PROJECTIVE AND CONFORMAL STRUCTURES IN GENERAL RELATIVITY

1. PROJECTIVE AND CONFORMAL STRUCTURES IN GENERAL RELATIVITY John Stachel Loops ’07, Morelia June 25-30, 2007

2. Work being done in collaboration with Mihaela Iftime

4. Goal Our goal is to contribute to the development of a background-independent, non-perturbative approach to quantization of the gravitational field based on the conformal and projective structures of space-time.

5. Outline of the Talk • What quantization is and is not • Measurement analysis and quantization • Processes are primary • Space-time structures • Conformal and projective structures • Variational principle based on them • Hopes for the future

6. Outline of the Talk • What quantization is and is not • Measurement analysis and quantization • Processes are primary • Space-time structures • Conformal and projective structures • Variational principle based on them • Hopes for the future

7. What is NOT Being Claimed Quantization only makes sense when applied to “fundamental” structures or entities.

8. The Mystique Surrounding Quantum Mechanics “Anything touched by this formalism thereby seems to be elevated– or should it be lowered?– to a fundamental ontological status. The very words ‘quantum mechanics’ conjure up visions of electrons, photons, baryons, mesons, neutrinos, quarks and other exotic building blocks of the universe.”

9. The Mystique Surrounding Quantum Mechanics (cont’d) “But the scope of the quantum mechanical formalism is by no means limited to such (presumed) fundamental particles. There is no restriction of principle on its application to any physical system. One could apply the formalism to sewing machines if there were any reason to do so!” (Stachel 1986)

10. What IS Quantization? Quantization is just a way accounting for the effects of h, the quantum of action, on any process undergone by some system: “fundamental” or “composite”

11. What IS Quantization? Quantization is just a way accounting for the effects of h, the quantum of action, on any process involving some system,– or rather on theoretical models of such a system-- “fundamental” or “composite”, in which the collective behavior of a set of more fundamental entities is quantized

12. The Quantum of Action "Anyone who is not dizzy after his first acquaintance with the quantum of action has not understood a word." Niels Bohr

13. “Atoms and Human Knowledge”--Niels Bohr 1957 “..an element of wholeness, so to speak, in the physical processes, a feature going far beyond the old doctrine of the restricted divisibility of matter. This element is called the universal quantum of action. It was discovered by Max Planck in the first year of this century and came to inaugurate a whole new epoch in physics and natural philosophy. We came to understand that the ordinary laws of physics, i.e., classical mechanics and electrodynamics, are idealizations that can only be applied in the analysis of phenomena in which the action involved at every stage is so large compared to the quantum that the latter can be completely disregarded.

14. Some Non-fundamental Quanta 1) quasi-particles: particle-like entities arising in certain systems of interacting particles, such as phonons and rotons in hydrodynamics (Landau 1941) 2) phenomenological photons: quantized electromagnetic waves in a homogeneous, isotropic dielectric (Ginzburg 1940)

15. Solid-state physics: A polariton laserLeonid V. Butov (Nature 447, 31 May 2007) At the foundation of modern quantum physics, waves in nature were divided into electromagnetic waves, such as the photon, and matter waves, such as the electron. Both can form a coherent state in which individual waves synchronize and combine. A coherent state of electromagnetic waves is known as a laser; a coherent state of matter waves is termed a Bose–Einstein condensate. But what if a particle is a mixture of an electromagnetic wave and matter? Can such particles form a coherent state? What does it look like?

16. Solid-state physics: A polariton laser (cont’d) If the microcavity is the right width, the energies of the cavity photon and the exciton can be made to match up. When this happens the two mix, forming a new particle. This is a combination of matter and electromagnetic waves — an exciton-polariton, or simply 'polariton'. These polaritons inherit some of the lightness of the cavity photons, and have masses much smaller than me.

17. Successful Quantization Successful quantization of some classical formalism does not mean that one has achieved a deeper understanding of reality– or better, an understanding of a deeper level of reality. It means that one has successfully understood the effects of the quantum of action on the phenomena (processes) described by the formalism

18. Peaceful Coexistence in QG Having passed beyond the quantum mystique, one is free to explore how to apply quantization techniques to various formulations of a theory without the need to single one out as the unique “right” one. One might say: “Let a hundred flowers blossom, let a hundred schools contend”

19. Three Morals of This Tale (1) If two such quantizations at different levels are carried out, one may then investigate the relation between them Example: Crenshaw demonstrates: “A limited equivalence between microscopic and macroscopic quantizations of the electromagntic field in a dielectric” [Phys. Rev. A 67 033805 (2003)]

20. Three Morals of This Tale (1 cont’d) If two such quantizations at the same level are carried out, one may also investigate the relation between them Example: the relation between loop quantization and usual field quantization of the electromagnetic field: If you “thicken” the loops, they are equivalent (Ashtekar and Rovelli 1992)

21. Three Morals of This Tale (2) The search for a method of quantizing space-time structures associated with the Einstein equations is distinct from: The search for an underlying theory of all “fundamental” interactions

22. Carlo Rovelli, 2004 I see no reason why a quantum theory of gravity should not be sought within a standard interpretation of quantum mechanics (whatever one prefers). …A common [argument] is that in the Copen-hagen interpretation the observer must be external, but it is not possible to be external from the gravitational field. I think that this argument is wrong; if it was correct it would apply to the Max-well field as well (Quantum Gravity, 370).

23. Carlo Rovelli, 2004 We can consistently use the Copenhagen interpretation to describe the interaction between a macroscopic classical apparatus and a quantum-gravitational phenomenon happening, say, in a small region of (macroscopic) spacetime. The fact that the notion of spacetime breaks down at short scale within this region does not prevent us from having the region interacting with an external Copenhagen observer (ibid.)

24. Three Morals of This Tale (3) An attempt to quantize the conformal and projective structures does not negate, and need not replace, attempts to quantize other space-time structures. Everything depends on the utility of the results in explaining some physical processes

25. Outline of the Talk • What quantization is and is not • Measureability analysis and quantization • Processes are primary • Space-time structures • Conformal and projective structures • Variational principle based on them • Hopes for the future

26. Commutation Relations One central method of taking into account the quantum of action is by means of introducing commutation relations between various particle (non- rel QM) or field (SR QFT) quantities into the formalism. But these commutation relations have more than a purely formal significance

27. Peter G. Bergmann Collaborator of Einstein Pioneer in study of quantization of “generally covariant” theories, including GR

28. Bergmann and Smith 1982 Measurability Analysis for the Linearized Gravitational Field “Measurability analysis identifies those dynamic field variables that are susceptible to observation and measurement (“observables”), and investigates to what extent limitations inherent in their experimental determination are consistent with the uncertainties predicted by the formal theory.”

29. Bergmann & Smith 1982 (cont’d) Measurability analysis of linearized GR, treated as massless spin 2 field, showed limits of co-measurability of “electric” and “magnetic” components of the linearized Riemann tensor coincide with limits of co-definability imposed by the covariant commutation relations.

30. Bergmann & Smith 1982 (cont’d) As in Bohr-Rosenfeld analysis of the co-measurability of electric and magnetic field components, rather than a test point particle, they had to use averages over test bodies occupying space-time regions, and with a similar treatment of the commutation relations.

31. Louis Crane ”Categorical Geometry and the Mathema-tical Founda-tions of Quantum Gravity” (2006)

32. Louis Crane:”Categorical Geometry and the Mathematical Foundationsof Quantum Gravity” (2006) “The ideal foundation for a quantum theory of gravity would begin with a description of a quantum mechanical measurement of some part of the geometry of some region; proceed to an analysis of the commutation relations between different observations, and then hypothesize a mathematical structure for space-time which would contain these relations and give general relativity in a classical limit. We do not know how to do this at present.”

33. Stachel”Prolegomena to any future QuantumGravity” (2007) “ ‘measurability analysis’… is based on ‘the relation between formalism and observation’; its aim is to shed light on the physical implications of any formalism: the possibility of formally defining any physically significant quantity should coincide with the possibility of measuring it in principle; i.e., by means of some idealized measurement procedure that is consistent with that formalism. Non-relativistic QM and special relativistic quantum electrodynamics, have both passed this test ; and its use in QG is discussed in Section 4.

34. Amelino-Camelia and Stachel”Measurement of the space-time interval between two events …” (2007) We share the point of view emphasized by Heisenberg and Bohr and Rosenfeld, that the limits of definability of a quantity within any formalism should coincide with the limits of measurability of that quantity for all conceivable (ideal) measurement procedures. For well-established theories, this criterion can be tested. For example, in spite of a serious challenge, source-free quantum electro-dynamics was shown to pass this test.

35. Amelino-Camelia and Stachel (cont’d) In the case of quantum gravity, our situation is rather the opposite. In the absence of a fully accepted, rigorous theory, exploration of the limits of measurability of various quantities can serve as a tool to provide clues in the search for such a theory: If we are fairly certain of the results of our measurability analysis, the proposed theory must be fully consistent with these results.”

36. Outline of the Talk • What quantization is and is not • Measurement analysis and quantization • Processes are primary • Space-time structures • Conformal and projective structures • Variational principle based on them • Hopes for the future

37. Lee Smolin “[R]elativity theory and quantum theory each ... tell us-- no, better, they scream at us- that our world is a history of processes. Motion and change are primary. Nothing is, except in a very approximate and temporary sense. How something is, or what its state is, is an illusion. It may be a useful illusion for some purposes, but if we want to think fundamentally we must not lose sight of the essential fact that 'is' is an illusion. So to speak the language of the new physics we must learn a vocabulary in which process is more important than, and prior to, stasis.“ (p. 53).

38. Carlo Rovelli, 2004 “The data from a local experiment (measurements, preparation, or just assumptions) must in fact refer to the state of the system on the entire bound-ary of a finite spacetime region. The field theoretical space ... is therefore the space of surfaces Σ [where Σ is a 3d surface bounding a finite spacetime region] and field configurations φ on Σ . Quantum dynamics can be expressed in terms of an [probability] amplitudeW[Σ , φ].

39. Rovelli, 2004 (cont’d) Following Feynman’s intuition, we can formally define W[Σ , φ] in terms of a sum over bulk field configurations that take the value φ on Σ. … Notice that the dependence of W[Σ, φ] on the geometry of Σ codes the spacetime position of the measuring apparatus. In fact, the relative position of the components of the apparatus is determined by their physical distance and the physical time elapsed between measurements, and these data are contained in the metric of Σ.”(p. 23).

40. Rovelli, 2004 (cont’d) Consider now a background independent theory. Diffeomorphism invariance implies immediately that W[Σ , φ] is independent of Σ ... Therefore in gravity W depends only on the boundary value of the fields. However, the fields include the gravitational field, and the gravitational field determines the spacetime geometry. Therefore the dependence of W on the fields is still sufficient to code the relative distance and time separation of the components of the measuring apparatus! (p. 23)

41. Rovelli, 2004 (cont’d) What is happening is that in background-dependent QFT we have two kinds of measurements: those that determine the distances of the parts of the apparatus and the time elapsed between measurements, and the actual measurements of the fields’ dynamical variables. In quantum gravity, instead, distances and time separations are on an equal footing with the dynamical fields. This is the core of the general relativistic revolution, and the key for back-ground- independent QFT

42. Outline of the Talk • What quantization is and is not • Measurement analysis and quantization • Processes are primary: Loops– but what kind? • Space-time structures • Conformal and projective structures • Variational principle based on them • Hopes for the future

43. Loops and Holonomies Loop integrals and holonomies may well be the most suitable set of observables for quantization. But: Loop integrals of a one-form are independent of all other space-time structures. Why confine the loops to space-like hypersurfaces?

44. Example: The Two Aharonov-Bohm Effects There are two Aharonov-Bohm effects: 1) Magnetic: involving space-like loops 2) Electric: involving time-like loops

45. Aharonov-Bohm Effect (cont’d) In the terms of modern differential geometry, the Aharonov-Bohm effect can be understood to be the holonomy of the complex-valued line bundle representing the electromagnetic field. The connection on the line bundle is given by the electromagnetic potential A, and thus the electromagnetic field strength is the curvature of the line bundle F=dA. The integral of A around a closed loop is the holonomy…

46. Electric Aharonov-Bohm effect Just as the phase of the wave function de-ends upon the magnetic vector potential, it also depends upon the scalar electric potential. By constructing a situation in which the electrostatic potential varies for two paths of a particle, through reg-ions of zero electric field, an observable Aharonov-Bohm interference phenomen-on from the phase shift has been predict-ed; again, the absence of an electric field means that, classically, there would be no effect.

47. Outline of the Talk • What quantization is and is not • Measurement analysis and quantization • Processes are primary • Space-time structures • Conformal and projective structures • Variational principle based on them • Hopes for the future

48. Space-time Structures There are a number of space-time structures that play an important role in the general theory of relativity.

49. Pseudo-Metric tensor field The chrono-geometry is represented mathematically by a pseudo-Riemannian metric tensor field on a four-dimensional manifold

50. Bernhard Riemann- From Globally to Locally Euclidean Assume that the spatial metric is locally Euclidean, but globally non flat: there is a curvature that varies from point to point. What kind of curvature?