Quantum Gravity, broadly construed, is a physical theory (still‘under construction’ after over 100 years) incorporatingboth the principles of general relativity and quantum theory. Such atheory is expected to be able to provide a satisfactory description ofthe microstructure of spacetime at the so-calledPlanck scale, at which all fundamental constants of the ingredient theories, \(c\)(the velocity of light in vacuo), \(\hslash\) (the reducedPlanck’s constant), and G (Newton’s constant), cometogether to form units of mass, length, and time. This scale is soremote from current experimental capabilities that the empiricaltesting of quantum gravity proposals along standard lines is renderednear-impossible, though there have been some recent developments thatsuggest the outlook might be more optimistic than previously surmised(see Carney, Stamp, and Taylor, 2022, for a review; Huggett,Linnemann, and Schneider, 2023, provides a pioneering philosophicalexamination of so-called “laboratory quantumgravity”).

In most, though not all, theories of quantum gravity, thegravitational field itself is also quantized. Since the contemporarytheory of gravity, general relativity, describes gravitation as thecurvature of spacetime by matter and energy, a quantization of gravityseemingly implies some sort of quantization of spacetime geometry:quantum spacetime. Insofar as all extant physical theories rely on aclassical (non-quantum) spacetime background, this presents not onlyextreme technical difficulties, but also profound methodological andontological challenges for the philosopher and the physicist. Thoughquantum gravity has been the subject of investigation by physicistsfor almost a century (see Rickles 2020), philosophers have only justbegun to investigate its philosophical implications.

• 1. Introduction
• 2. Gravity Meets Quantum Theory
• 3. Theoretical Frameworks
  • 3.1 String Theory
  • 3.2 Canonical and Loop Quantum Gravity
    • 3.2.1 Geometric variables
    • 3.2.2 Problem of time
    • 3.2.3 Ashtekar, loop, and other variables
  • 3.3 Other Approaches
• 4. Methodology
  • 4.1 Theory
  • 4.2 Experiment
• 5. Philosophical Issues
  • 5.1 Time
  • 5.2 Ontology
  • 5.3 Status of quantum theory
  • 5.4 The Planck Scale
  • 5.5 Background Structure
  • 5.6 Necessity of Quantization
• 6. Conclusion
• Bibliography
• Academic Tools
• Other Internet Resources
  • Unpublished, Online Manuscripts
  • Useful Web Resources
• Related Entries

1. Introduction

Dutch artist M.C. Escher’s elegant pictorial paradoxes areprized by many, not least by philosophers, physicists, andmathematicians. Some of his work, for exampleAscending andDescending, relies on optical illusion to depict what is actuallyan impossible situation. Other works are paradoxical in the broadsense, butnot impossible:Relativity depicts acoherent arrangement of objects, albeit an arrangement in which theforce of gravity operates in an unfamiliar fashion. (See theOther Internet Resources section below for images.) Quantum gravity itself may be like this:an unfamiliar yet coherent arrangement of familiar elements. Or it maybe more likeAscending and Descending, an impossibleconstruction which looks sensible in its local details but does notfit together into a coherent whole when using presently existingbuilding materials. If the latter is true, then the construction of aquantum theory of gravity may demand entirely unfamiliar elements.Whatever the final outcome, the situation at present is one of flux,with a great many competing approaches vying for the prize. However,it is also important to note that the prize is not always the same:string theorists seek a unified theory of all four interactions thathas the power of explaining such things as the numbers of generationsof elementary particles and other previous inexplicable properties.Other approaches are more modest, and seek only to bring generalrelativity in line with quantum theory, without necessarily invokingthe other interactions. Hence, theproblem of quantum gravitycan mean very different things to different researchers and whatconstitutes a possible solution to one group might not qualify as suchto another.

Given that quantum gravity does not yet exist as a working physicaltheory, one might legitimately question whether philosophers have anybusiness being involved at this stage. Certainly thephilosopher’s task will be somewhat different from that facedwhen dealing with a more-or-less settled body of theory such asclassical Newtonian mechanics, general relativity, or quantummechanics. In such cases, one typically proceeds by assuming thephysical soundness of the theory or theoretical framework and drawingout the ontological and perhaps epistemological consequences of thetheory, trying to understand what it is that the theory is telling usabout the nature of space, time, matter, causation, and so on.Theories of quantum gravity, on the other hand, are bedeviled by ahost of technical and conceptual problems, questions, and issues thatmake them largely unsuited to this kind of interpretive approach. Inthe case of string theory, there isn’t even really a‘theory’ to speak of, so much as several clues pointing towhat many hope will some day be an applicable, consistent physicaltheory. However, philosophers who have a taste for a broader and moreopen-ended form of inquiry will find much to think about, and it isentirely possible that future philosophers of physics will be facedwith problems of a very different flavour as a result of the peculiarnature of quantum gravity. Indeed, Tian Cao argues that quantumgravity offers up aunique opportunity for philosophers ofphysics, leaving them “with a good chance to make some positivecontributions, rather than just analysing philosophically whatphysicists have already established” (Cao, 2001, p. 138). Thissentiment has in fact been echoed by several physicists, not least byCarlo Rovelli (a central architect of the approach known as loopquantum gravity), who complains that he wishes philosophers would notrestrict themselves to “commenting and polishing the presentfragmentary physical theories, but would take the risk of trying tolookahead” (Rovelli, 1997, p. 182). This raises animportant point: though we think of general relativity and quantumtheory as ‘nice’ theories from the point of view ofphilosophical investigation, in a very real sense they are not thewhole story and break down at extreme scales. That is: we cannotignore the problem of quantum gravity.

2. Gravity Meets Quantum Theory

The difficulties in reconciling quantum theory and gravity into someform of quantum gravity come from theprima facieincompatibility of general relativity, Einstein’s relativistictheory of gravitation, and quantum field theory, the framework for thedescription of the other three forces (electromagnetism and the strongand weak nuclear interactions). Whence the incompatibility? Generalrelativity is described by Einstein’s equations, which amount toconstraints on the curvature of spacetime (the Einstein tensor on theleft-hand side) due to the presence of mass and other forms of energy,such as electromagnetic radiation (the stress-energy-momentum tensoron the right-hand side). (See John Baez’s webpages inOther Internet Resources for an excellent introduction.) In doing so, they manage to encompasstraditional, Newtonian gravitational phenomena such as the mutualattraction of two or more massive objects, while also predicting newphenomena such as the bending and red-shifting of light by theseobjects (which have been observed) and the existence of gravitationalradiation (until very recently, with the direct detection ofgravitational waves by LIGO, this was, of course, only indirectlyobserved via the decrease in the period of binary pulsars-see the1993 Physics Nobel Prize presentation speech by Carl Nordling.)

In general relativity, mass and energy are treated in a purelyclassical manner, where ‘classical’ means that physicalquantities such as the strengths and directions of various fields andthe positions and velocities of particles have definite values. Thesequantities are represented by tensor fields, sets of (real) numbersassociated with each spacetime point. For example, the stress, energy,and momentum \(T_{ab}(\boldsymbol{x},t)\) of the electromagnetic fieldat some point \((\boldsymbol{x},t)\), are functions of the threecomponents \(E_i, E_j, E_k, B_i, B_j, B_k\) of the electric andmagnetic fields \(\boldsymbol{E}\) and \(\boldsymbol{B}\) at thatpoint. These quantities in turn determine, via Einstein’sequations, an aspect of the ‘curvature’ of spacetime, aset of numbers \(G_{ab}(\boldsymbol{x},t)\) which is in turn afunction of the spacetime metric. The metric\(g_{ab}(\boldsymbol{x},t)\) is a set of numbers associated with eachpoint which gives the distance to neighboring points. A model of theworld according to general relativity consists of a spacetime manifoldwith a metric, the curvature of which is constrained by thestress-energy-momentum of the matter distribution. All physicalquantities — the value of the \(x\)-component of the electricfield at some point, the scalar curvature of spacetime at some point,and so on — have definite values, given by real (as opposed tocomplex or imaginary) numbers. Thus general relativity is a classicaltheory in the sense given above.

The problem is that our fundamental theories of matter and energy, thetheories describing the interactions of various particles via theelectromagnetic force and the strong and weak nuclear forces, are allquantum theories. Inquantum theories, these physical quantities do not in generalhave definitevalues. For example, in quantum mechanics, the position of an electronmay be specified with arbitrarily high accuracy only at the cost of aloss of specificity in the description of its momentum, hence itsvelocity. At the same time, in the quantum theory of theelectromagnetic field known as quantum electrodynamics (QED), theelectric and magnetic fields associated with the electron suffer anassociated uncertainty. In general, physical quantities are describedby a quantum state which gives a probability distribution over manydifferent values, and increased specificity (narrowing of thedistribution) of one property (e.g., position, electric field) givesrise to decreased specificity of its canonically conjugate property(e.g., momentum, magnetic field). This is an expression ofHeisenberg’sUncertainty Principle. In the context of quantum gravity the fluctuating geometry is knownas “spacetime foam”. Likewise, if one focusses in on thespatial geometry, it will not have a definite trajectory.

On the surface, the incompatibility between general relativity andquantum theory might seem rather trivial. Why not just follow themodel of QED and quantize the gravitational field, similar to the wayin which the electromagnetic field was quantized? This is more or lessthe path that was taken, but it encounters extraordinary difficulties.Some physicists consider these to be ‘merely’ technicaldifficulties, having to do with the non-renormalizability of thegravitational interaction and the consequent failure of theperturbative methods which have proven effective in ordinary quantumfield theories. However, these technical problems are closely relatedto a set of dauntingconceptual difficulties, of interest toboth physicists and philosophers.

The conceptual difficulties basically follow from the nature of thegravitational interaction, in particular the equivalence ofgravitational and inertial mass, which allows one to represent gravityas a property of spacetime itself, rather than as a field propagating\(in\) a (passive) spacetime background. When one attempts to quantizegravity, one is subjecting some of the properties of spacetime toquantum fluctuations. For example, in canonical quantizations ofgravity one isolates and then quantizes geometrical quantities(roughly the intrinsic and extrinsic curvature of three dimensionalspace) functioning as the position and momentum variables. Given theuncertainty principle and the probabilistic nature of quantum theory,one has a picture involving fluctuations of the geometry of space,much as the electric and magnetic fields fluctuate in QED. Butordinary quantum theory presupposes a well-defined classicalbackground against which todefine these fluctuations(Weinstein, 2001a, b), and so one runs into trouble not only in givinga mathematical characterization of the quantization procedure (how totake into account these fluctuations in the effective spacetimestructure?) but also in giving a conceptual and physical account ofthe theory that results, should one succeed. For example, afluctuating metric would seem to imply a fluctuating causal structureand spatiotemporal ordering of events, in which case, how is one todefine equal-time commutation relations in the quantum theory? (Seethe section on the Lagrangian formulation in the entry onquantum field theory.)

Cao (2001) believes that the conceptual nature of the problem demandsa conceptual resolution. He advocates what he calls ‘ontologicalsynthesis’. This approach asks for an analysis of theontological pictures of the two ingredient theories of quantumgravity, so that their consistency (the consistency of the resultingsynthesis) can be properly assessed. Ontology for Cao refers to theprimary, autonomous structures from which all other properties andrelations in a theory are constructed. A fairly simple inspection ofthe respective ontological constraints imposed by general relativityand quantum field theory reveals serious tension: general relativitydiscards the fixed kinematical structure of spacetime, so thatlocalization is rendered relational, but in quantum field theory afixed flat background is part of its ontological basis, from which thestandard features of the theory are derived. On the other hand, as wehave seen, quantum field theory involves quantum fluctuations in thevicinity of a point, while general relativity involves the use of asmooth point neighbourhood. Either way, in order to bring the twoontological bases together, some piece of either edifice must bedemolished. Cao proposes that the tension can best be resolved byfocussing firmly on thosesine qua non principles of therespective theories. Cao views the gravitational property of universalcoupling as essential, but notes that this does not requirecontinuity, so that the former could be retained while discarding thelatter, without rendering the framework inconsistent, thus allowingfor quantum theory’s violent fluctuations (Cao’s primecandidate for an essential quantum field theoretic concept). Likewise,he argues that quantum field theory requires a fixed background inorder to localize quantum fields and set up causal structure. But henotes that a relational account of localization could perform such afunction, with fields localized relative to each other. In so doing,one could envisage a diffeomorphism covariant quantum field theory(i.e. one that does not involve reference to fields localized atpoints of the spacetime manifold). The resulting synthesized entity (aviolently fluctuating, universally coupled quantum gravitationalfield) would then be what a quantum theory of gravity ought todescribe.

While such an approach sounds sensible enough on the surface, toactually put it into practice in the constructive stages oftheory-building (rather than a retrospective analysis of a completedtheory) is not going to be easy—though it has to be said, themethod Cao describes bears close resemblance to the way loop quantumgravity has developed. Lucien Hardy (2007) has developed a novelapproach to quantum gravity that shares features of Cao’ssuggestion, though the principles isolated are different fromCao’s. The causaloid approach is intended to provide aframework for quantum gravity theories, where idea is todevelop a general formalism that respects the key features of bothgeneral relativity, which he takes to be the dynamical(non-probabilistic) causal structure, and quantum theory, which hetakes to be the probabilistic (nondynamical) dynamics. The causaloid(of some theory) is an entity that encodes all that can be calculatedin the theory. Part of the problem here is that Cao’s (andHardy’s) approach assumes that the ontological principles holdat the Planck scale. However, it is perfectly possible that both ofthe input theories break down at higher energies. Not only that, thetechnical difficulties of setting up the kind of (physicallyrealistic) diffeomorphism-invariant quantum field theory he suggestshave so far proven to be an insurmountable challenge. One crucialaspect that is missing from Cao’s framework is a notion of whattheobservables might be. Of course, they must be relational,but this still leaves the problem very much open. (The idea of makingprogress by isolating appropriateprinciples of quantumgravity forms the basis of a special issue: Crowther and Rickles, eds,2014.)

We will look in more detail at how various conceptual andmethodological problems arise in two different research programsbelow. But first, we introduce some key features of the leadingresearch programs.

3. Theoretical Frameworks

All approaches to the problem of quantum gravity agree that somethingmust be said about the relationship between gravitation and quantizedmatter. These various approaches can be catalogued in various ways,depending on the relative weight assigned to general relativity andquantum field theory. Some approaches view general relativity as inneed of correction and quantum field theory as generally applicable,while others view quantum field theory as problematic and generalrelativity as having a more universal status. Still others view thetheories in a more even-handed manner, perhaps with both simplyamounting to distinct limits of a deeper theory. It has often beensuggested, since the earliest days of quantum gravity research, thatbringing quantum field theory and general relativity together mightserve to cure their respective singularity problems (the formerresulting from bad high frequency behaviour of fields; the latterresulting from certain kinds of gravitational collapse). This hopedoes seem to have been borne out in many of the current approaches.Roger Penrose has even argued that the joint consideration ofgravitation and quantum theory could resolve the infamous quantummeasurement problem (see Penrose 2001; see also the section on themeasurement problem in the entry onphilosophical issues in quantum theory). The basic idea of Penrose’s proposal is fairly simple to grasp:when there is wave-packet spreading of the centre of mass of somebody, there results a greater imprecision in the spacetime structureassociated with the spreading wave-packet, and this destroys thecoherence of the distant parts of the wave-function. There aredifficulties in distinguishing the gravitationally induced collapsethat Penrose proposes from the effective collapse induced by quantumtheory itself, thanks to decoherence—Joy Christian (2005) hassuggested that by observing oscillations in the flavor ratios ofneutrinos originating at cosmological distances one could eliminatethe confounding effects of environmental decoherence.

The two most popular approaches remain string theory and loop quantumgravity. The former is an example of an approach to quantum gravity inwhich the gravitational field is not quantized; rather, a distincttheory is quantized which happens to coincide with general relativity(as well as the other interactions) at low energies. The latter is anapproach involving (constrained) canonical quantization, albeit of aversion of general relativity based on a different choice of variablesthan the usual geometrodynamical, metric-based variables. We cover thebasic details of each of these in the following subsections.

3.1 String Theory

Known variously as string theory, superstring theory, and M-theory,this program (qua theory of quantum gravity) has its roots,indirectly, in the observation, dating back to at least the 1930s,that classical general relativity looks in many ways like the theoryof a massless ‘spin-two’ field propagating on the flatMinkowski spacetime of special relativity. [See Cappellietal. (eds.) 2012, and Gasperini and Maharana (eds.) 2008, forcollections of essays covering the early history of string theory;Rickles 2014 offers a conceptually-oriented history of the earlierdays of string theory; Rovelli 2001b (Other Internet Resources sectionbelow) and 2006 offer a capsule history, and Greene 2000 provides apopular account.] This observation led to early attempts to formulatea quantum theory of gravity by “quantizing” this spin-twotheory. However, it turned out that the theory is not perturbativelyrenormalizable, meaning that there are ineliminable infinities.Attempts to modify the classical theory to eliminate this problem ledto a different problem, non-unitarity, and so this general approachwas moribund until the mid-1970s, when it was discovered that a theoryof one-dimensional “strings” developed around 1970 toaccount for the strong interaction, actually provided a framework fora unified theory which included gravity, because one of the modes ofoscillation of the string corresponded to a massless spin-two particle(the ‘graviton’).

The original and still prominent idea behind string theory was toreplace the point particles of ordinary quantum field theory(particles like photons, electrons, etc) with one-dimensional extendedobjects called strings. (See Weingard, 2001 and Witten, 2001 foroverviews of the conceptual framework.) In the early development ofthe theory, it was recognized that construction of a consistentquantum theory of strings required that the strings “live”in a larger number of spatial dimensions than the observed three.String theories containing fermions as well as bosons must beformulated in nine space dimensions and one time dimension. Stringscan be open or closed, and have a characteristic tension and hencevibrational spectrum. The various modes of vibration correspond tovarious particles, one of which is the graviton (the hypotheticalmassless, spin-2 particle responsible for mediating gravitationalinteractions). The resulting theories have the advantage of beingperturbatively renormalizable. This means that perturbativecalculations are at least mathematically tractable. Since perturbationtheory is an almost indispensable tool for physicists, this is deemeda good thing.

String theory has undergone several mini-revolutions over the lastseveral years, one of which involved the discovery of various dualityrelations, mathematical transformations connecting, in this case, whatappear to be physically distinct string theories — type I, typeIIA, type IIB, (heterotic) SO(32) and (heterotic)\(\mathrm{E}_8\times\mathrm{E}_8\) — to one another and toeleven-dimensional supergravity (a particle theory). The discovery ofthese connections led to the conjecture that all of the stringtheories are really aspects of a single underlying theory, which wasgiven the name ‘M-theory’ (though M-theory is also usedmore specifically to describe the unknown theory of whicheleven-dimensional supergravity is the low energy limit). Therationale, according to one kind of duality (S-duality), is that onetheory at strong coupling (high energy description) is physicallyequivalent (in terms of physical symmetries, correlation functions andall observable content) to another theory at weak coupling (where alower energy means a more tractable description), and that if all thetheories are related to one another by dualities such as this, thenthey must all be aspects of some more fundamental theory. Thoughattempts have been made, there has been no successful formulation ofthis theory: its very existence, much less its nature, is stilllargely a matter of conjecture.

There has been some recent interest in dualities by philosophers,given their clear links to standard philosophical issues such asunderdetermination, conventionalism, and emergence/reduction. The linkcomes about because in a dual pair (of theories) one has a observableequivalence combined with what appears to be radical physical (andmathematical) differences. These differences can be as extreme asdescribing spacetimes of apparently different topological structures,including different numbers of dimensions. This has led somephysicists to speak of spacetimeemerging, depending on suchthings as the coupling strength governing physical interactions. Thiscan be seen most clearly in the context of the AdS/CFT duality inwhich a ten dimensional string theory is found to be observationallyequivalent (again covering physical symmetries, observables and theircorrelation functions) to a four dimensional gauge theory — thisis sometimes called a ‘gauge/gravity’ duality since thestring theory contains gravity (all string theories containgravitons) while the gauge theory does not. Since there is anequivalence between these descriptions, it makes sense to say thatneither is fundamental, and so (elements of) the spacetimes theyapparently describe are also not fundamental; thus implying that thespacetime we observe at low-energies is an emergent phenomenon —Vistarini 2013 is a recent discussion of spacetime emergence in stringtheory. One way to view such dual pairs is in terms of the twotheories (the gauge theory and a gravitational theory) being distinctclassical limits of a more all-encompassing quantum theory. In thiscase, the classical emergent structures also include the specificgauge symmetries and degrees of freedom of the limiting theories. Aproblem remains of making sense of the more fundamental theory (andthe associated physical structure it describes) from which thesespacetimes and gauge symmetries emerge.

Philosophically speaking, there is a large question mark over whetherthe dual pair should be seen as genuinely distinct in a physical senseor as mere notational variants of the same theory — talk of a“dictionary” relating the theories makes the latter morepalatable and suggests that the choice of physical interpretationmight be conventional. However, if we view the theories as notationalvariants, then our sense of theory-individuation is seeminglycompromised, since the dual pairs involve different dynamics anddegrees of freedom. (See Joseph Polchinski 2014, for a thoroughaccount of the various kinds of dualities along with some of theirinterpretive quirks; Rickles 2011 provides an early philosophicalexamination of string dualities; a useful, simplified guide is leBihan and Read, 2018; a more exhaustive study is de Haro andButterfield, forthcoming).

A further problem of string theory is that it appears to amount tomany distinct theories (albeit with dualities linking some of them,and so with the question of whether they are the same theory underdifferent representations) that form a ‘landscape’ oftheories. The problem faced is how we get from a vast space ofpossible string theories to the string theory that represents ourworld. This brings in issues of ‘anthropics,’ since itseems essential to include our own essential properties, as observersof the world, to narrow down the possibility space (see Kane 2021 forrecent developments in accessing our world from the landscape).

3.2 Canonical and Loop Quantum Gravity

Whereas (perturbative) string theory and other so-called‘covariant’ approaches view the curved spacetime ofgeneral relativity as an effective modification of a flat (or otherfixed) background geometry by a massless spin-two field, the canonicalquantum gravity program treats the full spacetime metric itself as akind of field, and attempts to quantize it directly without splittingit apart into a flat part and a perturbation. However, spacetimeitself is split apart into a stack of three dimensional slices (afoliation) on which is defined a spatial geometry. Technically, workin this camp proceeds by writing down general relativity in so-called‘canonical’ or ‘Hamiltonian’ form, since thereis a more-or-less clearcut way to quantize theories once they are putin this form (Kuchar, 1993; Belot & Earman, 2001). In a canonicaldescription, one chooses a particular set of configuration variables\(x_i\) and canonically conjugate momentum variables \(p_i\) whichdescribe the state of a system at some time, and can be encoded in aphase space. Then, one obtains the time-evolution of these variablesfrom the Hamiltonian \(H(x_i,p_i)\), which provides the physicallypossible motions in the phase space a family of curves. Quantizationproceeds by treating the configuration and momentum variables asoperators on a quantum state space (a Hilbert space) obeying certaincommutation relations analogous to the classical Poisson-bracketrelations, which effectively encode the quantum fuzziness associatedwith Heisenberg’s uncertainty principle. The Hamiltonianoperator, acting on quantum states, would then generate the dynamicalevolution.

When one attempts to write general relativity down in this way, onehas to contend with the existence ofconstraints on thecanonical variables that are inherited from the diffeomorphisminvariance of the spacetime formulation of the theory. The singletensorial equation that we see in standard presentations of theEinstein field equations is translated into 10 scalar equations in thecanonical formulation, with constraints accounting for four of theseequations (the remaining six are genuine evolutionary equations).Three of the constraints (known as the momentum or diffeomorphismconstraints) are responsible for shifting data tangential to theinitial surface and, thus, are related to the shift vector field. Theremaining constraint, known as the Hamiltonian (or scalar) constraint,is responsible for pushing data off the initial surface, and thus isrelated to the lapse function. If the constraints are not satisfied bythe canonical initial data then the development of the data withrespect to the evolution equations, will not generate a physicallypossible spacetime for choices of lapse and shift. However, when theconstraintsare satisfied then the various choices of lapseand shift will always grow the same 4D spacetime (that it, the samespacetime metric). However, to extract a notion of time from thisformulation demands that one first solve for the spacetime metric,followed by a singling out of a specific solution. This is a kind ofclassical problem of time in that since the spacetime geometry is adynamical variable, time is something that also must be solved for.Further, there is arbitrariness in the time variable as a result ofthe arbitrariness encoded in the constraints, stemming from the factthat time is essentially a freely chosen label of the threedimensional slices and so is not a physical parameter. However, onecan extract a time for each solution to the Einsteinequations by ‘deparametrizing’ the theory (i.e. isolatinga variable from within the phase space that is to play the role oftime). Below we see that things become more problematic in the shiftto quantum theory.

Although advocates of the canonical approach often accuse stringtheorists of relying too heavily on classical background spacetime,the canonical approach does something which is arguably quite similar,in that one begins with a theory that conceives time-evolution interms of evolving some data specified on ana priori givenspacelike surface, and then quantizing the theory. However, this doesnot imply any breaking of spacetime diffeomorphism invariance (orgeneral covariance) since the constraints that must be satisfied bythe data on the slice mean that the physical observables of the theorywill be independent of whatever foliation one chooses. However, theproblem is that if spacetime is quantized along these lines, theassumption (of evolving then quantizing) does not make sense inanything but an approximate way. That is, the evolution does notgenerate a classical spacetime! Rather, solutions will bewave-functions (solutions of some Schrödinger-type equation).This issue in particular is decidedly neglected in both the physicaland philosophical literature (but see Isham, 1993), and there is morethat might be said. We return to the issue of time in quantum gravitybelow.

3.2.1 Geometric variables

Early attempts at quantizing general relativity by Bergmann, Dirac,Peres, Wheeler, DeWitt and others in the 1950s and 1960s worked with aseemingly natural choice for configuration variables, namely geometricvariables \(g_{ij}\) corresponding to the various components of the‘three-metric’ describing the intrinsic geometry of thegiven spatial slice of spacetime. One can think about arriving at thisvia an arbitrary slicing of a 4-dimensional “block”universe by 3-dimensional spacelike hypersurfaces. The conjugatemomenta \(\pi_{ij}\) then effectively encode the time rate-of-changeof the metric, which, from the 4-dimensional perspective, is directlyrelated to the extrinsic curvature of the slice (meaning the curvaturerelative to the spacetime in which the slice is embedded). Thisapproach is known as ‘geometrodynamics’ since it viewsgeneral relativity as describing the dynamics of spatial geometry.

As mentioned above, in these geometric variables, as in any othercanonical formulation of general relativity, one is faced withconstraints, which encode the fact that the canonical variables cannotbe specified independently. A familiar example of a constraint isGauss’s law from ordinary electromagnetism, which states that,in the absence of charges, \(\nabla \cdot\boldsymbol{E}(\boldsymbol{x}) - 4\pi \varrho = 0\) at every point\(\boldsymbol{x}\). It means that the three components of the electricfield at every point must be chosen so as to satisfy this constraint,which in turn means that there are only two “true” degreesof freedom possessed by the electric field at any given point inspace. (Specifying two components of the electric field at every pointdictates the third component.) Thus, not all components of the Maxwellequations propagate the fields in a physical sense.

The constraints in electromagnetism may be viewed as stemming from the\(U(1)\) gauge invariance of Maxwell’s theory, while theconstraints of general relativity stem from the diffeomorphisminvariance of the theory.Diffeomorphism invariance means, informally, that one can take a solution of Einstein’sequations and drag it (meaning the metric and the matter fields)around on the spacetime manifold and obtain a mathematically distinctbut physically equivalent solution. The three‘supermomentum’ constraints in the canonical theoryreflect the freedom to drag the metric and matter fields around invarious directions on a given three-dimensional spacelikehypersurface, while the ‘super-Hamiltonian’ constraintreflects the freedom to drag the fields in the “time”direction, and so to the “next” hypersurface. (Eachconstraint applies at each point of the given spacelike hypersurface,so that there are actually \(4 \times \infty^3\) constraints: four foreach point.) In the classical (unquantized) canonical formulation ofgeneral relativity, the constraints do not pose any particularconceptual problems (though one does face a problem in definingsuitable observables that commute with the constraints, and thiscertainly has a conceptual flavour). One effectively chooses abackground space and time (via a choice of the lapse and shiftfunctions) “on the fly”, and one can be confident that thespacetime that results is independent of the particular choice.Effectively, different choices of these functions give rise todifferent choices of background against which to evolve theforeground. However, the constraints pose a serious problem (as muchconceptual as technical) when one moves to quantum theory.

3.2.2 Problem of time

All approaches to canonical quantum gravity face the so-called“problem of time” in one form or another (Kuchař(1992) and Isham (1993) are still excellent reviews; Rickles, 2006,offers a more philosophical guide). The problem stems from the factthat in preserving the diffeomorphism-invariance of general relativity— depriving the coordinates of the background manifold of anyphysical meaning — the “slices” of spacetime one isconsidering inevitably include time, just as they include space. Inthe canonical formulation, the diffeomorphism invariance is reflectedin the constraints, and the inclusion of what would ordinarily be a‘time’ variable in the data is reflected in the existenceof the super-Hamiltonian constraint. The difficulties presented bythis latter constraint constitute the problem of time.

Attempts to quantize general relativity in the canonical frameworkproceed by turning the canonical variables into operators on anappropriate state space (e.g., the space of square-integrablefunctions over three-metrics), and dealing somehow with theconstraints. When quantizing a theory with constraints, there are twopossible approaches. The approach usually adopted in gauge theories isto deal with the constraintsbefore quantization, so thatonly true degrees of freedom are promoted to operators when passing tothe quantum theory. There are a variety of ways of doing thisso-called ‘gauge fixing’, but they all involve removingthe extra degrees of freedom by imposing some special conditions. Ingeneral relativity, fixing a gauge is tantamount to specifying aparticular coordinate system with respect to which the“physical” data is described (spatial coordinates) andwith respect to which it evolves (time coordinate). This is difficultalready at the classical level, since the utility and, moreover, thevery tractability of any particular gauge generally depends on theproperties of the solution to the equations, which of course is whatone is trying to find in the first place. But in the quantum theory,one is faced with the additional concern that the resulting theory maywell not be independent of the choice of gauge. This is closelyrelated to the problem of identifying true, gauge-invariantobservables in the classical theory (Torre 2005, in the Other InternetResources section).

The preferred approach in canonical quantum gravity is to impose theconstraints after quantizing. In this ‘constraintquantization’ approach, due to Dirac, one treats the constraintsthemselves as operators \(A\), and demands that “physical”states \(\psi\) be those which are solutions to the resultingequations \(A \psi = 0\). The problem of time is associated with thesuper-Hamiltonian constraint, as mentioned above. Thesuper-Hamiltonian \(H\) is responsible for describing time-evolutionin the classical theory, yet its counterpart in theconstraint-quantized theory, \(H \psi = 0\), wouldprimafacie seem to indicate that the true physical states of thesystem do not evolve at all: there is no \(t\). Trying to understandhow, and in what sense, the quantum theory describes thetime-evolution of something, be it states or observables, is theessence of the problem of time (on which, more below).

3.2.3 Ashtekar, loop, and other variables

In geometrodynamics, all of the constraint equations are difficult tosolve (though the super-Hamiltonian constraint, known as theWheeler-DeWitt equation, is especially difficult), even in the absenceof particular boundary conditions. Lacking solutions, one does nothave a grip on what the true, physical states of the theory are, andone cannot hope to make much progress in the way of predictions. Thedifficulties associated with geometric variables are addressed by theprogram initiated by Ashtekar and developed by his collaborators (fora review and further references see Rovelli 2001b (Other InternetResources), 2001a). Ashtekar used a different set of variables, acomplexified ‘connection’ (rather than a three-metric) andits canonical conjugate, which made it simpler to solve theconstraints. This change of variables introduces an additionalconstraint into the theory (the Gauss law constraint generating SO(3)transformations) on account of the freedom to rotate the vectorswithout disturbing the metric. The program underwent furtherrefinements with the introduction of the loop transform, and furtherrefinements still when it was understood that equivalence classes ofloops could be identified with spin networks. One is able to recoverall the standard geometrical features of general relativity from thisformulation. (See Smolin (2001, 2004) for a popular introduction;Rovelli, 2004, offers a physically intuitive account; Thiemann, 2008,provides the mathematical underpinnings; Rickles, 2005, offers aphilosophically-oriented review.) Note that the problems of time andobservables afflict the loop approach just as they did the earliergeometrodynamical approach. The difference is that one has more(mathematical) control over the theory (and its quantization), interms of a definable inner product, a separable state space, and more.There is still a question mark over the construction of the fullphysical Hilbert space, since the solution of the Hamiltonianconstraint remains a problem. However, some progress is being made invarious directions, e.g. Thomas Thiemann’s master constraintprogramme (see Thiemann, 2006).

3.3 Other Approaches

Though the impression often painted of the research landscape inquantum gravity is an either/or situation between string theory andloop quantum gravity, in reality there are very many more options onthe table. Some (e.g., Callender and Huggett 2001, Wüthrich 2004(Other Internet Resources section); J. Mattingly 2005) have arguedthat semiclassical gravity, a theory in which matter is quantized butspacetime is classical, is at least coherent, though not quite anempirically viable option (we discuss this below). Other approachesinclude twistor theory (currently enjoying a revival in conjunctionwith string theory), Bohmian approaches (Goldstein & Teufel,2001), causal sets (see Sorkin 2003, in the Other Internet Resourcessection) in which the universe is described as a set of discreteevents along with a stipulation of their causal relations, and otherdiscrete approaches (see Loll 1998). Causal set theory has begun tostimulate some philosophical interest on account of the claims, byphysicists, to the effect that the theory embodies a notion ofobjective becoming or temporal passage based on the notion of the‘birth’ of spacetime atoms (see, e.g., Dowker 2014; for askeptical response, see Huggett 2014; Wüthrich, 2012, pursuesinstead the structuralist leanings of causal set theory).

Also of interest are arguments to the effect that gravity itself mayplay a role in quantum state reduction (Christian, 2001; Penrose,2001; also briefly discussed below). A fairly comprehensive overviewof the current approaches to quantum gravity can be found in Oriti(2009). In this entry we have chosen to focus upon those approachesthat are both the most actively pursuedand that havereceived most attention from philosophers. Let us now turn to severalmethodological and philosophical issues that arise quantum gravityresearch.

4. Methodology

Research in quantum gravity has always had a rather peculiar flavor,owing to both the technical and conceptual difficulty of the field andthe remoteness from experiment. Yoichiro Nambu (1985) wryly labelsresearch on quantum gravity “postmodern physics” onaccount of its experimental remoteness. Thus conventional notions ofthe close relationship between theory and experiment have but atenuous foothold, at best, in quantum gravity. However, since thereis a rudimentary ‘pecking order’ amongst thevarious approaches to quantum gravity, and since the history ofquantum gravity contains various fatalities, there clearly aresome methods of theory evaluation in operation, there areconstraints functioning in something like the way experiment andobservation function. Investigating these methods and constraintsconstitutes an open research problem for philosophers ofscience—for initial investigations along these lines, see JamesMattingly (2005a and 2009) and Rickles (2011). Audretsch (1981) arguesthat quantum gravity research conflicts with Kuhn’s account ofscientific development since it stems from the desire to unify (forreasons not based on any empirical tension) multiple paradigms, bothof which are well-confirmed and both of which make claims touniversality. One might easily question Audretsch’s focus ondirect empirical tensions here. Given, as he admits, both generalrelativity and quantum theory claim to beuniversal theories,any conceptual or formal tension that can be found to hold betweenthem must point to either or both theories being in error in theirclaims to universality—this is an empirical claim of sorts. Inthe context of string theory, Peter Galison (1995) argues thatmathematical constraints take the place of standard empiricalconstraints. James Cushing (1990) also considers some of the potentialmethodological implications of string theory (though he deals withstring theory in its earliest days, when it underwent a transitionfrom the dual resonance model of hadrons into a theory of quantumgravity). Dawid (2014) focuses in more detail on methodological issuesin string theory and defends the idea that string theory ischaracterised by a uniqueness claim (the no-alternatives argument)according to which string theory is theonly way to unifygravity and the other fundamental interactions, thus groundingphysicists’ strong belief in the theory; however, that is arather different problem (that of constructing a theory of everything)than the more restricted problem of quantum gravity — quantumgravity researchers from other approaches might simply reject the needfor such a unified theory (e.g., as opposed to a theory that iscompatible with the inclusion of other interactions).

4.1 Theory

As remarked in the introduction, there is no single, generallyagreed-upon body of theory in quantum gravity. The majority of thephysicists working in the field focus their attention on stringtheory, an ambitious program which aims at providing a unified theoryof all four interactions. A non-negligible minority work on what isnow called loop quantum gravity, the goal of which is simply toprovide a quantum theory of the gravitational interactionsimpliciter. There is also significant work in other areas,including approaches that don’t really involve the quantizationof a theory at all. [Good recent reviews of the theoretical landscapeinclude Carlip 2001, Smolin 2001 (Other Internet Resources sectionbelow), 2003, Penrose 2004, and Oriti, ed, 2009.] But there is no realconsensus, for at least two reasons.

The first reason is that it is extremely difficult to make anyconcrete predictions in these theories. String theory, in particular,is plagued by a lack of experimentally testable predictions because ofthe tremendous number of distinct ground or vacuum states in thetheory, with an absence of guiding principles for singling out thephysically significant ones (including our own). Though the stringcommunity prides itself on the dearth of free parameters in the theory(in contrast to the nineteen or so free parameters found in thestandard model of particle physics), the problem arguably resurfacesin the huge number of vacua associated with differentcompactifications of the nine space dimensions to the three weobserve. These vacua are either viewed as distinct string theories, orelse as solutions of one and the same theory (though some deeper,unknown theory, as mentioned above). Attempts to explain why we livein the particular vacuum that we do have recently given rise toappeals to the infamous anthropic principle (Susskind, 2003), wherebythe existence of humans (or observers) is invoked to, in some sense,“explain” the fact that we find ourselves in a particularworld by restricting the possible ground states to those that couldsupport such creatures in which we should expect our universe’sobserved features to be typical. (See Weinstein, 2006, for aphilosophical discussion of the usage of anthropic reasoning in stringtheory, including an ambiguity in the meaning of‘typicality’ in this context; Azhar, 2013, furtherdevelops this discussion.)

Loop quantum gravity is seemingly less plagued by a lack ofpredictions, and indeed it is often claimed that the discreteness ofarea and volume operators are concrete predictions of the theory, withpotentially testable consequences. Proponents of this approach arguethat this makes the theory more susceptible to falsification, and thusmore scientific (in the sense of Popper; see the entry onKarl Popper) than string theory (see Smolin 2006 for this line of argument).However, it is still quite unclear, in practice and even in principle,how one might actually observe these quantities. There have beenrecent suggestions that in order to probe the effects of the Planckscale (discreteness, or minimal length in particular) one needs tolook to the cosmological level for tiny violations of Lorentzinvariance. Rovelli and Speziale (2003) have argued that, in fact, theexistence of a minimal length does not imply a violation of theLorentz symmetry (a conclusion seconded by the proponents of thecausal set programme). Their argument turns on the fact that in thecontext of quantum theory, symmetries act on states (and so on meanvalues) rather than eigenvalues (representing the discrete quantitiesin the theory). However, in any case, there remains a question markover the theoretical status of the discreteness result which has beenshown to hold only for operators on thekinematical Hilbertspace, that is, for gauge-variant quantities. It is still an openquestion whether this result transfers to genuine observables (i.e.operators that satisfy all of the constraints and are defined on thephysical Hilbert space: that gauge-invariant quantities). SeeDittrich and Thiemann (2009) for a detailed investigation of theproblem and a possible resolution employing suitably gauge-fixed (bymatter) Dirac observables. Even if one overcomes this problem, andcould observe evidence of the discreteness of space, so manyapproaches involve such discreteness that one would face a furtherproblem in using this new data to decide between the discreteapproaches. For a philosophical discussion of this and related issues(including the question of whether the proposed discreteness breaksLorentz invariance), see Hagar (2009) — Hagar (2014) considersthese and related issues in a book-length treatment.

4.2 Experiment

The second reason for the absence of consensus is that there are noexperiments in quantum gravity, and little in the way of observationsthat might qualify as direct or indirect data or empirical evidence.This stems in part from the lack of theoreticalpredictions,since it is difficult to design an observational test of a theory ifone does not know where to look or what to look at. But it also stemsfrom the fact that most theories of quantum gravity appear to predictdepartures from classical relativity only at energy scales on theorder of \(10^{19}\) GeV. (By way of comparison, the proton-protoncollisions at Fermilab have an energy on the order of \(10^3\) GeV.)Whereas research in particle physics proceeds in large part byexamining the data collected in large particle accelerators, which areable to smash particles together at sufficiently high energies toprobe the properties of atomic nuclei in the fallout, gravity is soweak that there is no physically realistic way to do a comparableexperiment that would reveal properties at the energy scales at whichquantum gravitational effects are expected to beimportant—it would take a particle accelerator of galactic sizeto even approach the required energies. (In a little more detail, theweakness of gravity can be compared to the strength of theelectromagnetic interaction — cf. Callender and Huggett (eds.)2001, p. 4. An electron couples to the electromagnetic field with astrength of \(10^{-2}\), while the coupling of a mass to thegravitational field is \(10^{-22}\). Feynman (1963, p. 697) gives anexample that highlights this difference in magnitudes moredramatically by showing how the gravitational coupling between aproton and an electron in a hydrogen atom would shift thewave-function of an electron by just 43 arcseconds over a time periodof 100 times the age of the Universe! Hence, quantum gravity is moreof a theorist’s problem.)

Though progress is being made in trying to at least draw observationalconsequences of loop quantum gravity, a theory of quantum gravitywhich arguablydoes make predictions (Amelino-Camelia, 2003,in the Other Internet Resources section below; D. Mattingly, 2005), itis remarkable that the most notable “test” of quantumtheories of gravity imposed by the community to date involves aphenomenon which has never been observed, the so-called Hawkingradiation from black holes. Based on earlier work of Bekenstein (1973)and others, Hawking (1974) predicted that black holes would radiateenergy, and would do so in proportion to their gravitational“temperature,” which was in turn understood to beproportional to their mass, angular momentum, and charge. Associatedwith this temperature is an entropy (see the entry onthe philosophy of statistical mechanics), and one would expect a theory of quantum gravity to allow one tocalculate the entropy associated with a black hole of given mass,angular momentum, and charge, the entropy corresponding to the numberof quantum (micro-)states of the gravitational field having the samemass, charge, and angular momentum. (See Unruh, 2001, and referencestherein.) In their own ways, string theory and loop quantum gravityhave both passed the test of predicting an entropy for black holeswhich accords with Hawking’s calculation, using very differentmicroscopic degrees of freedom. String theory gets the number rightfor a not-particularly-physically-realistic subset of black holescalled near-extremal black holes, while loop quantum gravity gets itright for generic black holes, but only up to an overall constant.More recently, the causal set approach has also managed to derive thecorrect value.If the Hawking effect is real, then thisconsonance could be counted as evidence in favor of either or both/alltheories.

Erik Curiel (2001) has argued against the manner in which the abilityto derive the Bekenstein-Hawking result as a theorem of an approach isused asevidence for that approach in much the same way thatempirical evidence is used to justify a theory in normalcircumstances, say predicting the value of a well-confirmedexperimental result. It is true that black hole physics is used astesting ground for quantum gravity and the Bekenstein-Hawking resultdoes not have the status of an empirical fact. However, it is a strongdeduction from a framework thatis fairly mature, namelyquantum field theory on a curved spacetime background. In this sense,although it does not provide a constraint as strong as anexperimentally observed phenomenon, it might legitimately function asa constraint on possible theories. Constraints on theory constructioncome in a variety of shapes and sizes, and not all take the form ofempirical data — thought experiments are a case in point. In thecontext of quantum gravity it is especially important that one havesome agreed upon constraints to guide the construction. Without them,work would halt. It also seems reasonable to insist that a full theoryof quantum gravity be able to reproduce predictions of thesemi-classical theory of gravity, since this will be one of itspossible limits. Still, Curiel is right that researchers ought to berather more wary of attributing too much evidential weight to suchfeatures that remain empirically unconfirmed.

Curiel goes on to question, more generally, the ranking of approachesto quantum gravity given what he views as the absence of demonstratedscientific merit in any of them: elegance and consistency might wellbe merits of a scientific theory, but they do not count asscientific. (ibid, p. S437). However, this claim hinges onthe direct alignment of scientific merit and empirical clout; but thisrequires an argument, for it is far from obvious: from whence thisprescription? Surely if a theory is mathematically inconsistent thatsays something about itsphysical status too? Moreover, therelationship between experimental and observational data and theoriesis not a simple matter. Finally, it is perhaps too quick to say thatapproaches do not have empirical consequences. Already known empiricaldata can confirm the predictions of a theory; therefore, it is clearthat we can judge the extent to which the various contenders satisfythis old evidence, and how they do so. For example, string theory atleast has the potential of explaining why there are three generationsof elementary particles by invoking the Euler characteristic of thecompact spaces it employs—the Euler characteristic is equal totwice the number of generations (see Seifert, 2004, for details).Whatever one might think about string theory’s relationship withanthropic reasoning, we do have here a potential explanation of apreviously inexplicable piece of old empirical data, which ought tolend some credence to the theory. There is also the not inconsiderablefact that string theory is able to derive general relativity (and allthe physically observed facts that are associated with this theory) asa low energy feature. This is not anovel fact, but it is anphysical, empirical consequence of the theory nonetheless.

However, it should be noted, finally, that to date neither of the mainresearch programs has been shown to properly reproduce the world wesee at low energies. Indeed, it is a major challenge of loop quantumgravity to show that it indeed has general relativity as a low-energylimit, and a major challenge of string theory to show that it has thestandard model of particle physics plus general relativity as alow-energy limit. There are promising indications that both theoriesmight be able to overcome this challenge (see Thiemann for the loopquantum gravity case; for the string theoretic case, see Graña,2006). A similar problem faces causal set theory in the form of the‘inverse problem’, which roughly amounts to the difficultyof getting continuous manifolds (with their corresponding symmetries)from a fundamentally discrete theory (see Wallden, 2010, for a goodrecent review of causal sets, including a discussion of this problem,on which progress has also been made).

5. Philosophical Issues

Quantum gravity raises a number of difficult philosophical questions.To date, it is the ontological aspects of quantum gravity that haveattracted the most interest from philosophers, and it is these we willdiscuss in the first five sections below. In the final section,though, we will briefly discuss some further methodological andepistemological issues which arise.

First, however, let us discuss the extent to which ontologicalquestions are tied to a particular theoretical framework. In itscurrent stage of development, string theory unfortunately provideslittle indication of the more fundamental nature of space, time, andmatter. Despite the consideration of ever more exotic objects —strings, \(p\)-branes, D-branes, etc. — these objects are stillunderstood as propagating in a background spacetime. Since stringtheory is supposed to describe the emergence of classical spacetimefrom some underlying quantum structure, these objects are not to beregarded as truly fundamental. Rather, their status in string theoryis analogous to the status of particles in quantum field theory(Witten, 2001), which is to say that they are relevant descriptions ofthe fundamental physics only in situations in which there is abackground spacetime with appropriate symmetries. While this suggeststantalising links to issues of emergence, it is difficult to pursuethem without knowing the details of the more fundamental theory. Asalready mentioned, the duality relations between the various stringtheories suggest that they are all perturbative expansions of somemore fundamental, non-perturbative theory known as‘M-theory’ (Polchinski, 2002, see the Other InternetResources section below). This, presumably, is the most fundamentallevel, and understanding the theoretical framework at that level iscentral to understanding the underlying ontology of the theory (and sothe manner in which any other structures might emerge from it).‘Matrix theory’ is an attempt to do just this, to providea mathematical formulation of M-theory, but it remains highlyspeculative. Thus although string theory purports to be a fundamentaltheory, the ontological implications of the theory are still veryobscure — though this could be viewed as a challenge rather thana reason to ignore the theory.

Canonical quantum gravity, in its loop formulation or otherwise, hasto date been of greater interest to philosophers because it appears toconfront fundamental questions in a way that string theory, at leastin its perturbative guise, does not — certainly, it does so moreexplicitly and in language more amenable to philosophers. Whereasperturbative string theory treats spacetime in an essentiallyclassical way, canonical quantum gravity treats it asquantum-mechanical entity, at least to the extent of treating thegeometric structure (as opposed to, say, the topological ordifferential structure) as quantum-mechanical. Furthermore, many ofthe issues facing canonical quantum gravity are also firmly rooted inconceptual difficulties facing the classical theory, whichphilosophers are already well acquainted with (e.g. via thehole argument).

5.1 Time

As noted inSection 3.2.2 above, the treatment of time presents special difficulties incanonical quantum gravity, though they easily generalise to many otherapproaches to quantum gravity. These difficulties are connected withthe special role time plays in physics, and in quantum theory inparticular. Physical laws are, in general, laws of motion, of changefrom one time to another. They represent change in the form ofdifferential equations for the evolution of, as the case may be,classical or quantum states; the state represents the way the systemis at sometime, and the laws allow one to predict how itwill be in the future (or retrodict how it was in the past). It is notsurprising, then, that a theory of quantum spacetime would have aproblem of time, because there is no classical time against which toevolve the “state”. The problem is not so much that thespacetime is dynamical; there is no problem of time in classicalgeneral relativity (in the sense that a time variable is present).Rather, the problem is roughly that in quantizing the structure ofspacetime itself, the notion of a quantum state, representing thestructure of spacetime at some instant, and the notion of theevolution of the state, do not get any traction, since thereare no real “instants”. (In some approaches to canonicalgravity, one fixes a timebefore quantizing, and quantizesthe spatial portions of the metric only. This approach is not withoutits problems, however; see Isham (1993) for discussion and furtherreferences.)

One can ask whether the problem of time arising from the canonicalprogram tells us something deep and important about the nature oftime. Julian Barbour (2001a,b), for one, thinks that it tells us thattime is illusory (see also Earman, 2002, in this connection). It isargued that the fact that quantum states do not evolve under thesuper-Hamiltonian means that there is no change. However, it can alsobe argued (Weinstein, 1999a,b) that the super-Hamiltonian itselfshould not be expected to generate time-evolution; rather, one or more“true” Hamiltonians should play this role, thoughuncovering such Hamiltonians is no easy matter. (See Butterfield &Isham (1999) and Rovelli (2006) for further discussion.)

Bradley Monton (2006) has argued that a specific version of canonicalquantum gravity – that with a so-calledconstant meanextrinsic curvature [CMC] (or fixed) foliation – has thenecessary resources to render presentism (the view that all and onlypresently existing things exist) a live possibility (see the sectionon Presentism, Eternalism, and The Growing Universe Theory in theentry ontime for more on presentism). The reason is that with such a fixedfoliation one has at one’s disposal some spacelike hypersurfacethat contains a set of well-defined events that can be viewed throughthe lens of presentism, such that this set of events at thisparticular instant (or ‘thin-sandwich’) changes over time.Though he readily admits that CMC formulations are outmoded in thecontemporary theoretical landscape, he nonetheless insists that giventhe lack of experimental evidence one way or the other, it stands as aviable route to quantum gravity, and therefore presentism remains as apossible theory of time that is compatible with frontier theoreticalphysics. Christian Wüthrich (2010) takes Monton to task on avariety of both technical and non-technical grounds. He rightlyquestions Monton’s claim that the CMC approach really is anapproach to quantum gravity, in the same sense as stringtheory and loop quantum gravity. It is more of a piece of machinerythat is usedwithin a pre-existing approach (namely, thecanonical approach). He also questions Monton’s claim, inasmuchas it does constitute an approach of sorts, that it isviable. Simply not being ruled out on experimental groundsdoes not thereby render an approach viable. Besides, if anything hasthe prospect of saving presentism, then surely it is JulianBarbour’s position mentioned above. This at least has the addedbenefit of being a research programme that is being activelypursued.

A common claim that appears in many discussions of the problem of time(especially amongst philosophers) is that it isrestricted tocanonical formulations of general relativity, and has something to dowith the Hamiltonian formalism (see Rickles 2008a, pp. 340–1 formore details). The confusion lies in the apparently very differentways that time is treated in general relativity as standardlyformulated, and as it appears in a canonical, Hamiltonian formulation.In the former there is no preferred temporal frame, whereas the latterappears todemand such a frame in order to get off the ground(cf. Curiel, 2009, p. 59; Tim Maudlin (2004) tells a broadly similarstory).

However, this encodes several pieces of misinformation making it hardto make sense of the claim that general relativity and canonicaltheories cannot be “reconciled”. The canonical frameworkis simply a tool for constructing theories, and one that makesquantization an easier prospect. As a matter of historical fact thecanonical formulation of general relativity is a completed project,and has been carried out in a variety of ways, using compact spacesand non-compact spaces, and with a range of canonical variables. Ofcourse, general relativity, like Maxwell’s theory ofelectromagnetism, possesses gauge symmetries, so it is a constrainedtheory that results, and one must employ the method of constrainedHamiltonian systems. However, there is no question that generalrelativity is compatible with the canonical analysis of theories, andthe fact that time looks a little strange in this context is simplybecause the formalism is attempting to capture the dynamics of generalrelativity. In any case, the peculiar nature of general relativity andquantum gravity, with respect to the treatment of time, resurfaces inarguably the most covariant of approaches, the Feynman path-integralapproach. In this case that central task is to compute the amplitudefor going from an initial state to a final state (where these stateswill be given in terms of boundary data on a pair of initial and finalhypersurfaces). The computation of this propagator proceeds àla sum-over-histories: one counts to the number of possible spacetimesthat might interpolate between the initial and final hypersurfaces.However, one cannot get around the fact that general relativity is atheory with gauge freedom, and so whenever one has diffeomorphicinitial and final hypersurfaces, the propagator will be trivial.

A similar confusion can be found in discussions of the related problemof defining observables in canonical general relativity. The claimgets its traction from the fact that it is very difficult to constructobservables in canonical general relativity, while (apparently) it isrelatively straightforward in the standard Lagrangian description.(See, e.g., Curiel, 2009, pp. 59–60, for an explicit statementof this claim. Curiel cites a theorem of Torre, 1993, to the effectthat there can be no local observables in compact spacetimes, to arguethat the canonical formulation is defective somehow.) Again, thisrests on a misunderstanding over what the canonical formalism is andhow it is related to the standard spacetime formulation of generalrelativity. That there are no local observables is not an artefact ofcanonical general relativity. The notion that observables have to benon-local (in this case, relational) is a generic feature that resultsprecisely from the full spacetime diffeomorphism invariance of generalrelativity (and is, in fact, implicit in the theorem of Torrementioned earlier). It receives a particularly transparent descriptionin the context of the canonical approach because one can defineobservables as quantities that commute with all of the constraints.The same condition will hold for the four-dimensional versions, onlythey will have to be spacetime diffeomorphism invariant in that case.This will still rule out local observables since any quantitiesdefined at points or regions of the spacetime manifold will clearlyfail to be diffeomorphism invariant. Hence, the problems ofobservables (and the result that they must be either global orrelational in general relativity) is not a special feature of thecanonical formulation, but a generic feature of theories possessingdiffeomorphism invariance. As Ashtekar and Geroch point out,“[s]ince time is essentially a geometrical concept [in generalrelativity], its definition must be in terms of the metric. But themetric is also the dynamical variable, so the flow of time becomesintertwined with the flow of the dynamics of the system” (1974,p. 1215). A recent philosophical deep dive into the problem of time,which situates the quantum gravity specific version in a widerframework, is Gryb and Thébault, 2023.

5.2 Ontology

The problem of time is closely connected with a general puzzle aboutthe ontology associated with “quantum spacetime”. Quantumtheory in general resists any straightforward ontological reading, andthis goes double for quantumgravity. In quantum mechanics,one has particles, albeit with indefinite properties. In quantum fieldtheory, one again has particles (at least in suitably symmetricspacetimes), but these are secondary to the fields, which again arethings, albeit with indefinite properties. On the face of it, the onlydifference in quantum gravity is that spacetime itself becomes a kindof quantum field, and one would perhaps be inclined to say that theproperties of spacetime become indefinite. But space and timetraditionally play important roles in individuating objects and theirproperties—in fact a field is in some sense a set of propertiesof spacetime points — and so the quantization of such raisesreal problems for ontology.

One area that philosophers might profit from is in the investigationof the relational observables that appear to be necessitated bydiffeomorphism invariance. For example, since symmetries (such as thegauge symmetries associated with the constraints) come with quite alot of metaphysical baggage attached (as philosophers of physics knowfrom thehole argument), such a move involves philosophically weighty assumptions. Forexample, the presence of symmetries in a theory would appear to allowfor more possibilities than one without, so eradicating the symmetries(by solving the constraints and going to the reduced, physical phasespace) means eradicating a chunk of possibility space: in particular,one is eradicating states that are deemed to be physically equivalent,despite having some formal differences in terms of representaton.Hence, imposing the constraints involves some serious modalassumptions. Belot and Earman (2001) have argued that since thetraditional positions on the ontology of spacetime (relationalism andsubstantivalism) involve a commitment to a certain way of countingpossibilities, the decision to eliminate symmetries can have seriousimplications for the ontology one can then adopt. Further, if someparticular method (out of retaining or eliminating symmetries) wereshown to be successful in the quest for quantizing gravity, then, theyargue, one could have good scientific reasons for favouring one ofsubstantivalism or relationalism. (See Belot, 2011a, for more on thisargument; Rickles, 2008c, explicitly argues against the idea thatpossibility spaces have any relevance for spacetime ontology.)

In the loop quantum gravity program, the area and volume operatorshavediscrete spectra. Thus, like electron spins, they canonly take certain values. This suggests (but does not imply) thatspace itself has a discrete nature, and perhaps time as well(depending on how one resolves the problem of time). This in turnsuggests that space does not have the structure of a differentialmanifold, but rather that it only approximates such a manifold onlarge scales, or at low energies. A similar idea, that classicalspacetime is anemergent entity, can be found in severalapproaches to quantum gravity (see Butterfield and Isham, 1999 and2001, for a discussion of emergence in quantum gravity). Thepossibility that a continuous structure (with continuous symmetries)could emerge from a fundamentally discrete structure is a problem witha clear philosophical flavour —Huggett and Wüthrich, eds.,2013; Huggett, Matsubara, K., and Wüthrich, 2020; andWüthrich, Le Bihan, B., and Huggett, 2021 contain a wide varietyof papers investigating this issue, with their own contributionstending to focus on the notion of recovering ‘localbeables’ from such emergent theories.

5.3 Status of quantum theory

Whether or not spacetime is discrete, the quantization of spacetimeentails that our ordinary notion of the physical world, that of matterdistributed in space and time, is at best an approximation. This inturn implies that ordinary quantum theory, in which one calculatesprobabilities for events to occur in a given world, is inadequate as afundamental theory. As suggested in theIntroduction, this may present us with a vicious circle. At the very least, onemust almost certainly generalize the framework of quantum theory. Thisis an important driving force behind much of the effort in quantumcosmology to provide a well-defined version of themany-worlds orrelative-state interpretations. Much work in this area has adopted the so-called‘decoherent histories’ or ‘consistenthistories’ formalism, whereby quantum theories are understood tomake probabilistic predictions about entire (coarse-grained)‘histories’. Almost all of this work to date construeshistories to be histories of spatiotemporal events, and thuspresupposes a background spacetime; however, the incorporation of adynamical, quantized spacetime clearly drives much of thecosmology-inspired work in this area.

More generally, one might step outside the framework of canonical,loop quantum gravity, and ask why one should only quantize the metric.As pointed out by Isham (1994, 2002), it may well be that theextension of quantum theory to general relativity requires one toquantize, in some sense, not only the metric but also the underlyingdifferential structure and topology. This is somewhat unnatural fromthe standpoint where one begins with classical, canonical generalrelativity and proceeds to “quantize” (since thetopological structure, unlike the metric structure, is not representedby a classical variable). But one might well think that one shouldstart with the more fundamental, quantum theory, and then investigateunder which circumstances one gets something that looks like aclassical spacetime.

One final issue we might mention here is whether there is a conflictbetween the superposition principle and general relativity. Curielclaims that “[t]here exists no physical phenomenon wellcharacterized by experiment that cannot be accurately described by oneof the two theories, and no physical phenomenon that suggests that oneof the two is correct to the detriment of the other’saccuracy” (2001, p. S432). However, Roger Penrose (2004, Chapter30) has forcefully argued that the superposition principle can, insome circumstances, threaten the principle of general covariance,surely a core principle of general relativity! The idea is that if weprepare a lump of matter in a superposition of two position states(stationary in their ambient spacetime), \(\chi\) and \(\phi\), astate Penrose labels a “Schrödinger’s Lump”state, then the superposition is represented by: \(|\Psi \rangle =w|\chi \rangle + z|\phi \rangle\). Penrose then shows that astationary gravitational field does nothing to affect the fact thatany superposition of the (stationary) position states \(\chi\) and\(\phi\) will also be stationary. But then introducing thegravitational field of the lump itself raises a problem. Bythemselves, the components of the superposition would not seem toraise problems, and we can simply think of the field around thelocation associated with the lump’s states individually as beingnearly classical. Given the stationarity of the states \(\chi\) and\(\phi\), there will be a distinct Killing vector (i.e. a metricpreserving vector field) associated with each them. The problem thenarises: what of superpositions of these lump states? Are theystationary? Since the Killing vector fields of the two componentstationary states live on different spacetimes, with differentstructures, it seems we don’t have the invariant spatiotemporalstructure needed to answer the question. To try and say that thespacetime is really the same (the obvious answer) would conflict withgeneral covariance since then one would be supposing a robust notionof spacetime points which enables one to match up the two spacetimes.As we have seen above, Penrose’s proposed solution is toconsider such superpositions as generating a kind of geometricinstability which causes the collapse of the superposition.

Of course, one might question various moves in Penrose’sreasoning here (especially as regards the nature of the gravitationalfields of stationary quantum states), so there is clearly more to besaid. But there is potentially a conflict (and a measurable one atthat: see Penrose, 2002) between the superposition principle andprinciples of general relativity. Those with experience of thestandard quantum measurement problem will find much to interest themin this problem.

5.4 The Planck Scale

It is almost Gospel that quantum gravity is what happens when youreach the Planck scale. The standard refrain is that ‘somethingpeculiar’ happens to our concepts of space, time, and causalityat such such scales requiring radical revisions that must be describedby the quantum theory of gravity (see, e.g., Rovelli, 2007, p. 1287).However, the arguments underlying this orthodoxy have not beenrigorously examined. The usual motivation involves a dimensionalanalysis argument. The scales at which theories make their mark areset by the values of the fundamental constants. In this way theconstants demarcate the domains of applicability of theories: \(c\)tells us when specially relativistic effects will become apparent,\(\hslash\) tells us when quantum effects will become apparent, and Gtells us when gravitational effects will become apparent. As Planckwas able to demonstrate in 1899, these constants can be combined so asto uniquely determine a natural, absolute family of units that areindependent of all human and terrestrial baggage. The Planck lengthcan be written as \((G\hslash/c^3)^{\frac{1}{2}}\) and has the value\(10^{-33}\) in centimeters. Planck was not aware of the relevance ofthe scale set by the constants to the applicability of generalrelativity, of course, but Arthur Eddington seems to have been aware(though getting a different value as a result of using OsbornReynold’s determination for the finest grain believed possible),writing in the March edition ofNature in 1918:

From the combination of the fundamental constants, \(G, c\), and \(h\)it is possible to form a new fundamental unit of length\(\mathrm{L}_{\textit{min}} = 7 \times 10^{-28}\)cm. It seems to beinevitable that this length must play some role in any completeinterpretation of gravitation. ... In recent years great progress hasbeen made in knowledge of the excessively minute; but until we canappreciate details of structure down to the quadrillionth orquintillionth of a centimetre, the most sublime of all the forces ofNature remains outside the purview of the theories of physics.(Eddington, 1918, p. 36)

The idea that the Planck length amounts to aminimal lengthin nature follows from the argument that if distances smaller thanthis length are resolved (say in the measurement of the position of amass), then it would require energies concentrated in a region sosmall that a mini-black hole would form, taking the observed systemwith it – see Rovelli (2007, p. 1289) for this argument.Meschini (2007) is not convinced by such arguments, and doesn’tsee that the case for the relevance of the Planck scale to quantumgravity research has been properly made. He is suspicious of theclaims made on behalf of dimensional analysis. There is something toMeschini’s claims, for if the dimensional argument were truethen, without realising it, Planck would have stumbled upon thebeginnings of quantum gravity before either quantum field theory orgeneral relativity were devised! However, Meschini speculates that thefinal theory of quantum gravity “has nothing to do with one ormore of the above-mentioned constants” (p. 278). This seems toostrong a statement, since a core condition on a theory of quantumgravity will be to reduce to general relativity and quantum fieldtheory as we know it, according to limits involving these constants.Nonetheless, Meschini is surely right that the details of thesedimensional arguments, and the role of the Planck scale are callingout for a closer analysis.

5.5 Background Structure

In non-generally relativistic theories the spacetime metric is frozento a single value assignment for all times and all solutions: it ismodel independent. Of course, in general relativity the metric is whatone solves for: the metric is a dynamical variable, which implies thatthe geometry of spacetime is dynamical. This intuitive notion isbundled into the concept of background freedom, or backgroundindependence. In general, background independence is understood to bethe freedom of a theory from background structures, where the latteramount to some kind of absolute, non-dynamical objects in a theory.The extent to which their respective theories incorporate backgroundstructures has recently proven to be a divisive subject amongst stringtheorists and loop quantum gravity theorists and others. It is oftenclaimed that the central principle that distinguishes generalrelativity from other theories is its (manifest) backgroundindependence. But background independence is a slippery notion meaningdifferent things to different people. We face a series of questionswhen considering background independence: What, exactly, is it (beyondthe simple intuitive notion)? Why is it considered to be such animportant principle? What theories incorporate it? To whatextent do they incorporate it?

The debate between strings and loops on this matter is severelyhampered by the fact that there is no firm definition of backgroundindependence on the table and, therefore, the two camps are almostcertainly talking past each other when discussing this issue. It seemsprima facie reasonable to think that in order to reproduce amanifestly background independent theory like general relativity, aquantum theory of gravity should be background independent too, and sobackground independence has begun to function as a constraint onquantum gravity theories, in much the same way that renormalizabilityused to constrain the construction of quantum field theories.Advocates of loop quantum gravity often highlight the backgroundindependence of their theory as a virtue that it has over stringtheory. However, there is no proof of this implication, and aspects ofthe so-called ‘holographic principle’ seem to suggest thata background independent theory could be dual to a backgrounddependent theory (see the contributions to Biquard, ed., 2005).Furthermore, depending on how we define the intuitive notion ofbackground independence, and if ‘clues’ from the dualitysymmetries of M-theory are anything to go by, it looks like stringtheory might even bemore background independent than loopquantum gravity, for the dimensionality of spacetime becomes adynamical variable too (cf. Stelle, 2000, p. 7).

Indeed, various string theorists claim that their theory is backgroundindependent. In many cases it seems that they have a differentunderstanding of what this entails than loop quantum gravityresearchers—this takes us to the first, definitional, question.In particular some seem to think that the ability to place a generalmetric in the Lagrangian amounts to background independence. Thisfalls short of the mark for how the majority of physicists understandit, namely as a reactive dynamical coupling between spacetime andmatter. Though one can indeed place a variety of metrics in thestringy Lagrangian, one does not then vary the metric in the action.There is no interaction between the strings and the ambient spacetime.Indeed, this is not really distinct from quantum field theory of pointparticles in curved spacetimes: the same freedom to insert a generalmetric appears there too.

There is an alternative argument for the background independence ofstring theory that comes from the field theoretic formulation of thetheory: string field theory. The idea is that classical spacetimeemerges from the two dimensional conformal field theory on the stringsworldsheet. However, in this case one surely has to say somethingabout the target space, for the worldsheet metric takes on a metricinduced from the ambient target spacetime. Yet another argument forthe background independence of string theory might point to the factthat the dimensionality of spacetime in string theory has to satisfyan equation of motion (a consistency condition): this is how thedimensionality comes out (as 26 or 10, depending on whether oneimposes supersymmetry). One contender for the definition of backgroundindependence is a structure that is dynamical in the sense that onehas to solve equations of motion to get at its values. In this case wewould have extreme background independence stretching to the structureof the manifold itself. However, the problem with this is that thisstructure is the same in all models of the theory; yet we intuitivelyexpect background independent theories to be about structures that canvary across a theory’s models.

The issues here are clearly subtle and complex, and philosophers haveonly just begun to consider them. The central problem faced, as aphilosopher, when trying to make sense of claims such as these is thatthere is no solid, unproblematic definition of background structure(and therefore background independence and dependence) on the table.Without this, one simply cannot decide who is right; one cannot decidewhich theories are background independent and which are not. Hence, anurgent issue in both physics and the philosophy of physics is to workout exactly what is meant by ‘background independence’ ina way that satisfies all parties, that is formally correct, and thatsatisfies our intuitive notions of the concept. Until this isachieved, background independence cannot be helpfully used todistinguish the approaches, nor can we profitably discuss its merits.A serious attempt to define background independence in such a way asto make these tasks possible has been made by Domenico Giulini (2007).But Giulini admits that a general definition still eludes us. Thestumbling block might be that background independence simplyisn’t a formal property of theories at all. Gordon Belot (2011b)has recently argued that background independence is partly aninterpretive matter, and that one can have varying levels ofbackground independence (the latter notion is also defended by LeeSmolin, 2006). Rickles (2008b) argues that the place to seek a notionof background independence that can be put to use in the quantumgravitational context is by focusing on the kinds ofobservables that an approach employs, rather than squarely onproperties of the equations of motion. A recent, highly systematicapproach to the problem of background independence, circling throughthe various definitions of background structure and comparing them tothe various spacetime theories, is Read, 2023.

5.6 Necessity of Quantization

In earlier research on quantum gravity it was often supposed that ifthere was at least one quantum field in the world together with thegravitational field, then given the universal coupling of thegravitational field, it must follow that the quantization of the onefield somehow infects the gravitational field, implying that it mustnecessarily have quantum properties too. The arguments basicallyinvolve the consideration of a mass prepared in a superposition ofposition eigenstates. If the gravitational field remained classical(and, therefore, not constrained by the uncertainty relations) thenone could violate the uncertainty relations by simply makingmeasurements of the gravitational field, discovering the properties ofthe quantized matter to which it was coupled. However, all attempts atmaking this argument stick have so far failed, meaning that there isno logical necessity demanding that we quantize the gravitationalfield. Given that we also seemingly lack experimental reasons forquantization of the gravitational field (since we have not observedevidence of its quantum properties), several physicists (andphilosophers) have questioned the programme as it stands. It is, theyargue, a matter for experiment to decide, not logic. Note, however,that this does not mean that the project of quantum gravity itselfrests on unsteady ground: if there are quantum fields andgravitational fields in the world, then given the nature of gravity,we need to saysomething about the manner in which theyinteract. What is being questioned is whether this means that gravitycannot itself remain fundamentally classical while interacting withquantum fields. After all, as far as all our experiments show: gravityis classical and matter is quantum. This pessimistic argument isusually traced back to Rosenfeld, though he wavered somewhat on thematter (see DeWitt and Rickles, 2011, p. 164 and p. 170, forRosenfeld’s original arguments).

If it is to remain fundamentally classical, then there is the simplequestion of what such a classical gravitational field would couple to.The quantum properties? That seems problematic for the reasons givenabove. Moreover, given the form of the Einstein field equations, witha classical c-number on the left hand side, that would mean equating ac-number with a q-number (i.e. a quantum operator). The standard wayout of this problem is to couple the gravitational field instead tothe expectation value of the stress-energy tensor of some quantizedmatter field. The expectation value is a c-number. There have been avariety of arguments and no-go theorems against this so-calledsemi-classical gravitational theory, most of which replay the kind ofargument invoking violations of the uncertainty relations sketchedabove (see Eppley and Hannah 1977, and Page and Geilker 1981).Basically, the upshot of the Eppley and Hannah paper is that, giventhe coexistence of classical gravity and quantum fields, two thingscan happen upon a gravitational field measurement: on the one hand thequantum wavefunction could collapse, in which case there is momentumnon-conservation. On the other hand, the measurement could leave thequantum wavefunction in a coherent state, in which case signals can besent faster than light. Mattingly (2006) argues that, when properlyanalyzed, the thought experiments employed by Eppley and Hannahviolate basic physical principles involving the construction of theequipment that would be needed to make the necessary fieldmeasurements — however, while not viewing the originalsemi-classical approach as a viable option, Mattingly argues that anextension of the approach has the potential to reveal a viable theoryof micro-gravity (see Mattingly 2010 and 2014). Adrian Kent hasrecently argued that general hybrid classical/quantum theories(including those involving gravity) need not allow superluminalsignalling or violate relativity (Kent 2018).

A batch of new approaches based on analogies with condensed matterphysics and hydrodynamics point to another way in which gravity canescape quantization, though not in a truly fundamental sense.According to such approaches, gravity is emergent in the sense thatthe metric (or connection) variables, and other variables representinggravitational features, are collective variables that only appear atenergies away from the Planck scale. In other words, gravity is apurely macroscopic, low energy phenomenon and general relativity is aneffective field theory. This leaves the task of actually filling inthe details of the microscopic structure of spacetime (the‘atoms of spacetime’) out of which the low energycollective variables emerge (see Hu, 2009, for a conceptually orientedreview of such approaches; Crowther 2014 provides a detailedphilosophical analysis). As Rovelli notes (2007, p. 1304), the merefact that the gravitational field is an emergent, collective variabledoes not thereby imply an absence of quantum effects, and it ispossible that collective variables too are governed by quantumtheory.

Wüthrich (2005, pp. 779–80) has argued that the veryexistence of approaches to quantum gravity that do not involve thequantization of the gravitational field means that quantization of thegravitational field has to be acontingent matter. However,this seems to rest on a mistake. It might still be the case that thereare reasons of logical consistency forbidding the union of a classicaland quantum field even though there are entirely distinctnon-quantization approaches. For example, string theory does notquantize the gravitational field; however, it is clearly wrong to saythat the existence of this position would be ruled out if the variousno-go theorems outlawing hybrid classical-quantum theories were true.The fact that one can isolate states corresponding to gravitons in thestring spectrum stands quite independent from issues over theinteraction of classical and quantum field. The question of thenecessity of quantization (as a result of coupling a classicalgravitational field to quantum fields) should be held separate fromthe prospect of producing a quantum theory of gravity that does notinvolve gravitational field quantization, for both input theories, fordescribing the classical and quantum fields, could be fundamentallywrong at high energies, requiring entirely new principles. However, astronger argument against the impossibility hybrids is provided byJames Mattingly, who points out that since there are satisfiableaxioms for semiclassical theories, inconsistency cannot be establishedin general (2009, p. 381).

A recent proposal (Marletto and Vedral, 2017) has emerged showing howputting physical theories on entirely different information-theoreticfoundations (in this case constructor theory) might radicallytransform how we conceptualise the relationships between differenttheories, including hybrid classical and quantum systems. Theimplication in this case would be that the debate in question wouldhave to be re-considered in a more general light, independently ofspecific dynamical models.

6. Conclusion

Research on quantum gravity is beset by a combination of formal,experimental, and conceptual difficulties. It is inevitable that thequest for a quantum theory of gravity will continue – whetherfor reasons of necessity or not – and it seems that theresolution of the problem will require an equivalent combination offormal, experimental, and conceptual expertise. Given this, and giventhe central position quantum gravity research occupies in theoreticalphysics, it makes good sense for philosophers of physics (and generalphilosophers of science) to do their best to acquaint themselves withthe central details of the problem of quantum gravity and the mainapproaches that are seeking to crack the problem. Beyond this, quantumgravity research has the potential to invigorate several standardareas of philosophical inquiry, including our standard notions oftheory construction, selection and justification; the nature of space,time, matter, and causality, and it also introduces a new case studyin emergence, with entirely novel features.

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Unpublished, Online Manuscripts

• Amelino-Camelia, G., 2003, “Quantum gravity phenomenology”.
• Horowitz, G., 2000, “Quantum gravity at the turn of the millennium”.
• Polchinski, J., 2002, “M-theory: uncertainty and unification”.
• Rovelli, C., 2001b, “Notes for a brief history of quantum gravity”, (Centre De Physique Théorique, University of Marseilles).
• Greene, B., 1997, “String Theory on Calabi-Yau Manifolds”
• Smolin, L., 2003, “How far are we from the quantum theory of gravity?”.
• Susskind, L., 2003, “The anthropic landscape of string theory”.
• Sorkin, R., 2003, “Causal sets: discrete gravity”, (Notes for the Valdivia Summer School).
• Torre, C., 2005, “Observables for the polarized Gowdy model”.
• Wüthrich, C., 2004, “To quantize or not to quantize: fact and folklore in quantum gravity”.

Useful Web Resources

• Beyond Spacetime, (a philosophy of quantum gravity website maintained by Nick Huggettand Christian Wüthrich).
• Beyond Spacetime: Youtube Channel, (videos from conferences organised by the Beyond Spacetimeteam).
• Ascending and Descending, M.C. Escher.
• Relativity, M.C. Escher.
• The Meaning of Einstein’s Equations, by John Baez (University of California, Riverside).
• General relativity tutorial, also by John Baez (University of California, Riverside)
• Detection of gravitational radiation, 1993 Nobel prize presentation to Russell Hulse and JosephTaylor.
• Gravity In The Quantum World And The Cosmos (online videos), 33rd SLAC Summer Institute on Particle Physics.
• Quantum Gravity Videos, Perimeter Institute Recorded Seminar Archive.
• Luca Bombelli’s quantum gravity reading list.

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