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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
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Quantization of Gravity: the "Canonical Rite”
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Bohmian Mechanics
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Bohmian Quantum Gravity
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Quantization of Gravity: the "Canonical Rite”
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(3+1) decomposition of spacetime by means of
lapse and shift functions.
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Quantization of Gravity: the "Canonical Rite”
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The equations of motion are...
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...together with the constraints
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Quantization of Gravity: the "Canonical Rite”
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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
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Quantization of Gravity: the "Canonical Rite”
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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.
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Quantization of Gravity: the "Canonical Rite”
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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)
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Quantization of Gravity: the "Canonical Rite”
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Problem of Perennials:
What is the physical meaning of the wave
function?
Ontological Obscurity.
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Bohmian Mechanics
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Fundamental equations of Bohmian dynamics
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More explicitly:
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Bohmian Mechanics
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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).
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Bohmian Mechanics
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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.
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Bohmian Quantum Gravity
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Equations of Bohmian Quantum Gravity
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The explicit form of the “velocity field” is
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Bohmian Quantum Gravity
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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.
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Bohmian Quantum Gravity
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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.
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Bohmian Quantum Gravity
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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?
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Bohmian Quantum Gravity
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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
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Bohmian Quantum Gravity
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[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)
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BQG suggests a picture more clear and simple than
the one given above.
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