The main line of argumentation of this reviewer is that there are better approximations of full quantum gravity than the one I have used. This line of argumentation is invalid: The main point of the paper is an impossibility result. There cannot be any viable background-free quantum theory of gravity. The logic of such impossibility results is reverted: What is a cause of inaccuracy and error in other considerations – the use of approximations – increases the power of an impossibility result, and the weaker the approximation used, the stronger the impossibility result. Indeed, a theory which circumvents the impossibility result would have to differ from those covered by the result already in the domain of applicability of the weak approximation.
1. Appropriateness of the Use of the Newtonian Limit: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).
This is the first example of the wrong line of argumentation of the referee. But it is invalid for other reasons too:
Namely, before QED it was in fact Schrödinger's theory with Coulomb interaction in Galileian space-time which has given us a lot of insights into a possible full theory of quantum electrodynamics.
Even if full QED is already available, it is far away from being impossible to expect new insights from the consideration of some approximations like Coulomb interaction. In particular, the Bohm-Aharonov effect was clearly a new insight, but to understand it there was no necessity to consider QED – classical theory of the EM field was completely sufficient.
2. The Newtonian Limit Already Contains Fixed Background Spatial and Temporal Structures: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.… The hole argument no longer applies to this four-dimensional version of Newton's theory.
This point is completely irrelevant. The referee prefers Newton-Cartan theory as some sort of "true'' limit of GR, but in his reference 3 additional boundary conditions which appear sufficient to obtain Newtonian theory from GR resp. Newton-Cartan theory are described explicitly ("… if non-intersecting spacelike hypersurfaces covering the Galilean spacetime are required to asymptotically resemble Euclidean space''), so that NT is simply another, more restricted, but not a "wrong'' or "improper'' limit of GR.
Above theories – pure Newton as well as Newton-Cartan – have a fixed background and therefore no hole problem, thus, for the problems considered in the paper their difference is irrelevant.
Given that Newtonian theory can be obtained from Newton-Cartan theory under certain additional restrictions, but not reverse, there may be theories of quantum gravity having Newtonian theory, but not Newton-Cartan theory, as a limit. Such theories would be covered by an impossibility result based on Newtonian theory, but not if it would be based on Newton-Cartan theory. as not viable. Therefore, the use of Newtonian gravity makes our impossibility result stronger, and the suggestions of the referee to use, instead of Newtonian theory, Newton-Cartan theory have to be rejected.
3. Why bother With the Semi-Classical Theory When Newtonian Gravity has Already Been Quantized?:Here is the abstract of Reference 3:…
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 same argument applies, of course, to the suggestion to use the quantization of Newton-Cartan theory.
The point is invalid for other reasons too: We use semiclassical relativistic theory and full Newtonian quantum gravity as two different limiting cases of the case which interests us – superpositions of semiclassical states in the relativistic domain. Using quantum Newton-Cartan theory would only allow to replace full Newtonian quantum theory, but not semiclassical relativistic theory.
Then there is no reason at all to discuss the quantization of multi-particle Newtonian theory. It is well-known and unproblematic how to quantize it – simply multi-partice Schrödinger theory with Newtonian interaction potential. It is therefore not discussed, but used.
4. Appeal to the Superposition Principle: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.
Here, the referee has completely misunderstood the point of section 6. I do in no way note that the quantum hole problem disappears if one considers the whole process. Instead, what I have considered there is a vague hope that the problem may disappear if one considers the whole process, and argued that this hope is unreasonable, because it does not lead to a reasonable quantum theory for finite times.
My argument is not wrong. Instead, the counterargument of the referee is misguided, because the question is not if there exist some particular physical situations (like the one mentioned by the referee) where we can measure interference effects (without any doubt, we can) or use S-matrix theory to get some approximation of the results (which is the way how it is used) but that a theory which does not allow us to consider initial and final states containing superpositions can be consistent (if this is possible at all) only as an S-matrix-like theory.
Anyway, because the argument is irrelevant for the main point of the paper (which is the impossibility of a background-free quantum theory of gravity, not of vague hopes for developing theories which violate even basic quantum principles like the superposition principle) I have decided to omit it in the new version.
5. 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):…
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.
While the guess of the referee is simply wrong (it has never been the aim of this paper to support any claim of Logunov), it would not matter even if true: The line of argumentation of the referee should be simply rejected as anti-scientific. It is, of course, the free choice of the referee, as of every scientist, to ignore papers which argue in favour of unpopular theories like RTG. But to argue against the publication of papers because their results and arguments support unpopular theories would be the end of open scientific discussion. (It would be different if a paper would be based on theories widely considered to be empirically falsified or incorrect, but this is not even claimed.)
Even if one thinks that only papers supporting popular theories should be published, the argument is invalid: As mentioned in the article, the necessity of a background for quantum gravity supports not only RTG, but also string theory, which seems yet quite popular.
But if the author wants to defend it, he should 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.
I have no aim to defend RTG in any way. But it is obvious that the fixed Minkowski background which exists in RTG for all gravitational fields eliminates the hole argument, in its classical form as well as in its quantum version. For this purpose, one does not have to consider some special-relativistic limit, because the Minkowski background in RTG is defined for arbitrary gravitational fields and therefore eliminates the hole problem in general in the same way as the Minkowski background eliminates it in special relativity. This is well-known and trivial. That's why I see no necessity to consider this question in the paper.
It is also quite clear that in the Newtonian limit of general relativity (be it Newton-Cartan theory or classical Newtonian theory) we have a fixed background and, therefore, the hole argument does not work. But what would be the point of this? The fact remains that RTG and similar theories with fixed background have no quantum hole problem, but full GR has.
Indeed, in GR one can, at best, construct some fixed background which approximates the metric in some sense for weak fields. This construction will not be unique even for weak fields, and impossible in the general case. Even if one can fix the non-uniqueness in some way so that in the weak field limit the predictions of Newtonian theory for our experiment are recovered (how to do this is not clear) GR provides no hint why an objective observable like the one considered in our thought experiment should be computed using such an arbitrary approximative construction.
Last but not least, even if the hole problem would be solvable in GR for weak fields, the argument against GR remains valid for general fields. In RTG, GLET and string theory there is no quantum hole problem in the general situation, while there is one for GR.
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).
The referee has clearly misunderstood the difference between c-covariance and q-covariance, which has nothing to do with "only the spatial diffeomorphisms on the given foliation in the given background are relevant''. In my humble opinion, this suggests that the referee has not read the paper very carefully.
He has also misunderstood another important point: The experiment – the double slit experiment with the source particle gravitationally interacting with a test particle – is considered in different theoretical frameworks, and not in a single one.
All the description is based on Newtonian quantum mechanics and on not proven statements about its extension to the relativistic framework.
The referee does not seem to understand the role of thought experiments in the development of new physical theories. Based on his criterion, a paper by Einstein describing the elevator thought experiment and proposing the Einstein equivalence principle as a base for the future development of a relativistic theory of gravity should have been rejected as well: Such a paper could not have been based on some sort of "proven statements'' about relativistic gravity. If only papers based on "proven statements'' are acceptable, how could one develop a theory of quantum gravity? It seems, it should be developed in one hit in a single paper, without any intermediate steps not based on "proven statements''?
Nonetheless, while the requirement to base a paper about quantum gravity on proven statements about relativistic quantum gravity is clearly nonsensical, I have missed the quite obvious opportunity to formulate the impossibility result in form of a theorem. While only a question of style (it seems that any interested reader should understand that the point of the paper is that background-free theories of quantum gravity are impossible) this makes also more obvious that the use of weak(er) approximations is an advantage rather than a disadvantage and that other problems of background-free theories do not question our results.
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.
As can be easily observed, we need only the first case (two interacting particles) to make our point. If one of the particles is in a delta-like state, one particle theory with the corresponding potential given by the position of the other particle gives a reasonable approximation, but this is an unproblematic approximation inside the context of two-particle theory. In two-particle theory we can have superpositions of states with different delta-like wave functions for one particle.
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.
This is an example of an additional problem to be faced by a background-independent theory of quantum gravity. Once it makes the job of creating a background-free theory even more problematic, does not question our main result that such theories are not viable.
Let's note also that this problem is different from the problem we consider in our paper, thus, does not question the novelty of our results (which can be seen from the fact that our problem appears even in theories with a preferred foliation of spacetime).
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.
This argument should be rejected for the same reasons: It describes yet an additional problem of background-free theories of quantum gravity, thus, does not question our thesis that such theories are not viable.
By the way, the criterion of acceptability is clearly wrong. Whatever the theory of quantum gravity, there will be applications where a one-particle approximation, however inconsistent or non-unitary, will be sufficient as an approximation, simply because the effects of particle creation and destruction and the uncertainties related with the definition of the vacuum are small enough to be ignored.
Then, in the mentioned paper it is noted that the problem is an ultraviolet one. Thus, it may disappear in regularizations. (Similar inequivalence problems appear in field theory approaches to thermodynamics between states with different temperature, but disappear in the regularization defined by atomic theory.) Instead, the problem we have presented does not disappear in regularizations, thus, should be considered as more serious.
As a consequence most of the statements in the paper are based on unproven results. 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.
It seems, the referee should have rejected a paper about Einstein's elevator thougth experiment before the completion of GR as not being based on proven results about relativistic gravity.
Anyway, in the new version of the paper we present a formal impossibility theorem, thus, the point is irrelevant for the new version.