Gravity Meets Quantum Theory. 1946 is the year Bryce DeWitt entered Harvard graduate school. Quantum Gravity was his goal and remained his goal throughout his lifetime until the very end. The pursuit of Quantum Gravity requires a profound understanding of Quantum Physics and Gravitation Physics. As G. A. Vilkovisky commented , “Quantum Gravity is a combination of two words, and one should know both. Bryce understood this as nobody else, and this wisdom is completely unknown to many authors of the flux of papers that we see nowadays.” Oral History Transcript -- Drs. Bryce DeWitt & Cecile DeWitt-Morette This transcript is based on a tape-recorded interview deposited at the Center for History of Physics of the American Institute of Physics. aip.org/history/ohilist/23199.html The difficulties in reconciling quantum theory and gravity into some form of quantum gravity come from the prima facie incompatibility of general relativity, Einsteins relativistic theory of gravitation, and quantum field theory, the framework for the description of the other three forces (electromagnetism and the strong and weak nuclear interactions). Whence the incompatibility? General relativity is described by Einsteins equations, which amount to constraints on the curvature of spacetime (the Einstein tensor on the left-hand side) due to the presence of mass and other forms of energy, such as electromagnetic radiation (the stress-energy-momentum tensor on the right-hand side). In doing so, they manage to encompass traditional, Newtonian gravitational phenomena such as the mutual attraction of two or more massive objects, while also predicting new phenomena such as the bending and red-shifting of light by these objects (which have been observed) and the existence of gravitational radiation (which has to date only been indirectly observed via the decrease in the period of binary pulsars). (For the latter observation, see the 1993 Physics Nobel Prize presentation speech by Carl Nordling.) In general relativity, mass and energy are treated in a purely classical manner, where ‘classical’ means that physical quantities such as the strengths and directions of various fields and the positions and velocities of particles have definite values. These quantities are represented by tensor fields, sets of (real) numbers associated with each spacetime point. For example, the stress, energy, and momentum Tab(x,t) of the electromagnetic field at some point (x,t), are functions of the three components Ei, Ej, Ek, Bi, Bj, Bk of the electric and magnetic fields E and B at that point. These quantities in turn determine, via Einsteins equations, an aspect of the ‘curvature’ of spacetime, a set of numbers Gab(x,t) which is in turn a function of the spacetime metric. The metric gab(x,t) is a set of numbers associated with each point which gives the distance to neighboring points. A model of the world according to general relativity consists of a spacetime manifold with a metric, the curvature of which is constrained by the stress-energy-momentum of the matter distribution. All physical quantities — the value of the x-component of the electric field at some point, the scalar curvature of spacetime at some point, and so on — have definite values, given by real (as opposed to complex or imaginary) numbers. Thus general relativity is a classical theory in the sense given above. The problem is that our fundamental theories of matter and energy, the theories describing the interactions of various particles via the electromagnetic force and the strong and weak nuclear forces, are all quantum theories. In quantum theories, these physical quantities do not in general have definite values. For example, in quantum mechanics, the position of an electron may be specified with arbitrarily high accuracy only at the cost of a loss of specificity in the description of its momentum, hence its velocity. At the same time, in the quantum theory of the electromagnetic field known as quantum electrodynamics (QED), the electric and magnetic fields associated with the electron suffer an associated uncertainty. In general, physical quantities are described by a quantum state which gives a probability distribution over many different values, and increased specificity (narrowing of the distribution) of one property (e.g., position, electric field) gives rise to decreased specificity of its canonically conjugate property (e.g., momentum, magnetic field). This is an expression of Heisenbergs Uncertainty Principle. In the context of quantum gravity the fluctuating geometry is known as “spacetime foam”. On the surface, the incompatibility between general relativity and quantum theory might seem rather trivial. Why not just follow the model of QED and quantize the gravitational field, similar to the way in which the electromagnetic field was quantized? Just as we associate a quantum state of the electromagnetic field with the quantum state of electrically charged matter, we should, one might think, similarly just associate a quantum state of the gravitational field with the quantum state of both charged and uncharged matter. This is more or less the path that was taken, but it encounters extraordinary difficulties. Some physicists consider these to be technical difficulties, having to do with the non-renormalizability of the gravitational interaction and the consequent failure of the perturbative methods which have proven effective in ordinary quantum field theories. However, these technical problems are closely related to a set of daunting conceptual difficulties, of interest to both physicists and philosophers. The conceptual difficulties basically follow from the nature of the gravitational interaction, in particular the equivalence of gravitational and inertial mass, which allows one to represent gravity as a property of spacetime itself, rather than as a field propagating in a (passive) spacetime background. When one attempts to quantize gravity, one is subjecting some of the properties of spacetime to quantum fluctuations. For example, in canonical quantizations of gravity one isolates and then quantizes geometrical quantities (roughly the intrinsic and extrinsic curvature of three dimensional space) functioning as the position and momentum variables. Given the uncertainty principle and the probabilistic nature of quantum theory, one has a picture involving fluctuations of the geometry of space, much as the electric and magnetic fields fluctuate in QED. But ordinary quantum theory presupposes a well-defined classical background against which to define these fluctuations (Weinstein, 2001a, b), and so one runs into trouble not only in giving a mathematical characterization of the quantization procedure (how to take into account these fluctuations in the effective spacetime structure?) but also in giving a conceptual and physical account of the theory that results, should one succeed. For example, a fluctuating metric would seem to imply a fluctuating causal structure and spatiotemporal ordering of events, in which case, how is one to define equal-time commutation relations in the quantum theory? Cao (2001) believes that the conceptual nature of the problem demands a conceptual resolution. He advocates what he calls ‘ontological synthesis’. This approach asks for an analysis of the ontological pictures of the two ingredient theories of quantum gravity, so that their consistency (the consistency of the resulting synthesis) can be properly assessed. Cao proposes that the tension can best be resolved by focusing firmly on those sine qua non principles of the respective theories. Cao views the gravitational property of universal coupling as essential, but notes that this does not require continuity, so that the former could be retained while discarding the latter, without rendering the framework inconsistent, thus allowing for quantum theorys violent fluctuations (Caos prime candidate for an essential quantum field theoretic concept). Likewise, he argues that quantum field theory requires a fixed background in order to localize quantum fields and set up causal structure. But he notes that a relational account of localization could perform such a function, with fields localized relative to each other. In so doing, one could envisage a diffeomorphism covariant quantum field theory (i.e. one that does not involve reference to fields localized at points of the spacetime manifold). The resulting synthesized entity (a violently fluctuating, universally coupled quantum gravitational field) would then be what a quantum theory of gravity ought to describe. Cao, T.Y., 2001, “Prerequisites for a consistent framework of quantum gravity,” Studies in the History and Philosophy of Modern Physics, 32B: 181–204. While such an approach sounds sensible enough on the surface, to actually put it into practice in the constructive stages of theory-building (rather than a retrospective analysis of a completed theory) is not going to be easy—though it has to be said, the method Cao describes bears close resemblance to the way loop quantum gravity has developed. Lucien Hardy (2007) has developed a novel approach to quantum gravity that shares features of Caos suggestion, though the principles isolated are different from Caos. The causaloid approach is intended to provide a framework for quantum gravity theories, where idea is to develop a general formalism that respects the key features of both general relativity, which he takes to be the dynamical (nonprobabilistic) causal structure, and quantum theory, which he takes to be the probabilistic (nondynamical) dynamics. The causaloid (of some theory) is an entity that encodes all that can be calculated in the theory. Part of the problem here is that Caos (and Hardys) approach assumes that the ontological principles hold at the Planck scale. However, it is perfectly possible that both of the input theories break down at higher energies. Not only that, the technical difficulties of setting up the kind of (physically realistic) diffeomorphism-invariant quantum field theory he suggests have so far proven to be an insurmountable challenge. One crucial aspect that is missing from Caos framework is a notion of what the observables might be. Of course, they must be relational, but this still leaves the problem very much open. metanexus.net/essay/how-come-quantum-theory
Posted on: Sat, 24 May 2014 22:45:41 +0000
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