Dynamics for Canonical Loop Quantum Gravity

NEW DYNAMICS FOR CANONICAL LOOP QUANTUM GRAVITY

Mehdi Assanioussi

Faculty of Physics University of Warsaw

Dissertation submitted for the degree of Doctor of Philosophy Under the supervision of Prof. dr hab. Jerzy Lewandowski

December 2016

Declaration

I hereby declare that this thesis presents my original research work. Wherever contributions of others are involved, every eﬀort is made to indicate this clearly, with due reference to the literature, and acknowledgements of collaborative research and discussions. I also declare that the submitted thesis has not been the subject of proceedings resulting in the award of any other academic degree or diploma. The work was done under the supervision of professor dr hab. Jerzy Lewandowski at the Faculty of Physics, University of Warsaw.

Mehdi Assanioussi December 2016

Acknowledgements

I would like to express my deepest gratitude to my supervisor prof. Jerzy Lewandowski, for the freedom he gave me in my work, for his constant support to attend any conference I liked, and for his guidance and encouragement without which I would not have accomplished the work presented in this thesis. This journey would not have been as fruitful and exciting without multiple passionate and enlightening discussions I had with Emanuele Alesci, Ilkka Mäkinen, Andrea Dapor, Norbert Bodendorfer, Jędrzej Świeżewski and Wo- jciech Kamiński, which often lasted for many late evening hours and which helped enormously to shape my ideas the way they are now, and for that I am forever grateful. I also thank all my colleagues in the Institute of Theoretical Physics at the faculty of Physics, University of Warsaw, for the stimulating and convivial environment we shared. I would like to give special thanks to Simone Speziale who set me on the path of quantum gravity. I also thank prof. J. Fernando Barbero G. and prof. Hanno Sahlmann for assessing this dissertation. I would like to thank my parents for their unconditional support, unfading encouragement and all they sacriﬁced so that I could be where I am today. To both of my sisters, Ikhlas and Kenza, I want to address special thanks for their warm feelings and the immense joy they bring to my life. To my friends Taha, Faissal, Ghassane, Yasser, Saad, Kevin and Antoine, I am extremely fortunate to have met you and very grateful for your friendship and all the laughing with me and at me. A friendship which will always make me push forward through life. Last but not least, I would like to express my deepest gratitude and af- fection to Basia for her support, patience and all the wonderful moments we enjoyed together which helped me keep up the hard work.

« إن من الواجب على من يحقق في كتابات العلماء، إذا كان البحث عن الحقيقة هدفه، هو ٔان يستنكر جميع ما ٔيقراه، [...] وعليه ٔان يتشكك في نتائج دراسته ٔايضا حتى يتجنب الوقوع في ٔاي تحيز ٔاو تساهل. »

– ابن الهيثم، كتاب المناظر، 1021 –

« The duty of those who investigate the writ- ings of scientists, if learning the truth is their goal, is to disapprove all that they read, [...] They should also suspect their own critical ex- amination of it, so that they may avoid falling into either prejudice or leniency. »

– Alhazen, Book of Optics, 1021 –

Abstract

Canonical loop quantum gravity (LQG) is a canonical quantization of general relativity in its Hamiltonian formulation, which has successfully completed the construction of a kinematical Hilbert space and the implementation of the Gauss constraints and the spatial diffeomorphism constraints. In the first part of this thesis I give a general introduction to the Ashtekar– Barbero Hamiltonian formulation of general relativity and a detailed overview of the LQG program. In the second part, I present a new approach for quantizing the Hamiltoni- ans in various LQG models. The result allows to make a step forward toward improving the status of the dynamics in the theory. The construction is based on novel regularization procedures of the classical functionals, starting with the construction of a new geometrical operator, the curvature operator, related to the three–dimensional Ricci curvature. This new approach eventually leads to a diffeomorphism covariant scalar constraint operator in vacuum LQG with an anomaly–free constraints algebra. Moreover, thanks to the new prescription, it is possible to construct eligible symmetric extensions with the perspective to obtain self–adjoint extensions. In the third part, I expose how the new regularization is used to implement symmetric Hamiltonian operators in the context of two particular LQG deparametrized models, completing the quantization of the two models with a full quantum gravity sector. Finally, I present an approximation method based on time–independent perturbation theory, which was applied to the Hamilto- nian operators in those deparametrized models and where the perturbation parameter depends on the Barbero–Immirzi parameter.

Table of contents

I A centenary quest: Quantum gravity! 1

II On Dirac’s footsteps: canonical loop quantum gravity 9 II.1 Hamiltonian formulation of GR and Ashtekar-Barbero variables 9 II.1.1 ADM formulation of general relativity ...... 10 II.1.2 General relativity in Ashtekar-Barbero variables . . . . 14 II.2 Canonical quantization: LQG ...... 18 II.2.1 Holonomy-ﬂux algebra and kinematical Hilbert space . 19 II.2.2 Implementation of the Gauss and spatial diﬀeomorphism constraints ...... 25 II.2.3 On the dynamics in loop quantum gravity ...... 29

III Alternative dynamics in vacuum LQG 43 III.1 Vertex Hilbert space ...... 44 III.2 Curvature operator ...... 45 III.2.1 Regge calculus ...... 46 III.2.2 Construction of the curvature operator ...... 51 III.2.3 Properties of the curvature operator ...... 61 III.2.4 The Lorentzian part of the scalar constraint operator . 66 III.3 Euclidean operator ...... 67 III.3.1 Construction of the Euclidean operator ...... 68 III.3.2 Properties of the Euclidean operator ...... 73 III.4 Scalar constraint operator, quantum algebra and physical states 74 III.4.1 Quantum constraints algebra ...... 75 III.4.2 Physical states ...... 76

IV Towards new dynamics in LQG deparametrized models 79 IV.1 Emergent time in examples ...... 80 IV.1.1 Gravity coupled to a free Klein–Gordon ﬁeld ...... 80 IV.1.2 Gravity coupled to non–rotational dust ...... 83 xii Table of contents

IV.2 LQG quantization ...... 84 IV.2.1 Physical Hilbert spaces ...... 85 IV.2.2 Quantum dynamics ...... 85 IV.3 An approximation method for LQG dynamics ...... 95 IV.3.1 Perturbation theory with the Barbero–Immirzi parameter 96 IV.3.2 Numerical analysis of simple examples ...... 98

V Summary and outlook 105

References 109

Appendix A Key publications 117 Chapter I

A centenary quest: Quantum gravity!

When the theory of general relativity [1, 2] was proposed by Albert Einstein in 1915 as a relativistic theory of gravity, it was welcomed in the scientific community with less enthusiasm than it later proved to deserve. Despite its early successful predictions such as Mercury anomalous perihelion (1915) and the deflection of starlight by the Sun during the solar eclipse (1919), general relativity was for several years viewed as an intriguing theory with an unusual mathematical complexity, which stands aside from the generally accepted fields of theoretical physics at that time. Nevertheless, with the derivation of the first non–trivial solutions to the field equations of the theory and the emergence of Cosmology, many physicists and mathematicians of the time took interest in investigating the structure and implications of the theory. After almost two decades, general relativity started imposing itself as a revolutionizing theory and a cornerstone of modern theoretical physics. On one of his lecture in 1939, Paul Dirac adequately summarized the common perception about the theory then:

“What makes the theory of relativity so acceptable to physicists in spite of its going against the principle of simplicity is its great mathematical beauty. This is a quality which cannot be defined, any more than beauty in art can be defined, but which people who study mathematics usually have no difficulty in appreciating. [...] The restricted theory changed our ideas of space and time in a way that may be summarized by stating that the group of transformations to which the space-time continuum is subject must be changed from the Galilean group to the Lorentz group. [...] The general theory 2 A centenary quest: Quantum gravity!

of relativity involved another step of a rather similar character, although the increase in beauty this time is usually considered to be not quite so great as with the restricted theory, which results in the general theory being not quite so ﬁrmly believed in as the restricted theory.”

Few decades later will see the theory of relativity flourish in several aspects with new exact solutions, new mathematical formulations of the theory and better understanding of the dynamical nature of spacetime. Ironically, these achievements also exposed certain issues in the theory, namely the problem of observables and the problem of singularities present in cosmological and black hole solutions. In the meantime, another revolution was running: the rise of quantum mechanics. The base of quantum mechanics arose gradually from various experimental observations and theoretical descriptions of different phenomena thought originally to be independent. Some of the essential concepts were already developed in the nineteenth century with the wave theory of light, followed later by Max Planck’s hypothesis of exchange of energy in discrete amounts, the quanta. It is until the late 1920’s that the theory was consistently formulated and accepted as the standard theory of atoms and photons, with a new understanding of once trivial concepts, such as measurement and locality. Quantum mechanics was since then extended and successfully applied in many domains, setting itself at the base of uncountable theoretical and technological developments. The greatest consequent theoretical achievement is undeniably the extension of the principles of quantum mechanics to field theories, giving birth to the realm of quantum field theories (QFTs). QFTs stand today at the core of all modern physics with an incomparable success in the amount and accuracy of predictions, e.g the Standard model of elementary particles. However, a QFT is usually defined only perturbatively and it suffers from serious conceptual and technical problems, which in turn are translated into divergences in the calculations. One then needs to introduce renormalization procedures to tame those divergences and obtain finite results. Many physicists since then are not comfortable with the renormalization as a necessary concept, their perspective being that its presence indicates that the formulation of the QFTs is still incomplete. Richard Feynman commented this aspect in one of his books [3](1985):

“The shell game that we play [...] is technically called “renormalization”. But no matter how clever the word, it is what I would 3

call a dippy process! Having to resort to such hocus–pocus has pre- vented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It’s surprising that the theory still hasn’t been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.”

Renormalization is essentially viewed as a purely mathematical artifact. Integrals over the continuum span of energies diverge. One then places a cutoff on the fields by assuming that quanta cannot access energies above a very large fixed value. Effectively, this is equivalent to replacing the rigid continuous flat space by a fixed discrete structure, e.g. lattice, where wavelengths shorter than the regulator cannot exist. But at the moment one starts referring to the structure of space, an important question then comes up: what about gravity? General relativity is the theory of gravity where the very spacetime is continuous and dynamical! So instead of coming to rescue the theoretical framework of QFTs, it turns out that the principles of general relativity are totally inconsistent with the principles of standard QFTs. One then comes to the conclusion that the foundations of one, or both frameworks must be modified. This is where the idea of developing a quantum theory of gravity emerges as a possible solution. One could reasonably expect that a quantization of gravity might hold the key to consistently solve the issues present in both general relativity and standard QFTs. The idea of quantizing the gravitational field is actually almost as old as the general theory of relativity: in one of his papers [4] in 1916 Albert Einstein wrote:

“Nevertheless, due to the inner–atomic movement of electrons, atoms would have to radiate not only electro–magnetic but also gravitational energy, if only in tiny amounts. As this is hardly true in Nature, it appears that quantum theory would have to modify not only Maxwellian electrodynamics, but also the new theory of gravitation.”

Indeed, the first attempts to quantize the gravitational field date back to shortly after the formulation of general relativity. A century later, the research field known today as quantum gravity has largely developed but the challenge to define a consistent and complete quantum theory of gravity is still standing as strenuous and puzzling as ever. Various approaches, with different perspectives, exist today to attempt to overcome this challenge, here are some examples: 4 A centenary quest: Quantum gravity!

• String theory: it is a perturbative quantum ﬁeld theory with the main aim of unifying all fundamental interactions [5, 6], including quantum gravity. String theory is based on the idea of replacing point particles with strings characterized by certain quantum vibration modes. These modes determine the particles and their properties, e.g. spin. In case of gravity, a parti- cle of spin 2 is a perfect candidate for the graviton, that is the boson of the gravitational interaction which manifests in the linearized gravity approximation [1]. String theory requires the existence at least 10 spacetime dimensions, except of the classical 4 dimensions the non–observed dimensions are compactiﬁed. It additionally predicts the existence of a large new variety of particles which so far have not been observed.

• Causal dynamical triangulations: In this approach [7], one assumes that the fundamental structure of spacetime is discrete. More precisely, spacetime is not a smooth manifold but it is considered to be a piecewise linear manifold in the form of a triangulation. From the 3 + 1 perspective where space and time are distinguished, each spatial slice is a locally flat simplicial manifold labeled by a value of discrete time variable. The edges of the simplexes represent either a space–like or time–like segment. The theory is then defined in a path integral formulation, where the integration is over all possible configurations of triangulations which satisfy the condition that causality is preserved during the evolution. One then relies on numerical simulations to analyze the phase space for such spacetimes with a dimension higher than two.

• Causal sets: The approach is based on a theorem [8] which states that a spacetime is fully determined, up to a conformal factor, by its causal structure. One then uses a partially ordered set to describe spacetime, where the partial order is obtained from the causal structure of the considered spacetime [9].

• Asymptotic safety: This program is based on the idea that, although general relativity is a non–renormalizable theory, if one can find a non–trivial fixed point of the theory’s renormalization group flow, then one eventually could generalize the procedure of perturbative renormalization [10]. This a 5

general statement which could apply to any non–renormalizable quantum field theory. The investigation in this subject concerns the flow equation on the theory space, where each point represents one possible action for the considered field theory.

• Loop quantum gravity: Loop quantum gravity program (LQG) [11–14] is a canonical quantization, following Dirac’s program [15, 16] for constrained systems, of general relativity in its Hamiltonian formulation in terms of Ashtekar– Barbero variables [17, 18]. LQG succeeded in deﬁning a Hilbert space of kinematical quantum states, and implementing and solving the spatial diﬀeomorphism constraint. However, the scalar constraint is technically more involved and although several proposals for its implementation are available, the construction of a physical Hilbert space has not yet been achieved and the limit to recover general relativity within the theory is still unknown.

• Spin foam formalism: The Spin foam formalism [19, 20] is an approach based on a discretized and regularized path integral for quantum gravity. It was originally introduced as an alternative method to implement the dynamics in loop quantum gravity and deﬁne transition amplitudes for the kinematical quantum states. Today, the Spin foam program is considered to be a covariant quantization of the BF theory formulation of gravity. Though closely related, the Spin foam program, also called covariant loop quantum gravity, and the canonical loop quantum gravity program are considered to be independent approaches.

• Group field theory: Group field theory [21] is a quantum field theory where the field is defined on a group manifold. The quantum theory is set in a path integral formulation where the fields and the interaction terms carry a combina- torial and non–local character, distinguishing it from the standard local QFTs. It is a general formalism which could be related to several approaches to quantum gravity, including loop quantum gravity and causal dynamical triangulations, and it can play an important role in proposing new ways to formulate and investigate those theories [22].

There are still several other approaches as interesting as those mentioned above, but most of the approaches to quantum gravity fail so far to propose cur- 6 A centenary quest: Quantum gravity! rently falsifiable predictions. Although based on fundamentally different and sometimes complementary ideas, each approach presents a promising venue which deserves to be studied and developed further. My personal interest however lies in the canonical quantization of general relativity in general, and more specifically in loop quantum gravity. As mentioned above, the dynamical and semi–classical sectors of LQG are still unresolved issues. Hence, my research was focused on the particular issue of implementing the dynamics in LQG. More precisely, the central point of my work, presented in this thesis, was the construction and analysis of new Hamiltonian operators which satisfy certain criteria in various LQG models. The results can be briefly listed as follows:

1. Curvature operator: we introduced a new geometrical operator associated with an open region of a 3-manifold with the interpretation of mea- suring intrinsic curvature. The construction is based on an “external” regularization scheme using Regge calculus. We investigated some of its properties and checked its semi–classical behavior in some simple cases. The regularization scheme we adopt in this construction is for instance quite diﬀerent from the one used in the construction of the volume operator, because in this case the classical expression is written as the limit of a Regge-like discretization instead of introducing a Riemannian sum. This operator is used afterwards to deﬁne the Lorentzian part of LQG Hamiltonian.

2. New regularization of the Euclidean part of the Hamiltonian in Ashtekar– Barbero formulation: we develop a concrete regularization scheme where the curvature of the Ashtekar–Barbero connection is replaced by holonomies along closed loops attached to pairs of edges at a vertex of a spin network graph. By carefully specifying the properties of these special loops, the obtained Euclidean Hamiltonian operator is densely deﬁned.

3. Symmetric Hamiltonian operator in the LQG deparametrized model with a free scalar field: using the regularization schemes of the curvature operator and of the Euclidean part, we were able to implement the Hamil- tonian operator which governs the dynamics in the LQG deparametrized model with a free scalar field. A practical advantage of this Hamilto- nian operator is that the volume operator does not appear in it. This implies a considerable simplification of the calculation of the action of the Hamiltonian on spin network states. 7

4. Symmetric and anomaly free Hamiltonian constraint operator: we developed a concrete and explicit construction of a new scalar constraint operator for vacuum LQG using the same regularization procedure mentioned above. The operator is defined on the recently introduced space of partially diffeomorphism invariant states. Due to the properties of the special loops assignment and the curvature operator, the adjoint operator of the non–symmetric constraint operator is also densely defined. This fact opened up the possibility of introduce a symmetric scalar constraint operator as a suitable combination of the original operator and its adjoint. The algebra of the scalar constraint operators is anomaly free and a quantitative description of the structure of the kernel is available.

5. Approximation method to estimate the evolution in LQG deparametrized models: we proposed an approximation method for the dynamics by applying the standard time–independent perturbation theory of quantum mechanics to the Hamiltonian operators we derived. The perturbation parameter being determined by the Barbero–Immirzi parameter β and requires β ≫ 1. This method allows to deﬁne an approximate spectral decomposition of the Hamiltonian operators and ultimately to compute the evolution in a certain time interval.

The present thesis consists of five chapters. This first chapter gives a broad introduction to the problem of quantum gravity and an overview of certain approaches to solve it. The second chapter is centered on the essentials of canonical LQG exposed in two sections. The first section concerns the classical framework with a brief presentation of the ADM Hamiltonian formulation of general relativity then transitioning to the Ashtekar–Barbero connection formulation. The second section lays out the quantization program of LQG, starting with the construction of the kinematical Hilbert space, going through the implementation of the kinematical constraints and finally discussing the implementation of the dynamics. In the third chapter, composed of three sections, I present my work with my collaborators on a new approach to implement the scalar constraint in the vacuum LQG along with the analysis of various properties of the resulting operator [23, 24]. In the fourth chapter I expose the approach of deparametrization of gravity with a focus on two specific models, the free scalar field and the non–rotational dust. Then I expose the complete LQG quantization of the two models in addi- 8 A centenary quest: Quantum gravity! tion to my work with my collaborators on defining the Hamiltonian operators in both models [25, 24]. In the last section, I introduce an approximation method for the dynamics in deparametrized models, which consists of applying time independent perturbation theory to the Hamiltonian operators in both models [26]. In the last chapter, I conclude with a summary of the work realized, and an outlook of possible interesting directions that one could investigate for further insights and development in the still uncovered sectors of LQG. Chapter II

On Dirac’s footsteps: canonical loop quantum gravity

In this chapter we give an overview of two Hamiltonian formulations of general relativity, ADM formulation and Ashtekar–Barbero formulation, presented along the lines of [13]. As we will see, the Ashtekar–Barbero formulation, which could be viewed as an extension of ADM formulation, is more adapted to Dirac’s canonical quantization program. The mentioned quantization, which culminates in the loop quantum gravity (LQG) approach, is summarized in the second section of this chapter following roughly the presentation in [11]. In the last section, we discuss the quantum dynamics in LQG, in particular T. Thiemann’s proposal, then we ﬁnish with a review of the main issues and criticism concerning the implementation of the dynamics in LQG.

II.1 Hamiltonian formulation of GR and Ashtekar- Barbero variables

The standard action functional for general relativity takes the form

SGR = SEH + SM , (II.1.1) where SM is the matter action, while SEH is the Einstein–Hilbert (E–H) action for the gravitational ﬁeld which is represented by the spacetime metric variable gµν, plus a boundary term. SEH is expressed as ∫ √ ∫ √ 1 4 (4) 1 3 SEH [gµν] := d y |g|R + d x sgn(−h) |h|K. (II.1.2) 2k V k ∂V 10 On Dirac’s footsteps: canonical loop quantum gravity

Here, V is an arbitrary region of the four dimensional spacetime manifold M with a boundary ∂V , k = 8πG a constant, g is the determinant of the metric (4) gµν, R is the Ricci scalar curvature in V , h is the determinant of the induced metric on ∂V and K is the trace of the extrinsic curvature in ∂V . As shown in equation (II.1.2), the action SEH has in general two terms: the bulk term and 1 the boundary term . Its variation with respect to the metric gµν gives Einstein tensor Gµν which represents the gravitational part in Einstein equations, the equations of motion in general relativity:

1 G := R − g R = kT , (II.1.3) µν µν 2 µν µν where Rµν is the Ricci tensor and Tµν is the stress-energy tensor obtained from the variation of the matter action SM . For simplification, we neglect in what follows all the boundary contributions which could arise in the equations, including the term in (II.1.2), and we focus the analysis on the bulk terms only. The incorporation of boundary contributions in the results is always achievable and does not alter the main conclusions significantly. We then always assume that appropriate boundary conditions are satisfied and allow extending the region of integration from V to the whole spacetime manifold M (compact or otherwise).

II.1.1 ADM formulation of general relativity

In this section we briefly present the ADM Hamiltonian formulation of vacuum general relativity developed by Arnowitt, Deser and Misner in [27](1962). A Hamiltonian formulation of general relativity means necessarily introducing a splitting of spacetime into space and time. While this sounds like breaking the diffeomorphism invariance, it is not the case because the splitting is not fixed but kept arbitrary, allowing to recover the diffeomorphism invariance. It is though necessary to make a choice of the topology of the splitting. The assumption is to consider spacetime to be globally hyperbolic which is from classical physics perspective not much of a restriction. This assumption au- tomatically leads to the fact that the spacetime M has the topology R × Σ, where Σ is a fixed 3–dimensional manifold of arbitrary topology. Therefore we

1 The functional SEH may diverge when evaluated for certain solutions of the equations of motion, for instance in case of asymptotically flat spacetimes where the boundary is taken to infinity. It is then necessary to modify the boundary term by subtracting a counter–term in order to make SEH a well–defined functional [2]. However the counter–term does not modify the equations of motion of general relativity. II.1 Hamiltonian formulation of GR and Ashtekar-Barbero variables 11 are introducing a foliation of spacetime M into space–like hypersurfaces, one for each value of t ∈ R. Let us denote by nµ and N µ the vector fields normal and tangential to the hypersurfaces respectively. We then write

∂yµ ∂yµ = Nnµ + N µ = Nnµ + N a , (II.1.4) ∂t ∂xa where yµ are spacetime coordinates while xa are coordinates on a given space– like hypersurface Σ. The coeﬃcients N and N a are called the lapse and shift functions respectively. Without any loss of generality, we always assume that N > 0.

The spacetime metric gµν has then the form

2 a b b g00 = sN + qabN N , g0a = qabN , gab = qab , (II.1.5) where qab is the induced 3–metric from the spacetime metric on the hypersurface Σ. The Hamiltonian formalism requires considering 3–dimensional a quantities, therefore the spatial metric qab and the functions N and N are perfectly adequate for this task. The action SEH can be then re–expressed in terms of these quantities and their time derivatives (derivative with respect to t which we denote by ˙ ), the resulting form is called the ADM action ∫ ∫ √ ˙ a ˙ a 1 3 ab 2 SADM [qab, q˙ab,N, N,N , N ] = dt d x qN(R − s[KabK − K ]) . 2k R Σ (II.1.6)

In this equation q is the determinant of the metric qab, the scalar R is the Ricci scalar curvature in the spatial hypersurface Σ and qab is the inverse of the metric qab which are used to raise and lower spatial tensors indexes respectively. The symmetric tensor Kab is the extrinsic curvature tensor in Σ deﬁned as 1 K := (q ˙ − (L q) ) , (II.1.7) ab 2N ab N⃗ ab

⃗ a ab with L denoting the Lie derivative and N := {N }. The scalar K := q Kab is the trace of Kab.

In order to obtain the canonical Hamiltonian form of SADM , one performs the Legendre transform on the Lagrangian density (the integrand in (II.1.6)). 12 On Dirac’s footsteps: canonical loop quantum gravity

After some calculations and simpliﬁcations, one ends up with the expression ∫ ∫ ab a 3 ab a SADM [qab, p ,N,N ] = dt d x q˙abp − NC − N Ca , (II.1.8) R Σ

ab where the quantity p is the conjugate momentum to qab deﬁned as

∂S s √ pab := ADM = − q(Kab − qabK]) . (II.1.9) ∂q˙ab 2k

The functionals C and Ca are called the scalar constraints and the spatial diﬀeomorphism (or vector) constraints respectively. Their expressions are

C := − 2q ∇ pbc , (II.1.10) a ac b √ sk q C :=√ (q q − 2q q )pabpcd − R (II.1.11) q ab cd ac bd 2k where ∇ stands for the covariant derivative compatible with the metric qab. These constraints are secondary constraints which form the full Hamiltonian of vacuum general relativity. They are weighted by the lapse N and the a shift N which are now treated as Lagrange multipliers in the action SADM . This change in the interpretation of the lapse and shift functions is due to the fact that, while performing the Legendre transform, one obtains primary constraints which are a direct consequence of the absence of an explicit de- ˙ ˙ a pendence on the functions N and N in the expression of SADM (II.1.6). In

Dirac’s program [15, 16], and because the equations of motion for qab and pab are unaffected by those primary constraints, the lapse and shift functions become simply Lagrange multipliers. From the phase space perspective, the variables N and N a and their momenta become irrelevant and are therefore dropped. The final phase space of the theory is infinite dimensional, with a ab pair of variables (qab(x), p (x)) per every point x on the hypersurface Σ, and it carries the Poisson structure

′ ab cd ′ {qab(x), qcd(x )} = {p (x), p (x )} = 0 , (II.1.12a) { ab ′ } a b (3) ′ p (x), qcd(x ) = δ(cδd)δ (x, x ) . (II.1.12b)

In order to check whether there are any additional constraints to impose, one computes the Poisson brackets between the constraints (II.1.10) or equivalently between the smeared versions of the constraints as they appear in the II.1 Hamiltonian formulation of GR and Ashtekar-Barbero variables 13

Hamiltonian in equation (II.1.8), namely ∫ a 3 a Ca(N ) := d x N (x)Ca(x) , (II.1.13) ∫Σ C(N) := d3x N(x)C(x) . (II.1.14) Σ

One then ﬁnds

{ a b } − L ⃗ a Ca(N ),Cb(M ) = Ca([ M⃗ N] ) , (II.1.15a) { a } L C(N),Ca(M ) = C( M⃗ N) , (II.1.15b) ab {C(N),C(M)} = s Ca(q [NM,b − MN,b]) . (II.1.15c)

The equations (II.1.15) form what is called the Dirac algebra of the theory a and show that the scalar constraint C(N) and the vector constraint Ca(N ) are first class. This algebra is not a Lie algebra, this is because the structure coefficient in (II.1.15c) is not a constant but rather a function on the phase space, a structure function2. The constraint surface in the phase space is preserved under the transformations generated by the constraints (II.1.13): when the equations of mo- a tion are satisfied, in other words on the constraint surface (on–shell), Ca(N ) generates diffeomorphisms which preserve the hypersurface Σ while C(N) generates diffeomorphisms orthogonal to Σ. Therefore the full Hamiltonian a H := C(N) + Ca(N ) of vacuum general relativity manifests as a constraint which generates, on–shell, all diffeomorphisms of M . Generically, the Hamil- tonian H does not generate time translations and hence there is no notion of time in canonical general relativity. This peculiarity is inherent to any generally covariant theory and is known in general relativity as the problem of time. The issue is rather complex and has severe consequences in the quantum theory where one needs to define precise notions such as measurement and locality. There are several ideas and proposals in the literature, for instance [28, 29, 12, 30–33], on how one could resolve or circumvent this issue. We do not however present theses proposals here except of the so–called deparametrization approach which we discuss in chapter IV. Having a complete Hamiltonian formulation of general relativity, one would

2The presence of structure functions in the Dirac algebra may cause severe problems in the canonical quantizations of the constraints and the process of reduction to the physical Hilbert space. Those possible problems, described as anomalies in the quantum algebra, seem to not arise in LQG due to the methods used to implement the scalar and vector constraints as it is explained in section III.4. 14 On Dirac’s footsteps: canonical loop quantum gravity like to proceed with the canonical quantization of the theory. With ADM ab formulation, where the 3–metric qab and its conjugate momentum p are the phase space variables, it turns out that the canonical quantization program cannot be completed in a rigorous and fully background independent way due to various technical difficulties [13]. ADM formalism was the first Hamiltonian formulation of general relativity and it was very successful in developing new methods in order to investigate classical solutions, but it has unfortunately failed in clearing the path towards a quantum theory of gravity. It is a new Hamiltonian formulation, based on connection variables and introduced by A. Ashtekar [17](1986), which finally allowed to elegantly perform major steps in the canonical quantization of gravity, leading to what is nowadays known as the loop quantum gravity program [11–14]. The next section is a brief overview of this Hamiltonian formulation of general relativity in so–called Ashtekar– Barbero variables.

II.1.2 General relativity in Ashtekar-Barbero variables

The Hamiltonian formulation of general relativity in terms of Ashtekar–Barbero variables [17, 18, 34] can be understood as an extension of the ADM phase space along with a change of coordinates on this extended phase space through a canonical transformation. The construction can be carried out as follows. i ∈ { } 3 We ﬁrst introduce the co–triad ﬁeld ea (i 1, 2, 3 ), a so(3) algebra valued 1–form on the hypersurface Σ, related to the 3–metric as

i j qab = δijeaeb . (II.1.16)

This relation is invariant under local SO(3) transformations acting on the co– triad field. Since δij is the identity matrix, the position of the algebra indexes is of no importance and we most of the time omit writing the metric δij or its inverse in the equations. The co–triad field admits an inverse, the triad field ea defined by the equations eaei = δa and eaej = δj. We then introduce i √ i b√ b i a i a a | | a the densitized triad Ei := qei = det[E] ei which is a vector density of i weight one. Next we define the extrinsic curvature 1–form Ka by the following equation

i i − Kaeb := sKab . (II.1.17)

3Equivalently su(2) algebra valued, since the two algebras are isomorphic. II.1 Hamiltonian formulation of GR and Ashtekar-Barbero variables 15

i Because the tensor Kab is symmetric, Ka must satisfy

i i K[aeb] = 0 , (II.1.18) or equivalently ∫ ∫ i 1 3 i 1 3 i j a Gi(Λ ) := d x Λ (x)Gi(x) := d x Λ ϵijkKaEk = 0 . (II.1.19) k Σ k Σ where Λi is an arbitrary su(2) algebra valued function on Σ. ab The ADM variables (qab, p ) can be then expressed in terms of the quan- a i tities Ei and Ka in the following way

i i a j c b E E Ei Kc E E q = a b , pab = [i j] . (II.1.20) ab |det[E]| k |det[E]|

i a Considering the pair (Ka,Ei ) as the coordinates on the new extended phase space, the ADM phase space is recovered as a reduction of the extended phase space upon imposing the conditions (II.1.19). Moreover, implementing the Poisson brackets

{ i j ′ } { a b ′ } Ka(x),Kb (x ) = Ei (x),Ej (x ) = 0 , (II.1.21a) { i b ′ } i b (3) ′ Ka(x),Ej (x ) = kδjδaδ (x, x ) , (II.1.21b)

i 1 a one can show that the variables (Ka, k Ei ) are related to ADM variables ab (qab, p ) through a canonical transformation. The scalar and vector constraints in the new variables take the form ∫ −1 C (N a) = d3x N a[∇ (EbKi ) − ∇ (EbKi)] , (II.1.22) a k b i a a i b ∫ Σ i j a b √ s 2KaK E E C(N) = d3x N √ b [i j] − s |det[E]|R. (II.1.23) 2k Σ |det[E]|

We have then obtained a Hamiltonian formulation of general relativity, equivalent to the ADM formulation, with an extended phase space on which i 2 a the variables (Ka, k Ei ) form a pair of canonical coordinates and the constraint surface is recovered by solving the constraint (II.1.19), which generates inﬁnitesimal SO(3) rotations, the vector constraint (II.1.22) and the scalar constraint (II.1.23). The three constraints form a ﬁrst class constraints alge- 16 On Dirac’s footsteps: canonical loop quantum gravity bra and the full action of the system in the new variables is expressed as ∫ ∫ 1 i a a 3 ˙ i a − i a S[Ka,Ei ,N,N ] = dt d x KaEi (Λ Gi + NC + N Ca) . (II.1.24) R Σ k

i Note that, geometrically, the 1-form variable Ka is not a connection, i.e. it does not transform in the adjoint representation of so(3) ≃ so(2) under the action of SO(3) rotations generated by the constraints Gi. In order to obtain i a connection variable, we introduce the spin connection Γa on Σ compatible i ∇ with the co–triad ea, and which extends the spatial covariant derivative to su(2) algebra valued tensors

1 Γi := − ϵ eb (∂ ej − Γc ej) , and D ei := ∇ ei + ϵ Γj ek = 0 . a 2 ijk k a b ab c a b a c ijk a b (II.1.25)

i i Since the connection Γa depends only on ea, its inverse and its derivatives, we have

{ i b ′ } Γa(x),Ej (x ) = 0 . (II.1.26)

i It is then possible to introduce a (generalized) connection Aa on Σ deﬁned as

i i i Aa := Γa + βKa , (II.1.27) where β is an arbitrary real number called the Barbero–Immirzi parameter [35, 18]. This connection satisﬁes the commutation rules

{ i j ′ } Aa(x),Ab(x ) = 0 (II.1.28a) { i b ′ } i b (3) ′ Aa(x),Ej (x ) = kβδjδaδ (x, x ) . (II.1.28b)

i 1 a It is then clear that the pair of variables (Aa, kβ Ei ) forms a canonical set of coordinates on the extended phase space, they are related to the variables i 1 a i (Ka, k Ei ) via a canonical transformation. The connection Aa is called the Ashtekar–Barbero connection. i 1 a The advantage of the Ashtekar–Barbero variables (Aa, kβ Ei ) is that now we have a connection deﬁned on the hypersurface Σ which can play the role of a conﬁguration variable. Therefore general relativity is formulated in a form similar to Yang–Mills theories, with a Hamiltonian consisting of a sum of the II.1 Hamiltonian formulation of GR and Ashtekar-Barbero variables 17 following constraints ∫ 1 i 3 iD a Gi(Λ ) = d x Λ aEi , (II.1.29a) kβ ∫Σ 1 C (N a) = d3x N aEbF i , (II.1.29b) a kβ i ab Σ ∫ ( ) a b k √ 1 ϵijkE E F ( ) 3 √ i j ab − 2 | | C(N) = 2 d x N + 1 sβ det[E] R , 2skβ Σ | det[E]| (II.1.29c)

D i where a is the extended covariant derivative associated to the connection Aa, k k i j while Fab := 2∂[aAb] + ϵijkAaAb, is the curvature of this connection. Those constraints form a ﬁrst class constraints algebra

i ′j ′ k i {Gi(Λ ),Gj(Λ )} = Gk([Λ, Λ ] ) , {Gi(Λ ),C(N)} = 0 { i a } − L j { a } L Gi(Λ ),Ca(N ) = Gj( N⃗ Λ ) , C(N),Ca(M ) = C( M⃗ N) { a b } L ⃗ a { } a Ca(N ),Cb(M ) = Ca([ N⃗ M] ) , C(N),C(M) = s Ca(Z ) , (II.1.30)

a ab where Z := q [NM,b − MN,b] . To summarize, starting from the ADM Hamiltonian formalism, one can formulate general relativity in the form of a Yang–Mills theory where the conﬁgu- ration variable is the (real-valued) Ashtekar–Barbero connection, parametrized by the real Barbero–Immirzi parameter4 β. The main diﬀerence with respect to standard Yang–Mills theories is that the Hamiltonian is not a real Hamil- tonian which generates a time evolution, but it is a sum of constraints which i 1 a must be imposed on the Ashtekar–Barbero canonical variables (Aa, kβ Ei ): the Gauss constraint Gi (II.1.29a) which imposes SU(2) gauge invariance and reduces the (Ashtekar–Barbero) extended phase space to the ADM phase space, then the vector constraint Ca (II.1.29b) and the scalar constraint C (II.1.29c) which recover their interpretation as generators of spatial and orthogonal diffeomorphisms respectively once they are solved. The Poisson algebra of these

4Classically, the Barbero–Immirzi parameter can be allowed to be an arbitrary complex number without losing the equivalence between the ADM formulation and the Ashtekar– Barbero connection formulation. However, the canonical quantization of the theory with a complex connection could not be achieved so far. There are two reasons for this obstruction: on one hand, with a complex connection the internal gauge group becomes the non–compact group SL(2, C) (instead of SO(3)), and representation theory on such group as well as functional analysis on spaces of connections with non–compact groups are not developed enough to proceed with the quantization. On the other hand, with complex variables for general relativity, one obtains the so–called reality conditions for the metric and its momentum which are technically difficult to implement in the quantum theory. 18 On Dirac’s footsteps: canonical loop quantum gravity constraints is first class with structure functions. In the next section we present a condensed overview of the canonical quantization of the Ashtekar–Barbero Hamiltonian formulation of general relativity. The resulting theory is what is called the loop quantum gravity program. We reserve a short discussion at the end of that section to the question of implementing the scalar constraint C(N) in (II.1.29c) in LQG. In that discussion we focus on presenting, without going through too much technicalities, the old approach to define the quantum scalar constraint. We also discuss the main criticism towards this approach and the core motivation of our work presented in detail in chapters III and IV.

II.2 Canonical quantization: LQG

Naturally, the canonical quantization of general relativity requires the elabo- ration of a background–independent quantization procedure. While the quantization of the connection formulation presented above is very similar to standard quantum mechanics, the main difference arises from the presence of the infinite number of degrees of freedom, which for instance imposes certain modifications in the construction of the kinematical Hilbert space. Moreover, the distributional (singular) nature of the Poisson bracket (II.1.28b) between the Ashtekar–barbero variables entails to consider a specific set of smeared variables, which must still separate the points in the extended phase space. The form of the smearing however cannot be arbitrary: the chosen set of smeared variables must, on one hand, have closed Poisson brackets, on the other hand, must transform in a simple manner under SU(2) gauge transformations. Those restrictions are not particular to the Ashtekar–barbero connection formulation, they extend to any non–Abelian Yang–Mills theory. The solution to this issue is to choose the so–called holonomies (Wilson loops) and fluxes variables which we define later. These smeared variables generate a Lie algebra known as the holonomy–flux algebra [36–38]. The quantization strategy applied in LQG [11–14] can be outlined as follows:

1. The classical conﬁguration space A is chosen to be the space of all smooth i ⊗ a i su(2) algebra-valued 1-forms A := Aaτi dx , where Aa is the Ashtekar– 5 Barbero connection and τi ∈ su(2) is a basis of the algebra su(2);

5 (l) In this manuscript, we choose the basis τi = τi (i = 1, 2, 3) of the algebra su(2) in a II.2 Canonical quantization: LQG 19

2. One then considers the space Cyl of all cylindrical functions on A, i.e. functions of the variable A which have the form of a ﬁnite product of holonomies.

3. The Haar measure on SU(2) induces a natural measure on the space of cylindrical functions Cyl, which allows to deﬁne a natural Hermitian inner product on Cyl;

4. The kinematical Hilbert space Hkin is then deﬁned as the completion of the space Cyl with respect to the natural inner product on Cyl;

5. The holonomies and ﬂuxes admit a simple representation on Cyl, and

consequently on Hkin, as multiplicative and derivative operators respectively. Upon certain requirements, a phase space function can be pro-

moted to a well–defined operator on Hkin obtained through an adapted regularization procedure, the final operator is expressed as a function of holonomy and flux operators;

6. Finally, the constraints (II.1.29) are implemented by deﬁning either quantum constraint operators or group averaging procedures. One manages to solve the Gauss constraint and the vector constraint, however solving the quantum scalar constraint remains out of reach and the complete structure of the physical Hilbert space is so far unknown.

In the remaining parts of this section, we present in more detail the steps of the construction outlined above and we close with a discussion of few problems and criticism concerning the dynamics in LQG.

II.2.1 Holonomy-ﬂux algebra and kinematical Hilbert space

As mentioned above, it is necessary to introduce holonomies and ﬂuxes as classical variables on the extended phase space. Given a su(2) algebra-valued connection A, a holonomy he ∈ SU(2) of A along an oriented path e is deﬁned given representation l such that

( ) ( ) W 2 Tr τ (l) = 0, Tr τ (l)τ (l) = l δ , (II.2.1) i i k 3 ik √ where Wl := i l(l + 1)(2l + 1). 20 On Dirac’s footsteps: canonical loop quantum gravity as the solution to the equation

d h [A] = h [A]Ai (e(s))τ e˙a(s) , (II.2.2) ds e(s) e(s) a i where s ∈ [0, 1] denotes a parametrization of the path e. The holonomy he then takes the form ( ∫ )

he[A] = P exp − A , (II.2.3) e where P stands for the path ordering which orders the smallest path parameter to the left. The equation (II.2.3) shows that the smearing of the connection is made in one dimension which is natural for a 1-form. The holonomy transforms under SU(2) gauge transformations as

−1 he[Adg(A)] = g(e(0))he(A)g (e(1)) , (II.2.4) where g ∈ SU(2) and Ad is the adjoint action of the Lie group SU(2) on its algebra, and it satisﬁes the properties

− 1 ′ ′ he = he−1 , he◦e = hehe . (II.2.5)

In contrast, the flux is defined as a smearing in two dimensions, i.e. along a surfaces, of the densitized triad Ei . While it is quite natural to smear a vector density in two dimensions, it is in this case necessary in order to obtain a closed holonomy–flux Poisson Bracket algebra. Let S ⊂ Σ be an oriented 2-dimensional surface and ξ : S → su(2) be a smearing function which may involve holonomies. The flux PS,ξ corresponding to the vector density E is then defined as ∫ 1 b ∧ c i a PS,ξ := dx dx ϵabcξ (x)Ei (x) . (II.2.6) kβ S

The flux defined by equation (II.2.6) transforms in a non–local way under the action of SU(2) gauge transformations. From now on, we assume that the hypersurface Σ is analytic, and all curves and surfaces used to construct the holonomies and fluxes are analytic. The Poisson brackets algebra for holonomies and fluxes is highly non trivial to compute, it requires a certain regularization procedure of those classical variables and careful considerations relative to the intersections between the curves associated to holonomies and the surfaces associated to fluxes. While the Poisson bracket of two holonomies vanishes, the Poisson bracket of two II.2 Canonical quantization: LQG 21

fluxes does not vanish as one may naively expect. The reason is that fluxes incorporate vector fields acting on the space of holonomies, hence the Poisson bracket of fluxes is induced by the su(2) Lie–bracket of the vector fields. The 6 third Poisson bracket between a holonomy he and a flux PS,ξ, assuming that the curve e and the surface S intersect at most at one of the end points of the curve e, otherwise they do not intersect, can be expressed as follows { ζ(S, e) −h τ ξi(p) , if p = e(0); {h ,P } = × e i (II.2.7) e S,ξ i 2 ξ (p)τihe , if p = e(1), where   0 , if e and S do not intersect; ζ(S, e) = 1 , if e lies above S; (II.2.8)  −1 , if e lies below S.

After introducing the holonomies and fluxes, we define the classical configuration space A as the space of all smooth su(2) algebra-valued 1-forms i ⊗ a i ∈ A := Aaτi dx , where Aa is the Ashtekar–Barbero connection and τi su(2) is a basis of the algebra su(2). We then consider the space Cyl of all cylindrical functions on the classical configuration space A, i.e., complex valued functions depending on a su(2)-valued 1-form A through finitely many holonomies

∈ Ψ[A] Cyl, Ψ[A] = ψ(he1 [A], . . . , hen [A]) , (II.2.9) with a function ψ : SU(2)n → C. For a given cylindrical function, we call the curves associated to the considered holonomies edges and the set of edges the graph of the function. A cylindrical function with a graph Γ is said to be cylindrical with respect to the graph Γ. More precisely, a graph Γ is a ﬁnite set of compact 1-dimensional sub-manifolds of Σ, called the edges of Γ, such that every edge is either an embedded interval with boundary (an open edge with 2 end–points), or it is an embedded circle with a marked point (a closed edge with an end point), or an embedded circle. If an edge intersects any other edge of Γ it does so only at one or both of its endpoints. The end points of an edge are called vertices of the graph Γ and a vertex at which n edges meet is called a n–valent vertex or a vertex of valence n. Since we assume that the curves of the holonomies are analytic, the graphs associated to cylindrical functions

6More general cases of intersection between the curve and the surface can be treated by breaking them down to the special case shown above using the properties (II.2.5). 22 On Dirac’s footsteps: canonical loop quantum gravity are analytic at every point. We denote the space of all cylindrical functions with respect to a graph Γ by CylΓ. Note that a cylindrical function which is cylindrical with respect to a graph Γ is also cylindrical with respect to any other graph which contains Γ as a sub–graph. It follows that

⊆ ′ ⇒ ⊆ Γ Γ CylΓ CylΓ′ . (II.2.10)

Thus the space Cyl can be expressed as ∪ Cyl = CylΓ , (II.2.11) Γ where the union is over all possible analytic graphs. The space of all cylindrical functions Cyl turns out to be isomorphic to the space of square integrable functions on the space A of discontinuous su(2) algebra-valued 1-forms A, with respect to a measure induced from the Haar measure on SU(2). The space A represents an extension of the space A and it is usually called the quantum configuration space. Additionally, from the equation (II.2.7) it is clear that the fluxes form a family of vector fields on the space Cyl and they act as derivative operators on cylindrical functions. The algebra generated by smooth cylindrical functions and the fluxes acting on them form a Lie algebra called the holonomy–flux algebra [36–38]. A Hermitian inner product between cylindrical functions is introduced as follows: given two cylindrical functions Ψ and Ψ′, we define an embedded ′′ { ′′ ′′ } 7 graph Γ = e1, . . . , en′′ in Σ such that both functions can be written as

Ψ[A] = ψ(he′′ [A], . . . , he′′ [A]) , 1 n′′ ′ ′ (II.2.12) Ψ [A] = ψ (he′′ [A], . . . , he′′ [A]) . 1 n′′

In other words the functions Ψ and Ψ′ are cylindrical with respect to Γ′′. The scalar product between Ψ and Ψ′ is then deﬁned as ∫ ⟨Ψ|Ψ′⟩ := dµ ΨΨ′ (II.2.13) ∫ ′ := dg1 . . . dgn′′ ψ(g1, . . . , gn′′ ) ψ (g1, . . . , gn′′ ) , where dg stands for the Haar measure on SU(2). The scalar product between two cylindrical functions is independent of the chosen graph Γ′′, with respect

7The existence of Γ′′ is ensured by assuming the analyticity of Σ and of the edges of the graphs [11]. II.2 Canonical quantization: LQG 23 to which both functions are cylindrical. This independence is encoded in the measure dµ which is said to be a cylindrically consistent measure. For the moment, let us look more closely at the nature of a cylindrical function: each cylindrical function is associated to a graph Γ and to each edge of this graph is associated a SU(2) holonomy. A holonomy is a group element expressed in a chosen SU(2) representation j (positive half–integers), which (j) is the representation of the su(2) algebra generator τi (II.2.1) present in the definition of the holonomy. Holonomies can be then represented as square matrices8 and the product of holonomies, in the expression of the cylindrical function, means then that there is a contraction performed between the group elements. From the graph perspective, two holonomies are contracted with each other means that the edges associated to those two holonomies meet at a vertex of Γ. A holonomy with indexes contracted among each other means that this holonomy is associated to an embedded circle. The contraction of the matrices is then realized using SU(2) tensors, which are associated to the vertices of the graph Γ. To summarize, a cylindrical function can be fully characterized by a graph, a set of SU(2) representations called spins, associated to the edges of the graph, and a set of SU(2) tensors called intertwiners, associated to the vertices of the graph. Note that given a graph, the set of all possible sets of spins which can be associated to its edges is countable. Also, given a fixed set of spins associated to the edges, the space of possible SU(2) intertwiners is a finite dimensional vector space. With this characterization, the inner product (II.2.13) of two (normalized) cylindrical functions ΨΓ,{j},{ι} ′ and ΨΓ′,{j′},{ι′}, which do not contain trivial holonomies in their expressions, gives

′ ⟨ | ⟩ ′ ′ ′ ΨΓ,{j},{ι} ΨΓ′,{j′},{ι′} =δΓ,Γ δ{j},{j }δ{ι},{ι } , (II.2.14) where Γ, Γ′ represent the graphs, {j}, {j′} represent the sets of non–vanishing spins and {ι}, {ι′} represent the sets of SU(2) intertwiners. Now we can turn to deﬁning the kinematical Hilbert space of LQG. One ′ starts ﬁrst by introducing the Hilbert space H Γ associated to an analytic graph Γ

H ′ Γ := CylΓ , (II.2.15)

8 A holonomy he in the 0 representation is a trivial holonomy, i.e. he = 1. Therefore, given a graph Γ′ and a function Ψ in Cyl which is cylindrical with respect to a graph Γ ⊂ Γ′, Ψ can be made cylindrical with respect to the graph Γ′ by introducing trivial holonomies in the expression of Ψ such that those trivial holonomies are associated to the edges in Γ′\Γ. 24 On Dirac’s footsteps: canonical loop quantum gravity

namely, the Cauchy completion of the space CylΓ with respect to the inner product (II.2.13) on Cyl. This same inner product deﬁnes naturally a ′ ′ scalar product on H Γ. One then deﬁnes the Hilbert space HΓ ⊂ H Γ as the complete sub–space of cylindrical functions which do not contain trivial holonomies. In doing so, one has

′ Γ ≠ Γ ⇒ HΓ ⊥ HΓ′ . (II.2.16)

Finally, one defines the kinematical Hilbert space of LQG as ⊕ Hkin = HΓ , (II.2.17) Γ where the direct sum is over all possible analytic graphs. On each Hilbert space HΓ, one can introduce an orthonormal basis whose elements consist of normalized cylindrical functions associated to the graph Γ such that each of them is characterized by a specific choice of coloring, i.e. a specific set of spins {j} and SU(2) intertwiners {ι}. Therefore, we obtain an orthonormal basis on the kinematical Hilbert space Hkin called the spin–network basis and we call its elements spin–network functions. The holonomy–flux algebra induces the algebra of elementary quantum operators acting in the kinematical Hilbert space Hkin as follows: every cylindrical function Ψ is also a multiplication operator

(Ψ([A)Ψ′)[A] = Ψ[A]Ψ′[A] . (II.2.18)

The quantum flux operators are derivative operators obtained through the quantization of the classical flux defined in (II.2.6). The flux operators are essentially self–adjoint on Hkin and they are expressed as ℏ ∑ ∑ Pˆ = ξi(x) ζ(S, e)Jˆ , (II.2.19) S,ξ 2 x,e,i x∈S e where e runs through the germs9 beginning at a point x ∈ S, and ζ(S, e) is ˆ the coefficient defined in (II.2.7). The operator Jx,e,i is assigned to a pair (x, e) and is equal to the su(2) left–invariant vector field Le,i if x = e(0), or to the su(2) right–invariant vector field Re,i if x = e(1). The action of the algebra invariant vector fields on the function Ψ ∈ Cyl defined in (II.2.9), with for

9A germ beginning at a point x is the set of curves overlapping on a connected initial segment containing x. II.2 Canonical quantization: LQG 25

instance e1 belonging to the germ e, is given by

d d − L ϵτi R ϵτi e,iΨ = i ψ(hee , he2 , ...) , e,iΨ = i ψ(e he, he2 , ...) . dϵ ϵ=0 dϵ ϵ=0 (II.2.20)

In conclusion, the operators compatible with the LQG structure of Hkin are functions of the holonomy and flux operators. In order to be promoted to a well–defined operator on Hkin, every function on the extended phase space must first be expressed as a well–defined limit of a regularized functional which depends only on holonomies and fluxes, then each of those holonomies and fluxes is replaced by the corresponding quantum operator, and when ordered properly, those combined elementary operators lead to a final operator consistent with the original phase space function10.

II.2.2 Implementation of the Gauss and spatial diﬀeo- morphism constraints

II.2.2.1 SU(2)–gauge invariance

ˆ i The Gauss constraint operator Gi(Λ ) corresponding to the functional in equation (II.1.29a) can be easily deﬁned in terms of ﬂuxes. Given a cylindrical ˆ i function ΨΓ, Gi(Λ ) acts as ℏ ∑ ∑ Gˆ (Λi)Ψ = Λi(v) Jˆ Ψ . (II.2.21) i Γ 2 x,e,i Γ v∈Γ e

This operator is densely deﬁned and essentially self–adjoint on Hkin. Solv- ˆ i ing the constraint equation for Gi(Λ ) means constructing its kernel. As it happens, its kernel is identiﬁed with the space of gauge invariant cylindrical functions Ψ

Ψ(A) = Ψ(g−1Ag + g−1dg) , ∀ g ∈ C1(Σ,SU(2)) . (II.2.22)

We denote their algebra, a sub–algebra of Cyl, by CylG and the corresponding H G ⊂ H H G Hilbert space kin kin. A dense subspace of kin is spanned by the SU(2)– 10It turns out actually that with the described quantization procedure to promote classical phase space functions to quantum operators on Hkin, only functions of density weight one are potentially eligible to lead to well–defined operators. This is due to the fact that in the last step of the quantization procedure of a classical phase space function, one has to take the continuum limit, i.e. the regulator going to zero, however only functions of density weight one admit a well–defined and non–trivial continuum limit. 26 On Dirac’s footsteps: canonical loop quantum gravity gauge invariant spin–network functions. These are the spin–network functions with only SU(2)–gauge invariant intertwiners assigned to the vertices in their graphs. Then the Hilbert space of all gauge invariant states can be written as the orthogonal sum ⊕ H G H G kin = Γ , (II.2.23) Γ where now Γ ranges over all classes of closed analytic graphs11, for simplicity H G we call them admissible graphs, and Γ is the Hilbert space defined as the G completion of the space CylΓ spanned by the SU(2)–gauge invariant spin- network functions with the admissible graph Γ.

II.2.2.2 Implementing spatial diﬀeomorphisms

Let us now turn to the vector constraint (II.1.29b). Implementing the spatial diffeomorphism constraint is more subtle than the Gauss constraint. The diffeomorphisms we are interested in are the piecewise analytic diffeomorphism on the hypersurface Σ and we denote their group by Diff(Σ). Each element χ of Diff(Σ) maps Σ to itself and admits a natural action on the space Cyl: χ maps an admissible graph to another admissible graph and one can define its action on a holonomy as

χ(he) := hχ(e) . (II.2.24)

Hence χ deﬁnes a natural isomorphism on Cyl, and since the measure µ in (II.2.13) is diﬀeomorphism invariant, this isomorphism induces a unitary operator Uχ acting on Hkin as

ΨΓ ∈ HΓ ⊂ Hkin ,UχΨΓ := Ψχ(Γ) . (II.2.25)

Therefore the induced action of Diff(Σ) on Hkin preserves Hkin but does not preserve any of the sub–spaces HΓ. Going back to the spatial diffeomorphism constraint, imposing (II.1.29b) implies that we are primarily interested in the finite gauge transformations generated by the constraint (II.1.29b). Which means that in the quantum theory we are interested in states which are invariant under the action of all

11A closed graph is a graph with no one–valent vertices. Also, two graphs Γ and Γ′ belong to the same class if Γ′ can be obtained from Γ by a sequence of the following moves: splitting of an edge, connecting two edges, changing a orientation of an edge. II.2 Canonical quantization: LQG 27 the operators induced by the diffeomorphisms of Diff(Σ). However, since a generic Uχ, with χ ∈ Diff(Σ), does not preserve any of the sub–spaces HΓ which are orthogonal to each other, only the constant state Ψ = 1 ∈ Hkin is diffeomorphism invariant and the infinitesimal generator of a diffeomorphism χ fails to exist. To impose the vector constraint properly one has then to proceed in a rather different way and construct spatial diffeomorphism invariant states using the operators Uχ associated to finite diffeomorphisms. Those states must exist outside of the Hilbert space Hkin. The construction of the space of such states is achieved through a group averaging procedure [37]. The elements of each of the sub-spaces CylΓ are averaged with respect to all the diffeomorphisms in Diff(Σ) which map the graph Γ into other analytic graphs. But since Diff(Σ) is a non–compact group with no–known probability measure on it, we have to define the averaging in Cyl∗, the algebraic dual to Cyl. The final space is a Hilbert space of diffeomorphism invariant states with a scalar product naturally inherited from the scalar product on Hkin. The procedure goes as follows:

Given an admissible graph Γ, denote by TDiff(Σ)Γ the subset of Diff(Σ) consisting of all the diffeomorphisms χ which preserve every edge of Γ and its orientation. The operators induced by elements in TDiff(Σ)Γ are trivial on

HΓ. Let Diff(Σ)Γ be the set of diffeomorphisms in Diff(Σ) which preserve the graph Γ. Then we have that the quotient

GSΓ := Diff(Σ)Γ/TDiff(Σ)Γ , (II.2.26) forms the group of symmetries of the graph Γ which induce a non–trivial action on HΓ. We then start the construction of the spatial diffeomorphism invariant ∈ states by first averaging every state Ψ CylΓ with respect to the elements of

GSΓ ∑ 7→ 1 ΨΓ ΨΓ˜ := Uχ(ΨΓ) , (II.2.27) NΓ χ∈GSΓ where NΓ is the number of elements of GSΓ. Second, we deﬁne the averaging with respect to the remaining diﬀeomorphisms which do not preserve the graph

Γ. This averaging map, which we denote η, sends the states ΨΓ to the space 28 On Dirac’s footsteps: canonical loop quantum gravity

∗ ∈ Cyl , the dual space to Cyl. Given a state ΨΓ CylΓ, we have ∑ ∗ η(ΨΓ) = [Uχ(ΨΓ˜)] , (II.2.28)

χ∈Diﬀ(Σ)/Diﬀ(Σ)Γ

∗ ∈ ∗ 12 where [Uχ(ΨΓ˜)] Cyl is a dual state . The resulting η(ΨΓ) in (II.2.28) is a well–deﬁned linear functional

η(ΨΓ): Cyl → C (II.2.30) ′ ′ Ψ → η(ΨΓ)(Ψ ) , because given Ψ′ ∈ Cyl, only a ﬁnite set of terms in the sum contribute to ′ the number η(ΨΓ)(Ψ ). Note that the functional η(ΨΓ) is a distribution on A rather than a function. Hence we have deﬁned a map

∋ 7→ ∈ ∗ CylΓ ΨΓ η(ΨΓ) Cyl , (II.2.31) for every admissible graph Γ in the the decomposition (II.2.23). This map is then extended by linearity to the kinematical Hilbert space Hkin,

∗ η : Hkin −→ Cyl . (II.2.32)

Every state η(Ψ) ∈ Cyl∗ is diﬀeomorphism invariant

′ ′ η(Ψ)(Uχ(Ψ )) = η(Ψ)(Ψ ) , ∀χ ∈ Diﬀ(Σ) . (II.2.33)

∗ Hence, the space of spatial diffeomorphism invariant states, denoted CylDiff, is defined as

∗ CylDiﬀ := η(Cyl) . (II.2.34)

∗ The averaging procedure naturally induces a Hermitian inner product on CylDiﬀ

⟨η(Ψ)|η(Ψ′)⟩ := η(Ψ)(Ψ′) , (II.2.35)

∗ and the completion of CylDiff with respect to this inner product defines the ∗ 12To every state Ψ ∈ Cyl we can associate a dual state Ψ∗ ∈ Cyl defined by its action on functions in Hkin, namely

∗ ′ ′ Ψ :Ψ ∈ Hkin 7→ ⟨Ψ|Ψ ⟩ ∈ C . (II.2.29) II.2 Canonical quantization: LQG 29

Hilbert space of spatial diﬀeomorphism invariant states HDiﬀ,

H ∗ Diﬀ := CylDiﬀ . (II.2.36)

Finally, since the action of the unitary diffeomorphism operators Uχ is SU(2)– gauge invariant, one can define the Hilbert space of SU(2)–gauge invariant and spatial diffeomorphism invariant states as

H G G Diff := η(Cyl ) , (II.2.37) that is the Cauchy completion of the image of CylG with respect to the inner product (II.2.35). This space is spanned by the SU(2)–gauge invariant and spatial diffeomorphism invariant spin-network functions which are now labeled by spatial diffeomorphism equivalence classes of admissible graphs, denoted { } H G Γ , instead of graphs, and one can decompose Diff as ⊕ ⊕ H G H G ∩ H G Diff = {Γ} := η(Cyl Γ ) , (II.2.38) {Γ} {Γ}

H G where the graph Γ, labeling Γ , can be any element of its diﬀeomorphism class {Γ}.

II.2.3 On the dynamics in loop quantum gravity

We now reach the last step in the quantization of the Ashtekar–Barbero classical theory, that is the implementation of the scalar constraint (II.1.29c) and the construction of the physical Hilbert space. The scalar constraint is often called the Hamiltonian of the theory and generator of the dynamics, the reason is that from the spacetime perspective the Gauss and spatial diffeomorphism constraints generate gauge transformations at a fixed time, they are often called the kinematical constraints. Therefore, the question of defining the quantum dynamics resides in a proper implementation of the scalar constraint and defining the space of its Gauss and spatial diffeomorphism gauge invariant solutions which can be promoted to a Hilbert space, the physical Hilbert space. The fact that the algebra of the classical constraints (II.1.30) is not a Lie–algebra prevents one from using group averaging techniques in order to implement and solve the scalar constraint. One is then forced to proceed in a similar way to the Gauss constraint, namely deriving a well–defined operator from the classical expression (II.1.29c) through a precise regularization procedure, then trying to construct the elements of its kernel which are solu- 30 On Dirac’s footsteps: canonical loop quantum gravity tions to the Gauss and vector constraints and finally promote this solutions space to a Hilbert space. While several proposals for the scalar constraints operators are available [39, 24, 40] with a relatively accessible kernel space, the last step of constructing the physical Hilbert space is not yet achieved for certain technical reasons that we mention below. The first concrete proposal for a scalar constraint operator with a detailed analysis of its properties was made by T. Thiemann in a collection of papers [39, 41, 42]. This regularization procedure was inspired from many ideas which were proposed in the literature by various authors [43–45], and later on this first concrete procedure became a base to construct other regularization procedures that led to different eligible implementations of the quantum dynamics with relatively different properties of the resulting operators, among which was the Master constraint program which we briefly discuss in a paragraph of the following section. Nevertheless, the scalar constraint operator obtained from this old approach, as all other proposals, suffered various issues, some of which are the main motivation of our work presented in the two following chapters. In the following we briefly present the main steps in the old construction of a scalar constraint operator and a summary of its properties, we then give a very short overview of the Master constraint program and we conclude the section with some comments concerning the aforementioned proposals, in link with our motivation to build a different regularization procedure.

II.2.3.1 Thiemann’s scalar constraint operator

In this approach [39, 41, 42], one starts by writing the scalar constraint in (II.1.29c) in a slightly diﬀerent form, namely by expressing the Ricci scalar in terms of Ashtekar–Barbero variables. After some calculations, one ﬁrst obtains the expression ∫ C(N) = d3x N(x) C(x) (II.2.39) Σ ∫ ( ) a b 1 ϵijkE (x)E (x) 3 √ i j k − 2 m n = d x N(x) Fab(x) + (s β )ϵkmnKa Kb . 2k Σ | det[E](x)|

m The next step consists of expressing the 1–form Ka in terms√ of the variables m a | | Aa and Em, then getting rid of the singular factor 1/ det[E](x) . These two objectives are achieved using the following identities, usually referred to II.2 Canonical quantization: LQG 31 as Thiemann’s tricks ∫ 1 m { m } 3 i a { E } Ka = Aa (x),K ,K := d x KaEi = C (1),V (II.2.40) kβ Σ a b ϵijkE (x)E (x) 4 √ i j = ϵabc{Ak(x),V } , (II.2.41) | det[E](x)| kβ c where V is the volume of Σ, ∫ √ V := d3x | det[E](x)| , (II.2.42) Σ and CE(N) is a function on the phase space deﬁned as ∫ a b 1 ϵijkE (x)E (x) CE(N) = d3x N(x) √ i j F k (x) (II.2.43) 2k | | ab ∫Σ det[E](x) 2 3 abc k { k } = 2 d x N(x)ϵ Fab(x) Ac (x),V . k β Σ

C(N) then takes the form

C(N) = CE(N) + CL(N) , (II.2.44) where ∫ 2(s − β2) L 3 abc { k }{ m }{ n } C (N) := 4 3 d x N(x)ϵ ϵkmn Ac (x),V Aa (x),K Ab (x),K . k β Σ (II.2.45)

The procedure to promote this expression to a quantum operator breaks down roughly to the following steps:

• Deﬁne an operator Vˆ corresponding to the quantity V in (II.2.42);

• Deﬁne operators corresponding to the connection variable A and its cur- k ⊗ a ∧ b vature F := Fabτk dx dx ;

• Replacing every Poisson bracket in the expression by 1/iℏ times the commutator of the operators corresponding to the classical quantities in the Poisson brackets;

• Following the previous three steps one is able, up to certain regularization adjustments, to deﬁne an operator CˆE(N) corresponding to the quantity CE(N) and therefore an operator Kˆ corresponding to the quantity K; 32 On Dirac’s footsteps: canonical loop quantum gravity

• Deﬁne the operator CˆL(N) corresponding to the quantity CL(N) using holonomies and the operators Vˆ and Kˆ ;

• Finally one combines the operator CˆE(N) and the operator CˆL(N) to obtain a well–deﬁned operator Cˆ(N) corresponding to the scalar constraint C(N).

Clearly, what is concretely necessary to define the operator Cˆ(N) is to define the operators Vˆ and CÊ(N). Since the quantization of the volume is presented in section III.2.2.1, we focus here only on the details of constructing the operator CÊ(N). As mentioned before, the first step in the quantization of a function on the extended phase space is achieved by writing the function as a limit of a regularized functional which is expressed in terms of holonomies and fluxes. The connection variable A and its curvature F are regularized using holonomies as follows 1 1 A = (1 − h (A)) + O(ϵ) ,F = (1 − h (A)) + +O(ϵ′2) , (II.2.46) ϵ e ϵ′2 α where ϵ > 0 and ϵ′ > 0 are respectively the coordinate sizes of a curve e and a closed loop α, and 1 is the identity element on the group. One can then write ( ) 1 abc k { k } (l) (l) { (l) −1 } ϵ Fab(x) Ac (x),V = lim Tr hα hs hs ,V (E) , (II.2.47) ϵ→0 2 IJ K K Wl

(l) where Tr stands for the trace on the group,√ h represents a holonomy in a chosen SU(2) representation l, Wl := i l(l + 1)(2l + 1) is a normalization (l) factor resulting from the normalization of the basis τi (i = 1, 2, 3) of the su(2) algebra (II.2.1). The curve sK and the loop αIJ are so far chosen arbitrarily with a coordinate size ϵ. In order to define the full operator CÊ(N), one needs to elaborate a prescription to fix a consistent choice of those curves and loops. To this end, one first introduces a triangulation of the hypersurface Σ into tetrahedra ∆, characterized by a coordinate size ϵ, such that one can rewrite the integral in (II.2.43) as a limit of a Riemannian sum over the tetrahedra ∆, where each term in the sum is of the form of the regularized expression

(II.2.47). The choice of curves sK and loops αIJ in each of those terms is then appropriately adapted to each cell ∆. One then obtains II.2 Canonical quantization: LQG 33

E E C (N) = lim Cϵ (N) (II.2.48) ϵ→0 ∑ ∑ ( √ ) 2 (l) (l) (l) −1 := lim N(x∆) ϵIJK (∆)Tr h h {h , |q∆(E)|} , ϵ→0 k2βW 2 αIJ (∆) sK (∆) sK (∆) l ∆ I,J,K where the smearing function N(x) is replaced by its values at certain points x∆, ϵIJK (∆) is a regularization coefficient which depends on the structure of the tetrahedron ∆ and which vanishes if two of the labels I, J, K coincide, and q∆(E) is the square volume functional (see equation (III.2.20)). To finally obtain an operator defined on the space Cyl, one has to adapt the triangulation of Σ to each of the graphs in the decomposition (II.2.11) of Cyl. The procedure is quite technical and it is in many aspects similar to the procedure we use in section III.3 to define a new scalar constraint operator. We therefore do not develop it here and we simply display the final result then stress the main difference with our procedure, which lies in the assignment of the loops αIJ . Ê One obtains the quantum operator Cϵ (N) corresponding to the quantity E Cϵ (N) by simply replacing the holonomies and fluxes by the corresponding operators. Namely, given a cylindrical function ΨΓ with a graph Γ one has

2 ∑ 8 CÊ(N)Ψ := N(v) (II.2.49) ϵ Γ ℏ 2 2 i k βWl ∈ E(v) v Γ∑ ( ) − × ϵ(e ˙ , e˙ , e˙ )Tr h(l) h(l) [h(l) 1, Vˆ ] Ψ , I J K αIJ (ϵ) eK (ϵ) eK (ϵ) v Γ I,J,K where E(v) is the number of unordered triples of edges meeting at a vertex v. Ê The ordering of the holonomy and flux operators in the expression of Cϵ (N) is rather unique because any other ordering leads to an ill–defined operator (divergent action) on Cyl. The assignment of the loops is chosen such that

• each loop αIJ is associated to a triple of edges (eI , eJ , eK ) meeting at a vertex v in a diﬀeomorphism invariant way and it is oriented according

to the orientation of the triple (eI , eJ , eK );

• each loop αIJ partially overlaps with the wedge determined by the vertex

v and the pair of edges (eI , eJ ) and can be deﬁned as αIJ := sI ◦ sIJ ◦ sJ ,

where the segments sI and sJ are respectively segments of the edges eI

and eJ which contain the vertex v but not the end points of the edges 34 On Dirac’s footsteps: canonical loop quantum gravity

eI and eJ (sI ⊊ eI , sJ ⊊ eJ , sI ∩ sJ = v), and the segment sIJ , called

extraordinary edge, is an analytic curve linking the end point of sI with 13 the end point of sJ without intersecting any other edge of the graph Γ.

Because these loops partially overlap with the edges of the graph, by simple rules of spins re–coupling theory [46], the representations of the holonomies obtained in the image state and associated to the segments sI and sJ differ from the original representations j and j′ respectively associated to the edges eI and eJ in ΨΓ. The new representations associated to the segments sI and ′ sJ range in the intervals bounded by j l and j l respectively. It is therefore suitable to fix the representation of the holonomies associated to the loops αIJ to the minimal non–trivial one, that is the 1/2 representation. The last step in defining the operator CÊ(N) is to take the limit of the → ∗ regulator ϵ 0. As one cannot pass this operator to the space CylDiff, the limit ϵ → 0 must be taken in the space Cyl with an appropriate choice of (operator) topology. It turns out that one can define a suitable topology, called URST for Uniform Rovelli–Smolin Topology, in which the limit exist and is Ê ′ non–trivial. It is obtained by considering quantities of the form Ψ(Cϵ (N)Ψ ), ∈ ∗ ′ ∈ where Ψ CylDiff and Ψ Cyl, which prove to be independent of the value of the regulator ϵ. Therefore in the URST one obtains

Ê Ê Ê C (N) := lim Cϵ (N) = Cϵ (N) , (II.2.50) ϵ→0 0

Ê where ϵ0 is an arbitrary but fixed positive number. This final operator C (N) is defined on the space Cyl and not the dual space Cyl∗, then it can be extended by linearity to the kinematical Hilbert space Hkin. Finally, one can define the scalar constraint operator Cˆ(N) on the Hilbert space Hkin as

Cˆ(N) := CˆE(N) + CˆL(N) , (II.2.51) where the operator CˆL(N) is deﬁned through its action on a cylindrical function ΨΓ as

13The existence of such analytic curve is guaranteed by a rooting procedure (see [39]), which determines how the extraordinary edge is embedded with respect to the remaining edges of the graph. II.2 Canonical quantization: LQG 35

2(s − β2) ∑ 8 ∑ CˆL(N)Ψ := − N(v) ϵ(e ˙ , e˙ , e˙ ) (II.2.52) Γ iℏ3k4β3W 3 E(v) I J K ( 1/2 v∈Γ I,J,K ) × Tr h(1/2)[h(1/2) −1, Kˆ ]h(1/2)[h(1/2) −1, Vˆ ]h(1/2)[h(1/2) −1, Kˆ ] Ψ , eJ eJ eI eI eK eK Γ such that 1 Kˆ := [CˆE(1), Vˆ ] . (II.2.53) iℏ

• Properties of Thiemann’s scalar constraint operator: The operator Cˆ(N) is a non–symmetric densely deﬁned operator on the space

Hkin and it creates extraordinary edges with new 3–valent vertices at their end points. This scalar constraint operator is SU(2)–gauge invariant and therefore H G preserves the space kin. However, under the action of spatial diffeomorphisms, it transforms covariantly and therefore does not preserve spatial dif- H G feomorphism invariant states. Hence it cannot be defined on the space Diff. This is one of the motivations which led to the Master constraint program summarized in a following paragraph. Additionally, the scalar constraint operator Cˆ(N) is anomaly free on–shell in the sense that the commutator [Cˆ(N), Cˆ(M)], when acting on a state in Hkin, produces two diffeomorphism equivalent states, i.e. we have

G ′ ′ ∀ ∈ H ∀ ∈ H ˆ ˆ ′ Ψ Diﬀ , Ψ kin : Ψ([Cv, Cv ]Ψ ) = 0 . (II.2.54)

In other words, the commutator [Cˆ(N), Cˆ(M)] vanishes with respect to URST, but not on Hkin (off–shell). Since the classical scalar constraint functional is an observable, it is generally assumed that a quantum operator corresponding to it must be self–adjoint. However the operator in (II.2.51) is not even symmetric and it was argued in [47, 48, 39] that it is not necessary to have a self-adjoint constraint operator. The argument is that since one will be looking for the kernel of such operator, it may not be relevant to have it self-adjoint as long as zero belongs to its spectrum. It was also shown in [47, 48] that in the case of an open constraints algebra with structure functions, self–adjoint constraints operators may lead to anomalies in the quantum constraints algebras. The proof, however, is based on a specific choice of ordering of the operators corresponding to the structure functions and the constraints operators, and it does not generalizes to arbitrary ordering. Moreover, as discussed in section II.2.2.2, there is no 36 On Dirac’s footsteps: canonical loop quantum gravity operator associated to the spatial diffeomorphism constraint in LQG, therefore one practically cannot check fully the quantum constraints algebra off–shell. It then follows that one can consider a self–adjoint scalar constraint operator, on the same footing as the non–symmetric one, as long as it does not generate anomalies in the quantum constraints algebra on–shell. The question which remains is: is it possible to construct a symmetric operator, based on the regularization procedure summarized above, which could correspond to a valid quantization of the scalar constraint functional (II.2.39)? In order to answer this question one has to think of a symmetrization proce- ˆ dure, which would enable one to define a certain symmetric operator, Csym(N), as some kind of extension of the non–symmetric operator Cˆ(N). The most common approach, and the only one known to us, is to use the adjoint opera- tor14 Cˆ(N)† of the operator Cˆ(N). For instance, one can define the symmetric ˆ operator Csym(N) as

1 Cˆ (N) := (Cˆ(N) + Cˆ(N)†) , (II.2.55) sym 2 on a certain domain subset of Hkin. It turns out, however, that the domain of such operator is not dense in the Hilbert space Hkin. This is simply because ˆ † the adjoint operator C(N) is not densely defined on Hkin to begin with. ˆ † ˆ Hence neither the operator C(N) nor Csym(N) in (II.2.55) can be considered as eligible scalar constraint operators. The property of non–dense domain of the adjoint operator of Cˆ(N) is a direct consequence of the loops assignment in the old regularization. This obstruction in constructing a symmetric scalar constraint operator represents one of the criticism towards this regularization. We develop more on this point later in the last paragraph of this section. Note however that in the context of the Master constraint program, this same regularization procedure leads to a well-defined essentially self–adjoint Master H G constraint operator defined on Diff.

• The Master constraint program: The motivations for the Master constraint program [50, 51] are focused on tackling three main issues:

14Deﬁnition [49]: Let Tˆ be a densely deﬁned linear operator on a Hilbert space H . Let D(Tˆ†) be the set of φ ∈ H for which there is an η ∈ H with

(Tˆ ψ, φ) = (ψ, η) for all ψ ∈ D(Tˆ) .

For each such φ ∈ D(Tˆ†), we deﬁne Tˆ†φ = η. The operator Tˆ† is called the adjoint of Tˆ. II.2 Canonical quantization: LQG 37

1. The classical constraints algebra, with the scalar constraint C(N) (II.2.39), is not a Lie–algebra;

2. The scalar constraint operator is not self–adjoint;

H G 3. The scalar constraint operator cannot be defined on Diff. The idea is to first define a functional on the extended phase space which could replace the scalar constraint functional C(N). Such functional has, on one hand, to reduce the phase space to the same constraint surface as the functional C(N), on the other hand, to define the same set of weak observables on the constraint surface as C(N). Furthermore, one wants the classical algebra of the new functional, together with the Gauss and vector constraints, to form a Lie–algebra. Such functional, called the Master constraint M, was introduce for the first time in [50] and it takes the form ∫ C(x)2 M := d3x √ . (II.2.56) Σ | det[E](x)|

Imposing the constraint M = 0 is equivalent to imposing the constraint C(x) = 0, moreover one has

i {Gi(Λ ), M} = 0 (II.2.57a) a {M,Ca(M )} = 0 (II.2.57b) {M, M} = 0 . (II.2.57c)

The algebra involving the functional M is then trivial which fulfills the Lie– algebra requirement. The quantization of M can be realized following the same prescription followed in order to quantize C(N). The presence of C(x)2 in the definition of M allows to implement a natural symmetric ordering of the elementary operators. The final operator M^ is defined directly on HDiff. It is SU(2)–gauge H G invariant and essentially self–adjoint on Diff. Hence the quantum constraint algebra of M^ is anomaly free (on–shell).

• Comments about the old construction

• Self–adjointness of the scalar constraint operator: The property of non–dense domain of the adjoint operator of Cˆ(N) is a direct consequence of the loops assignment in the old regularization. A short explanation of this conclusion goes as follows: recall that loops in 38 On Dirac’s footsteps: canonical loop quantum gravity

this regularization overlap partially with the edges of a given graph. This overlap implies that the SU(2) representations associated to the overlap segments are shifted by 1/2. Consequently, given a spin–network state with one of the initial edges in its graph carrying a spin 1/2, one of the spin–network components of the image state, under action of Cˆ(N), has an edge with spin equal to zero. Cylindrical consistency however implies that a state with an edge with spin zero is equivalent to the same state without that edge. The removal of an edge is therefore a casual consequence of the action of the operator Cˆ(N) on certain spin–network states. Naturally, by removing an edge of a graph, one loses all diffeomorphism invariant information concerning the geometrical disposition of that edge at the corresponding vertex and its neighborhood. In other words, there are infinitely many diffeomorphism inequivalent configurations at a given vertex which produce the same graph–component with a removed segment. Let us illustrate this analysis in the following example. Consider two spin–network states consisting of one 4–valent vertex each, with the four edges carrying spins 1/2. We take the first state with a graph where the tangent vectors to the edges at the vertex are pairwise independent, and the second state with a graph where two edges are tangent at the vertex up to a fixed order. The action of CÊ(N) can be illustrated as follows

CˆE(N) = + + ...

= + ... + + ...,

CˆE(N) = + ...,

(II.2.58) II.2 Canonical quantization: LQG 39

The two initial states are diﬀeomorphically inequivalent but their images share certain components in the spin–network basis. Hence, the same component is generated for other (inﬁnitely many) orthogonal initial states. This particular aspect of the action of the operator Cˆ(N), which is the

existence of infinitely many diffeomorphism inequivalent states in Hkin which can be mapped to the same configuration with a certain edge removed, causes that a spin–network with the late configuration cannot belong to the domain of the adjoint operator Cˆ(N)†. Hence the domain ˆ † ˆ † of C(N) is not dense in Hkin and consequently discarding C(N) , or a possible symmetric extension using Cˆ(N)†, as an eligible operator for the quantum scalar constraint.

• Constraints algebra: The way one defines the anomaly freeness of the quantum constraints algebra is rather unsatisfactory. As we mentioned before, one can verify the closure of the algebra only on–shell, that is after solving the spatial diffeomorphism constraint. This is of course due to the absence of an operator representation of the generators of spatial diffeomorphisms. In the case of a scalar constraint operator, the closure of the commutator of two scalar constraint operator was shown to hold on a larger space than H G the space Diff, called the habitat space [52], and it is so far not clear if this result should be considered as an indication that the regularization procedure of the scalar constraint is somehow incorrect. The argument is that this regularization forces the algebra to close without a need to be on–shell, which could mean that the scalar constraint operator induces a smaller physical Hilbert space than it should. For this reason, there were many works [53–56] realized by M. Varadarajan, A. Laddha and al. in the direction of defining a new regularization procedure in which the a constraints Ca(N ) and C(N) would be implemented as operators on the same footing. In the case of the Master constraint operator, the operator M^ is directly

defined on the space HDiff and no off–shell closure of the algebra can be considered.

• Solutions and physical Hilbert space: In both cases of the scalar constraint operator and the Master constraint operator, there exist systematical methods to generate solutions to the 40 On Dirac’s footsteps: canonical loop quantum gravity

H G quantum constraints. One looks for solutions directly in the space Diﬀ satisfying

Ψ(Cˆ(N)Φ) = 0 or M^ Ψ = 0 , (II.2.59)

∈ H G ∈ H G where Ψ Diff and Φ kin. The set of all solutions Ψ forms the space of physical states. However, the systematical procedure which follows from the equations (II.2.59) provide only a partial control on the general form of the solutions, explicit general expressions for the solutions are not available and final form of the space of physical states is still unknown. There are also hints which point to the fact that physically interesting solutions may exist only in the form of non–normalizable states. If this is indeed the case, one would have to define an extension of the scalar

product on HDiﬀ which would allow to construct a larger Hilbert space of solutions to all quantum constraints, that is, the physical Hilbert space.

• Locality: From the expression of the the scalar constraint operator, one can see that this operator, as well as the Master constraint operator, act on the vertices of a given graph in a rather independent way. The constraint operator acts on each vertex separately, then the independent vertex contributions are summed to give the final state. This locality of the action of the operator is one of the main criticism against the regularization procedure used to define the scalar constraint operator. The claim is that such local action on a given spin–network state would not allow dynamical correlations between different vertices of the state and their close neighborhoods. This perspective is relatively inspired from lattice and discrete field theories. However, in our opinion, such view is too naive. The reason is that the notion of locality in lattice and discrete field theories is strongly tied to the fact that there is a fixed choice of background, which is totally opposite to the situation in LQG where the physical states are, on one hand, averaged with respect to spatial diffeomorphisms, on the other hand, they are generically in the form of a superposition of spin–network states. Hence, the notion of locality, as present in lattice field theories, is without meaning in LQG. Our view would be that one has to define, and then observe, locality through the dynamics of certain quantum observables which would be defined on possibly a physical Hilbert space. More arguments opposing this naive notion of locality can be found in [13]. II.2 Canonical quantization: LQG 41

• Ambiguities: One can argue that the old regularization procedure carries several ambiguities. Among them, three points are of central importance:

– The loop assignment: the choice of prescription to attach the additional loop may seem to be completely arbitrary, however the im- position of diffeomorphism invariance and the existence of the limit with respect to the regulator is highly restrictive. Generically, one can reduce the ambiguity of assignment to three cases: i) the loop overlaps completely with the original graph; ii) the loop overlaps partially with the original graph; iii) the loop intersects the graph at exactly one point. The first case can already be discarded by invoking the condition that one has to take the limit of the regulator, although in certain contexts, such as algebraic quantum gravity, one could drop this requirement. In the strict canonical approach however, one is left with options ii) and iii). With the two requirements mentioned above, option ii) is most consistently realized in the old prescription, nevertheless one still has a certain freedom in adjusting the diffeomorphism invariant characteristics of the extraordinary edge. Option iii) was actually the first formal proposal for the regularization of the scalar constraint to be considered in the literature [43], but there was so far no concrete implementation of such proposal till recently [25, 24]. This is partly the object of our work elaborated with our collaborators and presented in chapters III and IV of this dissertation. – The choice of representation: there is indeed no conclusive argument which would fix the representation on the added loops. The naive semi–classical analysis of the operator Cˆ(N), or M^ , does not show any dependence on the choice of the representation on the loops. Certain arguments, coming from 2 + 1 gravity and ordinary QFT, were presented in [13] arguing that spin 1/2 is a rather natural choice. However, one has to consider that perhaps only a, yet to be developed, complete semi–classical regime in the full theory could induce a selection principle determining the proper spin on the loops. – The ordering of the elementary operators in CÊ: in a canonical quantization of a classical phase space function, one always has to wonder about the ordering of the elementary operators in the final 42 On Dirac’s footsteps: canonical loop quantum gravity

expression of the quantum operator. Surprisingly and luckily, in the case of the operator CÊ, the regularization leaves one with a unique choice of ordering, which is the one presented above. The reason is that any other choice of ordering leads to an ill–defined operator. However, this is not the case for the operator CˆL (II.2.52). Interestingly, if one manages to implement a symmetric operator CÊ, one ends up with a natural symmetric ordering for CˆL.

This concludes our list of comments concerning the old regularization procedure. In the following chapter, we present our proposal to implement the scalar constraint in LQG. It is the first concrete implementation of the option iii) discussed above relative to the assignment of the loops created by the operator CÊ. While our proposal does not escape certain criticisms present also in the old approach, the new construction has certain advantages with respect to the later. For instance, this new regularization allows the construction of symmetric and densely defined constraint operator(s) for the theory, a first step towards possibly defining a self–adjoint constraint operator. Moreover, thanks to a work [57] by J. Lewandowski and H. Sahlmann, a symmetric constraint operator can be defined on a subspace of the dual space Cyl∗. This subspace can be seen as an intermediate space between Cyl∗ and the Hilbert space

HDiff. In fact, this space is a Hilbert space on which the commutator of two scalar constraint operators, defined using our prescription, vanishes. Another major difference with respect to the old construction is how one defines the operator CˆL. While the old approach is based on the classical identities involving Poisson brackets, in our construction we introduce an independent quantization of the 3–dimensional Ricci scalar leading to what we call the curvature operator [23]. As we explain below, the new scalar constraint operator turns out to have various interesting properties which open new avenues in the treatment and the understanding of the dynamics in LQG. Chapter III

Alternative dynamics in vacuum LQG

In this chapter we present a new regularization procedure for the vacuum scalar constraint in LQG. We detail the various steps of the construction of the quantum operator along with a short analysis of its properties. The starting point is the expression1 ∫ ( ) a b k √ N(x) ϵijkE (x)E (x)F (x) ( ) 3 √i j ab − 2 | | C(N) = d x 2 + 1 sβ det[E](x) R(x) , Σ 2skβ | det[E](x)| (III.0.1)

In the case of s = 1 (the spacetime signature + + + +), the choice β = 1 kills the second term (and corresponds to the original self–dual Ashtekar variables). For that reason we call the first term the Euclidean part, and we call the second term the Lorentzian part. As mentioned in the previous chapter, a quantum operator corresponding to the scalar constraint C(N) in (III.0.1) would not preserve the Hilbert space H G Diff of spatial diffeomorphism invariant states. This is because of the presence of the lapse function N in the expression. In other words, an operator Cˆ(N) is not invariant under spatial diffeomorphisms. This fact raises serious difficulties in the treatment of relevant questions such as self–adjointness, spectral resolution and anomaly–freeness of the constraints algebra. A solution to this issue is presented in the following section.

1The idea of considering this particular form of the scalar constraint for quantization in LQG was ﬁrst suggested in [58]. 44 Alternative dynamics in vacuum LQG

III.1 Vertex Hilbert space

An interesting proposal to go around the issue mentioned above is to introduce a new Hilbert space [57]. The idea of the new Hilbert space, called the vertex H G Hilbert space and denoted vtx, is to construct an intermediate space of partial H G solutions to the vector constraints from elements of the Hilbert space kin. H G H G Similarly to the construction of Diﬀ, the construction of vtx is achieved by group averaging method applied to the elements of each of the sub–spaces G CylΓ , but this time the averaging is made with respect to all the piecewise analytic diﬀeomorphisms which preserve the vertices of a given graph.

Given a graph Γ and the corresponding space of cylindrical functions CylΓ, denote by Diff(Σ)Vert(Γ) the group of piecewise analytic diffeomorphisms which preserve the analyticity of Γ and act trivially in the set of its vertices Vert(Γ). H G −→ H G The maps Γ kin obtained by the elements of Diff(Σ)Vert(Γ) are in one to one correspondence with the elements of the quotient

DΓ := Diff(Σ)Vert(Γ)/TDiff(Σ)Γ , (III.1.1) where TDiff(Σ)Γ is the set of diffeomorphisms in Diff(Σ) which preserve the graph Γ. Since DΓ is a non–compact set and we do not know any probability measure on it, we define the averaging in Cyl∗, the algebraic dual to Cyl. Given ∈ ∈ ∗ a cylindrical function ΨΓ CylΓ, we introduce the element ηV (Ψ) Cyl as ∑ ∗ ηV (ΨΓ) = [Uχ(ΨΓ˜)] , (III.1.2) χ∈DΓ where ΨΓ˜ is the averaged state defined by analogy with (II.2.27). The resulting ηV (ΨΓ) is a well–defined linear functional on Cyl because, similarly to the map η of section II.2.2.2, given Ψ′ ∈ Cyl only a finite set of terms in the sum ′ contribute to the number ηV (ΨΓ)(Ψ ). Hence we have defined a map

∋ 7→ ∈ ∗ CylΓ ΨΓ ηV (ΨΓ) Cyl , (III.1.3) for every graph Γ. The map ηV can be extended by linearity to the Hilbert H H G space kin. Then one deﬁnes the vertex Hilbert space vtx as the completion

H G ∩ H G vtx := ηV (Cyl kin) , (III.1.4) III.2 Curvature operator 45 under the norm induced by the natural scalar product

′ ′ ⟨ηV (Ψ)|ηV (Ψ )⟩ := ηV (Ψ)(Ψ ) . (III.1.5)

H G The Hilbert space vtx admits an orthogonal decomposition that is reminiscent of (II.2.23): Let FS(Σ) be the set of finite points subsets of Σ. Then ⊕ H G H G vtx = V , (III.1.6) ∈ V ⊕FS(Σ) H G H G V := [Γ] , (III.1.7) [Γ]∈[Γ(V )] where Γ(V ) is the set of graphs Γ with vertices set V = Vert(Γ), [Γ(V )] is the set of Diff(Σ)V –equivalence classes [Γ] of the graphs Γ in Γ(V ). Additionally, H G vtx carries a natural action of Diff(Σ), which we also denote by U and it is defined as

Uχη(Ψ) := ηV (UχΨ), χ ∈ Diﬀ(Σ) . (III.1.8)

H G H G A short calculation shows that Uχ is unitary and maps V to χ(V ) in the decomposition (III.1.6). H G Each space [Γ] consists of Diff(Σ)Vert(Γ) invariant elements. In this sense, they are partial solutions to the quantum vector constraint. They can be turned into full solutions of the quantum vector constraint by a similar averaging with respect to the remaining Diff(Σ)/Diff(Σ)Vert(Γ) and obtain the space H G Diff of spatial diffeomorphism invariant states defined in section II.2.2.2. Having defined the vertex Hilbert space, we develop in what follows our proposal for the quantization of the scalar constraint (III.0.1).

III.2 Curvature operator

We start with the implementation of the Lorentzian part of (III.0.1), namely ∫ 1 − sβ2 √ L 3 | | C (N) := 2 d x N(x) det[E](x) R(x) . (III.2.1) 2skβ Σ

The quantization of this classical functional, performed in [23], is achieved through an external regularization based on the Regge approximation [59] of the 3d Einstein–Hilbert (E–H) action. In the following, we present briefly the theory of Regge calculus, we discuss the generalization of the 3D Regge action to arbitrary piecewise flat decomposition and we highlight some results 46 Alternative dynamics in vacuum LQG about the convergence of the discrete action to the continuum one. Then we present the construction of the 3D scalar curvature operator and we expose the adopted regularization scheme. Finally, we discuss some properties of the final curvature operator and its semi–classical (large spins limit) behavior.

III.2.1 Regge calculus

Regge calculus [59–61] is a discrete approximation of general relativity which approximates spaces with smooth curvature by piecewise ﬂat spaces: given a n–dimensional Riemannian manifold Σ, we consider a simplicial decomposition

C∆ approximating Σ where we assume that curvature lies only on the hinges of

C∆, namely on its n−2 simplices. In this context, Regge derived the simplicial equivalent of the E–H action: ∫ ∫ √ √ ∑ n n SEH = −gR d x → SR = −gR d x = 2 ϵhVh , (III.2.2) C Σ ∆ h where the sum extends to all the hinges h with measure Vh and deﬁcit angle

ϵh: ( ) ∑ ∑ 2π − sh − sh ϵh = 2π θh = θh if the hinge h is not on the boundary λh sh sh (III.2.3) ( ) ∑ ∑ 2π − sh − sh ϵh = π θh = θh if the hinge h is on the boundary. λh sh sh

sh θh is the dihedral angle at the hinge h and the sum extends to all the simplices sh sharing the hinge h. The coeﬃcient λh is the number of simplices sharing the hinge h or twice this number if the hinge is respectively in the bulk, or on the boundary of the triangulation. Using simplices for the decomposition implies that both Vh(lab) and ϵh(lab) are functions of the hinges lengths lab joining two sites a and b of C∆. Equation (III.2.2) can also be written in another form which, as we will see later, is more adapted to our quantization scheme: ∫ ( ) ( ) 1 √ ∑ ∑ 2π ∑ ∑ 2π − n Vsh − sh Vs − s gR d x = h θh = h θh , 2 C∆ λh λh h sh s h∈s (III.2.4) where in the last equality the ﬁrst sum is over simplices s of C∆ while the second is over the hinges h in each simplex. III.2 Curvature operator 47

As we are interested in the Hamiltonian constraint of the 4–dimensional theory, we consider only spaces of dimension n = 3. Therefore the expression we want to quantize is: ∫ ( ) 1 √ ∑ ∑ 2π − 3 s − s gR d x = Lh θh , (III.2.5) 2 C λh ∆ s h∈s

s where Lh is the length of the hinge h belonging to the simplex s. Also, if we think only about computing the integral of the scalar curvature, the formula (III.2.5) can be extended to arbitrary piecewise ﬂat cellular decompositions as presented below. This is an important step in our approach to construct the operator as the reason will be clear later.

III.2.1.1 From simplicial decompositions to arbitrary piecewise ﬂat cellular decompositions:

This part is about generalizing the classical Regge expression for the integrated scalar curvature. It is important∫ √ to point out that we are only interested in computing the quantity |g|R dx3 for an arbitrary piecewise flat cellular decomposition of space. This does not mean that we are building a generalization of Regge calculus since we don’t derive any equations of motion. Let us first introduce the definition of a cellular decomposition: a cellular decomposition C of a space Σ is a disjoint union (partition) of open cells of varying dimension satisfying the following conditions:

i) An n–dimensional open cell is a topological space which is homeomorphic to the n–dimensional open ball;

ii) The boundary of the closure of an n–dimensional cell is contained in a ﬁnite union of cells of lower dimension.

In 3D Regge calculus we consider a simplicial decomposition of a 3D manifold which is a special cellular decomposition. Using the ϵ–cone structure [59] we induce a flat manifold with localized conical defects. Those conical defects lie only on the 1–simplices and encode curvature. Thereby it can be proven that the scalar curvature is distributional and proportional to the deficit angles carried by the 1–simplices. Then by integration over the entire space one gets equation (III.2.2) (see [60]). This construction is independent from the choice of the simplicial decomposition: the same expression would hold for arbitrary piecewise flat cellular decompositions, i.e decompositions such that the space inside each 3–cell is flat. The difference being that the deficit angle along one 48 Alternative dynamics in vacuum LQG

hinge, though still constant, is not determined by the hinges lengths lab. On arbitrary piecewise ﬂat decompositions the lengths do not form a complete set of variables for the theory and more parameters such as the angles are needed. The ﬁnal expression of the integrated scalar curvature in the general case can be written as

∫ ( ) 1 √ ∑ ∑ ∑ 2π − 3 c − c gR d x = Lhϵh = Lh θh , (III.2.6) 2 C λh h∈C c∈C h∈c where the ﬁrst sum now is over the 3–cells c and λh is the number of 3–cells sharing the hinge h (if it’s not on the boundary of C ). Now we express the classical lengths and dihedral angles in terms of the densitized triad (electric ﬁeld). Given a curve γ embedded in a 3–manifold Σ:

γ : [0, 1] → Σ s → γa(s) ,

the length Lγ of the curve in terms of the electric ﬁeld Ei is: ∫ 1 √ i i Lγ(E) = ds G (s)G (s) , (III.2.7) 0 where 1 ijk b c a ϵ ϵabcE E γ˙ (s) Gi(s) = 2 √ j k . (III.2.8) | det[E](x)|

a a a dγa(s) In (III.2.8) the Ei’s are evaluated at x = γ (s) and γ˙ (s) = ds .

To deﬁne the dihedral angle, we consider two surfaces S1 and S2 intersecting in the curve γ. The dihedral angle between those two surfaces is then: ∫ ( [ ]) 1 b 1 c 2 E nb(S , s)E nc(S , s) θ12(E) = ds π − arccos j j , (III.2.9) γ b 1 | c 2 | 0 Ej nb(S , s) Eknc(S , s) √ b k b k c k k where Ej nb(S , s) = δijEi nb(S , s)Ej nc(S , s) and nb(S , s) is the normal 2 one form on the surface Sk. We can therefore express Regge action in terms of the densitized triad as

2The normals are always taken to be inwards. III.2 Curvature operator 49 follows: ∫ √ 3 SR = −gR d x C ( ) ∑ ∑ 2π c − c =2 Lh(E) θh(E) λh c∈C h∈c ∫ √ ∑ ∑ 1 1 ijk b c a 1 ij′k′ b′ c′ a′ ϵ ϵabcE E γ˙ (s) ϵ ϵa′b′c′ E ′ E ′ γ˙ (s) =2 ds 2 √ j k 2 √ j k | | | | c∈C γ(s)=:h∈c 0 det[E](x) det[E](x) ∫ ( [ ]) 1 b 1 c 2 2π E nb(S , s)E nc(S , s) × ds − π + arccos j j . b 1 | c 2 | 0 λh Ej nb(S , s) Eknc(S , s) (III.2.10)

Equation (III.2.10) is the classical formula that we adopt to express the integrated scalar curvature and it is the basis of our construction to deﬁne a curvature operator.

III.2.1.2 On the convergence of Regge action:

The question of convergence of Regge action to the E–H action and the rela- tionship between the discrete scheme and the corresponding continuum theory is of crucial importance. There have been extensive studies on this aspect of Regge calculus. In particular, it’s possible to derive the Regge action from the E–H one [60] and it was shown [62] that given any lattice, regular or not, the deviation of Regge action from its continuum limit can be expressed as a power series in l2, where l is the typical length of the lattice. This proves that Regge action approaches the E–H action when the typical length goes to 0,

lim SR = SEH , (III.2.11) l→0 provided that certain general boundary conditions are satisfied. Moreover this convergence result can be generalized to some non–simplicial decompositions. For example if we consider a polyhedral decomposition (or any decomposition with flat hinges), the result is recovered by invoking the simple argument that such decomposition can always be refined using simplices and therefore inducing a simplicial decomposition where the additional hinges carry null deficit angles. For more general decompositions where for instance the hinges are not straight lines, the result is not straightforward. However there exists at least a class of such decompositions for which the convergence holds. A simple example is to consider a decomposition where the hinges are arcs of 50 Alternative dynamics in vacuum LQG

˘ n circles such that the length lab in R of each arc, connecting two sites a and b n of the lattice, is proportional to the Euclidean distance lab in R between the same sites ˘ lab = ξ.lab , (III.2.12) with a proportionality constant ξ that is the same for all hinges. Let Ξ be the decomposition with arcs as hinges and characterized by the constant ξ, and C∆ the decomposition with straight lines connecting the same sites. Note 3 that in Ξ, two sites can be connected by any number ωab of hinges with equal ˘k k lengths lab. Thereby we can generate the set of deficit angles ϵ˘ab for Ξ using k C the deficit angles ϵab of ∆ such that for every two connected sites a and b we have 4 k∑=ωab ˘k k lab.ϵ˘ab = lab.ϵab , (III.2.13) k=1 where k labels the different arcs connecting the two sites a and b. Hence we can write ∑ ∑ ˘ SR(Ξ) = 2 lh.ϵ˘h = 2 lh.ϵh = SR(C∆) , (III.2.14)

h∈Ξ h∈C∆ where the index h labels the hinges of the decomposition. This shows that, from any polyhedral decomposition of space, we can construct an equivalent piecewise flat decomposition characterized by a positive number ξ (larger than 1) where straight hinges are replaced by arcs. Then, by keeping the coefficient ξ constant in the refinement process, the convergence result can be recovered in this particular non–simplicial case. This example suggests that Regge action written for a non–simplicial decomposition, and specially with non straight hinges, could converge to the continuum action. Since the convergence of the expression (III.2.6) is crucial for the construction and the interpretation of the operator introduced in this work, we have to restrict ourselves to only cellular decompositions allowing this convergence result5. Therefore in the rest of the paper the term cellular decomposition will

3The number of hinges could be infinite but we exclude this case. Latter on we will see that the prescription we are considering in the regularization implies that the set of hinges linking two sites forms a 3–cell when the number of those hinges exceeds one. 4Such choice is always possible since in the general case the deficit angles are not determined by the lengths as it was in the simplicial case. For instance we can take ϵ˘k = ϵab ab ξ.ωab 5As long as we don’t give a general proof of the convergence result for general decompositions, we may expect that this result doesn’t hold for all decompositions. But for our construction, specifically in the regularization scheme, it is enough it exists one class of decompositions with non straight hinges which imply the convergence. III.2 Curvature operator 51 refer to a cellular decomposition for which Regge action converges to the E–H action.

III.2.2 Construction of the curvature operator

The construction of the operator [23] is in two parts: ﬁrst we write an approximate expression of the classical integral by implementing a cellular decomposition C ϵ of the 3d manifold, characterized by a regulator ϵ. Second, we promote the regularized expression to an operator, which after taking the regulator limit, leads to a background independent operator acting on the kinematical Hilbert space Hkin. A preliminary step in the construction is to match Regge calculus context with LQG framework. This is achieved by invoking the duality between spin–networks and quanta of space, that allows to describe for example spin– networks in terms of quantum polyhedra [63–65]. The idea is that given a graph, we build a dual cellular decomposition which we then use to regularize the classical expression (III.2.10) for the curvature. Consider a cellular decomposition C ϵ of the manifold Σ. The size of the cells is assumed to be controlled by the regulator ϵ, in such a way that the ′ ϵ coordinate size ϵ∆′ of each cell ∆ ∈ C satisﬁes ϵ∆′ < ϵ. We can then write the integral in (III.2.1) as a limit of a Riemannian sum over the cells ∆′, ∫ √ ∑ ∫ √ 3 3 d x N(x) |det[E]|R = lim N(x∆′ ) d x |det[E]|R, (III.2.15) ϵ→0 ′ Σ ∆′∈C ϵ ∆

′ where on the right–hand side x∆′ denotes any point inside ∆ . ′ Next we decompose each cell ∆ into c∆′ closed cells ∆, forming a cellular ′ ∆–decomposition of ∆ , where a cell ∆ has a boundary formed by√ a number n∆ of 2–surfaces (faces). We then approximate the integral of |det[E]|R in (III.2.15) by a regularized Regge action for the ∆–decomposition of ∆′, obtaining ∫ √ ∑ ∑ 3 d x N(x) |det[E]|R = lim N(x∆′ ) R∆(E) . (III.2.16) ϵ→0 Σ ∆′∈C ϵ ∆⊂∆′ 52 Alternative dynamics in vacuum LQG

The functional R∆(E) is deﬁned on the classical phase space as v u u 1 1 ∑ ϵijkP I P J ϵij′k′ P I ′ P J ′ t 4 2 S∆,u,j S∆,u,k 2 S∆,u,j S∆,u,k R∆(E) := 2 (kβ) √ √ |q (E)| |q (E)| u⊂∂∆ ( ∆ [ ∆]) 2π PSI ,kPSJ ,k × − π + arccos ∆,u ∆,u , (III.2.17) λu 2|P I ||P J | S∆,u S∆,u where we use the following notation:

I – given a cell ∆, the index I = 1, ..., n∆ labels the surfaces (faces) S∆ forming the boundary ∂∆ of the cell ∆ and u labels the hinges on that boundary (the 1–skeleton of the cell);

I J – the symbols S∆,u and S∆,u stand for the two surfaces in ∂∆ that intersect at u;

a I – P I represents the flux of the field E across the S , defined in (II.2.6), S∆,i i ∆ with

S = SI , and ξ(x) := h τ h−1 , (III.2.18) ∆ p∆(x) i p∆(x)

and √

|P I | := P I P I ; (III.2.19) S∆,u S∆,u,k S∆,u,k

– q∆(E) is the square volume functional [66–69]

(kβ)3 ∑ q∆(E) := κIJK (∆)ϵijkPSI ,iPSJ ,jPSK ,k , (III.2.20) 3! ∆ ∆ ∆ I,J,K

with κIJK (∆) a regularization coeﬃcient, depending on the shape of the cell ∆, which vanishes if two of the labels I, J, K coincide;

– ﬁnally λu is a ﬁxed integer parameter corresponding to the number of cells sharing the hinge u in the decomposition C ϵ.

In the limit ϵ → 0, equivalent to ϵ∆ → 0 for every cell ∆, each term in the sum deﬁning the functional R∆(E) (III.2.17), when rescaled by the coordinate length lu ∝ ϵ∆ of the edge u on the boundary of ∆, namely III.2 Curvature operator 53

v u u 1 1 k2β2 ϵijkPSI ,jPSJ ,k ϵij′k′ PSI ,j′ PSJ ,k′ t 2 √ ∆,u ∆,u 2 √ ∆,u ∆,u lu |q (E)| |q (E)| ( ∆ [ ∆ ]) 2π PSI ,kPSJ ,k × − π + arccos ∆,u ∆,u , (III.2.21) λu 2|P I ||P J | S∆,u S∆,u approximates the function L (E)( 2π − θ (E)), where L (E) and θ (E) are u λu u u u respectively the length of the hinge u and the dihedral angle at u in ∆ expressed in terms of densitized triads (III.2.7) and (III.2.9).

The sum over the cells ∆ of the functional R∆(E) corresponds to the reg- ′ ularized Regge∫ action√ [59] in 3d on ∆ , which is by itself an approximation of 3 | | the function ∆′ d x det[E] R as we have shown in section III.2.1.2. To continue the calculation from equation (III.2.16), we assume that the cells ∆ are chosen such that we obtain the same contributions R∆(E) from each cell ∆, up to higher order corrections in ϵ∆′ (equivalently, up to higher order corrections in ϵ). Hence each sum over the cells ∆ becomes the number ˜ of cells c∆′ times the contribution of the cell ∆, chosen to be the cell containing the point x∆′ at which the smearing lapse function N is evaluated. In this way we obtain ∫ √ ∑ 3 d x N(x) |det[E]|R = lim N(x ˜ ⊂ ′ ) c∆′ R ˜ (E) . (III.2.22) ϵ→0 ∆ ∆ ∆ Σ ∆′∈C ϵ

Let us now introduce the approximation

1 ∑ N(x∆˜ ⊂∆′ ) = N(x∆) , (III.2.23) c∆′ ∆⊂∆′ which is an averaging of the values of the lapse function N inside the cell ∆′, and which can be seen as a better approximation of the value of the function N inside the cell ∆′, in the sense that we are probing the function N in several ∈ ∩ ′ points x∆ ∆ ∆ instead of one point x∆˜ . Inserting equation (III.2.23) in equation (III.2.22), we come to the result we are looking for: ∫ √ ∑ ∑ 3 d x N(x) |det[E]|R = lim N(x∆)R∆(E) ϵ→0 Σ ′ ′ ∆∑∈C ϵ ∆⊂∆ = lim N(x∆)R∆(E) , (III.2.24) ϵ→0 ∆∈C ϵ 54 Alternative dynamics in vacuum LQG where the last step is achieved by combining the two sums over ∆′ and ∆.

III.2.2.1 The curvature operator:

Before promoting the expression in (III.2.24) to an operator, we proceed with an analysis of the term ϵijkP I P J appearing in equation (III.2.17). This S∆,u,j S∆,u,k term approximates the classical function

a b c 1 2 ϵijkϵabcEj Eku˙ (s) = lim (kβ) ϵijkPSI ,jPSJ ,k , (III.2.25) ϵ →0 4 ∆,u ∆,u ∆ ϵ∆ where s is parameterizing the curve u.

Considering a cylindrical function ΨΓ ∈ Cyl with a graph Γ = (e1, ..., en), a b c a the straightforward quantization of ϵijkϵabcEj Eku˙ (s) by replacing Ej with ℏ δ j induces the factor i δAa(x)

a b ′ ϵabce˙I (t)e ˙J (t ) , (III.2.26) in the formal action of the operator on ΨΓ. This factor vanishes unless

′ e˙I (t) ∦ e˙J (t ) . (III.2.27)

′ which means that the edges eI (t) and eJ (t ) are different (I ≠ J) and transversal at their intersection point. In order to pass this property to the quantum ′ operator, we introduce the coefficient κIJ (∆), defined in the following, in the expression of R∆(E) and we write v u ∑ u 1 1 ′ ′ ϵijkPSI ,jPSJ ,k ϵij k PSI ,j′ PSJ ,k′ ′ 2 2 t 2 ∆,u ∆,u 2 ∆,u ∆,u R∆(E) := 2 κ (∆)k β √ √ IJ |q (E)| |q (E)| u⊂∂∆ ( [ ∆ ]) ∆ 2π PSI ,kPSJ ,k × − π + arccos ∆,u ∆,u . (III.2.28) λu 2|P I ||P J | S∆,u S∆,u

In order to promote this expression to an operator, we ﬁrst need to set some requirements on the decomposition C ϵ of Σ, so that we adapt it to the G G functions in Cyl : given a ΨΓ in Cyl with a graph Γ = (e1, ..., en) with the set of vertices Vert(Γ) = (v1, ..., vm), the requirements are as follows:

i. each cell ∆ contains at most one vertex of the graph Γ;

I ii. each closed and connected 2–cell (face) S∆ on the boundary of a cell ∆, containing a vertex of Γ, is punctured exactly by one edge of the graph Γ. III.2 Curvature operator 55

The intersection is transversal and belongs to the interior of the edge;

iii. if two faces on the boundary of a cell ∆ intersect, then their intersection is a connected 1-cell;

iv. if v ∈Vert(Γ) and v ∈ ∆, then

– x∆ = v, ′ – κIJ (∆) is not zero only for edges eI and eJ of Γ meeting transversally at v and puncturing two adjacent faces on the boundary of ∆;

v. if ∆ does not contain an edge of Γ but it contains a segment of an edge then, by splitting the edge and reorienting its segments suitably, we turn that case into the case of ∆ containing a 2–valent vertex.

We call the decomposition C ϵ a covering decomposition of the graph Γ. Having the quantum operators corresponding to the ﬂuxes, we now deﬁne the quantum operator corresponding to R∆(E) as ∑ √ √ √ ˆ ′ 2 2 d−1 ˆ ˆ ˆ ˆ d−1 R∆ := κ (∆)k β V ϵijkP I P J ϵij′k′ P I ′ P J ′ V IJ ∆ S∆,u,j S∆,u,k S∆,u,j S∆,u,k ∆ u⊂∂(∆ [ ]) ˆ ˆ 2π PSI ,kPSJ ,k × − π + arccos ∆,u ∆,u , (III.2.29) λu 2Pˆ I Pˆ I Pˆ J Pˆ J S∆,u,m S∆,u,m S∆,u,l S∆,u,l and hence the operator corresponding to the expression (III.2.24)

∫ \√ ∑ 3 ˆ ˆ d x N(x) |det[E]|R := lim RC ϵ := lim N(x∆)R∆ , (III.2.30) ϵ→0 ϵ→0 Σ ∆∈C ϵ

ˆ where we introduced the volume operator V∆ deﬁned as v u √ u 3 3 ∑ ˆ t k β ˆ ˆ ˆ V∆ := qˆ∆(E) := κIJK (∆)ϵijkPSI ,iPSJ ,jPSK ,k . (III.2.31) 3! ∆ ∆ ∆ I,J,K

ˆ −1 The proper inverse V∆ of the volume operator does not exist because the volume operator has a non–empty kernel. However, by properly restricting the ˆ domain of V∆, we get an invertible operator for which we can deﬁne an inverse d−1 volume operator V∆ and then extend maximally its domain [70]. Considering the geometrical interpretation of such operator, the inverse volume operator must satisfy the following two conditions: 56 Alternative dynamics in vacuum LQG

– it acts only at the vertices of the spin–network graph and it annihilates states annihilated by the volume operator;

– it has the same eigenstates of the volume operator, with non–vanishing eigenvalues equal to the inverse of the corresponding non–vanishing eigenvalues of the volume operator;

Such operator exists and can be introduced as6

d−1 ˆ 2 2 6 −1 ˆ V := lim(V∆ + ϵ l ) V∆ . (III.2.33) ∆ ϵ→0

Where l is a constant which has the dimension of length. This is known as Tikhonov regularization [71] and the limit is well–defined. The result is a d−1 ˆ hermitian operator V∆ which commutes with V∆ and admits a self–adjoint extension on the Hilbert space Hkin. ˆ The operators ordering in our definition (III.2.29) of R∆ is specifically ˆ∆ ˆ∆ chosen so that it guarantees that the operators Lu,IJ and θu,IJ , that would correspond to the classical lengths [70] and dihedral angles [72] respectively, ˆ make the building blocks of the operator R∆(E) which can be written in the form ( ) ∑ 2π ˆ ′ ˆ∆ − ˆ∆ R∆ := 2 κIJ (∆) Lu,IJ θu,IJ , (III.2.34) λu u⊂∂∆

ˆ∆ ˆ∆ and that each of the operators Lu,IJ and θu,IJ admits a self–adjoint extension on Hkin.

Considering a function ΨΓ in the space CylΓ, thanks to the regularization detailed above we have { ˆ ˆ RvΨΓ if ∆ contains a vertex v of Γ ; R∆ ΨΓ = (III.2.35) 0 if ∆ does not contain a vertex v of Γ ;

6 ˆ ∑ In other words, given the spectral decomposition of the volume operator V = | ⟩⟨ | i vi vi vi , we have { −1 d− v |v ⟩ if v ≠ 0, V 1 |v ⟩ = i i i (III.2.32) ∆ i 0 otherwise. III.2 Curvature operator 57 where √ √ √ ℏ2 2 2 ∑ k β ′ d− d− Rˆ := κ (∆) V 1 (ϵ Jˆ Jˆ )(ϵ ′ ′ Jˆ ′ Jˆ ′ ) V 1 v 4 IJ v ijk v,eI ,j v,eJ ,k ij k v,eI ,j v,eJ ,k v  I,J   2π Jˆ Jˆ ×  − π + arccos √ v,eI ,k √v,eJ ,k  , λv,eI ,eJ ˆ ˆ ˆ ˆ Jv,eI ,mJv,eI ,m Jv,eJ ,lJv,eJ ,l (III.2.36)

such that eI runs through the set of edges meeting at the vertex v, λv,eI ,eJ =

λu is the integer parameter associated to the pair (eI , eJ ) dual to the faces intersecting at the hinge u, and

d−1 ˆ 2 2 6 −1 ˆ Vv := lim(Vv + ϵ l ) Vv , (III.2.37) ϵ→0 where v u u ℏ3k3β3 ∑ Vˆ := t κ (∆) ϵ Jˆ Jˆ Jˆ , (III.2.38) v 48 IJK ijk x,eI ,i x,eJ ,j x,eK ,k I,J,K

ˆ Although the action of the operator R∆ is well–defined in the space CylΓ, this operator still carry a dependence on the chosen covering decomposition C ϵ ′ through the coefficients κIJ (∆), κIJK (∆) (present in the volume operator) ′ and λv,eI ,eJ . To remove the dependence in κIJ (∆) and κIJK (∆), we need to ˆ ˆ appropriately average V∆ and R∆ over the relevant background structures [23], âv ˆ and use the resulting operators R∆ to define the operator RC ϵ corresponding to (III.2.30), then finally take the limit of the regulator ϵ → 0. On one hand, the averaging procedure for the volume operator was realized in [66–69] and leads to reducing κIJK (∆) to mainly an overall (undetermined) ˆ positive constant κ0. The operator Vv in (III.2.38) becomes v u u ℏ3k3β3 ∑ Vˆ := κ t ϵ(e ˙ , e˙ , e˙ ) ϵ Jˆ Jˆ Jˆ , (III.2.39) v 0 48 I J K ijk x,eI ,i x,eJ ,j x,eK ,k I,J,K where ϵ(e ˙I , e˙J , e˙K ) := sgn[det(e ˙I , e˙J , e˙K )], with e˙I ’s being the tangent vectors to the edge eI ’s at the vertex v. On the other hand, the averaging procedure to eliminate the decompo- ′ sition dependence in κIJ (∆) is achieved as follows: from the characteristics of a covering decomposition C ϵ, we can deduce that the 3–cells forming C ϵ 58 Alternative dynamics in vacuum LQG are isomorphic to spherical polyhedra verifying the requirement (ii.) for the decomposition C ϵ. It is important to note that for a fixed number of faces n, such spherical polyhedra regroup in a finite number of classes. A class is defined by the number of hinges forming the boundary of each face. For instance, for n = 3 we have only one class which can be represented by the 3–hosohedron7, for n = 4 we have the 4–hosohedron, the spherical tetrahedron and a third class obtained by taking a 4–hosohedron and replacing one of its vertices by two connected vertices. These classes can be represented by planar graphs (see figure III.1) similar to Schlegel diagrams for polytopes [73, 74], which are obtained by choosing a face of the polyhedron and projecting all the other faces on it as viewed from above. Labeling the faces I, J, K . . . , each class defines, up to permutations of the labels, the adjacency rules for the faces. Each permutation of the labels of faces defines a configuration. More precisely, if we consider a class and represent each hinge in it as IJ (I ≠ J), using the labels of the two faces intersecting at this hinge, a configuration is one permutation of faces labels on the full set of hinges contained in the class. This can be represented as a set of labeled hinges {IJ, KL, MN, . . . }. Considering the case n = 4 as an example, label the faces 1, 2, 3, 4 and the hinges by IJ, I ≠ J ∈ {1, 2, 3, 4}. Then the 4–hosohedron (a class) is represented by {12, 23, 34, 41} or by any other configuration obtained by permuting the labels of faces. The number of inequivalent configurations for a certain class is of course finite, for the 4–hosohedron it is 3. The tetrahedron is represented by a unique configuration {12, 13, 14, 23, 24, 34}, every permutation of the labels maps a configuration to itself. It is clear that those two classes are defining different configurations hence selecting different hinges. The fact that the number of inequivalent configurations, we denote it Nconf, is always finite allows us to define an averaging procedure over those configurations associated to a 3–cell with a fixed number of faces n, or in other words to a vertex of Γ with a given valence n.

For a n–valent vertex v of a given graph Γ, a pair of edges (eI , eJ ) appears only in a subset of the full set of configurations, therefore we have a number of appearances Napp of a pair of edges in the set of configurations, this number depends only on the valence of the vertex v. Thus we can define a coefficient ′ κv depending only on the valence of v

′ Napp ≤ κv = 1 , (III.2.40) Nconf 7A n–hosohedron [73] is a tessellation of n areas on a spherical surface such that each area is bounded by two circular arcs and all areas share the same two vertices. III.2 Curvature operator 59

Fig. III.1 Classes of spherical polyhedron with 4 faces, from the left to the right: the 4–hosohedron, the class with 3 vertices and the spherical tetrahedron. which is the same for all pairs of edges meeting at the vertex v. Hence, using C ϵ ′ the requirement (iii.) on , we can deﬁne the average of κIJ (∆) as

′av ′ κIJ := κv ϵ (e ˙I , e˙J ) , (III.2.41) where ϵ (e ˙I , e˙J ) is 0 if e˙I and e˙J are linearly dependent or 1 otherwise. Now we still need to deal with the ambiguity related to the coeﬃcients

λv,eI ,eJ . Classically the coefficients λv,eI ,eJ (= λu) are associated to hinges and they come from the sharing conditions on the hinges: given a hinge u , we ϵ specify all the 3–cells of C containing this hinge on their boundaries and λu is then the number of those 3–cells. Following from our construction of the operator, those coefficients are totally arbitrary, they depend on the choice of the covering decomposition but there is no information in the spin–network states that could fix them. They are free parameters. Of course one could always define a prescription which determines the values of these parameters.

Nevertheless, the choice of the λv,eI ,eJ ’s could control different interesting fea- tures of the final operator, for instance the locality of the operator: we could ask that the λv,eI ,eJ ’s equals the valence of the vertex v, which would make the operator ultra–local as it depends only on the properties at the vertex v. Or we could ask that a λv,eI ,eJ is equal to the number of vertices forming the smallest loop in Γ containing the pair of edges (eI , eJ ), making the operator local, and so on. Note that both the AQG framework [75–77] and the proposal in [78] for a continuous formulation of the LQG phase space, require the assignment not only of the abstract graph Γ on which the spin–networks are defined, but also a choice of an embedding with a further choice of a dual graph Γ∗: this assignment in our case would correspond to a unique choice of λv,eI ,eJ ’s and would remove the ambiguity. Therefore, it is clear that the freedom in the choice of the coefficients λv,eI ,eJ ’s is due to the non–uniqueness of the covering decomposition dual to a graph. On one hand, this tells us that these coefficients can be used to control the locality properties of the operator. In this 60 Alternative dynamics in vacuum LQG sense the information given by a one cell containing a vertex is not enough to define unambiguously the curvature operator in the region containing only that vertex. This picture is analogue to the parallel transport on an infinitesimal closed loop as a way to prob curvature classically in a point on a manifold: the loop is a non local object allowing to explore a very small neighborhood of the relevant point. In the same way, we need to explore the structure around each non empty cell to know the coefficients λv,eI ,eJ ’s. On the other hand, the non–uniqueness of a covering decomposition can be fixed by introducing any consistent prescription and a priori the only criterion available to favor a choice over another is the semi–classical limit, but it appears that at least the global behavior of the large spin limit is not affected by this choice (section III.2.3.4).

Assuming that we fix the coefficients λv,eI ,eJ for every point x of Σ, we can define the operator ( ) ∑ 2π âv ′ ˆ − ˆ Rx,ϵ :=2κx Lx,IJ θx,IJ λx,eI ,eJ I,J √ √ √ ℏ2 2 2 ′ ∑ k β κx d− d− := ϵ (e ˙ , e˙ ) V 1 (ϵ Jˆ Jˆ )(ϵ ′ ′ Jˆ ′ Jˆ ′ ) V 1 4 I J x ijk x,eI ,j x,eJ ,k ij k x,eI ,j x,eJ ,k x  I,J   2π Jˆ Jˆ ×  − π + arccos √ x,eI ,k √x,eJ ,k  , λx,eI ,eJ ˆ ˆ ˆ ˆ Jx,eI ,mJx,eI ,m Jx,eJ ,lJx,eJ ,l (III.2.42)

ˆ ˆ where Lx,IJ and θx,IJ are respectively the length [70] and dihedral angle [72] operators, and v u u ℏ3k3β3 ∑ Vˆ := κ t ϵ(e ˙ , e˙ , e˙ ) ϵ Jˆ Jˆ Jˆ , (III.2.43) x 0 48 I J K ijk x,eI ,i x,eJ ,j x,eK ,k I,J,K

d−1 with Vx defined as in (III.2.37) and eI runs through the set of germs starting âv at the point x. The action of Rx,ϵ on a cylindrical function ΨΓ vanishes when x ̸∈ Γ. Notice that there is no dependence on the regulator ϵ in the expression of âv → Rx,ϵ, therefore the limit ϵ 0 is trivial. The resulting operator, which we sim- âv ply denote Rx , equals the right hand side of equation (III.2.42). Hence we can define the final curvature operator Rˆ(N), independent of any decomposition III.2 Curvature operator 61

C ϵ, and acting in the space Cyl of cylindrical functions as

∫ \√ ∑ 3 | | ˆ ˆav d x N(x) det[E] R := R(N) := N(x)Rx . (III.2.44) Σ x∈Σ

A ﬁnal remark concerning the action of the operator Rˆ(N) is that, since ˆav ˆ the expression of Rx in (III.2.42) contains only J operators, the curvature ˆ ⊂ operator R(N) preserves each of the subspaces CylΓ Cyl. Schematically, given a state with a graph Γ, the operator Rˆ(N) acts at a vertex x of Γ carrying an intertwiner i as follows:

CˆL(N) = . (III.2.45)

We then say that the curvature operator Rˆ(N) is a graph preserving operator.

III.2.3 Properties of the curvature operator

III.2.3.1 SU(2)–gauge and diﬀeomorphism transformations:

Since the Jˆ operators are SU(2)–gauge invariant, the curvature operator Rˆ(N) is SU(2)–gauge invariant and is naturally defined on the space of SU(2)–gauge invariant cylindrical functions CylG. Also, thanks to the averaging procedure, we can introduce the operator Rˆ∗(N) defined by duality ∗ on the Hilbert space H G vtx which is preserved under its action. Consequently it defines a diffeomorphism covariant operator. In case the smearing function N is taken to be a constant on Σ, the operator Rˆ∗(N = const.) becomes diffeomorphism invariant H G which means that it preserves the Hilbert space Diff.

III.2.3.2 Self–adjoint curvature operator:

The curvature operator defined in equation (III.2.44), and consequently the operator Rˆ∗(N), are not symmetric operators. This is clear from the definition âv of the operator Rx in (III.2.42) because the length operators do not commute with the angle operators. It is possible though to introduce a symmetric cur- ˆ∗ ˆ∗ vature operator, which we denote Rsym(N), using the operator R (N) and its adjoint operator Rˆ∗(N)†. Both operators Rˆ∗(N) and Rˆ∗(N)† are densely de- 62 Alternative dynamics in vacuum LQG

8 H G ˆ∗ ˆ∗ ﬁned and closable on vtx. We deﬁne Rsym(N) as a combination of R (N) and Rˆ∗(N)†, namely

ˆ∗ D ˆ∗ D ˆ∗ ∩ D ˆ∗ † ⊂ H G → H G Rsym(N): (Rsym(N)) := (R (N)) (R (N) ) vtx vtx (III.2.46) 1 Rˆ∗ (N) := (Rˆ∗(N) + Rˆ∗(N)†) , sym 2 where D stands for the domain of the operator in the argument. Furthermore, using the graph preserving nature of the operator, it is straightforward to show ˆ∗ that the operator Rsym(N) is essentially self–adjoint. In the following we present certain numerical simulations exposing the par- ˆ∗ tial spectrum and expectation values of the operator Rsym(N) in speciﬁc cases.

III.2.3.3 Spectrum of the curvature operator:

In ﬁgures III.3 and III.4 we report the eigenvalues of the symmetric curvature ˆ∗ operator Rsym(N = 1) respectively in the case of a 4–valent vertex with all spins equal j1 = j2 = j3 = j4 = j0, and for the geometry dual to a loop of three 4–valent vertices with equal internal spins (labeling the links forming the loop) and equal external spins (see ﬁgure III.2).

Fig. III.2 : Examples of tested conﬁgurations: 4–valent vertex with equal spins j0 and three connected 4–valent vertices with equal internal spins j and equal external spins j′.

8We keep the same notation for the operators and their closures. III.2 Curvature operator 63

Ρ 140

120

100

j 2 4 6 8 10 0 ˆ∗ Fig. III.3 : Spectrum ρ of the curvature operator Rsym(N = 1) in the case of a regular

4–valent vertex plotted as a function of the spin j0. The parameters λx,eI ,eJ are ﬁxed to 1. −2 1 ℏ 2 Units ( kβκ0 ) are used.

1 2 j 3 4 5

0 Ρ 20

2 1 4 3 5 j¢

ˆ∗ Fig. III.4 : Spectrum ρ of the curvature operator Rsym(N = 1) in the case of the internal geometry in a conﬁguration of three 4–valent vertices plotted as a function of the spin ′ j (internal spin) and j (external spin). The parameters λx,eI ,eJ are ﬁxed to 3. Units −2 1 ℏ 2 ( kβκ0 ) are used.

III.2.3.4 Semi–classical properties: ˆ∗ It is important to stress that in the case of the curvature operator Rsym(N = 1), the semi–classical limit (large spins limit) does not mean the continuous limit but rather a discrete limit which is classical Regge calculus. In ﬁgure III.5 we report the expectation values of the curvature operator ˆ∗ Rsym(N = 1) on Livine–Speziale coherent states [79] in the case of a regular

4–valent vertex as a function of the spin j0. Livine–Speziale coherent states are SU(2) invariant intertwiners, obtained by group averaging of SU(2) coherent states [80] which minimize the uncer- 64 Alternative dynamics in vacuum LQG tainty ∆ = |⟨J⃗ 2⟩ − ⟨J⃗⟩2| in the direction of the angular momentum. The SU(2) coherent states are constructed from the highest weight state through the group action and they are labeled by the spin j and a unit vector nˆ deﬁn- ing a direction on the sphere S2. A Livine–Speziale coherent state can be decomposed in the conventional basis of intertwiners9 as

∑ ∏V ∑ ι ...ι − |j, nˆ⟩ = a (ˆn ) C 1 V 3 |j . . . j , ι . . . ι − ⟩ (III.2.47) 0 mi i m1...mV 1 V 1 V 3 m1...mV i=1 ι1...ιV −3

ι1...ιV −3 where Cm1...mV are the (generalized) Clebsch–Gordan coefficients and ami (ˆni) are the coefficients defining a coherent state associated to one spin ji in terms of the spin basis. Using recoupling theory, these generalized coefficients [46] can always be decomposed into sums of products of conventional (3–valent) Clebsch–Gordan coefficients.

XR\

j 1 2 3 4 5 6 7 0 ⟨ ⟩ ˆ∗ Fig. III.5 : Expectation values R of Rsym(N = 1) on Livine–Speziale coherent states plotted as a function of the spin: case of a regular 4–valent vertex with the parameters −2 1 ℏ 2 λx,eI ,eJ ﬁxed√ to 1. Units ( kβκ0 ) are used. The curve is a ﬁt with a square root function 6, 57 + 12, 87 j0.

In Figures III.6 and III.7 we report the expectation values of the curvature operator on Rovelli–Speziale [81] semi–classical state, as a function of the spin, respectively for the conﬁguration of a regular 4–valent vertex and for the internal geometry in the conﬁguration of three 4–valent vertices with equal internal spins and equal external spins.

9 The conventional intertwiner basis states |j1 . . . jV , i1 . . . iV −3⟩ for a vertex of valence V , H j1 ⊗ · · · ⊗ jV j are basis elements of the Hilbert space j1,··· ,jV = Inv[V V ] with V being the Hilbert space corresponding to the irreducible representations of SU(2) with spin j. Those states are labeled by V − 3 quantum numbers depending on the coupling of the external legs. We indicate with |jik⟩ the basis state of a V –valent intertwiner with the spins ji and jk coupled together and arbitrary couplings for the spins left. III.2 Curvature operator 65

Rovelli–Speziale semi–classical tetrahedron is a semi–classical quantum state corresponding to the classical geometry of the tetrahedron determined by the areas A1,...,A4 of its faces and two dihedral angles θ12, θ34 between A1 and 7 A2 respectively A3 and A4. It is deﬁned as a state in the intertwiner basis

|j12⟩ ∑ | ⟩ ψ = cj12 j12 (III.2.48)

j12

with coeﬃcients cj12 such that

< ∆θˆ > < θˆ >→ θ ; ij → 0 (III.2.49) ij ij ˆ < θij > in the large scale limit, for all ij. The large scale limit considered here is taken when all spins are large.

The expression of the coeﬃcients cj12 satisfying the requirements is: { } − 2 1 −(j12 j0) cj12 (j0, k0) = 1 exp + iϕ(j0, k0)j12 (III.2.50) 4 4σ (2πσj12 ) j12 where j0 and k0 are given real numbers respectively linked to θ12 and θ34 through the following equations:

2 2 2 2 2 2 j0 = 2j1j2 cos θ12 + j1 + j2 ; k0 = 2j3j4 cos θ34 + j3 + j4 (III.2.51)

σj12 is the variance which is appropriately fixed and the phase ϕ(j0, k0) is the dihedral angle to j0 in an auxiliary tetrahedron related to the asymptotic of the 6j symbol performing the change of coupling in the intertwiner basis (see [81]). For a classical regular tetrahedron, using the expression (III.2.2) for Regge action, the integrated classical curvature scales linearly in terms of the length of its hinges because the angles do not change in the equilateral configuration when the length is rescaled. This means that the integrated classical curvature scales as the square root function of the area of a face. In figures III.5 and ˆ∗ III.6, we see that the expectation values of Rsym(N = 1) on coherent states and semi–classical (regular) tetrahedron for large spins scales as a square root function of the spin, this matches nicely the semi–classical evolution we expect. In the second case, represented in III.7, in which the state is picked on the configuration where three identical tetrahedra are glued together (see figure 66 Alternative dynamics in vacuum LQG

XR\

j 1 2 3 4 5 6 7 0 ⟨ ⟩ ˆ∗ Fig. III.6 : Expectation values R of Rsym(N = 1) as a function of the spin on a Rovelli– Speziale semi–classical state representing the configuration of a regular 4–valent vertex with the parameters λ fixed to 1. The curve is a fit with a square root function 6, 55 + √ x,eI ,eJ −2 1 ℏ 2 12, 77 j0. Units ( kβκ0 ) are used.

XR\

0.2

j 1 2 3 4 5

-0.2

-0.4

⟨ ⟩ ˆ∗ Fig. III.7 : Expectation values R of Rsym(N = 1) as a function of the spin on a Rovelli– Speziale semi–classical state representing the conﬁguration of three 4–valent vertices (three tetrahedra glued together) with equal internal spins j and equal external spins j′ (determined −2 1 ℏ 2 by j), with the parameters λx,eI ,eJ ﬁxed to 3. Units ( kβκ0 ) are used.

III.2), we can notice that the expectation values approach zero as the value of the spins increase which means that the conﬁguration in the considered region is close to the ﬂat geometry which is also the semi–classical behavior we expect.

III.2.4 The Lorentzian part of the scalar constraint operator

From equation (III.2.1), it is obvious that the operator which corresponds to the Lorentzian part of the vacuum (smeared) scalar constraint, is proportional to (one of) the curvature operator(s) defined above [23, 24], with a smearing function corresponding to the lapse function. The ambiguity lies in the choice III.3 Euclidean operator 67 among the non–symmetric operators, Rˆ∗(N) and Rˆ∗(N)†, and the symmetric ˆ∗ extensions such as Rsym(N) analyzed above. In the continuum semi–classical limit, all those choices should tend to coincide, but they differ significantly in the high quantum regime. We now introduce the following notation for the three eligible operators for the Lorentzian part of the scalar constraint

1 − sβ2 CˆL(N) := Rˆ∗(N) (III.2.52) 2skβ2 1 − sβ2 CˆL(N)† := Rˆ∗(N)† (III.2.53) 2skβ2 1 − sβ2 CˆL (N) := Rˆ∗ (N) (III.2.54) sym 2skβ2 sym (III.2.55)

In section III.4 we discuss certain consequences that those three diﬀerent choices of operators induce on the quantum algebra of the constraints and on the physical states [24].

III.3 Euclidean operator

Now we turn to the implementation of the Euclidean part of the scalar constraint (III.0.1), namely ∫ ϵ Ea(x)Eb(x)F k (x) E 1 3 ijk √i j ab C (N) := 2 d x N(x) . (III.3.1) 2skβ Σ | det[E](x)|

We ﬁrst rewrite CE(N) using Thiemann’s trick in order to suppress its non– singular form. The identity we use is

a b ϵijkE (x)E (x) 4 √ i j = ϵabc{Ak(x),V } , (III.3.2) | det[E](x)| kβ c where V is the volume of the hypersurface Σ, ∫ √ V := d3x | det[E](x)| . (III.3.3) Σ

The Euclidean functional CE(N) takes the form ∫ 2 E 3 abc k { k } C (N) = 2 3 d x N(x)ϵ Fab(x) Ac (x),V . (III.3.4) sk β Σ 68 Alternative dynamics in vacuum LQG

III.3.1 Construction of the Euclidean operator

The expression (III.3.4) is regularized via approximation of the integral by a Riemannian sum over a decomposition C ϵ of Σ, with again ϵ being a parameter characterizing the size of the cells ∆ in C ϵ. Similarly to the curvature operator defined above, the smearing function N(x) is replaced by its values at certain points x∆, one point inside each cell ∆, and the connection coefficients are replaced by holonomies along open curves sI (∆) while the curvature coefficients by the holonomies along loops αIJ (∆) ∑ ∑ E 2 C ϵ (N) = N(x ) ϵ (∆) C sk2β3W 2 ∆ IJK l ∆∈C ϵ I,J,K ( √ ) − × Tr h(l) h(l) {h(l) 1 , q (E)} , (III.3.5) αIJ (∆) sK (∆) sK (∆) ∆

(l) where√ h represents a holonomy in a chosen SU(2) representation l, Wl = 10 i l(l + 1)(2l + 1) is a normalization factor , ϵIJK (∆) is a regularization coefficient which depends on the structure of the cell ∆ and which vanishes if two of the labels I, J, K coincide, and q∆(E) is the square volume functional defined in (III.2.20). The curves sK (∆) and loops αIJ (∆) are assigned to each cell ∆ and summed over such that the functional in (III.3.5) converges to CE(N) in the limit ϵ → 0. The first step of the quantization is to define in Cyl a partition dependent Ê quantum operator CC ϵ (N). As mentioned before, such operator will not have a limit when ϵ → 0. Still, by duality we could possibly obtain a well–defined H G operator on vtx that carries the diffeomorphism covariance property of the classical constraint. It is then necessary to adapt our partition independently to each graph Γ associated to a subspace CylΓ of Cyl. Our prescription is as follows:

• C ϵ is a triangulation, i.e. each cell ∆ is a tetrahedron;

• each tetrahedron ∆ has at most one vertex of the graph Γ as one of its vertices;

• each vertex v of the graph Γ coincides with a vertex of a tetrahedron ∆v

10The representation l is left arbitrary in our construction. In representation l, we remind (l) the reader that we choose a basis τi (i = 1, 2, 3) of su(2), satisfying ( ) ( ) W 2 Tr τ (l) = 0, Tr τ (l)τ (l) = l δ . i i k 3 ik III.3 Euclidean operator 69

and x∆v = v;

• if v is a vertex of Γ, then

i – v is a vertex of nv tetrahedra ∆v saturating the neighborhood of v (i.e the tetrahedra meet at v and compose a closed neighborhood centered at v);

i – the edges of the tetrahedra ∆v saturating the neighborhood of v do not overlap with the edges of Γ meeting at v, except for one IJK IJK tetrahedron, which we call ∆v . The tetrahedron ∆v is adapted

to one chosen ordered triple of edges (eI , eJ , eK ) meeting at v, i.e., IJK the edges (sI , sJ , sK ) of ∆v meeting at v are segments of the

edges (eI , eJ , eK ) of the graph Γ but do not coincide with them;

– to the ordered triple of edges (sI , sJ , sK ) meeting at v there are assigned three loops (αIJ , αJK , αKI ) oriented according to the order

of the triple (sI , sJ , sK ); – A loop αIJ veriﬁes the following conditions: i. αIJ is an analytic curve; ii. αIJ lies in a surface deﬁned through a canonical choice of coor-

dinates adapted to the edges (sI , sJ , sK ) and does not intersect the graph Γ at any point except at v; IJ iii. α is tangent to the two edges eI and eJ of the graph Γ at

the vertex v up to orders kI + 1 and kJ + 1 respectively, where

kI (≥ 0) and kJ (≥ 0) are respectively the orders of tangentiality 11 of eI and eJ at the node ; IJK iv. Denote by sIJ the edge of ∆v that links the edges (sI , sJ ) IJK to form a triangle of the the tetrahedron ∆v . The shape of IJ the loop α marries the shape of the triangle (sI , sJ , sIJ ) as good as possible;

This prescription for the adapted partition is twofold: The ﬁrst part, which contains all the requirements except the conditions on the loops, coincides with some of the requirements on the partition in the old approach to regularize the scalar constraint discussed in section II.2.3. Additionally, in the old approach the number nv is set to be equal to 8 for any vertex v of the graph thanks to a speciﬁc procedure to construct the saturating structure around v. We could

11 The order of tangentiality of an edge eI incident at a vertex v is the highest order of tangentiality of the edge eI with the remaining edges incident at v [25]. 70 Alternative dynamics in vacuum LQG

adopt the same procedure to fix nv but it is a priori possible to keep it as a free parameter that is the same for all vertices, hence we drop the v label in rest of the manuscript. The second part of the above prescription is about the conditions on the loop structure. Those conditions are different from the conditions in the old IJ construction which makes the loop α coincides with the triangle (sI , sJ , sIJ ) IJK of ∆v . The whole prescription is diffeomorphism invariant and it makes a loop assigned to a pair of edges unique up to diffeomorphisms. The surfaces in which the loops lie are chosen using a prescription that was introduced in

[45] and presented in [39]: ﬁrst, to each pair of links eI and eJ incident at v, we deﬁne an adapted frame in a small enough neighborhood of v. Then we require that the loop αIJ , associated to the pair (eI , eJ ), lies in the coordinate plane spanned by the edges eI and eJ . The choice of the adapted frame is based on the following lemma:

Let e and e′ be two distinct analytic curves intersecting only at their starting point v. Then there exist parameterizations of these curves, a number δ > 0, and an analytic diﬀeomorphism such that, in the corresponding frame, the curves are given by

(a) e(t) = (t, 0, 0), e′(t) = (0, t, 0), t ∈ [0, δ] if their tangents are linearly independent at v, (b) e(t) = (t, 0, 0), e′(t) = (t, tn, 0), t ∈ [0, δ] for some n ≥ 2 if their tangents are co–linear at v.

We will call the associated frame a frame adapted to e, e′.

Second, we define a diffeomorphism invariant prescription for the topology of the routing of the loop αIJ . In other words, the plane in which the loop lies should be chosen in a way which is diffeomorphism invariant, and which does not cause the loop to intersect the graph Γ at any point different from the vertex v. The choice that αIJ lies in a small enough neighborhood of v guarantees that the loop cannot intersect any edge of Γ except the edges incident at the vertex v. Then the routing of the loop in that neighborhood is achieved through the prescription given in [39] and which we do not repeat here. This concludes the assignment of the surfaces containing the loops. The remaining conditions, first introduced in [25], make each loop uniquely assigned to a given pair. III.3 Euclidean operator 71

As we will see later, those conditions all together allow to introduce a densely defined adjoint operator of the non–symmetric scalar constraint oper- ator12, thereby providing a way to define a symmetric constraint operator (the key condition is that as in [11, 57, 25] the loops do not overlap with the given graph). In the rest of the manuscript we refer to those loops as special loops. Having the adapted partition, we straightforwardly quantize√ the expression in (III.3.5) by replacing the Poisson bracket of h−1 and q (E) with 1/iℏ sK (∆) ∆ ˆ times the commutator of the corresponding operators, taking for Vv the volume operator defined in (III.2.43). ∈ Considering a state ΨΓ CylΓ, the resulting operator acts as ∑ ∑ ∑ E 2 Cˆ ϵ (N)Ψ := N(v) ϵ (∆) C Γ ℏ 2 3 2 IJK i sk β Wl ϵ ∆∈C (v∈∆∩Γ I,J,K ) − × Tr h(l) h(l) [h(l) 1 , Vˆ ] Ψ . (III.3.6) αIJ (∆) sK (∆) sK (∆) v Γ

Notice that a specific ordering of the operators has been chosen: the commutator precedes the holonomy around the loop. This is the only choice which leads to a well–defined quantum operator. At this stage, the operator defined in (III.3.6) still depends on the triangulation C ϵ. The dependence on the triangulation is removed in three steps:

(...) a) Denote by B(v) the closed region formed by the n tetrahedra ∆v of C ϵ saturating a vertex v. Here (...) contains the labels of the edges intersecting at v and defining a specific tetrahedron. Classically, as we take the limit ϵ → 0 in the sense of refining the adapted triangulation ′ C ϵ to another adapted triangulation C ϵ such that ϵ′ < ϵ, we have ∫ ∫ ≈ n , (III.3.7) B IJK (v) ∆v

the label IJK refers to one speciﬁc tetrahedron of B(v). In other words, the integral over B(v) converges to n times the integral over any tetrahedron of B(v) as we take the limit ϵ → 0. For the operator in (III.3.6), this translates as

12As mentioned in the previous chapter, in case of the old construction [39], the kinematical adjoint operator of the non–symmetric scalar constraint operator is not densely deﬁned on ∗ Hkin and the dual operator is not densely deﬁned on Cyl . 72 Alternative dynamics in vacuum LQG

∑ ∑ E 2n IJK Cˆ ϵ (N)Ψ := N(v)ϵ (∆ ) C Γ iℏsk2β3W 2 IJK v l v∈Γ∩C ϵ IJK ∈C ϵ ( ∆v ) × (l) (l) (l) −1 ˆ Tr h IJK h IJK [h IJK , Vv] ΨΓ . (III.3.8) αIJ (∆v ) sK (∆v ) sK (∆v )

b) A triangulation C ϵ selects at each vertex v of a graph Γ a unique triple

of edges (eI , eJ , eK ) meeting at v and associates to it the coeﬃcient IJK ϵIJK (∆v ). In order to remove this triangulation dependence from the operator, it is enough to average at each vertex v over the classes of triangulations that select diﬀerent triples meeting at v. Therefore the operator would contain contributions from all possible triples meeting at the same node and, in a similar way to the volume operator, we obtain

2 ∑ ∑ CˆE(N)Ψ := N(v) κ ϵ(e ˙ , e˙ , e˙ ) ϵ Γ ℏ 2 3 2 v I J K i sk β Wl v∈Γ ( I,J,K ) − × Tr h(l) h(l) [h(l) 1, Vˆ ] Ψ , (III.3.9) αIJ (ϵ) eK (ϵ) eK (ϵ) v Γ

where now I, J, K run through all triples of edges of the graph Γ meeting

at the vertex v and κv := n/E(v), E(v) being the number of unordered triples of edges meeting at v (hence E(v) depends only on the graph Γ). Notice that due to the presence of the volume operator in its expression, Ê Cϵ,v annihilates two–valent vertices and vertices which have degenerate Ê differential graph structure. Therefore the action of the operator Cϵ (N) on a cylindrical function is always finite and one writes ∑ Ê Ê Cϵ (N) := N(x)Cϵ,x x∈Σ 2 ∑ ∑ := N(x) κ ϵ(e ˙ , e˙ , e˙ ) ℏ 2 3 2 x I J K i sk β Wl ∈ x Σ ( I,J,K ) − × Tr h(l) h(l) [h(l) 1, Vˆ ] , (III.3.10) αIJ (ϵ) eK (ϵ) eK (ϵ) x

where eI runs through the set of germs starting at the point x and the

loops αIJ represent the new edges to be added at the same point x.

c) The only dependence on the triangulation left is in ϵ. We then need to → Ê take the limit ϵ 0. As we have mentioned above, in this limit Cϵ (N) does not converge to any well–defined operator in the space Cyl. The Ê way around this problem is to first pass the operator Cϵ (N) to the space III.3 Euclidean operator 73

∗ ηV (Cyl) ⊂ Cyl by duality, then take the limit [57]. The convergence is ensured and the ﬁnal operator is then deﬁned as

[ ]∗ ˆE ˆE C (N) := lim Cϵ (N) , (III.3.11) ϵ→0

∗ acting in the space ηV (Cyl) ⊂ Cyl .

III.3.2 Properties of the Euclidean operator

III.3.2.1 Gauge and diﬀeomorphism transformations:

ˆ(l) ˆ Since the operators h and Jx,e,i are both SU(2)–gauge invariant, the operator CÊ(N) is SU(2)–gauge invariant and passes naturally to the Hilbert space H G vtx. Thanks to the averaging procedure and the specific prescription of the special loops in the regularization, the operator CÊ(N) is densely defined on H G the Hilbert space vtx and it is diffeomorphism covariant. It maps its domain D Ê ⊂ H G H G (C (N)) vtx to another subset of vtx. In case the smearing function N is taken to be a constant on Σ, the operator CÊ(N = const.) becomes H G diffeomorphism invariant which means that it preserves the Hilbert space Diff. The operator CÊ(N) does not preserve the graph structure as it removes special loops at the vertices13. Schematically, given a state with an admissible graph Γ, the operator CÊ(N) defined in (III.3.11) modifies the graph Γ at a vertex x carrying an intertwiner i as follows:

CˆE(N) = . (III.3.12)

We then say that the operator CˆE(N) is a graph changing operator.

III.3.2.2 Symmetric Euclidean operator:

The Euclidean operator CÊ(N) defined in equation (III.3.11) is not a symmetric operator. However, similarly to the curvature operator, it is possible to Ê introduce a symmetric Euclidean operator, which we denote Csym(N), through a combination of the operator CÊ(N) and its adjoint operator CÊ(N)†. For

13 ˆE G The operator Cϵ (N) is regularized in the space Cyl and it changes the graph of a state by adding special loops at the vertices. Therefore, the dual operator CˆE(N) acting H G vtx is removing special loops at the vertices. 74 Alternative dynamics in vacuum LQG instance, a simple choice of symmetric extension is

Ê D Ê D Ê ∩ D Ê † ⊂ H G → H G Csym(N): (Csym(N)) := (C (N)) (C (N) ) vtx vtx 1 CÊ (N) := (CÊ(N) + CÊ(N)†) . (III.3.13) sym 2

Ê Ê † Ê Note that the operators C (N), C (N) and Csym(N) above are densely 14 H G defined and closable on vtx. They are all equally suitable candidates for the Euclidean part of the scalar constraint operator. In each case, the general structure of the kernel of the scalar constraint operator is known on a qualitative level and, interestingly, in the case of the operator CÊ(N), the kernel of the scalar constraint operator becomes more tractable with respect to the spin–network basis. Those aspects, along with the question of the quantum constraints algebra, are discussed in the following section.

III.4 Scalar constraint operator, quantum algebra and physical states

Having defined the two operator parts of the scalar constraint (III.0.1), we can now introduce the complete non–symmetric scalar constraint operator as ∑ ∑ ˆ Ê ˆL Ê ˆL ˆ C(N) := C (N) + C (N) = N(x)(Cx + Cx ) =: N(x)Cx . (III.4.1) x∈Σ x∈Σ

D ˆ ⊂ H G It is deﬁned on a dense domain (C(N)) vtx and maps it to a subset of H G vtx. One can also consider the adjoint operator

Cˆ(N)† := (CˆE(N) + CˆL(N))† . (III.4.2)

The adjoint operator Cˆ(N)† is closed and also densely deﬁned on D(Cˆ†(N)), hence the operator Cˆ(N) is closable15 and (Cˆ(N)†)† = Cˆ(N). If the implementation of the scalar constraint is appropriate, then in the semi–classical limit of the theory the expectation values of the operator and its adjoint should coincide, up to small quantum corrections. Hence, the operator Cˆ(N)† could stand as the quantum scalar constraint operator in the theory on the same footing as the operator Cˆ(N). Therefore, in the rest of this manuscript we consider the closure of Cˆ(N) and Cˆ†(N) as being the non–symmetric scalar

14We keep the same notation for the operators and their closures. 15We keep the same notation for Cˆ(N) and its closure. III.4 Scalar constraint operator, quantum algebra and physical states 75 constraint operators at our disposal. Following rigorously the canonical quantization, one can argue that, since the classical scalar constraint functional is an (weak) observable, the quantum operator corresponding to it must be self–adjoint. In our case, we can realize the ﬁrst necessary step which is to construct a symmetric scalar constraint operator that is closable and densely deﬁned, namely16

1 Cˆ (N) := (Cˆ(N) + Cˆ†(N)) , (III.4.3) sym 2

ˆ ˆ ˆ† with D(Csym(N)) = D(C(N)) ∩ D(C (N)). But other symmetrizations could also be valid.

III.4.1 Quantum constraints algebra

Now we turn to the question of the quantum constraints algebra. Let us for the moment assume that our constraint operator is Cˆ(N). We then can make a short calculation to check if this operator does not induce anomalies in the quantum constraints algebra. For two scalar constraint operators the ∈ H G calculation goes as follows: given a state ΨΓ vtx, we have ∑ ˆ ˆ ′ ˆ ˆ [C(N), C(M)]ΨΓ = N(v)M(v )[Cv, Cv′ ]ΨΓ . (III.4.4) v,v′∈Γ

Because the regularization used to construct the operator is local with respect to each node, the commutator

ˆ ˆ ′ [Cv, Cv′ ]ΨΓ = 0 , ∀ v ≠ v , (III.4.5) hence ∑ ˆ ˆ ˆ ˆ [C(N), C(M)]ΨΓ = N(v)M(v)[Cv, Cv]ΨΓ . (III.4.6) v∈Γ

H G ˆ ˆ In the space vtx the commutator [Cv, Cv] also vanishes for the reason that the H G two terms of the last commutator, when acting on a state in kin (before taking the limits of the regulators III.3.11), produce two diﬀeomorphism equivalent

16A proof of self–adjointness, or not, of this operator is still missing. 76 Alternative dynamics in vacuum LQG

17 H G states, therefore the commutator vanishes on any state in vtx

[Cˆ(N), Cˆ(M)] = 0 . (III.4.7)

When it comes to the algebra with respect to the other constraints we already know that, on one hand, the operator Cˆ(N) preserves the SU(2)– gauge invariance as it should, on the other hand and as we mentioned before, a diffeomorphism constraint operator does not exist in this representation and the only thing we could check is whether Cˆ(N) is covariant with respect to the action of diffeomorphisms. The calculation and the result are not different than in the case of the old constraint operator and we find that indeed the operator Cˆ(N) is diffeomorphism covariant:

−1 ˆ ˆ ∗ ∀ ∈ ∞ Uf [C(N)]Uf = C(f N) , f Diﬀ (Σ) . (III.4.8)

Therefore we conclude that the scalar constraint operator Cˆ(N) is anomaly free, partially on–shell. Similarly, one can show that the operators Cˆ(N)† and ˆ Csym(N) are also anomaly free.

III.4.2 Physical states

The structure of the kernels of the operators Cˆ(N) and Cˆ†(N), equivalently H G the states in vtx which satisfy

Cˆ(N)Ψ = 0 , or Cˆ†(N)Ψ = 0 , (III.4.9) can to some extent be described quite easily and the elements can be computed following a systematical (recursive) procedure. The properties we know so far of the kernel elements can be summarized as follows:

• every state which is in the kernel of the volume operator Vˆ and has coplanar edges at all the vertices of its graph, is in the kernels of Cˆ(N) and Cˆ†(N);

• thanks to the graph preserving property of the Lorentzian part, the set of states of non–zero volume18 in the kernel of Cˆ(N) contains an inﬁnite number of states which have the form of ﬁnite linear combinations of 17 H G The commutator also vanishes with respect to URST (topology) on kin [44, 13] similarly to the old constraint operator. 18By a state of non–zero volume we mean any state which is not in the kernel of the volume operator. III.4 Scalar constraint operator, quantum algebra and physical states 77

spin–network states with diﬀerent graphs and simple examples can be straightforwardly derived;

• generic states of non–zero volume which are in the kernel of Cˆ(N)† have the form of inﬁnite linear combinations of spin–network states with different graphs;

• states of non–zero volume with graphs which do not contain special loops are not in the kernel of Cˆ†(N).

ˆ With those properties, one can deduce that the kernel of Csym(N) has the following structure:

• every state which is in the kernel of the volume operator Vˆ and has ˆ coplanar edges at all the veritices of its graph, is in the kernel of Csym(N); ˆ • generic states of non–zero volume which are in the kernel of Csym(N) have the form of inﬁnite linear combinations of spin–network states with diﬀerent graphs;

• states of non–zero volume with graphs which do not contain special loops ˆ are not in the kernel of Csym(N).

As we mentioned before, the construction of physical states is achieved via averaging the states in the kernel of the chosen scalar constraint operator, H G subset of vtx, with respect to the remaining diﬀeomorphisms of Diﬀ(Σ).

Chapter IV

Towards new dynamics in LQG deparametrized models

As presented in chapter II, the Hamiltonian formulation of general relativity encodes the dynamics in constraints. In this case, quantities such as the metric are not observable. What is observable is the geometry, that is quantities which are invariant under the action of diffeomorphisms. These are called observables. Dynamically, this translates in a frozen picture where there is no time flow nor evolution of physical quantities. This specific aspect raises several serious issues in the interpretation of a quantum theory of gravity, as one fails to make the link to the experimental setup with definite instants of time. One of the approaches to circumvent this problem of time is deparametrization of gravity [82, 83, 44, 85, 84, 86–91]. This approach however carries the drawback of choosing a specific global reference frame to parametrize either time or both space and time, hence the description and interpretation of the physics derived within the framework would be tied to this choice of frame. Nevertheless, this approach turns out to be technically very efficient in constructing the quantum theory and bypass the difficulties encountered in the case with the standard constraints. In many cases, it is possible to define complete quantum theories where gravity is fully quantized [92, 90, 25]. Those models then become a very rich playground to test the many technical steps of the quantization procedures along with the development and analysis of new methods and ideas to answer even more complex questions concerning the semi–classical and coherent states, the quantum observables and the continuum limit of the quantum theory. In this chapter we present a short overview of two models where only 80 Towards new dynamics in LQG deparametrized models deparametrization of the scalar constraint with respect to a scalar field is performed. In the first model the scalar field is a free Klein–Gordon field [44, 92, 90, 25], while in the second model it is an irrotational dust field [89–91]. The main difference between the two models is the final form of the physical Hamiltonian which dictates the dynamics of the gravitational degrees of freedom with respect to the relational time provided by the considered scalar field. We later proceed with the LQG quantization of the two models and define the complete quantum theories. In both models the spatial diffeomorphism constraints are solved on the quantum level. Finally, we close the chapter with a section exposing a new approximation method for the Hamiltonian operators, which is implemented in order to analyze the physical evolution in the considered deparametrized models.

IV.1 Emergent time in examples

IV.1.1 Gravity coupled to a free Klein–Gordon ﬁeld

The Hamiltonian formulation of the model of gravity minimally coupled to a free Klein–Gordon scalar ﬁeld φ(x), whose action ∫ √ √ 3 (4) 1 µν SSF = dtd x |g|R − |g| g φ,µφ,ν , (IV.1.1) M 2

i is set as a fully constrained system for the canonical variables Aa(x) and φ(x) a and their conjugate momenta Ei (x) and π(x) respectively. The analysis shows that the vector constraints Ca(x) and the scalar constraints C(x) in this model 1 gr are expressed in terms of the vacuum gravity constraints (II.1.29), Ca (x) and Cgr(x), and the scalar ﬁeld variables as follows

gr Ca(x) = Ca (x) + π(x)φ,a(x) , (IV.1.2)

2 √ gr 1 π (x) 1 a b C(x) = C (x) + √ + E (x)E (x)φ,a(x)φ,b(x) | det[E](x)| , 2 | det[E]| 2 i i (IV.1.3) while the Gauss constraints remains unchanged with respect to the vacuum i a case (II.1.29a). A solution by points in the phase space (Aa,Ei , φ, π) must 1We use the superscript gr to indicate that we refer to the vacuum gravity constraints. IV.1 Emergent time in examples 81 satisfy all the constraints:

i G (x) = 0 Ca(x) = 0 C(x) = 0 . (IV.1.4)

The deparametrization procedure starts by assuming that the constraints in (IV.1.4) are satisfied. We first solve the vector constraints (IV.1.2) for the gradient of the scalar field

Cgr φ = − a , (IV.1.5) ,a π and then use this condition in (IV.1.3) to obtain

1 π2(x) 1 CgrCgr √ C(x) = Cgr(x) + √ + Ea(x)Eb(x) a b | det[E](x)| . 2 | det[E]| 2 i i π2 (IV.1.6)

The new scalar constraints in (IV.1.6) do not explicitly depend on the scalar ﬁeld φ, we say that the scalar constraints are deparametrized. The next step is to solve equation (IV.1.6) for π, this gives: ( ) √ √ 2 | | − gr gr 2 − a b gr gr π = det[E](x) C (C ) Ei (x)Ei (x)Ca Cb . (IV.1.7)

Notice that in this case it is necessary on the constraint surface to have2

Cgr(x) ≤ 0 . (IV.1.8)

The sign ambiguity in (IV.1.7) amounts to treating diﬀerent regions of the phase space, namely for + and − respectively

2 ≥ ≤ a b | | π / Ei (x)Ei (x)φ,a(x)φ,b(x) det[E](x) . (IV.1.9)

We choose the phase space region corresponding to + and ≥. It contains spatially homogeneous spacetimes useful in cosmology. Then, the scalar constraints can be rewritten in an equivalent form as

C′(x) = π(x) ∓ h (x), (IV.1.10) √ SF √ √ √ − | | gr | | gr 2 − a b gr gr hSF := det[E] C + det[E] (C ) Ei Ei Ca Cb . (IV.1.11)

2Note that this condition on the gravitational scalar constraints must also be implemented in the quantum theory. 82 Towards new dynamics in LQG deparametrized models

We also restrict ourselves to the case of

π(x) ≥ 0 , (IV.1.12) although technically there is no problem in admitting both signs in the quantum theory.

Note that the constraints C′ commute strongly,

{C′(x),C′(y)} = 0 , (IV.1.13) this in turn implies [85, 87] that

{hSF (x), hSF (y)} = 0 . (IV.1.14)

Furthermore, a Dirac observable O on the phase space would satisfy

i ′ {O,G (x)} = {O,Ca(x)} = {O,C (x)} = 0 . (IV.1.15)

The vanishing of the ﬁrst and second Poisson brackets induce gauge invariance and spatial diﬀeomorphism invariance respectively. The vanishing of the third Poisson bracket is equivalent to writing

dO = {O, π(x)} = {O, h (x)} , (IV.1.16) dφ(x) SF which in an integrated non–singular form becomes ∫ ∫ ∫ O 3 d 3 3 d x = {O, d x π(x)} = {O, d x hSF (x)} . (IV.1.17) Σ dφ(x) Σ Σ ∫ 3 This equation shows precisely how the quantity Σ d x hSF (x) arises as a physical Hamiltonian in the reference frame where the scalar ﬁeld φ is used as a clock. Spacetime is then foliated with slices of constant values of the scalar ﬁeld which becomes the emergent physical time. The choice of the time gauge t = φ, where t denotes the time coordinate, implies

φ˙ = 1 , φ,a = 0 . (IV.1.18)

Moreover, the dynamical preservation of this gauge condition forces the lapse function N to be ﬁxed as N = 1. IV.1 Emergent time in examples 83

∫ 3 Finally, notice that Σ d x hSF (x) is a functional of the gravitational variables only, hence all the redundant degrees of freedom in the scalar constraints (IV.1.10) are absorbed in the scalar ﬁeld φ. The dynamics of the system is then promoted from imposing constraints, to describing evolution of the gravitational degrees of freedom with respect to the physical time set by the scalar ﬁeld.

IV.1.2 Gravity coupled to non–rotational dust

The model of gravity coupled to non–rotational dust was first introduced in [89] and is a special case of a more general model introduced and analyzed in [84]. The action of the non–rotational dust model is of the form ∫ √ √ 3 (4) 1 µν SD = dtd x |g|R − M |g|(g ρ,µρ,ν + 1) , (IV.1.19) M 2 where ρ represents the scalar dust field and M is a Lagrange multiplier. Notice that the difference between the dust Lagrangian (IV.1.19) and the scalar field one in (IV.1.1) stands from adding a potential term in the dust Lagrangian density of the system which is very similar to a cosmological constant term. While the Gauss and the spatial diffeomorphism constraints are identical to the massless scalar field case, the mentioned difference appears explicitly in the scalar constraints of the theory, namely

1 Π2(x) C(x) =Cgr(x) + √ 2M | det[E]| M √ + | det[E]|(Ea(x)Eb(x)ρ (x)ρ (x) + 1) , (IV.1.20) 2 i i ,a ,b where Π is the conjugate momentum to the dust ﬁeld ρ given by

M √ Π = | det[E]|(ρ ˙ − N aρ ) (IV.1.21) N ,a Additionally, one obtains a second class constraint imposed on M of the form

2 2 Π M = ab (IV.1.22) | det[E]|(1 + q ρ,aρ,b)

At this point, we can already implement the time gauge t = ρ which in- 84 Towards new dynamics in LQG deparametrized models duces, similarly to the scalar ﬁeld case, the following conditions

ρ˙ = 1 , ρ,a = 0 ,N = 1 . (IV.1.23)

Using these conditions along with equation (IV.1.21) to solve the second class constraints (IV.1.22), we obtain the ﬁnal expression of M

Π M = √ . (IV.1.24) | det[E]|

Replacing M in (IV.1.20) by its explicit form, and using the diffeomorphism constraints as in the scalar field case, we obtain the new simplified scalar constraints

′ gr C (x) = Π(x) + C =: Π(x) − hD(x) . (IV.1.25)

′ The constraints C commute strongly as well as the functionals hD which are proportional to the gravitational scalar constraints [85, 87], hence they depend on the gravitational variables only. Therefore, in this time gauge, a Dirac observable O on the gravity phase space would satisfy

i {O,G (x)} = {O,Ca(x)} , (IV.1.26) and would evolve according to the equation ∫ ∫ ∫ O 3 d 3 3 d x = {O, d x π(x)} = {O, d x hD(x)} , (IV.1.27) Σ dρ(x) Σ Σ ∫ 3 where Σ d x hD(x) arises as the physical Hamiltonian in the reference frame where the dust ﬁeld ρ is used as a clock. Spacetime is then foliated with slices of constant values of the dust ﬁeld, which is now the emergent physical time in the model.

IV.2 LQG quantization

The LQG quantization of the above deparametrized models [92, 90] can be formally outlined in two points:

• The physical Hilbert space is the space of quantum states of the matter free gravity which satisfy the quantum vector constraints and the quantum Gauss constraints. IV.2 LQG quantization 85

• The dynamics is deﬁned by a Schrödinger like equation

d i Ψ = − Hˆ Ψ , (IV.2.1) dT ℏ where T is a parameter of the transformation φ 7→ φ + T or ρ 7→ ρ + T . ˆ The Hamiltonian operator∫ H is the quantum operator corresponding to the classical observable d3x h (x) in the case of the scalar ﬁeld model, ∫ Σ SF 3 and Σ d x hD(x) in the case of the non–rotational dust model.

The concrete quantization of both models is presented in the following.

IV.2.1 Physical Hilbert spaces

In both models, the kinematical Hilbert space is exactly Hkin, which is the Hilbert space of vacuum LQG defined as the completion of the space Cyl of cylindrical functions (section II.2.1). The Gauss and spatial diffeomorphism constraints are then implemented and solved in the same way as in the case of the vacuum theory discussed in H G section II.2.2. The resulting space is the Hilbert space Diff of SU(2)–gauge invariant and spatial diffeomorphism invariant states with a scalar product H H G induced from the scalar product on kin. Therefore, the space Diff is the physical Hilbert space in both quantum models.

IV.2.2 Quantum dynamics

The last ingredient to complete the LQG quantization program of the two models is to deﬁne quantum Hamiltonian operators which would generate the quantum dynamics through a Schrödinger–like equation

d iℏ |ψ⟩ = Hˆ |ψ⟩ , (IV.2.2) dT

| ⟩ ∈ H G for any state ψ Diff. This task can be achieved in a satisfactory man- ∫ner through a careful regularization∫ [25, 24] of the classical expressions of 3 3 Σ d x hSF (x) =: HSF and Σ d x hD(x) =: HD, in a way very similar to the procedure adopted for the scalar constraints and presented in chapter III. 86 Towards new dynamics in LQG deparametrized models

IV.2.2.1 Scalar ﬁeld deparametrized model Hamiltonian

We recall the expression of the Hamiltonian in the deparametrized model of (Lorentzian) gravity coupled to a scalar ﬁeld √ ∫ √ √ √ 3 − | | gr | | gr 2 − a b gr gr HSF := d x det[E] C + det[E] (C ) Ei Ei Ca Cb . Σ (IV.2.3)

A preliminary step is to assume that, since the physical Hilbert space in the H G ab gr gr model is Diff, a proper operator corresponding to the term q Ca Cb in (IV.2.3) would vanish on spatial diﬀeomorphism states. Therefore, considering gr the sign condition (IV.1.8) on C , we could reasonably reduce the form of HSF to ∫ √ √ 3 gr HSF := d x −2 | det[E]|C (IV.2.4) ∫Σ √ 1 ( ) 3 a b k 2 | | = d x 2 ϵijkEi (x)Ej (x)Fab(x) + (1 + β ) det[E] R(x) , Σ kβ where we obtain again two terms under the square root which we could refer to k as the Euclidean part, containing the curvature Fab, and the Lorentzian part containing the Ricci scalar. In order to deﬁne the corresponding operator, we need to consider how to regularize and quantize an expression of the form ∫ √ d3x a2(x) + b2(x) , (IV.2.5) Σ

i a where a(x) and b(x) are functionals of the ﬁelds Aa and Ei . Introducing a decomposition C ϵ of the manifold Σ into cells ∆, the integral can be approximated as √ ∫ (∫ ) (∫ ) √ ∑ 2 2 3 2 2 3 3 d x a (x) + b (x) = d x a(x) + d x b(x) + O(ϵ∆) , Σ ∆ ∆ ∆ (IV.2.6)

3 where∫ for every cell ∆∫, ϵ∆ denotes the coordinate volume of ∆. If the inte- 3 3 grals ∆ d x a(x) and ∆ d x b(x) can be quantized as well-deﬁned operators, ∫equation√ (IV.2.6) then shows how to deﬁne the operator corresponding to 3 2 2 Σ d x a (x) + b (x). Equation (IV.2.6) is the basis of our construction of the operator corresponding to HSF . IV.2 LQG quantization 87

We proceed with the regularization of the two parts separately and along the same lines as the scalar vacuum constraints in chapter III. We start with the quantization of the Euclidean part of our Hamiltonian (see (IV.2.4)).

i) In equation (IV.2.6), the role of a(x) is now played by the function √ 1 a b k √ ϵijkE (x)E (x)F (x) . (IV.2.7) kβ2 i j ab

Consequently, we consider the quantization of the integral ∫ √ E 3 a b k CSF := d x ϵijkEi (x)Ej (x)Fab(x) , (IV.2.8)

where in the following we consider the domain of integration to be the whole hypersurface Σ. The restriction to a sub–region such as a cell ∆ is straightforward. According to the general framework of LQG presented previously, we need

to express the integral in terms of holonomies he and ﬂuxes PS,i. Considering a decomposition C ϵ of the manifold Σ, we can then write ∑ √∑ E ′ k (l) CSF = lim kβ ϵIJ (∆) ϵijk PSI ,iPSJ ,j(h IJ ) . (IV.2.9) ϵ→0 ∆ ∆ α∆ ∆∈C ϵ I,J

where we use the short notation for the holonomy term ( ) k (l) 3 (l) k (l) (h IJ ) := Tr D (hαIJ )(τ ) . (IV.2.10) α∆ 2 ∆ Wl √ ′ with again Wl = i l(l + 1)(2l + 1). The factor ϵIJ (∆) is a regularization coeﬃcient which depends on the structure of the cell ∆ and which vanishes if the labels I,J coincide.

′ k (l) Each term ϵIJ (∆) ϵijk PSI ,iPSJ ,jϵabc(h IJ ) gives rise to a well–deﬁned op- ∆ ∆ α∆ erator acting on Cyl,

′ ˆk (l) ˆ ˆ ϵIJ (∆) ϵijk (h IJ ) PSI ,iPSJ ,j , (IV.2.11) α∆ ∆ ∆

with a unique ordering of the operators such that the holonomy is on the 88 Towards new dynamics in LQG deparametrized models

extreme left3. In this way we obtain an operator on Cyl ∑ ∑ ′ ˆk (l) ˆ ˆ ϵIJ (∆) ϵijk (h IJ ) PSI ,iPSJ ,j , (IV.2.12) α∆ ∆ ∆ ∆∈C ϵ I,J

which depends on the decomposition C ϵ. However, as we reﬁne the decomposition C ϵ, the operator family does not converge to any operator in Cyl, similarly to the case of the Euclidean part of the scalar constraints operator discussed in section III.3.1. The solution is to consider the dual operator of G the regulated operator in (IV.2.12) in the space ηV (Cyl ), deﬁned through

the averaging map ηV introduced in section III, then take the limit of the regulator. This is a consistent approach since the final end is to define the Hamiltonian operator on the physical Hilbert space in the theory, that is H G → Diff. The limit as ϵ 0 exists upon similar conditions on the decomposition C ϵ imposed in the case of the Euclidean part of the scalar constraints operator, namely: Given a graph Γ with a set of vertices Vert(Γ), the conditions on the decomposition C ϵ are as follows:

– each cell ∆ contains at most one vertex of the graph Γ; – if v ∈Vert(Γ) and v ∈ ∆, then I ⊂ * to each edge eI there is assigned a surface S∆ ∂∆ intersecting the edge transversally (there may be surfaces in ∂∆ not intersecting any edge);

* to each ordered pair of edges eI and eJ meeting transversally at IJ v there is assigned a special loop (see section III.3.1) α∆ oriented ′ ′ according to the order of the pair (eI , eJ ); ′ * for edges eI and eJ of Γ meeting transversally at v ϵIJ (∆) is not zero; ′ * for edges eI and eJ of Γ meeting tangentially at v ϵIJ (∆) = 0;

* for edges eI and eJ of Γ meeting transversally at v the corre-

sponding loop αIJ , in the limit ϵ → 0, is shrank to v in the same diﬀeomorphism invariant way as in the case of the Euclidean part of the scalar constraints operator; ′ ′′ – the value of non vanishing ϵIJ is an overall constant κv, depending on the valence of the vertex but independent of ∆,I,J, multiplied by

3As mentioned before, an ordering with the holonomy operator on the extreme right is inadmissible as it leads to an ill deﬁned operator. IV.2 LQG quantization 89

the factor ϵ (e ˙I , e˙J ) which is 0 if e˙I and e˙J are linearly dependent or 1 otherwise..

It follows that we can deﬁne a family of operators, labeled by the regulator

ϵ, on a dense subset of Hkin and which corresponds to the Euclidean part of the Hamiltonian. The limit ϵ → 0 is then taken for the dual operator G deﬁned on ηV (Cyl ), resulting in the operator

ℏ2 ∑ ∑ [ ]∗ ′′ ˆk (l) ˆ ˆ κx lim ϵ (e ˙I , e˙J ) ϵijk (h IJ ) Jx,eI ,iJx,eJ ,j , (IV.2.13) 4 ϵ→0 αϵ x∈Σ I,J

where eI runs through the set of germs starting at the point x and the loops

αIJ represent the new edges to be added at the same point x. G The operator is then extended to a symmetric operator on the space ηV (Cyl ). Ê G One then defines the Euclidean part operator CSF on the space ηV (Cyl ) by introducing the square root, obtaining ∑ Ê Ê CSF := CSF,x . (IV.2.14) x∈Σ

where √ [ ] ℏkβ . ∑ ∗ . ˆE . ′′ ˆk (l) ˆ ˆ . CSF,x := . κx lim ϵ (e ˙I , e˙J ) ϵijk (h IJ ) Jx,eI ,iJx,eJ ,j . . 2 ϵ→0 αϵ I,J (IV.2.15)

. . which depends on the chosen symmetric extension, symbolized by . ... This H G operator is naturally passed to the physical Hilbert space Diﬀ

CÊ :D(CÊ ) ⊂ H G −→ H G SF ∑SF Diff Diff Ê Ê CSF := CSF,x . (IV.2.16) x∈Σ

D Ê H G where the domain (CSF ) is dense in Diff. In virtue of (IV.2.15), The Ê Euclidean part operator CSF is graph changing. Note that one needs to introduce self–adjoint positive operator under the Ê square root in (IV.2.15) in order for the operators CSF,x to be well–defined. However, a symmetric operator for the Euclidean term is not necessary to construct the complete Hamiltonian operator, for which only the square root of the sum of the squares of the Euclidean and Lorentzian terms needs 90 Towards new dynamics in LQG deparametrized models

to be deﬁned instead. ii) Now we turn to the Lorentzian part of the Hamiltonian operator. Following the strategy of quantization indicated by equation (IV.2.6), we introduce a second operator corresponding to ∫ √ L 3 | | HSF := d x det[E] R, (IV.2.17) Σ √ where the function |det[E]| R plays the role of b(x) in (IV.2.6). We proceed in a similar way to the quantization of the curvature operator introduced in section III.2. Consider a cellular decomposition C ϵ of the manifold Σ characterized by a regulator ϵ. We can then write the integral (IV.2.17) as a limit of a Riemannian sum over the cells ∆′, ∫ √√ √ L 3 | | | | HSF = d x det[E] det[E] R Σ √( )( ) ∑ ∫ √ ∫ √ = lim d3x |det[E]| d3x |det[E]|R , ϵ→0 ′ ′ ∆′∈C ϵ ∆ ∆ (IV.2.18)

′ Next we decompose each cell ∆ into c∆′ closed cells ∆, where a cell ∆ has a boundary formed by√ a number n∆ of 2-surfaces (faces). Then we | | approximate the integral√ of det[E] by a Riemannian sum over the cells ∆, and the integral of |det[E]|R by a regularized Regge action for an appropriate ∆-decomposition of ∆′, obtaining v u( )( ) ∑ u ∑ √ ∑ L t HSF = lim q∆(E) R∆(E) , (IV.2.19) ϵ→0 ∆′∈C ϵ ∆⊂∆′ ∆⊂∆′

where we use the same notation as in the regularization of the curvature operator in section III.2.2. To continue the calculation from equation (IV.2.19), we assume that the

cells ∆ are chosen such that we obtain the same contributions q∆(E) and

R∆(E) from each cell ∆, up to higher order corrections in ϵ∆′ (equivalently, up to higher order corrections in ϵ). IV.2 LQG quantization 91

In this way we obtain ∑ ∑ √√ L HSF = lim q∆(E) R∆(E) ϵ→0 ∆′∈C ϵ ∆⊂∆′ ∑ √√ = lim q∆(E) R∆(E) , (IV.2.20) ϵ→0 ∆∈C ϵ

where last step is achieved by combining the two sums over ∆′ and ∆.

Notice(√ that the) expression of R∆(E) in (III.2.17) contains an overall factor −1 of√ q∆(E) . This leads to a crucial√ simpliﬁcation in the expression of q∆(E) R∆(E), namely, the factors of q∆(E) are canceled: √ ∑ √ 2 2 q∆(E) R∆(E) = k β ϵijkP I P J ϵij′k′ P I ′ P J ′ S∆,u,j S∆,u,k S∆,u,j S∆,u,k u⊂∂∆ ( [ ]) 2π PSI ,kPSJ ,k × − π + arccos ∆,u ∆,u . λu 2|P I ||P J | S∆,u S∆,u (IV.2.21)

In the quantum theory, this simpliﬁcation implies that the volume operator will be absent from the Lorentzian part, and consequently from the whole Hamiltonian operator. The absence of the volume is an important technical advantage in the calculation of the action of the ﬁnal Hamiltonian operator.

Having the operators corresponding to P I s’, we are now able to deﬁne S∆,u,j the quantum operator corresponding to q∆(E) R∆(E) as ∑ √ \ 2 2 ′ ˆ ˆ ˆ ˆ q∆(E) R∆(E) := k β κ ϵijkP I P J ϵij′k′ P I ′ P J ′ ∆IJ S∆,u,j S∆,u,k S∆,u,j S∆,u,k u⊂∂∆ ( [ ]) ˆ ˆ 2π PSI ,kPSJ ,k × − π + arccos ∆,u ∆,u . λu 2Pˆ I Pˆ I Pˆ J Pˆ J S∆,u,m S∆,u,m S∆,u,l S∆,u,l (IV.2.22)

By imposing the same conditions on the cellular decomposition C ϵ as in the case of the curvature operator (section III.2), then taking the limit of the

regulator ϵ → 0 in Hkin, one obtains the operator 92 Towards new dynamics in LQG deparametrized models

∑ ˆL ˆL CSF,kin := CSF,x,kin x∈Σ [ √ ∑ ℏ2 2 2 ′ ∑ k β κx := ϵ (e ˙ , e˙ ) (ϵ Jˆ Jˆ )(ϵ ′ ′ Jˆ ′ Jˆ ′ ) 4 I J ijk x,eI ,j x,eJ ,k ij k x,eI ,j x,eJ ,k x∈Σ I,J    1 2 2π Jˆ Jˆ ×  − π + arccos √ x,eI ,k √x,eJ ,k  , λx,eI ,eJ ˆ ˆ ˆ ˆ Jx,eI ,mJx,eI ,m Jx,eJ ,lJx,eJ ,l (IV.2.23)

′ where λx,eI ,eJ is an integer parameter, κx is the averaging coeﬃcient which

appears in the curvature operator, eI runs through the set of germs starting

at the point x and ϵ (e ˙I , e˙J ) is zero if e˙I and e˙J are linearly dependent or one otherwise. Notice that there is no ordering ambiguity in the operator ˆL CSF,kin, the operator is essentially self-adjoint and it is graph preserving. ˆL The square root present in the definition of the operator CSF,x,kin could be ˆL understood here as taking the square root of the operator CSF,x,kin restricted to the positive part of its spectrum. However this restriction is not necessary to construct the complete Hamiltonian operator, for which the square root of the sum of the squares of the Euclidean and Lorentzian terms needs to be defined instead. ˆL The operator CSF,kin is clearly SU(2)–gauge invariant and also diffeomorphism invariant, hence we can define by duality the essentially self–adjoint H G Lorentzian part operator on Diff as

ˆL D ˆL ⊂ H G −→ H G CSF : (CSF ) Diﬀ Diﬀ [ ]∗ ∑ ˆL ˆL ˆL CSF := CSF,kin = CSF,x , (IV.2.24) x∈Σ

[ ]∗ ˆL ˆL where CSF,x := CSF,x,kin . iii) Finally, we arrive to the point where we can define the physical Hamiltonian operator in the quantum scalar field deparametrized model, corresponding to the quantization of the Hamiltonian functional in (IV.2.3). The physical Hamiltonian operator is required to be self-adjoint on some non-trivial domain in order for it to generate unitary evolution of the quantum system and for its spectra to admit a physical interpretation. A first step toward achieving self-adjointness of the Hamiltonian operator is to construct a sym- IV.2 LQG quantization 93

metric operator, the proposal we consider in our analysis in section IV.3 is ( ) ˆ D ˆ ⊂ H G −→ H G HSF : HSF Diﬀ Diﬀ √ ∑ ˆ 1 ˆ ˆ HSF := √ CSF,x + CSF,x . (IV.2.25) 2 x∈Σ 4kβ

where

Cˆ :=CÊ + (1 + β2)CˆL SF,x SF,sym,x SF,sym,x ( ) [ ]† [ ]† Ê 2 Ê 2 2 ˆL 2 ˆL 2 :=(CSF,x) + (CSF,x) + (1 + β ) (CSF,x) + (CSF,x) . (IV.2.26)

ˆ ˆ The combination CSF,x+ CSF,x was introduced to impose the sign condition (IV.1.8) on the operator under the square root. The existence of self-adjoint ˆ extensions for the Hamiltonian operator HSF is still an open question.

IV.2.2.2 Non-rotational dust deparametrized model Hamiltonian

At this point, the Hamiltonian operator in the case of non-rotational dust model can be defined in a straightforward manner. Recall that the expression of the Hamiltonian in the deparametrized model of (Lorentzian) gravity coupled to non-rotational dust is ∫ 3 gr HD := − d x C (x) . (IV.2.27) Σ where Cgr(x) are the scalar constraints in vacuum gravity (III.0.1). Hence, the quantization of the functional HD coincides exactly with the quantization of the scalar constraints in vacuum gravity with the only difference that the lapse (smearing) function N is taken to be a constant equal to 1. Therefore we can define the quantum Hamiltonian operator in the non-rotational dust deparametrized model as ( ) ˆ D ˆ ⊂ H G −→ H G HD : HD Diff Diff ˆ ˆ HD := −Csym(N = 1) , (IV.2.28)

ˆ where Csym(N) is deﬁned in equation (III.4.3) with a Lorentzian signature s = −1. Here, we are of course obliged to choose a symmetric operator in order to look for a self-adjoint extension which guarantees unitary dynamics. 94 Towards new dynamics in LQG deparametrized models

Another way to define the Hamiltonian operator in the case of non-rotational dust model is possible. It is based on the observation that, in the case of the scalar field deparametrized√ model, we were able to define an operator corre- gr sponding to the functional det[E]C and we can write the Hamiltonian HD as ∫ ∫ √ 3 gr 3 1 gr HD = − d x C = − d x √ det[E]C . (IV.2.29) Σ Σ det[E]

We can then regularize this expression using a decomposition C ϵ and we obtain ( ) ∑ ∫ √ ( 1 ) 3 gr HD = − lim ∫ √ d x det[E]C . (IV.2.30) ϵ→0 ′ ′ ϵ 3 ∆ ∆ ∈C ∆′ d x det[E]

From this point we can proceed in a similar way to defining the Hamiltonian in the scalar field case and the result of the quantization can be summarized as follows ∫ √ 1 3 gr ˆ ( ) d−1 d x det[E]C → CSF and ∫ √ → V , 3 ∆′ d x det[E] (IV.2.31) and we define the Hamiltonian operator as ( ) ˆ D ˆ ⊂ H G −→ H G HD : HD Diff Diff √ √ ∑ 1 Hˆ := − Vd−1Cˆ Vd−1 , (IV.2.32) D 4kβ2 SF,x x∈Σ

ˆ ˆ where CSF,x was deﬁned in (IV.2.26). The operator HD in (IV.2.32) is symmetric and densely deﬁned. It is also the operator that we consider in our analysis of time evolution in the dust model as part of section IV.3.

In summary, the construction of the physical Hilbert space in both deparametrized models presented above, along with the successful implementation of the Hamiltonian functionals as well–defined symmetric quantum operators on this space, complete the LQG quantization program and introduce two full quantum deparametrized models of gravity. One then naturally turn to the question of testing those theories. An important aspect of this question is to obtain a sufficient control on the quantum dynamics. More specifically, we need to be able to compute the evolution of relevant physical states and ob- IV.3 An approximation method for LQG dynamics 95 servables with a relatively good precision. In the following section, we present an approximation method introduced in [26], which we apply to the physical Hamiltonian operators in the case of the deparametrized models defined above. We then use it to analyze the generated time evolution in some examples with specific physical states. This work represents the first steps towards achieving the goal of understanding and controlling the new dynamics exposed in this dissertation.

IV.3 An approximation method for LQG dynamics

ˆ The complete spectral decomposition of the two Hamiltonian operators HSF ˆ in (IV.2.25) and HD in (IV.2.32), densely defined on the non–separable phys- H G ical Hilbert space Diff, has not been achieved so far. It is then imperative to develop and use approximate methods in the analysis of the dynamics they induce. A straightforward approach is to consider the expansion of the evolution operator in powers of the time parameter and introduce a truncation of the expansion at a certain fixed order of time. In this case the evolution operator reduces to a finite sum of terms where each one is calculated through a finite number of successive actions of the considered Hamiltonian operator. Such truncation forms a valid approximation of the full time evolution when the time interval under consideration is relatively small. Nevertheless, this method would not be enough to be able to deal with the Hamiltonian op- ˆ erator HSF present in the deparametrized model with a massless scalar field (IV.2.25). This is because of the presence of the square root in the expression ˆ of HSF , which requires an access to the spectral decomposition of the operator under the square root. An interesting approach to this problem, developed in [26] and presented below, is to use standard time–independent perturbation theory of quantum mechanics to introduce a perturbative expansion of the Hamiltonian, the small perturbation parameter being determined by the Barbero–Immirzi parameter β. This method allows us to define an approxi- ˆ mate spectral decomposition of the operator HSF and hence to compute the evolution without need of a truncation with respect to the time parameter. 96 Towards new dynamics in LQG deparametrized models

IV.3.1 Perturbation theory with the Barbero–Immirzi parameter ˆ Recall that the expression of the operator CSF,x in (IV.2.26) is of the form ( ) 1 Cˆ := (1 + β2) CˆL + CˆE . (IV.3.1) SF,x SF,sym,x 1 + β2 SF,sym,x

ˆL Since the operator CSF,sym,x is graph preserving and acts locally on the vertices of the graph without changing the SU(2) representations [23, 25], its spectral decomposition breaks down to stable finite dimensional blocks, similarly to the volume operator. Each block corresponds to the Hilbert space of a fixed graph with fixed coloring (spins) and takes the form of a tensor product over the vertices of stable sub–blocks, each representing a separate intertwiners space assigned to each vertex of the colored graph. Given a colored graph, the dimension of each intertwiner space is then fixed, hence one can proceed with ˆL the diagonalization of the (essentially self–adjoint) operator CSF,sym,x. Having the spectral decomposition of this operator, the idea is to treat the Euclidean 2 Ê part operator 1/(1 + β )CSF,sym,x as a perturbation to the Lorentzian part ˆL 2 4 operator CSF,sym,x, with 1/(1 + β ) being the perturbation parameter . This means that we will assume that the Barbero–Immirzi parameter is significantly large, β ≫ 1, large enough so that the perturbative expansion in 1/(1 + β2) gives a good approximation for the eigenstates5 and eigenvalues of the Hamiltonian. The condition for this is that the corrections to the eigenvalues are small compared to the spectral gaps present in the discrete spectrum of the unperturbed operator (in this case, the Lorentzian part of the (squared) Hamiltonian). The procedure is then as follows:

Given an intertwiner space Iv of dimension dv associated to a vertex v of a given colored graph Γ, the Lorentzian part operator is put in a diagonal form

∑dv ˆL | ⟩⟨ | CSF,sym,v = λi(v) λi(v) λi(v) . (IV.3.2) i=1

4 ˆL Ê Note that the two operators CSF,sym,x and CSF,sym,x contain a certain dependence on β in their definitions, however this dependence is of the same order in both operators and hence does not affect the approximation. 5Since we expect to be dealing with unbounded operators, it is not clear to us yet if, given a fixed value of β, the perturbative expansion would be valid for all eigenstates of ˆ ˆ H G HSF or HD on Diff. IV.3 An approximation method for LQG dynamics 97

For β ≫ 1, we can write

1 ∑dv CˆL + CˆE = λ′ (v) |λ′ (v)⟩⟨λ′ (v)| , (IV.3.3) SF,sym,v 1 + β2 SF,sym,v i i i i=1 where 1 λ′ (v) = λ (v) + ⟨λ (v)| CˆE |λ (v)⟩ (IV.3.4) i i 1 + β2 i SF,sym,v i

( ) ′ 2 2 ∑dv ˆE 1 ⟨λi(v)| C |λk(v)⟩ ( ) SF,sym,v O 2 −3 + 2 + (1 + β ) , 1 + β λi(v) − λk(v) k=1 λk≠ λi and

[ ′ ] ∑dv ⟨λ (v)| CÊ |λ (v)⟩ | ′ ⟩ | ⟩ 1 k SF,sym,v i | ⟩ λi(v) = λi(v) + 2 λk(v) (IV.3.5) 1 + β λi(v) − λk(v) k=1 λ ≠ λ [ k i ] ( ) ′ 2 2 ∑dv ⟨λ (v)| CÊ |λ (v)⟩ 1 −1 i SF,sym,v k | ⟩ + 2 2 λi(v) 1 + β 2 (λi(v) − λk(v)) k=1 λk≠ λi ( ) [ ′ ] 2 ∑dv ⟨λ (v)| CÊ |λ (v)⟩⟨λ (v)| CÊ |λ (v)⟩ 1 k SF,sym,v l l SF,sym,v i | ⟩ + 2 λk(v) 1 + β (λi(v) − λk(v))(λi(v) − λl(v)) k=1, l=1 λk≠ λi λl≠ λi ( ) [ ′ ] 2 ∑dv ⟨λ (v)| CÊ |λ (v)⟩⟨λ (v)| CÊ |λ (v)⟩ − 1 i SF,sym,v i k SF,sym,v i | ⟩ 2 2 λk(v) 1 + β (λi(v) − λk(v)) k=1 ̸ ( λk)=λi + O (1 + β2)−3 .

The sums in (IV.3.4) and (IV.3.5) are over the eigenstates of the Lorentzian part in the new intertwiner spaces at v, which together contain the image of I ′ the space v by the Euclidean part. The upper limit of the summation dv is then the finite sum of dimensions of those new intertwiner spaces at the vertex v. Because the Euclidean part does not preserve each of the stable subspaces of the Lorentzian part separately, as it modifies the graph structure at the vertex v, the first order correction to the eigenvalue λi in (IV.3.4) and the last sum in (IV.3.5) vanish, as they are proportional to expectation values of the Euclidean part on eigenvectors of the Lorentzian part. An important remark is that in case there is degeneracy in the spectrum of the Lorentzian part, which is the case in the examples presented below, one has to choose the proper 98 Towards new dynamics in LQG deparametrized models eigenvectors to which the corrected eigenvectors converge when 1/(1 + β2) goes to zero [26], equivalently β goes to infinity. It is then straightforward to obtain the explicit expression of the square root operator and the evolution operator: √ ∑dv √ ˆ ˆ ′ | ′ ⟩⟨ ′ | CSF,v + CSF,v := λi(v) λi(v) λi(v) , (IV.3.6) i=1 λ′ ≥0 i √ ( ) ( √ ) i ∏ i 1 + β2 exp − φHˆ := exp − φ Cˆ + Cˆ ℏ SF ℏ 4kβ2 SF,x SF,x x∈Σ∩Γ ( √ ) ∏ ∑dx i (1 + β2) = exp − φ λ′ (x) |λ′ (x)⟩⟨λ′ (x)| , ℏ 2kβ2 i i i x∈Σ∩Γ i=1 ′ ≥ λi 0 (IV.3.7)

It follows that given an operator A and an initial state |Ψ0⟩, the state at time T is given by |Ψ(T )⟩ = USF (T ) |Ψ0⟩ and the expectation value ⟨A(T )⟩ is computed as

⟨A(T )⟩ = ⟨Ψ(T )| A |Ψ(T )⟩ (IV.3.8) √ d ( ( √ )) ∏ ∑x i (1 + β2) √ ⟨ = exp − T λ′ − λ′ ⟨Ψ |λ′ ⟩ λ′ A |λ′ ⟩⟨λ′ |Ψ ⟩ . ℏ 16πGβ2 i j 0 j j i i 0 x∈Σ i,j=1 ′ ≥ λi 0 ′ ≥ λj 0

ˆ All that was mentioned above for the operator HSF can be similarly applied ˆ to HD in the dust model. In the following, we analyze the proposed pertubative expansion (the β– expansion) for both models discussed above, through numerical simulations of the evolution of expectation values of the volume and the curvature operators on speciﬁc test states.

IV.3.2 Numerical analysis of simple examples

In this section we present examples of the evolution of the expectation values of the volume operator and the curvature operator (III.2.46) (with N = 1), in both deparametrized models discussed above, using the perturbative expansion up to second order with specific values of β. We consider initial states corresponding to eigenvectors of the volume operator with a graph consisting IV.3 An approximation method for LQG dynamics 99 of a single non–degenerate 4–valent vertex where all spins j are equal and fixed as j = 2 or j = 10. Although the volume operator has degenerate subspaces, the evolution of the expectation value of the operators considered below is independent of the choice of the eigenvector in the corresponding degenerate subspace. In all the calculations, we fix the constants in the operators as follows

k = ℏ = κ0 = 1 , α = 3 , (IV.3.9) where κ0 is the averaging constant present in the definition of the volume operator (III.2.39)[67]. Moreover, we fix to 1/2 the SU(2)–representations of the holonomies associated to the special loops created or annihilated by the Euclidean parts of the considered Hamiltonians. The parameter T in the graphics stands for the standard time given either by the scalar field or the dust field depending on the considered case, while the parameters T ′ and T ′′ in the embedded graphics stand for the rescaled times given by

√ 1 + β2 T ′ := 1 + β2 T,T ′′ := T. (IV.3.10) β3/2

The reason to consider these rescaled time parameters is explained later in the comments.

• Perturbation theory in the scalar ﬁeld model:

– Eigenvectors with spins j = 2:

XV\Β32

0.60

0.59

0.58 Β=10

0.57 Β=20 0.573 0.6 0.56 Β=50 0.572

0.55 0.571 0.58

0.54 T T' 0.002 0.004 10 20 T 0.5 1.0 1.5 2.0

Fig. IV.1 Evolution of the expectation value ⟨V ⟩ of the volume operator with an initial eigenvector with eigenvalue v = 0.5730. 100 Towards new dynamics in LQG deparametrized models

XR\Β12 16.0

15.5

15.0 Β=10 Β=20 14.5 15.2

15.5 Β=50 14.0 15

15 13.5 T T' 0.01 0.02 10 20

T 0.5 1.0 1.5 2.0

Fig. IV.2 Evolution of the expectation value ⟨R⟩ of the curvature operator with an initial eigenvector with eigenvalue v = 0.8725.

– Eigenvectors with spins j = 10:

XV\Β32

8.0

7.5 Β=25 7.0 8 Β=50 6.5 8 7 Β=100

6.0 7.5 6

5.5 T T' 0.001 0.002 2 4 T 0.02 0.04 0.06 0.08 0.10

Fig. IV.3 Evolution of the expectation value ⟨V ⟩ of the volume operator with an initial eigenvector with eigenvalue v = 8.3177.

XR\Β12

32 Β=25 Β=50 30 32 34 Β=100

28 31 32

26 T T' 0.001 0.002 2 4 T 0.05 0.10 0.15 0.20 0.25 0.30

Fig. IV.4 Evolution of the expectation value ⟨R⟩ of the curvature operator with an initial eigenvector with eigenvalue v = 5.1078. IV.3 An approximation method for LQG dynamics 101

• Perturbation theory in the dust ﬁeld model:

– Eigenvectors with spins j = 2:

XV\Β32

0.86

0.84 Β=10 0.82 Β=20 0.872 0.86 0.80 Β=50 0.87

0.868 0.78 0.82 T T'' 0.005 0.01 1 2 0.76 T 0.2 0.4 0.6 0.8 1.0

Fig. IV.5 Evolution of the expectation value ⟨V ⟩ of the volume operator with an initial eigenvector with eigenvalue v = 0.8725.

XR\Β12

18.2

Β=10 18.0 18.3 18.3 Β=20 18.2 18.2 Β=50 17.8 18.1 18.1

T T'' 0.01 0.02 2 4 17.6

T 0.1 0.2 0.3 0.4 0.5

Fig. IV.6 Evolution of the expectation value ⟨R⟩ of the curvature operator with an initial eigenvector with eigenvalue v = 0.5730. 102 Towards new dynamics in LQG deparametrized models

– Eigenvectors with spins j = 10:

XV\Β32

4 Β=50

3 Β=100 5.1 5 Β=200 2 5.09 4

1 T T'' 0.001 0.002 2 4 T 0.2 0.4 0.6 0.8 1.0

Fig. IV.7 Evolution of the expectation value ⟨V ⟩ of the volume operator with an initial eigenvector with eigenvalue v = 5.1078.

XR\Β12

28 Β=50 Β=100 30 30 26 Β=200 28 28

24 T T'' 0.001 0.002 2 4

T 0.2 0.4 0.6 0.8 1.0

Fig. IV.8 Evolution of the expectation value ⟨R⟩ of the curvature operator with an initial eigenvector with eigenvalue v = 8.3177.

General comments:

• Since the initial states we are considering in this numerical analysis correspond to specific spin–network states, and the evolution operators in this approximation contain a finite order of the graph changing Euclidean operators, the expectation values of the volume and curvature operators are both bounded through the time evolution of those states. One expects that in order to obtain unbounded expectation values in this approximation, one should consider initial states which take the form of infinite linear combination of spin–network states.

• The embedded graphics on the left of each ﬁgure display the evolution on a very small time interval. One can notice that the expectation values IV.3 An approximation method for LQG dynamics 103

at T = 0 do not coincide. The reason is that in our analysis we ﬁrst compute the expectation values as functions of time using the expression in (IV.3.7), then we evaluate the resulting functions in a certain time interval. More explicitly, given a state |ψ⟩ we have

∑d | ⟩ | ′ ⟩⟨ ′ | ⟩ ̸ | ⟩ ψ(T = 0) = λi λi ψ = ψ , (IV.3.11) i=1

where we obtain a projector on the space generated by the corrected | ′ ⟩ | ⟩ eigenvectors λi and not the initial eigenvectors λi of the considered Hamiltonian operator. In general, this projector can coincide with the identity operator at most6 on the zeroth order in the β–expansion, hence the discrepancy in the expectation values at T = 0 for diﬀerent values of the Barbero–Immirzi parameter β. This discrepancy at T = 0 also indicates the deviation of the norm of the corrected eigenvectors from unity. For instance, the large deviation of the expectation values of the curvature operators at T = 0 with β = 10 in ﬁgure IV.2 is due to the large norm of the corrected eigenvectors, which implies that the β = 10 value of the Immirzi–Barbero parameter is not good enough to make sense of the perturbation method.

• In the case of the scalar ﬁeld model, ﬁgures IV.1, IV.2, IV.3, IV.4, when the perturbation from the Euclidean part is small, one can notice on the plots a periodic evolution of the expectation values of the volume and curvature operators. This periodicity seems to manifest for all eigenvectors of the volume operator, independently of the intertwiner space. This strongly indicates the presence of a certain symmetry between the vol- ˆ ume operator, the curvature operator and the Lorentzian operator CSF . We leave this question for future investigations.

• Figures IV.6 and IV.8, for the expectation value of the curvature operator in the dust model, show practically constant curves for values of β larger than 20. This is expected because when the perturbation is small, the dust model Hamiltonian reduces to almost the curvature operator itself, hence the constant expectation value.

6In the case of the scalar ﬁeld model, in addition to the corrections arising from the perturbation, the diﬀerence in general could also depend on the sign of the spectrum of the Lorentzian part operator since we have the condition of positivity as expressed in (IV.3.7). However, in our examples the spectrum of the Lorentzian part operators is positive on the associated intertwiners spaces. 104 Towards new dynamics in LQG deparametrized models

• Finally, the embedded graphics on the right of each figure display the evolution with respect to the rescaled time. Comparing those graphics to the graphics of the evolution with respect to the standard time exhibits how the overall factors depending on β in the evolution operators affect the phases in the evolution curves. Those overall factors are obtained by factorizing out all the dependence on β in the Lorentzian and Euclidean parts of the Hamiltonian operator, i.e one write the Hamiltonian in the form ( ) 1 Hˆ = f(β) CˆL + CÊ , (IV.3.12) 0 1 + β2 0

ˆL ˆE such that C0 and C√0 are independent of β. f(β) is then the rescaling 2 ˆ 2 3/2 ˆ factor which equals 1 + β for HSF and (1 + β )/β for HD.

The simulations presented above show that the proposed β–approximation can indeed be applied fully and consistently in a certain sector of the physical Hilbert space in both deparametrized models. Therefore, this perturbation method presents itself as a promising tool in the investigation of the dynamics in LQG models. A very interesting outcome is the periodic character of the evolution of the expectation values of the volume and curvature operators in the deparametrized model with a free Klein–Gordon scalar ﬁeld. While it is a rather unexpected result, it is clearly reminiscent to the particular physical Hamiltonian present in the model, and the choice of the volume and curvature operators as observables. This indeed suggests the presence of a special relation between the spectral decompositions of the mentioned operators which is yet to be understood. As a future work, we would like to establish more accurately to which extent one could apply this approximation with respect to the admissible range of the Barbero–Immirzi parameter β and the choice of initial states. Chapter V

Summary and outlook

Canonical loop quantum gravity has come a long way since the formulation of Ashtekar–Barbero variables. It has successfully completed the construction of a kinematical Hilbert space and the implementation of the Gauss constraints and the spatial diffeomorphism constraints in the quantum theory, leading to HG a gauge and spatial diffeomorphism invariant Hilbert space Diff. Nevertheless LQG suffers from several issues and the theory is not yet complete. Among these issues are the treatment of the scalar constraint and the quantum constraints algebra off–shell. Historically, the first rigorous proposal for a scalar constraint operator was introduced by T. Thiemann in [39], based on some concepts suggested by C. Rovelli and L. Smolin in [44]. The proposed operator is not symmetric and is defined on the kinematical Hilbert space. Generically one cannot pass such operator to the space of spatial diffeomorphism invariant states due to the presence of the lapse function in the constraint. At the end, even if the general structure of the solutions to the quantum scalar constraint is known, a concrete well–defined physical Hilbert space is still unavailable. Regardless of the several attempts to define such Hilbert space [52], this specific obstacle has led to new research directions, in particular the master constraint program, the algebraic quantum gravity program, the LQG deparametrized models and the spinfoam program in the covariant framework. The question of the quantum constraints algebra off–shell remains a conceptual problem and a downside in the implementation of the constraints in LQG. The reason is, as we mentioned above, the absence of a consistent representation of the spatial diffeomorphism generators on the kinematical Hilbert space of LQG. However, novel promising ideas are emerging [53–56] which propose alternative quantization strategies of the constraints such that the 106 Summary and outlook quantum Dirac algebra can be computed off–shell. Recently, a particular development in the direction of improving the status H G of LQG dynamics was the construction of the Hilbert space vtx of partially diffeomorphism invariant states [57]. This space opened the room to new possibilities to define a symmetric constraint operator, making discussions of self-adjoint extensions and spectral analysis more accessible. In this thesis, we presented an alternative proposal for quantizing the scalar H G constraint which would exactly lead to define an operator preserving vtx. The construction of the Euclidean part of the constraint operator uses a regularization based on the assignment of special loops [25], while for the Lorentzian part of the constraint we use the curvature operator introduced in [23]. The resulting non-symmetric operator Cˆ(N) is densely defined on the Hilbert space H G vtx. It is diffeomorphism covariant and its algebra is anomaly–free. Addi- tionally, thanks to the properties of the special loops, the adjoint Cˆ†(N) is H G also a densely defined operator on vtx, which is not the case with the old scalar constraint operator. It became then possible to construct symmetric ˆ ˆ constraint operators, Csym(N), as combinations of the operators C(N) and Cˆ†(N), with the possibility to obtain self–adjoint extensions. Note that the ˆ ˆ† ˆ operators C(N), C (N) and Csym(N) are all equally suitable candidates for the scalar constraint operator in loop quantum gravity. The freedom of choice between different eligible scalar constraint operator should be regarded as a quantization ambiguity that can be fixed only through a solid semi–classical analysis of the dynamics in the theory. In each case, the general structure of the kernel of the constraint operator is known on a qualitative level, as outlined in section III.4.2. In addition to defining a new scalar constraint operator, the regularization we propose was successfully used to implement symmetric Hamiltonian operators in the context of quantum deparametrized models. Though one cannot claim that the quantization is entirely successful, until one has a proof of self– adjointness of the Hamiltonian operators, it is nevertheless complete and one has at disposition two models with a full quantum gravity sector. This opens new grounds to investigate the quantum dynamics of the theory and test LQG construction methods. Moreover, since the dynamics is cast in the form of a time evolution of quantum states of geometry, one can focus directly on the question of controlling the quantum dynamics for a better understanding of the evolution of semi–classical states describing e.g. cosmological spacetimes. Having this in mind, thanks to the graph–preserving nature of the curvature operator and consequently the Lorentzian parts of the Hamiltonian operators, 107 we have shown that one could apply time–independent perturbation theory to the dynamics in the two models, where the perturbation parameter depends on the Barbero–Immirzi parameter. Adopting this approach, we were able to elaborate a highly accurate numerical analysis of the evolution generated by the given Hamiltonian operators on several test states. With the progress presented here, one can think of several improvements to develop those models. A first key point in this direction is to extend them to include quantum matter fields, ultimately coupling the Standard Model matter to gravity and quantize the theory. One can proceed with the quantization of the full Hamiltonian operator following the polymer quantization of LQG, which would require a very careful treatment of the added matter. A similar exercise was performed in [93] considering a scalar field with a potential term. In fact a particular representation for the matter fields coupled to gravity (without deparametrization) was already derived in LQG [94], and the dynamics of the fully constrained theory was encoded in a Master constraint [50]. However there are still many aspects of that proposal which need to be clarified and tested, along with the possibility of obtaining new representations for the matter sector which would not require a Master constraint, and would allow the existence of a self–adjoint physical Hamiltonian operator. If this program is achieved properly, it would be then of great interest to study the quantum interactions between gravity and matter degrees of freedom, to investigate the possible modifications in the dynamics of matter fields which emerge in highly quantum regimes of geometry, then to look into possible relations between such models and the standard quantum field theory on fixed background. This last point may highly contribute in grasping some understanding concerning the construction of the semi-classical limit in LQG. At that point, one could also work on the comparison of different eligible candidates for the Hamiltonian operator in a deparametrized model from the perspective of certain properties, mainly locality, in the sense of dynamical correlations between the vertices in the graph of a given state, and entanglement: on one hand between matter degrees of freedom and gravity degrees of freedom, on the other hand between the fields degrees of freedom in two different regions of the space-like hypersurface. A second key point in the direction of improving LQG models is to be able to identify interesting states, e.g. states which encode a certain meaning of spacetime symmetries. Those states would in a sense represent different quantum cosmological sectors labeled by classical cosmological solutions in general relativity. A possible mechanism to construct such states is what is called a 108 Summary and outlook quantum symmetry reduction. Based on ideas in the work of J. Engle et al. [95] and a related work of N. Bodendorferet al. [96, 97], one starts by identifying constraints defined on the classical phase space of general relativity which are compatible with, or even equivalent to, requiring certain classical spacetime symmetries characteristic of specific cosmological solutions. One then quantizes those functionals and imposes them as constraints equations on the quantum states. The space of solutions to these equations would be interpreted as the sector of the Hilbert space which represents the corresponding classical symmetries. Identifying such subspaces would allow not only to interpret the dynamics of observables in such sectors, but also would open new perspectives in the direction of linking the full theory Hilbert space to the mini–(and midi– )superspace models such as loop quantum cosmology [98–104] and quantum reduced loop gravity [105–109]. All together, this program could bring very useful insights in the direction of constructing the semi–classical limit in LQG. Coming back to the unsolved issues in LQG, the probably most important challenge can be formulated in the following question: How one recovers general relativity from LQG, if at all possible? While this is directly related to the procedure the constraints of general relativity are implemented in LQG and how they are solved, its most critical aspect lies in how to recover the notion of the so familiar continuous spacetime. This aspect is known as the problem of the continuum limit in LQG. Over the years, there has been considerable work to develop and examine various ideas tackling this problem and very interesting progress has been made, for instance through the group field theories program [110, 111, 22, 112]. In summary, loop quantum gravity is more than ever an exciting and promising route towards realizing the long awaited revolution of a theory of quantum gravity. Hard obstacles are yet to be overcome and the route might as well be a dead end. It is rather clear that the riddle of quantum gravity will still elude the physicists for a long time to come, and LQG stands today as an approach with flourishing opportunities for further improvement and development. The work is then owed to continue on all various approaches to define a quantum theory of gravity, with the hope that every day new ideas will arise to get us closer to the dream, as once Richard Feynman put it:

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Appendix A

Key publications

In this appendix, I list the papers on which this thesis is based, numbered as they appear in the bibliography.

[23] E. Alesci, M. Assanioussi, J. Lewandowski, Curvature operator for loop quantum gravity, Phys. Rev. D 89, 124017 (2014), [arXiv:1403.3190].

[24] M. Assanioussi, J. Lewandowski, I. Mäkinen, New scalar constraint operator for loop quantum gravity, Phys. Rev. D 92, 044042 (2015), [arXiv:1506.00299].

[25] E. Alesci, M. Assanioussi, J. Lewandowski, I. Mäkinen, Hamiltonian operator for loop quantum gravity coupled to a scalar ﬁeld, Phys. Rev. D 91, 124067 (2015), [arXiv:1504.02068].

[26] M. Assanioussi, J. Lewandowski, I. Mäkinen, Time evolution in deparametrized models of loop quantum gravity, to appear soon.