"A quantum variant of Einstein's hole argument"

Looking at Newtonian limit of general relativity (GR) for insights into a possible general-relativistic quantum theory of gravity, is a little like saying that you can get insights into quantum electrodynamics by looking at the Coulomb interaction in Galilei space-time (i.e., the pre-special relativistic space-time, with the Galilei group as its preferred symmetry group, and the Galilei transformations between the coordinate systems of any pair of preferred inertial frames of reference).

If one does want to look at this limit of GR, then one must do it right. GR is based on compatible pseudo-metric (chrono-geometry) and affine connection (inertio-gravitational field). The compatibility conditions uniquely fix the relation between metric and connection; this implies that, if the inertio-gravitational field is dynamical, then so must be the pseudo-metric. The dynamical nature of the pseudo-metric is what leads to the conclusion of the hole argument: the points of the space-time manifold are only individuated dynamically, i.e., by the choice of a solution to the Einstein field equations. But in the Newtonian limit, performed properly in four-dimensional form (see e.g. Trautman1, Ehlers2, Christian3, etc.), leads to a four-dimensional formulation of Newtonian gravity. "According to this theory, space is flat, time is absolute with instantaneous causal influences, and the degenerate 'metric ' structure of spacetime remains fixed with two mutually orthogonal non-dynamical metrics, one spatial and the other temporal. The spacetime according to this theory is, nevertheless, curved, duly respecting the principle of equivalence, and the non-metric gravitational connection-field is dynamical in the sense that it is determined by matter distributions" (cited from Christian-- see fuller quotation in point 3 below).

In a little more detail: The inertio-gravitational structure (and thus the equivalence principle) is incorporated into this four-dimensional version of Newton's theory by means of this connection field: the difference between inertial and gravitational effects is no longer absolute, but depends on the acceleration of the frame of reference. In the Newtonian limit, this structure remains dynamical (i.e., subject to field equations based on the Ricci tensor), but the chrono-geometry becomes a fixed, non-dynamical structure-- or rather two such fixed structures: 1) the Newtonian absolute time, and 2) a three-geometry with fixed Euclidean metric. The compatibility conditions between the chrono-geometry and the inertio-gravitational field no longer fix the latter uniquely once the former is given; just enough freedom is left to introduce non-flat connections that are the four-dimensional equivalents of Newtonian gravitational fields.

There is now a class of preferred chron-geometrical frames of reference, wider than the class of inertial frames: all non-rotating frames, linearly accelerated with respect to each other are included. Thus any event (point of space-time) can be individuated by reference to its position in the fixed chronometry (the absolute time of the event) and the fixed Euclidean geometry of any one of the preferred frames (the position of the event with respect to some set of three spatial Cartesian coordinates). Of course, its position w.r.t. any other of the class of preferred frames can then be determined by an obvious generalization of the Galilei transformations. The hole argument no longer applies to this four-dimensional version of Newton's theory.

Here is the abstract of Reference 3:

"Cartan's spacetime reformulation of the Newtonian theory of gravity is a generally covariant Galilean-relativistic limit-form of Einstein's theory of gravity known as the Newton-Cartan theory. According to this theory, space is flat, time is absolute with instantaneous causal influences, and the degenerate 'metric ' structure of spacetime remains fixed with two mutually orthogonal non-dynamical metrics, one spatial and the other temporal. The spacetime according to this theory is, nevertheless, curved, duly respecting the principle of equivalence, and the non-metric gravitational connection-field is dynamical in the sense that it is determined by matter distributions. Here, this generallycovariant but Galilean-relativistic theory of gravity with a possible non-zero cosmological constant, viewed as a parameterized gauge theory of a gravitational vector-potential minimally coupled to a complex Schrödinger-field (bosonic or fermionic), is successfully cast ? for the first time ? into a manifestly covariant Lagrangian form. Then, exploiting the fact that Newton-Cartan spacetime is intrinsically globally-hyperbolic with a fixed causal structure, the theory is recast both into a constraint-free Hamiltonian form in 3+1dimensions and into a manifestly covariant reduced phase-space form with non-degenerate symplectic structure in 4-dimensions. Next, this Newton-Cartan-Schroedinger system is non-perturbatively quantized using the standard C*-algebraic technique combined with the geometric procedure of manifestly covariant phase-space quantization. The ensuing unitary quantum field theory of Newtonian gravity coupled to Galilean-relativistic matter is not only generally-covariant, but also exactly soluble and ? thanks to the immutable causal structure of the Newton-Cartan spacetime ? free of all conceptual and mathematical difficulties usually encountered in quantizing Einstein's theory of gravity. Consequently, the resulting theory of quantized Newton-Cartan-Schrödinger system constitutes a perfectly consistent Galilean-relativistic sector of the elusive full quantum theory of gravity coupled to relativistic matter, regardless of what ultimate form the latter theory eventually takes."

So, if the author wants to discuss the quantization of Newtonian theory, he should discuss the fully quantized version, rather than confining himself to semi-classical arguments, which (although I shall go into detail about only one point-- see the following point 4) are dubious in themselves.

The measurable quantity in any quantum process is the square of the sum of the partial probability amplitudes over all paths in space-time that are indistinguishable in the given process. In Section 6 of his paper, the author notes that, if one considers an entire process, his superposition problem disappears, because the initial preparation and the final measurement produce and measure, respectively, a definite initial and final state. But, as Feynman emphasized, the aim of any quantum theory is to enable the calculation of the total probability amplitude for such a process (the corresponding probability being the square of the absolute value of the amplitude, of course).

So what must be summed up are not intermediate "states" or wave functions, but the partial amplitudes for all of the alternate ways ("paths in space time" in this case) between the initial and final states that are indistinguishable by means of the given experimental arrangement. The author's argument that this approach forces one into "some sort of S-matrix theory, which does not allow us to compute anything for finite distances" is simply wrong. To take his own double slit example, one can place the monochromatic source of particles (i.e., prepared so that all are traveling at the same speed) at some finite distance in front of the screen with the two slits, and some sort of plate on which to register the particles at a finite distance behind the screen. As long as the distances between source and screen and screen and measuring plate are large compared with the wave length of the particle beam, an interference pattern can be measured on the plate.

The aim of this paper seems to be to provide further arguments for the claim of A. Logunov's "relativistic theory of gravitation" that Einstein's general theory requires a background Minkowski space-time metric for its physical interpretation. Here is an excerpt from Luganov's Preface to The Theory of Gravity, translated into English by G. Pontecorvo (Nauka, 2001):

"The hypothesis underlying RTG asserts that the gravitational field, like all other physical fields, develops in Minkowski space, while the source of this field is the conserved energy-momentum tensor of matter, including the gravitational field itself. This approach permits constructing, in a unique manner, the theory of the gravitational field as a gauge theory. Here, there arises an effective Riemannian space, which literally has a field nature. In GRT the space is considered to be Riemannian owing to the presence of matter, so gravity is considered a consequence of space-time exhibiting curvature. The RTG gravitational field has spins 2 and 0 and represents a physical field in the Faraday-Maxwell spirit. The complete set of RTG equations follows directly from the least action principle. Since all physical fields develop in Minkowski space, all fundamental principles of physics ? the integral conservation laws of energy-momentum and of angular momentum ? are strictly obeyed in RTG. In the theory the Mach principle is realized: an inertial system is determined by the distribution of matter. Unlike GRT, acceleration has an absolute sense. Inertial and gravitational forces are separated, and they differ in their nature."

Logunov's theory has been discredited many times, going back at least as far as Ya B Zel'dovich, Leonid P Grishchuk, "The general theory of relativity is correct!" SOV PHYS USPEKHI, 1988, 31 (7), 666-671; and is hardly taken seriously by most workers in general relativity.

But if the author wants to defend it, he shoukd consider the special-relativistic limit of general relativity, i.e., the theory linearized around Minkowski space-time- and show how this eliminates the hole argument. If he does so, I expect that he will find that the same arguments work for the background space-time structure of the Newtonian limit of general relativity.

1 See, e.g., Andrzej Trautman, W. Kopczynski, Spacetime and Gravitation (John Wiley and Sons 1992).

2 Ehlers calls such space-times "general nonrelativistic spacetimes." See, e.g., Jürgen Ehlers, "The Nature and Structure of Spacetime" in Jagdesh Mehra (ed.), The Physicist's Concept of Nature (D. Reidel 1973), pp. 71-91, and "Survey of general relativity theory", in Israel, Werner (ed), Relativity, Astrophysics and Cosmology (D. Reidel 1973), pp. 1-125.

3 Joy Christian, "Exactly soluble sector of quantum gravity (2008)" (http://preprints.cern.ch/archive/electronic/gr-qc/9701/9701013.pdf). An earlier version appeared in "Physical Review."

The paper is not acceptable for publication in Foundations of Physics.

After a discussion on the hole argument and on the open problems in defining observables in generally covariant theories like Einstein general relativity, the author refuses the notion of background independence in favour of a preferred physical background by invoking a semiclassical quantum theory including gravity as a one-particle approximation of QFT on a fixed curved background with a fixed foliation of spacetime. In this framework he considers a source particle gravitationally interacting with a test article and studies a double slit type of experiment involving a superposition of different gravitational fields. Then he defines a certain transition probability as an obersable to be used to corroborate his thesis that only the spatial diffeomorphisms on the given foliation in the given background are relevant (c-covariance against q-covariance).

All the description is based on Newtonian quantum mechanics and on not proven statements about its extension to the relativistic framework.

1) In Newton gravity we have either two particles interacting through the action-at-a-distance Newton gravitational potential or a particle in an external gravitational Newton potential. The first quantization of both systems is done only on the particle variables. In the second case we can have superposition of different states in the same external gravitational potential, but not superposition of states with different external gravitational potentials.

2) The discussion of the double slit experiment is based on the non-relativistic theory of entanglement. Before applying it to gravity, i.e. to general relativity, one has to find its extension to special relativity, where the non-local properties of the Poincare' group and the associated complications in the definition of the relativistic center of mass are clearly incompatible with the description of composite quantum systems as tensor products of the Hilbert spaces of the components. Behind these problems there is the conventional nature of the choice of how to make clock synchronization for the definition of a instantaneous 3-space. This problem becomes worse in general relativity, where the equivalence principle forbids the existence of global inertial frames.

3) If we give a foliation of Minkowski spacetime, i.e. a global non-inertial frame, and we try to quantize the massive Klein-Gordon field with the Tomonoga-Schwinger method, we get generically "non-unitary evolution" (the Torre-Varadarajan no-go theorem, Class.Quantum Grav. 16 (1999) 2651). In curved spacetimes it is even worst, because the given foliation must also be such to admit Fourier transform on its leaves (needed to define the Fock space). Therefore, there is no acceptable formulation of QFT to be used for semiclassicl gravity in a fixed foliation in the one-particle approximation.

As a consequence most of the statements in the paper are based on unproven rusults. The only available partial options are either background-independent loop quantum gravity (a type of relationism) or QFT on the fixed Minkowski background (needed for Fock space) in the family of harmonic gauges with gravity reduced to a spin-2 massless particle (the graviton). At this stage of development both of them do not allow a discussion of the double-slit experiment in the terms used in the paper.

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