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Planck-scale Metaphysics An Appraisal of Bohmian Quantum Gravity Antonio Vassallo Instytut Filozofii, Uniwersytet Warszawski Section de Philosophie, Université de Lausanne Summary ● Quantization of Gravity: the "Canonical Rite” ● Bohmian Mechanics ● Bohmian Quantum Gravity 2 Quantization of Gravity: the "Canonical Rite” ● (3+1) decomposition of spacetime by means of lapse and shift functions. 3 Quantization of Gravity: the "Canonical Rite” ● The equations of motion are... ● ...together with the constraints 4 Quantization of Gravity: the "Canonical Rite” ● ● ● Replace the phase space Γ with the (Hilbert) space of complex functions on Q which are square-integrable with respect to some physically relevant measure μ: L2(Q,μ). The configuration space Q is the space of 3dimensional Riemannian metrics definable on a 3surface embedded in spacetime. Replace canonical variables and algebra with the “quantum” counterparts 5 Quantization of Gravity: the "Canonical Rite” ● ● Implement the classical constraints to restrictions on the physically allowed wave functions All the wave functions that satisfy both conditions (should) form the physical Hilbert space of the theory. 6 Quantization of Gravity: the "Canonical Rite” ● Problem(s) of Spacetime: “If one assigns the intrinsic 3-geometry, one cannot also specify the extrinsic curvature. The uncertainty principle thus deprives one of any way whatsoever to predict, or even give meaning to, 'the deterministic classical history of space evolving in time'.” (C.W. Misner, K.S. Thorne, J.A. Wheeler – Gravitation. W.H. Freeman and Company, 1973, pp. 1182-3) “The quantum fluctuations in the geometry are nevertheless inescapable, if one is to believe the quantum principle and Einstein's theory.” (Ibid., p. 1193) 7 Quantization of Gravity: the "Canonical Rite” ● ● ● Problem of Perennials: What is the physical meaning of the wave function? Ontological Obscurity. 8 Bohmian Mechanics ● Fundamental equations of Bohmian dynamics ● More explicitly: 9 Bohmian Mechanics ● ● ● ● ● Bohmian mechanics is empirically equivalent to standard quantum mechanics. It is consistent with the experimental violations of Bell's inequalities. It is a deterministic but non-local theory of local beables. It faces no measurement problem. It deprives of metaphysical import the “Dirac picture” of quantum mechanics (naïve realism about operators). 10 Bohmian Mechanics ● ● It is ontologically clear (But what about the wave function?). Tensions with special relativity: either backwards causation or preferred foliations are required for a Bohmian theory in Minkowski spacetime. 11 Bohmian Quantum Gravity ● Equations of Bohmian Quantum Gravity ● The explicit form of the “velocity field” is 12 Bohmian Quantum Gravity ● ● Reply to the Problem of Spacetime: A solution of the Wheeler-DeWitt equation on an (initial) surface generates the evolution of the 3-metric along all the leaves of the foliation. This evolution is well-defined and deterministic at all scales (no fluctuations). Reply to the Problem of Perennials: It stems from a misguided metaphysical reading of the operators formalism. 13 Bohmian Quantum Gravity ● ● ● ● Problem: The dynamics depends, in general, on the foliation chosen: (Different foliation→Different 4-space). Possible solution (1): Who cares? Possible solution (2): Diminish as much as possible the dynamical relevance of the foliation. Possible solution (3): Consider a (physically justified) subset of solutions of the WheelerDeWitt equation that generate generally covariant velocity fields. 14 Bohmian Quantum Gravity ● ● ● BQG is a background-independent theory. “Positions” are part of the dynamics. The configuration space plays a preeminent role. Can we still make sense of a “particle” ontology? 15 Bohmian Quantum Gravity ● ● What are the ingredients of BQG? Local beables Universal weave Weaving law What does the theory talk about? What are the local beables in the quantum gravitational case? Chunks of space + “adjacency” relation Prespacial entities + “twining” relation 16 Bohmian Quantum Gravity ● [I]t is reasonable to expect that at small scale, space is a quantum superposition of weave states. […] Therefore the picture of physical space suggested [is that of] a quantum probabilistic cloud of lattices.” (C. Rovelli – Quantum Gravity. CUP, 2004, p.193) ● BQG suggests a picture more clear and simple than the one given above. 17 “Ooo2” template for OpenOffice.org Impress ©2005 Chih-Hao Tsai http://technology.chtsai.org/impress/ 18