Study of gravitational theories through hamiltonian methods and consistent deformations

Resumen

General Relativity, motivated by Einstein's ideas and developed in a series of papers that culminated in the gravitational field equations, is the most accurate classical theory of gravitation we have so far. It entailed a huge change of paradigm in the conception of the nature of the universe, by replacing the rigid Galilean space and time by a dynamical, (in general curved) spacetime. The newborn theory had a critical success in explaining the precession of the perihelion of Mercury, the deflection of light by the Sun and, later on, the gravitational redshift experienced by light. However, even the astonishing success of General Relativity had a chink: singularities. They appeared, loosely, as points where spacetime breaks down or has an `end'. The two most famous instances of singularities are, perhaps, the one in Schwarzschild black holes and the initial singularity of Friedman-Lemaître-Robertson-Walker cosmological models. Although the formalism allows such situations, singularities are largely regarded as undesirable and signal our lack of understanding of the more profound physics needed to completely understand these physical situations. In a sense, the theory itself strongly suggests that something is still missing and nicely tells us where to look. There was some hope of resolving this conundrum with the help of (also very recent) ideas in Quantum Mechanics. In particular, since General Relativity is a field theory, one would like to find a relativistic Quantum Field Theory for the gravitational field (or as usually said, to quantize gravity), along the lines successfully followed to understand other field theories, in particular, in Quantum Electrodynamics, which eventually gave rise to the celebrated Standard Model of Particle Physics. However, as it was conceived, Quantum Field Theory frontally clashes with General Relativity. Firstly, the privileged Minkowski spacetime is used as a background. Worse, the perturbative techniques that allowed to extract the relevant physical information from path integrals by employing Feynmann diagrams completely fails when trying to quantize gravity. The main obstacle for the development of the traditional quantization programme for General Relativity is its non-renormalizability. This issue arises because the coupling constant of the theory, Newton's constant in this case, is dimensionful. This fact is made apparent in the one loop corrections, which diverge and can not be absorbed. A possible explanation for this failure is that usual perturbative treatments in physics assume that the spacetime is a continuum at all scales. This works for particle physics, since the scales of particles are much larger than the Planck length, the scale in which quantum gravity effects should manifest themselves. However, for a theory of quantum gravity, this hypothesis may be completely wrong and there is no reason a priori to assume it. This fact motivated the development of non-perturbative theories for quantum gravity that can account for the micro structure of spacetime, with the benefit that this approach completely avoids the non-renormalizability issue. Among the plethora of proposals, two of them have had the largest impact and a number of results over the years; these competing approaches are String Theory and Loop Quantum Gravity. Although String Theory is not just a theory of quantum gravity (in fact, it is far more ambitious), it has paid close attention to how gravity manifests itself in the whole of the theory, and has produced numerous results on black holes and cosmology. On the other hand, Loop Quantum Gravity is a very natural continuation of the ideas that arose in the development of General Relativity. This approach has also achieved some results on black holes and in cosmology under the name Loop Quantum Cosmology, a quantization of the cosmological sector of General Relativity. However, after more than 30 years of intense scientific activity, no one has been able to achieve a full theory of quantum gravity yet. In view of this fact, it may be an interesting possibility to pursue perturbative approaches again. Evidently, since the traditional method does not work, we need to come up with new ideas to deal with the problem. A possible solution is described in the thesis. As a review, the thesis first explores Hamiltonian methods from the perspective introduced by Gotay, Nester and Hinds (called the GNH algorithm), which has some advantages over the usual Dirac method, namely that it is global, more geometrically transparent and more computationally economic. The Hamiltonian equation (and hence its solution, i.e., the Hamiltonian vector field) plays a central role in the GNH algorithm. Indeed, in contrast to Dirac's method, in which the Hamiltonian vector field is of secondary importance (usually the focus is on the constraints), in the GNH approach both are equally relevant. This vector field determines the evolution of the system, which must be guaranteed to be stable, namely, to remain within the the phase space submanifold where the dynamics takes place itself. In the GNH algorithm this is translated into tangency conditions of the Hamiltonian vector field to the constraints. Note that in Dirac's method this is also required by the on-shell vanishing of Poisson brackets with the total Hamiltonian. From the original GNH algorithm stem two other Hamiltonian techniques that are explored. The first achieves a Hamiltonian description in the tangent bundle which is completely equivalent to that in the cotangent bundle. The construction is canonical once a choice of Lagrangian is made (just as in the usual treatment), and the GNH algorithm can be applied with one difference from the cotangent bundle case. The peculiarity of the tangent bundle is that one must additionally require the so-called second order condition on the Hamiltonian vector field to ensure that the resulting integral curves are the lifts of curves in the original configuration space. The second is a streamlined method that produces the same results as the GNH algorithm before the tangency conditions are implemented. It is based on the equations of motion: once they are derived, it is possible to obtain the equations for the Hamiltonian vector field and necessary conditions for the constraints. This method is useful because, in some cases, it helps finding the constraints by suitably rewriting the equations of motion in equivalent forms. Using these methods, a selection of well known gravitational theories is presented and analyzed, with the addition of the scalar field. The scalar field does not describe any gravitational phenomena. It is nevertheless interesting to study because even though it is one of the simplest, it already exhibits the mathematical subtleties appearing in (infinite dimensional) field theories, making it a perfect example to study and understand the funcional analytic details in the GNH algorithm. The next action studied is Chern-Simons. This action is very important both in geometry and in physics. Geometrically, it is closely related to invariants of manifolds. Physically, it can be rewritten in a plethora of ways in order to obtain many (if not all) gravitational theories in three dimensions. The Husain-Kuchař action is also reviewed. This 4-dimensional model has as dynamical fields three linearly independent 1-forms and a connection. The fact that a 1-form is missing in order to form a non-degenerate coframe has interesting implications. It turns out that the metrics this model produces are degenerate, but restricting to the leaves of the foliation, one can define a non-degenerate 3-metric that does not evolve in the Hamiltonian sense. As a consequence, this model is describing 3-geometries (i.e., 3-metrics modulo diffeomorphisms), which is very interesting from a geometrical point of view. The Husain-Kuchař model is also relevant physically because it is a theory of gravity that does not have the hard-to-quantize Hamiltonian constraint present in General Relativity. This makes it a perfect toy model for the purpose of quantization in the context of Loop Quantum Gravity. Finally, the well-known Cartan-Palatini action for General Relativity is studied and a summarised version of the GNH procedure is given. Then, the thesis proceeds to present its original content used towards the resolution of two main problems. On the one hand, an alternative action for the Euclidean version of anti-self-dual General Relativity is studied in detail, in particular, the GNH algorithm is used to find its Hamiltonian formulation (i.e., Hamiltonian vector field and constraints). The interest for self-dual formulations of General Relativity originated a long time ago. Plebański introduced in the late 70s a formulation for General Relativity where self-dual 2-forms played a central role, which was later seen to be equivalent to Ashtekar’s formulation. Also, the vacuum Einstein equations have been found to be the condition that the curvature of a self-dual connection is self-dual. Moreover, Petrov’s classification of spaces heavily relies on self-duality. It is clear that self-dual connections have a privileged spot in the variational setting for General Relativity. With that in mind, it is worth paying some attention to the Euclidean version of self-dual formulations of General Relativity, which are also independently interesting for several reasons. The path integral formulation for the quantization of this type of theories yields a Gaussian-like probability distribution, which can be computed through a perturbative expansion. Then, one can attempt to perform a Wick rotation hoping to recover the Lorentzian version. Also, Donaldson invariants in 4-manifolds are interesting from a mathematical point of view and are connected to Yang-Mills theories. The studied action is the result of plugging the concrete expression of anti-self-dual connections into the Cartan-Palatini action with tetrad and connection variables. In this way, one obtains an action composed by two terms: the first describes the Husain-Kuchař model while the second is linear in the 0-th element of the coframe. By doing this, the SO(4) gauge symmetry is naturally split in two SO(3) components, one of which is not so explicitly manifest in the sense that it is not directly reflected by the internal indices of the fields used to write the action. It appears then, that although one manages to split the symmetry in smaller groups, which are easier to handle, their complexity still shows up somewhere. The solution for the Hamiltonian vector field (from which all the information can be extracted) depends on 10 arbitrary functions: these correspond to the SO(4) and diffeomorphism symmetries. In their explicit expressions one can see which parameter corresponds to each symmetry. On the other hand, a new perturbative scheme for field theories has been proposed and its viability has been analysed in a variety of interesting gravitational theories. The core idea for this procedure was proposed by Smolin in the early 90s as a new perturbative framework. His proposal was based on considering Newton’s constant G as a coupling constant appearing in the definition of the curvature and the covariant differential. Then, he used similar expressions with a self-dual connection and found that the self-dual Cartan-Palatini action can be written as the sum of a lower order term, independent of G, and a coupling term proportional to G. This idea can be readily compared to the traditional perturbative approach in which the metric is written as the Minkowski metric plus a perturbation, leading to an action with a ‘free’ part (i.e. quadratic in the fields) describing massless, spin 2 particles, known as gravitons, propagating in a Minkowski background geometry and interaction terms proportional to powers of G. However, in Smolin's original paper there are several issues that remain unresolved. First, it is not clear why this is the right way to introduce G. More importantly, it is not physically clear what taking G to 0 means or even how to use it as a perturbative parameter since it is a dimensionful quantity. The ambiguities disappear by using internally Abelianized versions of the theories as the starting point of a perturbative expansion, i.e., replacing the internal symmetry group of the action by an Abelian group of the same dimension. In terms of the Lie algebra, this has the effect of setting all the Lie brackets to 0, which removes the quadratic terms in the connection from the action. This greatly simplifies the resulting equations of motion and has allowed us to solve them directly in the cases at hand without the need to resort to Hamiltonian methods. A first requirement for the perturbative scheme to work is that the unperturbed theory be integrable and the solution be expressible in a reasonable form in order to serve as the starting point of the perturbative expansion. However, this is not enough. An equally important condition is that the perturbation is a consistent deformation of the base action (i.e., the deformation introduced to the action preserves the number of gauge symmetries, although their expressions may change). This is equivalent to saying that the perturbation is regular rather than singular. Indeed, for regular perturbations one can write an expansion in power series, while for singular perturbations this is not possible. Note that in the singular case, perturbative schemes are still available, however they require different ad hoc approaches for the particular problem at hand that may be difficult to find. Only when these two conditions hold it is reasonable to pursue a perturbative scheme along these lines. A particularly nice feature of the method discussed in the thesis is that it does not need any auxiliary background objects, which are fundamental in the traditional approach to perturbative gravity, namely, the Minkowski metric. Also, each term of the perturbative expansion of the action is covariant, which may be beneficial since there is no breaking of the diffeomorphism symmetry (which as mentioned before, might lead to unexpected unpleasant effects). These two facts combined make this approach very reasonable for a perturbative scheme for gravitational theories. The consistent deformations method is applied to the gravitational theories presented in the thesis. This includes Chern-Simons, Husain-Kuchař, the Euclidean version of anti-self-dual General Relativity and Cartan-Palatini actions. As necessary steps, their symmetries and their internally abelianized versions are studied. In particular, the equations of motion of the internally abelianized models are solved, in some cases, in spacetimes of arbitrary topologies. The result obtained is that, in the Chern-Simons, Husain-Kuchař and the Euclidean version of anti-self-dual General Relativity cases, the full action is a consistent deformation of its internally abelianized version and a perturbative scheme can be set up. Unfortunately, in the actions relevant for General Relativity, namely the Cartan-Palatini and the Holst action, the scheme does not work, since the resulting deformation is not consistent. This is due to the fact that General Relativity has two local degrees of freedom, while its internally Abelianized version (also called the U(1)^3 model) is topological, i.e., it has no local degrees of freedom. In this sense, the model for (anti-self-dual) Euclidean General Relativity is interesting, since it actually is a consistent deformation of its internally Abelianized version, while Euclidean Cartan-Palatini is not. This motivates the search of an action that yields Lorentzian General Relativity and to which the consistent deformation procedure can be applied. It is somewhat surprising that, given two different actions for the same physical theory, one can be consistently deformed from its internal Abelianization while the other can not. It is also very surprising that precisely the actions that yield the physically relevant gravitational theory are not consistent deformations of their internal Abelianizations, while other simpler actions are. It would be worth investigating if there is some physical property of a theory that determines whether this process works or not. Obtaining such a characterization could open the possibility for new actions admitting consistent deformations. An also very interesting (but also very difficult) problem would be to find all the possible consistent deformations of a given action. The appropriate context to deal with this problem is cohomology and the use of the BRST symmetry. Then, the task is equivalent to computing some cohomology groups, which is in general hard to do.