Towards general relativity through parametrized theories

Resumen

The satisfactory quantization of general relativity (GR) is one of the most important open problems in theoretical physics. Its resolution is crucial to understand the physics hidden at high energies and small scales such as the behavior of black holes or the Big Bang itself. The fundamental feature that distinguishes general relativity from the rest of field theories is the absence of a background geometrical structure, like the Minkowski metric in the usual field theories, which suggests that new methods are necessary to tackle this problem. It was probably Einstein the first who, through heuristic reasonings, pointed out that the new theory of relativity had to be modified by quantum effects. He did so as early as 1916 in his first paper about gravitational radiation [23]. Later on, several physicists like Oskar Klein, Rosenfeld, Fierz, or Pauli arrived at the same conclusion. However, contrary to Einstein who moved on to believe in the necessity of a new brand unifying theory, they mostly considered that similar arguments as the ones applied for electromagnetism would suffice. In fact Rosenfeld wrote the first papers on quantum gravity by applying some Pauli’s quantization ideas to linearized GR [35]. Soon after that, the relation of this theory with the linear spin-two quantum field theory was discovered. It is now believed that Matvei Pretrovich Bronstein, a young Russian physicist, was the first person to realize that the quantization of the (non-linear) gravitational field required a special treatment due to its unique features [20]. His ideas, nonetheless, were doomed to oblivion as he was arrested and executed in 1938 during the Great Purge in the Soviet Union. It seems that the French physicist Jacques Salomon was the only one outside the Soviet Union to acknowledge and develop his ideas, although sadly he was executed in 1942 by the Nazis during the German occupation of France. For further details and references I strongly recommend to the reader the excellent historical reviews of Stachel [37] and Rovelli [36]. In fact the latter contains a brief history of quantum gravity up to the beginning of the 21st century, of which I will mention some of its landmarks in the following. During the late 40’s and early 50’s Bergmann and his collaborators started to study the phase space quantization of non-linear field theories and the observables in GR [18,19]. At the same time, Dirac developed his procedure to deal with general constrained Hamiltonian systems [21,22]. However, its application to the Hamiltonian formulation of GR was quite unclear. At this point Gupta introduced some of the elements necessary for the perturbative quantization of GR [25]. In particular, he pinpointed out the necessity to have a background metric. Years later, in the 70’s, t’Hooft and Veltman proved that such approach was unsuccessful due to the non-renormazibility of the resulting theory [40]. In the meantime, several physicists like Feynman, DeWitt, Wheeler, or Penrose worked to solve this problem, already considered at that point as a Herculean task. Probably, the most important result found in the context of GR at that moment was due to Arnowit, Deser, and Misner who, following the program started by Dirac, obtained the Hamiltonian formulation of GR using the so-called ADM variables [3]. The following milestone of this very brief history was the discovery by Hawking of black hole radiation [27]. He used some techniques developed in the context of QFT in curved space-times to show that a spherical black hole with mass M emits thermal radiation at a temperature T =hc^3/(16π^2 k G M) This was in agreement with some earlier observations by Bekenstein who found a formal analogy between thermodynamics and black holes [15–17]. Hawking proved that, in fact, this correspondence was also physical. The formalization of black hole thermodynamics implies the existence of some kind of entropy S (the Bekenstein-Hawking entropy), which is related to the area A according to S=A/(4 l^2) This result, that has been derived in several alternative ways, is now used as a consistency condition for candidate quantum gravity theories. The first of them, proposed in the 70’s and 80’s, were supergravity [24], higher derivative gravity [38], and the connection formulation of GR [4]. From those seminal ideas two important candidates emerged: string theory and Loop Quantum Gravity. Both theories were able to derive, almost at the same time in the late 90’s, the Bekestein-Hawking radiation law [5,39]. As a final remark of this quick historical review, I would like to mention the quantization of 2 + 1 gravity carried out in 1988 by Witten [42]. The aim of this thesis can be neatly summarized in the following sentence Boundaries, GNH, and parametrized theories. It takes three to tango. Let us proceed to explain each term and why we can take advantage of their interrelation. I. Boundaries The world is full of boundaries. They do matter. For instance, the objects that we use everyday are limited by their “edges”. Also, in the case of sound and electromagnetic waves, which are essential for our lives, their behavior depends strongly on the shape of the walls that they encounter. Then, it is natural to include boundaries in our physical models to faithfully represent our reality. However, boundaries are tricky. If we are located within the bulk of an object, when we arrive at the boundary there is a sudden change in dimensionality as it is reduced by 1. This is probably the origin of many of the difficulties that boundaries pose but, on the other hand, their presence leads to more fruitful and interesting theories. The best example is probably the general theory of relativity, where boundaries are essential to understand black holes or infinity. In particular it seems mandatory to have a clean description of GR with boundaries in order to give a proper explanation of black hole entropy. This is precisely one of our main goals: to understand the role of boundaries in some field theories in order to apply what we learn to GR. II. GNH algorithm The Dirac’s “algorithm” allows us to deal with singular mechanical system in which some constrains (functional relations between dynamical variables) must be preserved by its evolution. Despite its success in dealing with finite dimensional systems, its application to field theories is not as clean, specially in the presence of boundaries. The GNH algorithm was developed to generalize and simplify Dirac’s algorithm, and it does so by relying on geometric methods. This in turn provides a clearer understanding of the procedure. Besides, boundaries can be included without any conceptual change in the algorithm, although in the case of field theories some care must be exercised due to functional analytic issues. Therefore, another important goal of this thesis is the following: to obtain the dynamics of several field theories through the GNH algorithm. III. Parametrized theories The absence of background geometric objects in GR makes this theory invariant under diffeomorphisms. This group is infinite dimensional, which renders this field theory much more complicated than the usual ones. It is then desirable to have some simpler dynamical models which are also diff-invariant and use them to understand, for instance, the application of the GNH algorithm or the role of boundaries in the theory. This is exactly what it is achieved through parametrization, a procedure that introduces diff-invariance into any theory involving background geometric objects. Regarding this issue, the main goal is: to understand the role of the diff-invariance in theories simpler than GR and its relation with other gauge symmetries they may possess. IV. Interplay of the three Those three elements together conform the main thread of the thesis. After an introduction of the basic mathematicalal concepts, we study first some field theories with boundaries using the GNH algorithm which will serve, in turn, to underline the importance of some functional analytic subtleties that must be taken into account in more complicated models. Then, we proceed to introduce parametrized theories and develop the simplest case: parametrized classical mechanics. It is useful as a warm-up for the study, that we develop later, of parametrized electromagnetism with boundaries, the revisiting of the parametrized scalar field (with a detailed description of the behavior at the boundary of this simpler theory), and the study of the parametrized Maxwell-Chern-Simons theory with boundaries. Finally, to tie up the thesis, the last chapter is devoted to the theory that served as a beacon since the beginning: the general theory of relativity. We apply what we have learned with the aforementioned theories to the Hamiltonian formulation of GR to derive, in a geometric and easy way, the ADM formulation. We also look at another interesting model: unimodular gravity. Let us end this introduction by summarizing the state of the art of the research about the three aforementioned nuclear concepts. I. Boundaries Boundaries are nowadays some sort of trending topic in physics. Many interesting results have been obtained in many contexts like condensed matter (topological insulators), quantum computation, and of course general relativity. Focusing on the topic of this thesis, we found some contributions that are worth mentioning. First, the Hamiltonian formulation of the parametrized scalar field in bounded spatial regions has been discussed in [2]. The authors of that paper encountered some difficulties when Robin boundary conditions were imposed and left it as an open question the existence of the the canonical formalism in such case. It is interesting to mention that, with the proper geometric approach, we answered affirmatively this question in [7], as we explain in the thesis. Boundaries are of capital importance in GR and have got a lot of attention for decades. They can show very pathological behaviors even at the topological level [33] and its role in the study of black holes is crucial. Of particular interest is the notion of isolated horizons, a quasi-local concept that seems more suited than the event horizon (a global concept) for the Hamiltonian formulation of black holes. For more details about this topic, we recommend the reader the review of Ashtekar and Krishnan [6]. From the properties of isolated horizons, several ideas emerged in the context of LQG that made it possible, among other things, to perform a rigorous state counting to compute the black hole entropy [1,14]. In fact, although not related to the contents of this thesis, we have recently contributed to this area in [12], where we study the spectrum of the area operator in LQG providing a very accurate description of the distribution of its eigenvalues. II. GNH algorithm The GNH method has been extensively studied by mathematicians in areas related to sym plectic geometry, mechanics, or Hamiltonian reduction. Nonetheless, they are mostly interested in the technical details and extensions, and not so much in its application to physically relevant examples. Physicists, on the other hand, seem to be comfortable with the good old Dirac algorithm although we know that it can crash badly in some interesting examples. Probably the first extensive use of the GNH algorithm for non-trivial physical example in field theories was [13], where a careful study of the scalar and electromagnetic field in the presence of boundaries is carried out. In fact this paper was the origin of my thesis and has lead to several publications [7–9,11], where the GNH algorithm plays, thanks to its clear geometric interpretation, a central role in the understanding of the problems at hand. III. Parametrized theories Introduced by Dirac in the 50’s, parametrized theories regained relevance with some works of Kuchař, Isham, Hájíček, and Torre [26,28,29,41] in the boundaryless case. Also, we have already mentioned the work [2], where the parametrized scalar field with different boundary conditions was considered. There has been also some interest in the the context of LQG to explore the role of diffeomorphisms [31,32]. Parametrized field theories are of great importance because they are diff-invariant on one hand, and can be solved in typical examples, on the other. An interesting problem that crops up in this setting is to understand the interplay between diff-invariance and other “more standard” gauge symmetries. Several authors have worked on this topic [30,34,41] with inconclusive results. As we have been able to show, it is actually possible to completely understand how diff-invariance and ordinary gauge symmetries interact by studying the parametrized EM field [10]. To this end we have relied once again on the geometric GNH algoritm as we explain in the thesis. We provide a brief but deatiled description of the space of embeddings as well as a discussion of the meaning of the GNH algorithm. Both topics are well known for people working on those fields, but are not so well known for the general audience. Our presentation is accesible but withou lossing rigor. Based on our papers [7,8,11], we began to think about boundaries and how they can be used to measure what happens in the bulk. To do so we studied what seemed to be a very simple case: a string with a mass attached to each end. This system is, however, surprisingly rich and hard to deal with. The inclusion of the masses prevents us from using the standard Sturm-Liouville theory which suggests that novel ideas are needed. In fact, we circumvented this problem by relying on non trivial measures and their associated Radon-Nikodym derivatives. The main result of this chapter is that the Fock space of the whole system, that we obtained thanks to the GNH algorithm, is not of the form H syst ⊗H mes with a factor associated with the boundary and a factor associated with the bulk. This suggests some sort of strong entanglement of the boundary and the bulk, which should come as no surprise after a short reflection because at the classical level the positions of the masses are completely determined by the configuration of the string (continuity conditions). Besides, we showed that we can define some dynamics over the boundary with the help of the trace operators. Such dynamics is not unitary, which is to be expected in the same way that the energy is conserved in the whole system but not on the subsystems. We end up this section studying the unitary implementation of the scalar field with standard boundary conditions obtaining a complete characterization of the possible unitary evolutions through foliations. This result generalizes the ones known for boundaryless systems. The following chapter is devoted to motivate and describe in detail the parametrization procedure. The main idea is to include diffeomorphisms as variables in such a way that the resulting theory is diff-invariant. The importance of this procedure has been pinpointed out several times, but it is worth it to make it explicit once again. First, let us mention that it is a generalization of the unparametrized theory which allows us to describe any field theory in an arbitrary foliation. Second, it is a perfect generator of toy-models for GR. Finally, some algebraic quantization methods can be applied in this simpler context with the hope to learn something about the quantization of the full GR. After the description of the parametrized theories the chapter continues with the warm-up exercise of studying parametrized classical mechanics. We carefully developed the ideas and the implementation of the GNH algorithm, outlining the steps where due care is needed in the infinite dimensional case. With our hand and mind loosened up with the aforementioned preparatory exercise, we are ready to jump into the parametrized electromagnetic field with boundaries. Our study generalizes our work [10] as in the thesis we include boundaries. The most important result of this part is the identification of sectors, closely related to the Gauss law of electromagnetism, where a bifurcation appears in the dynamics. The same bifurcation is identified at the boundary although, in this case, its careful study is much trickier and not so interesting for our purposes. Nonetheless, a discussion of similar nature can be carried out in a simpler example that we also consider: the parametrized scalar field with boundaries [9]. We obtain a complete characterization of the sectors associated with the behavior of the boundaries. The consistency of the dynamics forces some additional conditions that can be explicitly derived for some concrete examples. The next natural step is to study some theories with interesting and non-standard behaviors at the boundary. We did so with the parametrized Maxwell-Chern-Simons theory, a generalization of parametrized EM in 2 + 1 dimensions. We developed, using the same methods as in the previous chapters, the Hamiltonian formulation of the theory and identified the different boundary conditions that naturally appear. This is important in order to understand some features that are supposed to play a role in the quantum description of those theories. We end up this part of the thesis with an analog study of Chern-Simons theory. The last part of the thesis is devoted to the general theory of gravitation. We derive, using the full machinery developed in the thesis, the Hamiltonian formulation of GR. It obviously coincides with the well known ADM formulation but our novel approach through the GNH algorithm is simpler to use and easier to interpret. After that, we proceed to study the Hamiltonian formulation of unimodular gravity, which is much less known but useful in several contexts because it provides an interesting perspective on the problem of time and the origin of the cosmological constant.