\section{What is even the problem?}
Developing a theory of quantum gravity is, from a pragmatic point of view, a horribly silly enterprise of prophylaxing: we develop a theory for a regime with which neither normally do not deal but with which will — for all what we know — not deal with long into the future (if at all).
But then again we are driven towards a theory of quantum gravity from the finding that in various ways what we have as quantum gravity — low energy quantum gravity — is not fully satisfactory: it is not predictive from around the Planck scale anymore (more on the question of where exactly in the next section); and indeed even becomes inconsistent — in the sense that unitary gets violated. So we are theoretically motivated to develop a more fundamental theory.
But then there is also the idea that even if we cannot immediately test it we are curious about what physical scenarios there could be, say, for the inside of a black hole; or the (supposed) beginning of the universe.
I have already taken a stance then on what the problem of quantum gravity is to some extent: i.e., develop a better UV behave theory than low-energy quantum gravity is (go beyond the frontiers). (One might say this is already a bit too fast: gravity might be classical after all.)
Here are some prominent voices from physics on what to take to be a theory of quantum gravity. Carlo Rovelli says that \begin{quote} Quantum gravity is the name given to the tentative theory combining general relativity and quantum mechanics. It is a theory that is not yet complete, and that is still very much under construction. \cite{rovelli2004quantum} \end{quote}
Similarly, we find Ashtekar saying: \begin{quote} A quantum theory of gravity is expected to provide a satisfactory framework for understanding the physics of the very early universe and of black holes. The belief is that this theory will combine the principles of quantum mechanics with those of general relativity in a consistent and coherent manner. \cite{ashtekar1991lectures} \end{quote}
An explicit demand on `accomodation’ is given by de Wit: \begin{quote} A quantum theory of gravity must, at the very least, reproduce general relativity in a classical limit and accommodate the quantum field theory of other interactions.”\cite{dewitt2003global} \end{quote}
Kiefer brings a definition via analogy (where it is not prima facie, as so often with analogies, how far this analogy should be taken — it might just be understood in a very limited fashion, perhaps then seeing string theory as something else): \begin{quote} Quantum gravity is the attempt to construct a theory that describes the gravitational interaction in terms of quantum theory, just as quantum electrodynamics describes the electromagnetic interaction. \cite{kiefer2012quantum} \end{quote}
Jacobson on the other hand gives an open definition \begin{quote} Quantum gravity is an unfinished project to understand the quantum behavior of the gravitational field and spacetime geometry. It could require entirely new principles beyond those of current quantum field theory.”\cite{jacobson2003intro} \end{quote}
%It could be interesting to compare conception of the problem of QG based on approach/community.
What about philosophers? Huggett and Wüthrich write:
\begin{quote} the ultimate need for a theory that in some way unifies QM and GR—a theory of quantum gravity (QG)— arises from the existence of phenomena in our universe in which the domains of the two overlap. There are other, more theoretical, arguments for such a theory and concerning the form it should take. For a critical evaluation of these, arguing that overlap is best reason to seek quantum gravity, and that empirical considerations best dictate its form see Callender and Huggett (2001); for further discussion W ̈uthrich (2005). \cite{HuggettWuthrich} \end{quote}
Wallace catches up on the compatibility definition: \begin{quote} What is a theory of ‘quantum gravity’ ? It is widely described1 as any physical theory which combines general relativity and quantum theory; or, alternatively, as any physical theory that has classical general relativity and quantum theory as limiting cases or partial approximations. \end{quote} Out like this, one walks right into LEQG.\footnote{Wallace’s take is justified given how also in philosophy of physics the combined way is common: Butterfield and Isham (2001, p.33): “by ‘quantum gravity’ we mean any approach to the problem of combining (or in some way ‘reconciling’) quantum theory with general relativity”; Rovelli (2008): “Quantum Gravity is the name given to any theory that describes gravity in the regimes where quantum effects cannot be disregarded”; or Rickles (2008, p.262): “Quantum gravity involves the unification of the principles of quantum theory and general relativity”.}
The way how Wallace cashes out the combination definition shows that there is more to the QG one searches. I think it is fair to characterize this more as a UV better behaved theory, at the very least. What is not clear though, why such a theory is expected to be fundamental (as often implicitly assumed); or why such a theory should go along with unification in some strong sense (as praised about string theory).
Other than demanding a UV-better theory, one might of course demand a fundamental theory of gravity (but it is not clear that this exists in an interesting sense); or whatever final (and thus fundamental) theory (it is not clear that this must exist; but arguably string theory could be seen as a candidate). I think this goal of fundamentality makes at least prima facie sense.
Of course, one could have some desiderata on what the fundamental or final theory should look like (that it offers some explanatorily attractive sense of unification). But desiderata are already at some other level. \footnote{Salimkhani has nicely worked out that unification can happen parasitic to empirical or consistency demands.}
\section{Where is the problem?}
Jacobs argues that none of the standard arguments for the Planck length as immediately physically significant (dimensional analysis, black hole argument, generalised microscope, effective field theory regime due to non-renormalisabielity) is straightforwardly conclusive. The assumption Casper identifies for the EFT scenario argument (which he nevertheless takes most impressive) should sound rather harmless for physicists: namely that the same units in the quantum action are used (hbar) for gravity as for matter — an assumption dubbed by him `action-universality’. As Jacobs spells out here: \begin{quote} The assessment of Action-Universality itself would require an account of the nature of fundamental constants. For example, one could think of hbar as like c, which is a fundamental feature of the spacetime arena in which physical fields evolve. We would therefore expect any relativistic theory to feature the same constant. (Even this is not so clear, as bimetric theories of gravity feature multiple metrics and hence multiple local upper bounds on two-way speeds. The possibility of such theories is one reason to reject Action-Universality.) But one could equally think of hbar as more like G, which is the coupling constant of a particular force. Different forces have different coupling constants. If hbaris like this, then it is conceivable that gravity has a different constant of action than other forces, in which case Action-Universality is false. For as far as I am aware, there is no extant account of the metaphysical nature of a constant such as hbar that could help us decide this question. (p. 18) \end{quote} The first option here seems just the standard way of thinking. The other possibility seems to be a physical possibility of the character of a logical possibility: given that the whole idea of quantisation is suggested and backed by analogy, it is just very plausible to take this to be the case about hbar too (or at the very least one would want to prima facie pursue such a theoretical ansatz as opposed to any other one, all other things being equal).
The most important point is that well-definedness in a concrete sense breaks done (loss of unitarity) eventually — not just productivity. It is in this sense that we should not be happy with the EFT view (LEQG). (LEQG is an argument against the spin-2 background view: I get much more from GR than from the background-relative spin-2 approach.)
The most important point is that well-definedness in a concrete sense breaks done (loss of unitarity) eventually — not just productivity. It is in this sense that we should not be happy with the EFT view (LEQG). (LEQG is an argument against the spin-2 background view: I get much more from GR than from the background-relative spin-2 approach.)
There are deeper regimes associated with LEQG than you would think though.\footnote{ \begin{quote} At present, we have little evidence supporting applications of LEQG in the perturbative/coherent regime, which is significant because it is this regime which distinguishes unitary quantum gravity from variants in which superpositions of mass distributions cause wavefunction collapse, as has been advocated by D ́ıosi (1987, 2014), Penrose (1969, 1996, 2014), Stamp (2015) and others. For that very reason there has been considerable interest in probing that regime, at least nonrelativistically, by attempting to place a relatively-massive object in a superposition and to test the coherence of that superposition via interference experiments. Those experiments are challenging, but not impossible with current technology: recent experimental work by Donadi et al (2021) appears to rule out at least the simplest form of the Di ́osi-Penrose model of gravitational collapse, and in doing so to corrobate at least a part of the perturbative/coherent nonrelativistic regime. We can optimistically hope for LEQG to be more systematically confirmed — or falsified — in that regime in the comparatively near future, with recent proposals for experiments on self-gravitating entangled systems (Marletto and Vedral 2017; Christodoulou and Rovelli 2018). Probing more strongly relativistic aspects of the perturbative/incoherent regime looks far more challenging (performing the two-slit experiment with gravitons, for instance, is ridiculously beyond any current or foreseeable gravity-wave-detection technology) so that it seems likely any tests of this regime will have to be indirect, probably via cosmology.And no breakdown of this kind can hope to be exhaustive: no doubt the theory will sooner or later be put to the test in regimes that defeat my categorization. \end{quote} }
It is illuminating how Jacobs spells out what kind of boundary the Planck length could possibly be: as an upper boundary (quantum gravity becomes only relevant at this or smaller scales); as an exact boundary (quantum gravity becomes more or less relevant at this scale, not much more below or above); or as a lower boundary (quantum gravity becomes relevant at latest at this scale). While he basically assigns many arguments to be just after the lower boundary (clearly the dimensional analysis case has not enough information to say anything), he does not discuss too explicitly how to assess what he takes to be the most convincing case — that from being an effective field theory: but he does mention that the corrections on the EFT view come in at or below the Planck length; so it should be seen as an upper boundary case (again, speaking in terms of distance scales now). In fact, one should expect this to be a strict upper boundary as the real issue is not loss of productivity (which happens already at that scale) but loss of unitarity (which happens at some lower scales).
I take here to lie a sort of conundrum then: is there something about the need of quantum gravity we do not see then on this supposed best case for taking the physical relevance of the Planck length seriously? This is in so far confusing as that everything would seem model-able on the EFT view. (But maybe this is just a hint then that gravity is not an effective field theory after all. So the argument is after all not that good.)\footnote{Maybe consider Schneider here?}
What is also a bit strange: Jacobs (rightly) points out the epistemic nature of the Planck-lenght arguments from black holes, and generalised Heisenberg microscope — but does not mention this for his preferred renormalisabielity option. Naively, one might that is just renormalisabielity (and so predictivity!) at stake here which is epistemic. But then note: the Planck scale is not the scale when the EFT breaks down — it is the scale when the EFT corrections come in. (We could wonder though whether GR is really best conceived of as an EFT. See \cite{Schneider} for some dissenting view.)
Jacobs focuses on the Planck length scale; and practice usually focuses on the Planck energy scale. Interestingely, one can also consider the Planck mass, as it has been done more recently. Naively, one would not expect some difference here. But then again the Planck mass stands out because its value is as such much closer to known regimes:
\begin{quote} Puzzling is the fact that | unlike Planck length and Planck energy, mPlanck falls within a very reachable physical domain: micrograms. It has long been hard to see what sort of quantum gravity effect can happen at the scale of the weight of a human hair. \end{quote}
Can the Planck mass be behind a whole avenue of quantum phenomenology that challenges that quantum effects can only be relevant at unreachable scales? As Huggett et al., write, indeed so:
\begin{quote} According to lore in the philosophy of QG, the problem of quantum gravity is (very nearly) purely one for the theoreticians. It is just too difficult to hope for discriminating signatures of QG in data, because the relevant empirical regimes far exceed our capacities for experimentation (in high energy physics) or direct detection (in astrophysics and cosmology). But this lore is misleading of fundamental physics practice today. In recent decades, and to wide acclaim in the surrounding discipline, a range of empirical testing strategies have been pursued within the arena of quantum gravity phenomenology, as proposed means of gaining significant, increased empirical traction on the problem of QG. \end{quote}
However, as also stressed by Huggett, Linnemann and Schneider, the case is more intricate than you would think. Depending on the paradigm one finds oneself in, one will either see the gravitationally-induced entanglement experiments that work at the relevant mass Planck length — provided the right outcome — as providing evidence about the quantum nature of gravity or not. These are paradigms in so far as that the decisive assumptions at play cannot be decided between based on rational arguments; it requires a push whether to think in one or the other but no specific argument seems to prefer one over the other
But what kind of paradigm could it be that suddenly the Planck mass comes into focus with no corresponding Planck length or energy. (Something a bit lighter than a hair could be a Planck mass — but without being of the Planck length or measured with Planck energy — how can it relevant for QG? Note that already Planck length and momentum have not immediately the same interpretations: length directly pertains to the object (at least naively); energy is meant to describe the energy of light needed to measure out that very object. So in that sense it is not odd if mass is some distinct property again.) It cannot be that simple, as also noted by Held.\footnote{\begin{quote} It is misleading to expect quantum gravitational effects if only one of these scales is reached. For instance, we surpass the Planck mass, MPlanck ∼ 22 μg, in most everyday processes. To expect quantum gravity, all scales involved in a physical process must become Planckian. Put differently, one can identify the following non-Planckian limits: gravity is negligible whenever length mass ≫ G c2, quantum effects are negligible whenever length × mass ≫̵ h. Both of the above limits are tremendously successful descriptions of all measured physical processes in each of the respective regimes of physics. Their unification into a quantum theory of gravity remains a century-long challenge for fundamental theoretical physics. \end{quote}}
Developing a theory of quantum gravity is, from a pragmatic point of view, a horribly silly enterprise of prophylaxing: we develop a theory for a regime with which neither normally do not deal but with which will — for all what we know — not deal with long into the future (if at all).
But then again we are driven towards a theory of quantum gravity from the finding that in various ways what we have as quantum gravity — low energy quantum gravity — is not fully satisfactory: it is not predictive from around the Planck scale anymore (more on the question of where exactly in the next section); and indeed even becomes inconsistent — in the sense that unitary gets violated. So we are theoretically motivated to develop a more fundamental theory.
But then there is also the idea that even if we cannot immediately test it we are curious about what physical scenarios there could be, say, for the inside of a black hole; or the (supposed) beginning of the universe.
I have already taken a stance then on what the problem of quantum gravity is to some extent: i.e., develop a better UV behave theory than low-energy quantum gravity is (go beyond the frontiers). (One might say this is already a bit too fast: gravity might be classical after all.)
Here are some prominent voices from physics on what to take to be a theory of quantum gravity. Carlo Rovelli says that \begin{quote} Quantum gravity is the name given to the tentative theory combining general relativity and quantum mechanics. It is a theory that is not yet complete, and that is still very much under construction. \cite{rovelli2004quantum} \end{quote}
Similarly, we find Ashtekar saying: \begin{quote} A quantum theory of gravity is expected to provide a satisfactory framework for understanding the physics of the very early universe and of black holes. The belief is that this theory will combine the principles of quantum mechanics with those of general relativity in a consistent and coherent manner. \cite{ashtekar1991lectures} \end{quote}
An explicit demand on `accomodation’ is given by de Wit: \begin{quote} A quantum theory of gravity must, at the very least, reproduce general relativity in a classical limit and accommodate the quantum field theory of other interactions.”\cite{dewitt2003global} \end{quote}
Kiefer brings a definition via analogy (where it is not prima facie, as so often with analogies, how far this analogy should be taken — it might just be understood in a very limited fashion, perhaps then seeing string theory as something else): \begin{quote} Quantum gravity is the attempt to construct a theory that describes the gravitational interaction in terms of quantum theory, just as quantum electrodynamics describes the electromagnetic interaction. \cite{kiefer2012quantum} \end{quote}
Jacobson on the other hand gives an open definition \begin{quote} Quantum gravity is an unfinished project to understand the quantum behavior of the gravitational field and spacetime geometry. It could require entirely new principles beyond those of current quantum field theory.”\cite{jacobson2003intro} \end{quote}
%It could be interesting to compare conception of the problem of QG based on approach/community.
What about philosophers? Huggett and Wüthrich write:
\begin{quote} the ultimate need for a theory that in some way unifies QM and GR—a theory of quantum gravity (QG)— arises from the existence of phenomena in our universe in which the domains of the two overlap. There are other, more theoretical, arguments for such a theory and concerning the form it should take. For a critical evaluation of these, arguing that overlap is best reason to seek quantum gravity, and that empirical considerations best dictate its form see Callender and Huggett (2001); for further discussion W ̈uthrich (2005). \cite{HuggettWuthrich} \end{quote}
Wallace catches up on the compatibility definition: \begin{quote} What is a theory of ‘quantum gravity’ ? It is widely described1 as any physical theory which combines general relativity and quantum theory; or, alternatively, as any physical theory that has classical general relativity and quantum theory as limiting cases or partial approximations. \end{quote} Out like this, one walks right into LEQG.\footnote{Wallace’s take is justified given how also in philosophy of physics the combined way is common: Butterfield and Isham (2001, p.33): “by ‘quantum gravity’ we mean any approach to the problem of combining (or in some way ‘reconciling’) quantum theory with general relativity”; Rovelli (2008): “Quantum Gravity is the name given to any theory that describes gravity in the regimes where quantum effects cannot be disregarded”; or Rickles (2008, p.262): “Quantum gravity involves the unification of the principles of quantum theory and general relativity”.}
The way how Wallace cashes out the combination definition shows that there is more to the QG one searches. I think it is fair to characterize this more as a UV better behaved theory, at the very least. What is not clear though, why such a theory is expected to be fundamental (as often implicitly assumed); or why such a theory should go along with unification in some strong sense (as praised about string theory).
Other than demanding a UV-better theory, one might of course demand a fundamental theory of gravity (but it is not clear that this exists in an interesting sense); or whatever final (and thus fundamental) theory (it is not clear that this must exist; but arguably string theory could be seen as a candidate). I think this goal of fundamentality makes at least prima facie sense.
Of course, one could have some desiderata on what the fundamental or final theory should look like (that it offers some explanatorily attractive sense of unification). But desiderata are already at some other level. \footnote{Salimkhani has nicely worked out that unification can happen parasitic to empirical or consistency demands.}
\section{Where is the problem?}
Jacobs argues that none of the standard arguments for the Planck length as immediately physically significant (dimensional analysis, black hole argument, generalised microscope, effective field theory regime due to non-renormalisabielity) is straightforwardly conclusive. The assumption Casper identifies for the EFT scenario argument (which he nevertheless takes most impressive) should sound rather harmless for physicists: namely that the same units in the quantum action are used (hbar) for gravity as for matter — an assumption dubbed by him `action-universality’. As Jacobs spells out here: \begin{quote} The assessment of Action-Universality itself would require an account of the nature of fundamental constants. For example, one could think of hbar as like c, which is a fundamental feature of the spacetime arena in which physical fields evolve. We would therefore expect any relativistic theory to feature the same constant. (Even this is not so clear, as bimetric theories of gravity feature multiple metrics and hence multiple local upper bounds on two-way speeds. The possibility of such theories is one reason to reject Action-Universality.) But one could equally think of hbar as more like G, which is the coupling constant of a particular force. Different forces have different coupling constants. If hbaris like this, then it is conceivable that gravity has a different constant of action than other forces, in which case Action-Universality is false. For as far as I am aware, there is no extant account of the metaphysical nature of a constant such as hbar that could help us decide this question. (p. 18) \end{quote} The first option here seems just the standard way of thinking. The other possibility seems to be a physical possibility of the character of a logical possibility: given that the whole idea of quantisation is suggested and backed by analogy, it is just very plausible to take this to be the case about hbar too (or at the very least one would want to prima facie pursue such a theoretical ansatz as opposed to any other one, all other things being equal).
The most important point is that well-definedness in a concrete sense breaks done (loss of unitarity) eventually — not just productivity. It is in this sense that we should not be happy with the EFT view (LEQG). (LEQG is an argument against the spin-2 background view: I get much more from GR than from the background-relative spin-2 approach.)
The most important point is that well-definedness in a concrete sense breaks done (loss of unitarity) eventually — not just productivity. It is in this sense that we should not be happy with the EFT view (LEQG). (LEQG is an argument against the spin-2 background view: I get much more from GR than from the background-relative spin-2 approach.)
There are deeper regimes associated with LEQG than you would think though.\footnote{ \begin{quote} At present, we have little evidence supporting applications of LEQG in the perturbative/coherent regime, which is significant because it is this regime which distinguishes unitary quantum gravity from variants in which superpositions of mass distributions cause wavefunction collapse, as has been advocated by D ́ıosi (1987, 2014), Penrose (1969, 1996, 2014), Stamp (2015) and others. For that very reason there has been considerable interest in probing that regime, at least nonrelativistically, by attempting to place a relatively-massive object in a superposition and to test the coherence of that superposition via interference experiments. Those experiments are challenging, but not impossible with current technology: recent experimental work by Donadi et al (2021) appears to rule out at least the simplest form of the Di ́osi-Penrose model of gravitational collapse, and in doing so to corrobate at least a part of the perturbative/coherent nonrelativistic regime. We can optimistically hope for LEQG to be more systematically confirmed — or falsified — in that regime in the comparatively near future, with recent proposals for experiments on self-gravitating entangled systems (Marletto and Vedral 2017; Christodoulou and Rovelli 2018). Probing more strongly relativistic aspects of the perturbative/incoherent regime looks far more challenging (performing the two-slit experiment with gravitons, for instance, is ridiculously beyond any current or foreseeable gravity-wave-detection technology) so that it seems likely any tests of this regime will have to be indirect, probably via cosmology.And no breakdown of this kind can hope to be exhaustive: no doubt the theory will sooner or later be put to the test in regimes that defeat my categorization. \end{quote} }
It is illuminating how Jacobs spells out what kind of boundary the Planck length could possibly be: as an upper boundary (quantum gravity becomes only relevant at this or smaller scales); as an exact boundary (quantum gravity becomes more or less relevant at this scale, not much more below or above); or as a lower boundary (quantum gravity becomes relevant at latest at this scale). While he basically assigns many arguments to be just after the lower boundary (clearly the dimensional analysis case has not enough information to say anything), he does not discuss too explicitly how to assess what he takes to be the most convincing case — that from being an effective field theory: but he does mention that the corrections on the EFT view come in at or below the Planck length; so it should be seen as an upper boundary case (again, speaking in terms of distance scales now). In fact, one should expect this to be a strict upper boundary as the real issue is not loss of productivity (which happens already at that scale) but loss of unitarity (which happens at some lower scales).
I take here to lie a sort of conundrum then: is there something about the need of quantum gravity we do not see then on this supposed best case for taking the physical relevance of the Planck length seriously? This is in so far confusing as that everything would seem model-able on the EFT view. (But maybe this is just a hint then that gravity is not an effective field theory after all. So the argument is after all not that good.)\footnote{Maybe consider Schneider here?}
What is also a bit strange: Jacobs (rightly) points out the epistemic nature of the Planck-lenght arguments from black holes, and generalised Heisenberg microscope — but does not mention this for his preferred renormalisabielity option. Naively, one might that is just renormalisabielity (and so predictivity!) at stake here which is epistemic. But then note: the Planck scale is not the scale when the EFT breaks down — it is the scale when the EFT corrections come in. (We could wonder though whether GR is really best conceived of as an EFT. See \cite{Schneider} for some dissenting view.)
Jacobs focuses on the Planck length scale; and practice usually focuses on the Planck energy scale. Interestingely, one can also consider the Planck mass, as it has been done more recently. Naively, one would not expect some difference here. But then again the Planck mass stands out because its value is as such much closer to known regimes:
\begin{quote} Puzzling is the fact that | unlike Planck length and Planck energy, mPlanck falls within a very reachable physical domain: micrograms. It has long been hard to see what sort of quantum gravity effect can happen at the scale of the weight of a human hair. \end{quote}
Can the Planck mass be behind a whole avenue of quantum phenomenology that challenges that quantum effects can only be relevant at unreachable scales? As Huggett et al., write, indeed so:
\begin{quote} According to lore in the philosophy of QG, the problem of quantum gravity is (very nearly) purely one for the theoreticians. It is just too difficult to hope for discriminating signatures of QG in data, because the relevant empirical regimes far exceed our capacities for experimentation (in high energy physics) or direct detection (in astrophysics and cosmology). But this lore is misleading of fundamental physics practice today. In recent decades, and to wide acclaim in the surrounding discipline, a range of empirical testing strategies have been pursued within the arena of quantum gravity phenomenology, as proposed means of gaining significant, increased empirical traction on the problem of QG. \end{quote}
However, as also stressed by Huggett, Linnemann and Schneider, the case is more intricate than you would think. Depending on the paradigm one finds oneself in, one will either see the gravitationally-induced entanglement experiments that work at the relevant mass Planck length — provided the right outcome — as providing evidence about the quantum nature of gravity or not. These are paradigms in so far as that the decisive assumptions at play cannot be decided between based on rational arguments; it requires a push whether to think in one or the other but no specific argument seems to prefer one over the other
But what kind of paradigm could it be that suddenly the Planck mass comes into focus with no corresponding Planck length or energy. (Something a bit lighter than a hair could be a Planck mass — but without being of the Planck length or measured with Planck energy — how can it relevant for QG? Note that already Planck length and momentum have not immediately the same interpretations: length directly pertains to the object (at least naively); energy is meant to describe the energy of light needed to measure out that very object. So in that sense it is not odd if mass is some distinct property again.) It cannot be that simple, as also noted by Held.\footnote{\begin{quote} It is misleading to expect quantum gravitational effects if only one of these scales is reached. For instance, we surpass the Planck mass, MPlanck ∼ 22 μg, in most everyday processes. To expect quantum gravity, all scales involved in a physical process must become Planckian. Put differently, one can identify the following non-Planckian limits: gravity is negligible whenever length mass ≫ G c2, quantum effects are negligible whenever length × mass ≫̵ h. Both of the above limits are tremendously successful descriptions of all measured physical processes in each of the respective regimes of physics. Their unification into a quantum theory of gravity remains a century-long challenge for fundamental theoretical physics. \end{quote}}