Physics Reports vol.346

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Reports Physics Vol Philosophy Education Book pinch compression electrically conducting filament magnetic forces. These methods have been applied also to GR, but many difficulties arise in this case due to the fact that Einsteins theory cannot be formulated in terms of a quantum field theory on a fixed Minkowski background. To be more specific, in GR the geometry of the background spacetime cannot be given a priori: spacetime is the dynamical variable itself. In order to introduce the notions of causality, time, and evolution, one must first solve the equations of motion and then build the spacetime.

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For example, in order to know if particular initial conditions will give rise to a black hole, it is necessary to fully evolve them by solving the Einstein equations. Then, taking into account the causal structure of the obtained solution, one has to study the asymptotic metric at future null infinity in order to assess whether it is related, in the causal past, with those initial conditions.

This problem becomes intractable at the quantum level.

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Due to the Uncertainty Principle, in non-relativistic Quantum Mechanics, particles do not move along well-defined trajectories and one can only calculate the probability amplitude t , x that a measurement at time t detects a particle around the spatial point x. Similarly, in Quantum Gravity, the evolution of an initial state does not provide a specific spacetime.

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In the absence of a spacetime, how is it possible to introduce basic concepts such as causality, time, elements of the scattering matrix, or black holes? The canonical and covariant approaches provide different answers to these questions. The canonical approach is based on the Hamiltonian formulation of GR and aims at using the canonical quantization procedure. The canonical commutation relations are the same that lead to the Uncertainty Principle; in fact, the commutation of certain operators on a spatial three- manifold of constant time is imposed, and this three-manifold is fixed in order to preserve the notion of causality.

In the limit of asymptotically flat spacetime, the motion generated by the Hamiltonian must be interpreted as temporal evolution in other words, when the background becomes the Minkowski spacetime, the Hamiltonian operator assumes again its role as the generator of translations. The canonical approach preserves the geometric features of GR without the need to introduce perturbative methods [].


The covariant approach, instead, employs Quantum Field Theory concepts and methods. The basic idea is that the problems mentioned above can be easily circumvented by splitting the metric g into a kinematical part usually flat and a dynamical part h , as in. The geometry of the background is given by the flat metric tensor and is the same as in Special Relativity and ordinary Quantum Field Theory, which allows one to define the concepts of causality, time, and scattering.

The quantization procedure is then applied to the dynamical field, considered as a small deviation of the metric from the Minkowski background metric. Quanta are discovered to be particles with spin two, called gravitons, which propagate in flat spacetime and are defined by h. Substituting the metric g into the HilbertEinstein action, it follows that the Lagrangian of the gravitational sector contains a sum whose terms represent, at different orders of approximation, the interaction of gravitons and, eventually, terms describing mattergraviton interaction if matter is present.

Such terms are analyzed by using the standard techniques of perturbative Quantum Field Theory. These quantization programs were both pursued during the s and s. In the canonical approach, Arnowitt et al. In this Hamiltonian formalism, the canonical variables are the three-metric on the spatial submanifolds obtained by foliating the four-dimensional manifold note that this foliation is arbitrary.

The Einstein equations give constraints between the three- metrics and their conjugate momenta and the evolution equation for these fields, known as the WheelerDeWitt WDW equation. In this way, GR is interpreted as the dynamical theory of the three-geometries geometrodynamics. The main difficulties arising from this approach are that the quantum equations involve products of operators defined at the same spacetime point and, in addition, they entail the construction of distributions whose physical meaning is unclear.

In any case, the main problem is the absence of a Hilbert space of states and, as consequence, a probabilistic interpretation of the quantities calculated is missing. The covariant quantization approach is closer to the known physics of particles and fields in the sense that it has been possible to extend the perturbative methods of QED to gravitation. This has allowed the analysis of the mutual interaction between gravitons and of the mattergraviton interactions.

The formulation of Feynman rules for gravitons and the demonstration that the theory might be unitary at every order of the expansion was achieved by DeWitt [40]. Further progress was achieved with YangMills theories, which describe the strong, weak, and electromagnetic interactions of quarks and leptons by means of symmetries. Such theories are renormalizable because it is possible to give the fermions a mass through the mechanism of Spontaneous Symmetry Breaking.

Then, it is natural to attempt to consider gravitation as a YangMills theory in the covariant perturbation approach and check whether it is renormalizable.

Our Ignorance About Gravity

Due to the non- renormalizability of gravity at different orders, its validity is restricted only to the low-energy domain, i. This implies that the full unknown theory of gravity has to be invoked near or at the Planck era and that, sufficiently far from the Planck scale, GR and its first loop corrections describe the gravitational interactions.

In this context, it makes sense to add higher order terms to the HilbertEinstein action as we will do in the second part of this Report. Besides, if the free parameters are chosen appropriately, the theory has a better ultraviolet behavior and is asymptotically free. Nevertheless, the Hamiltonian of these theories is not bounded from below and they are unstable.

In particular, unitarity is violated and probability is not conserved. An alternative approach to the search for a theory of Quantum Gravity comes from the study of the Electroweak interaction. In this approach, gravity is treated neglecting the other fundamental interactions. The unification of the Electromagnetic and the weak interactions suggests that it might be possible to obtain a consistent theory when gravitation is coupled to some kind of matter.

This is the basic idea of Supergravity [42]. In this class of theories, the divergences due to the bosons gravitons in this case are canceled exactly by those due to the fermions, leading to a renormalized theory of gravity. Unfortunately, this scheme works only at the two-loop level and for mattergravity couplings. The Hamiltonian is positive-definite and the theory turns out to be unitary. But, including higher order loops, the infinities re-appear, destroying the renormalizability of the theory. Perturbative methods are also used in String Theories.

In this case, the approach is completely different from the previous one because the concept of particle is replaced by that of an extended object, the fundamental string. The usual physical particles, including the spin two graviton, correspond to excitations of the string. The theory has only one free parameter the string tension and the interaction couplings are determined uniquely. It follows that string theory contains all fundamental physics and it is therefore considered as a candidate for the Theory of Everything.

String Theory seems to be unitary and the perturbative series converges implying finite terms. This property follows from the fact that strings are intrinsically extended objects, so that ultraviolet divergencies coming from small scales or from large transfer impulses, are naturally cured. In other words, the natural cutoff is given by the string length, which is of Planck size lPl. At scales larger than lPl , the effective string action can be rewritten in terms of non-massive vibrational modes, i. This eventually leads to an effective theory of gravity non-minimally coupled with scalar fields, the so-called dilaton fields. To conclude, let us summarize the previous considerations:. In the quantization program for gravity, two approaches have been used: the covariant approach and the perturbative approach. They do not lead to a definitive theory of Quantum Gravity.

In the low-energy regime with respect to the Planck energy at large scales, GR can be generalized by introducing into the HilbertEinstein action terms of higher order in the curvature invariants and non-minimal couplings between matter and gravity. These lead, at the one-loop level, to a consistent and renormalizable theory.

Physics Reports vol.346 Physics Reports vol.346
Physics Reports vol.346 Physics Reports vol.346
Physics Reports vol.346 Physics Reports vol.346
Physics Reports vol.346 Physics Reports vol.346
Physics Reports vol.346 Physics Reports vol.346
Physics Reports vol.346 Physics Reports vol.346

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