Phenomenological Quantum Gravity

These notes summarize a set of lectures on phenomenological quantum gravity which one of us delivered and the other attended with great diligence. They cover an assortment of topics on the border between theoretical quantum gravity and observational anomalies. Specifically, we review non-linear relativity in its relation to loop quantum gravity and high energy cosmic rays. Although we follow a pedagogic approach we include an open section on unsolved problems, presented as exercises for the student. We also review varying constant models: the Brans-Dicke theory, the Bekenstein varying α model, and several more radical ideas. We show how they make contact with strange high-redshift data, and perhaps other cosmological puzzles. We conclude with a few remaining observational puzzles which have failed to make contact with quantum gravity, but who knows... We would like to thank Mario Novello for organizing an excellent school in Mangaratiba, in direct competition with a very fine beach indeed. WHY QUANTUM GRAVITY? The subject of quantum gravity emerged as part of the unification program that led to electromagnetism and the electroweak model. We’d like to unify all forces of Nature. Forces other than gravity are certainly of a quantum nature. Thus we cannot hope to have a fully unified theory before quantizing gravity. To come clean about it right from the start, we should stress that there is no compelling experimental reason for quantizing gravity. For all we know, gravity could stand alone with respect to all other forces, and simply be exactly classical in all regimes. There is no evidence at all that the gravitational field ever becomes quantum 1. Yet this hasn’t deterred a large number of physicists from devoting lifetimes to this pursuit. Assaults on the problem currently follow two main trends: string/M theory [1, 2] and loop quantum gravity [3, 4]. Both have merits and deficiencies, commented extensively elsewhere. As a poor third we mention Regge-calculus (and lattice techniques), non-commutative geometry, and several other methods none of which has fared better or worse than the two main strands. This course is not about those theories. Rather it’s about the question: Where might experiment fit into these theoretical efforts of quantizing gravity? A middle ground has recently emerged – phenomenological quantum gravity. The requirements are simple: a phenomenological formalism must provide a believable approximation limit for more sophisticated approaches; it must also make clear contact with experimental anomalies that don’t fit into our current understanding of the world. The following argument illustrates what we mean by this. When physicists find themselves at a loss they often turn to dimensional analysis. Following this simplistic philosophy we estimate the scales where quantum gravity effects may become relevant by building quantities with dimensions of energy, length and time from h̄ (the quantum), c (relativity) and G (gravity). These are called the Planck energy EP, the Planck length lP and the Planck time tP. For instance Ep = √ h̄c5/G ≈ 1.2× 1019GeV ≈ 2.2× 10−5g. Quantum gravitational effects are expected to kick in for energies above EP or lengths and durations smaller than lp and tp. Beware: dimensional analysis can be too naive. We expect quantization of gravity to take the form of a theory in which space and time are discretized. General relativity is a theory of curved space-time. Thus, quantum gravity should quantize not only curvature but actual 1 Exercise for the student: discuss the gravitational field of a photon undergoing the double slit experiment. Could you collapse the wave function by measuring the gravitational field? Lay out all possibilities in the form of thought-experiments. You may find it interesting to make contact with the old problem of how classical and quantum systems interact. Then repeat the exercise with a double slit experiment where gravitons are used instead of photons. durations and lengths. Here quantization means to replace a continuum by a discrete structure. This may be done, as a first approximation, just as Planck did in the first quantum theory, when he simply proposed that the energy of a harmonic oscillator would be a multiple of a fixed energy, h̄ω . The same could be true for space and time: space is granular, time has an atomic structure. This is, however, an approximation. As we know, what actually happens in quantum mechanics is that observables are replaced by operators with a discrete spectrum, whose eigenvalues represent the possible outcomes of a measurement. Similarly, in loop quantum gravity, area and volume become operators with a discrete spectrum and the geometry becomes a quantum state (a spin foam, more specifically). Still, we may use the “Planck-style” of quantization as a basis for a phenomenological model. The point of this simplified quantization approach is that now we have a springboard for contact with experiment. The argument is based on a paradox similar in flavor to the Zeno paradox in ancient Greek philosophy (which incidentally concerned the absurd apparent conflict between discrete and continuum). Whatever quantum gravity may turn out to be, it is expected to agree with special relativity when the gravitational field is weak or absent, and also with all experiments probing the nature of space-time on scales much larger than lP (or energy scales smaller than EP). The granules of space-time should be invisible unless we examine these scales with a powerful “microscope”. This immediately gives rise to a simple question: In whose reference frame is lP the threshold for new phenomena? For suppose that there is a physical length scale which measures the size of spatial structures in quantum space-times, such as the discrete area and volume predicted by loop quantum gravity. Then if this scale is lP in one inertial reference frame, special relativity suggests it will be different in another observer’s frame – a straightforward implication of Lorentz-Fitzgerald contraction, easily derived from the Lorentz transformations. In other words the border between classical and quantum gravity is not invariant or well defined. Similar arguments can be made with energy and time. There are two obvious answers to the problem. On the one hand, Lorentz transformations may be correct on all scales, such that the Planck length is sensitive to Lorentz contraction. In this case, quantum gravity picks up a preferred frame in which the Planck length is the border between classical and quantum gravity. On the other hand, it could be that quantum gravity is the ether wind, and that all other effects baffle the Michelson-Morley experiment. A distinct possibility is for quantum gravity to respect the principle of relativity, but require a revision of the Lorentz transformations at extreme scales. Such transformations should leave the Planck length invariant. A toy model for this is to let the speed of light be length dependent, and go to infinity for lp. A more sophisticated version is non-linear relativity, also called doubly special relativity (DSR), developed in the next section. This is the prototype of the argument leading to phenomenological quantum gravity. The strength of this last approach is that a relation to observations is quickly obtained in this way. Indeed, DSR explains ultra high energy cosmic ray anomalies. This illustrates what is meant by phenomenological quantum gravity. The theoretical problem is too hard. Perhaps it needs a bit of fresh air, called experiment. A simplified formalism could then be set up with the flavors of attempts at a full solution. That is, such a formalism can act as a target for low-energy approximations to the the full solution, and can also make immediate contact with experiment. A bridge between theory and experiment has been set up. NONLINEAR RELATIVITY "Doubly Special Relativity" (DSR) [6, 7, 8, 9] is a semiclassical theory, formulated in flat space-time, yet significant at extremely high energy scales. It is based on a non-linear extension of the laws of Special Relativity [8]. Just as Einstein’s relativity theory is "special" because it holds invariant the speed of light (c) as a fundamental relativistic scale, non-linear relativity is "doubly special" because it fixes, in addition, a relativistic energy scale. We stress the word "relativistic" in order to highlight what type of fundamental scale we are dealing with. For instance, h̄ is not a relativistic scale since it does not affect the transformations between inertial observers like c does at very high velocities. The new fixed energy scale will play a role, independent of c, in relativistic transformations at very high energies. Part of the justification for the introduction of an invariant energy scale into Special Relativity can be found in the lineage of Einstein’s theory. Galileo’s original expression for the energy of a fundamental particle, E = p2/2m, is linearly invariant under the transformations he defined circa 1600 A.D.. These Galilean Transformations describe the relativity of inertial motion. They state that position and time coordinates measured in a "primed" frame moving at velocity v in the x-direction with respect to a lab frame at rest are expressed as x′ = x− vt and t ′ = t, respectively. A length l, a time interval t and the velocity of a particle moving at speed v1 in the moving frame would then be written v1 = v1 − v (1) l′ = l (2) t ′ = t. (3) Roughly 200 years later Maxwell formulated his equations describing electricity and magnetism and introduced the notion of a constant value for the speed of light. In order for this invariance to hold whilst satisfying Galilean transformations, the notion of a “preferred observer” had to be introduced. The preference was made manifest by introducing a uniform cosmic background called the “ether”. In other words, the relativity of inertial observers was lost. While Michaelson and Morley worked to prove the constancy of the speed of light proposed by Maxwell, Einstein’s work in the early 1900s demolished the concept of the ether by unifying the previously disjoint notions of time and space in new laws describing the relativity of inertial motion. His discovery of the equivalence of mass and energy led him to a revised definition of particle energy, the quadratic "dispersion relation" E2 = p2c2 +m2c4. This was no longer, however, invariant under Galilean transformations. In order to reestablish observer independent laws, new relativistic transformations, the Lorentz transformations, had to be formulated. With respect to a frame at rest, a particle will have coordinates in a frame moving at velocity v in the x-direction given by, t ′ = γ(t − vx c2 ) (4) x′ = γ(x− vt) (5) y′ = y (6) z′ = z, (7)