aa r X i v : . [ g r- q c ] A p r Space, Time, Matter in Quantum Gravity
Claus Kiefer
Abstract
The concepts of space, time, and matter are of central importance in anytheory of the gravitational field. Here I discuss the role that these concepts might playin quantum theories of gravity. To be concrete, I will focus on the most conservativeapproach, which is quantum geometrodynamics. It turns out that spacetime is absentat the most fundamental level and emerges only in an appropriate limit. It is expectedthat the dynamics of matter can only be understood from a fundamental quantumtheory of all interactions.
In his famous habilitation colloquium on June 10, 1854, Bernhard Riemann con-cluded
The question of the validity of the hypotheses of geometry in the infinitely small is boundup with the question of the ground of the metric relations of space.. . . Either therefore thereality which underlies space must form a discrete manifoldness, or we must seek the groundof its metric relations outside it, in binding forces which act upon it. . . . This leads us intothe domain of another science, of physic, into which the object of this work does not allowus to go to-day. (Riemann (1868); translated by William Kingdon Clifford 1873) Claus KieferInstitute for Theoretical Physics, University of Cologne, Zülpicher Straße 77a, 50937 Köln, Germanye-mail: [email protected] The German original reads (Jost (2013), p. 43): “Die Frage über die Gültigkeit der Voraussetzungender Geometrie im Unendlichkleinen hängt zusammen mit der Frage nach dem innern Grunde derMassverhältnisse des Raumes. . . . Es muss also entweder das dem Raume zu Grunde liegendeWirkliche eine discrete Mannigfaltigkeit bilden, oder der Grund der Massverhältnisse ausserhalb,in darauf wirkenden bindenden Kräften, gesucht werden. . . . Es führt dies hinüber in das Gebiet einerandern Wissenschaft, in das Gebiet der Physik, welches wohl die Natur der heutigen Veranlassungnicht zu betreten erlaubt.” The English translation can be found in Jost (2016). For the role ofClifford in the development of these ideas, see e.g. Giulini (2018). 1 Claus Kiefer
Riemann’s pioneering ideas are important for at least two reasons. First, althoughRiemann did not take into account the time dimension, his ideas led to the math-ematical formalism that enabled Albert Einstein to formulate his theory of generalrelativity (GR) in 1915. In GR, gravity is understood as the manifestation of adynamical geometry of space and time, which are unified into a four-dimensionalspacetime.Second, as is clear from the sentences quoted above, matter and geometry areno longer imagined as independent from each other; the metric now depends on the“binding forces which act upon it”. The metrical field is no longer given rigidly onceand for all, but stands in causal dependence on matter. This idea is at the core of inGR.In his commentary on Riemann’s text from 1919, Hermann Weyl emphasized theunification of geometry and field theory in physics,
For geometry, here the same step happened that Faraday and Maxwell performed withinphysics, in particular electricity theory, which was done by the transition from an action-at-a-distance to a local-action theory: carrying out the principle to understand the world fromits behaviour in the infinitely small. See Jost (2013), p. 45. Riemann’s approach turned out to be much more powerful than alternative ideason the foundation of geometry, for example those of Hermann von Helmholtz, see e.g.Jost (2016), p. 119. Helmholtz starts from experience and postulates the possibilityof free motion of bodies. As he can prove mathematically, for this free motion aspace with constant curvature is required. From the later perspective of GR, thisturns out to be too narrow. Riemann’s idea, that bodies can carry geometry withthem, is realized in GR, which allows spaces, in fact spacetimes, to have arbitrarycurvature, as determined by the Einstein field equations. These equations read R µν − g µν R + Λ g µν = κ T µν . (1)Here, g µν denotes the spacetime metric, R µν the Ricci tensor, and R the Ricciscalar. Non-gravitational degrees of freedom (for simplicity called ‘matter’) aredescribed by a symmetric energy–momentum tensor T µν ; it obeys the covariantconservation law T νµν ; = . (2)It is important to emphasize that this is not a standard conservation law (with a partialinstead of a covariant derivative) from which a conserved current and charge canbe derived. If the energy–momentum tensor obeys the dominant energy condition “Für die Geometrie geschah hier der gleiche Schritt, den F araday und Maxwell innerhalb derPhysik, speziell der Elektrizitätslehre, vollzogen durch den Übergang von der Fernwirkungs- zurNahewirkungstheorie: das Prinzip, die Welt aus ihrem Verhalten im Unendlichkleinen zu verstehen,gelangt zur Durchführung.” The title of Helmholtz’s article, “Ueber die Thatsachen, die der Geometrie zu Grunde liegen”(“On the Facts which Lie at the Bases of Geometry”), makes a dig at the title of Riemann’s work. This is only possible in the presence of a symmetry, as expressed by a Killing vector.pace, Time, Matter in Quantum Gravity 3 (energy densities dominate over pressures), causality is implemented in the sensethat no influence from outside the lightcone can enter its inside.There are two free parameters in the gravitational sector: κ and Λ . From theNewtonian limit, one can identify κ = π G / c , (3)with G the gravitational (Newton) constant and c the speed of light. In 1917, Einsteinhad recognized that another free parameter is allowed – the cosmological constant Λ which has the physical dimension of an inverse length squared. From observationswe find the value Λ ≈ . × − m − ≈ . ( Gpc ) − . The relation of this value tonaive estimates from quantum field theory is an open question.The Einstein field equations (1) describe a non-linear interaction between geom-etry and matter. In this sense, T µν must not be interpreted as the source from whichthe metric is determined. For the description of matter, the metric is also needed,since it enters the field equations for matter as well as the equation of motion for testbodies given by Ü x µ + Γ µαβ Û x α Û x β = . (4)Here, Γ µαβ are the components of the Levi–Civita connection, which is determined bythe metric, and the dots denote derivatives with respect to proper time (for timelikegeodesics) or with respect to an affine parameter (null geodesics). Equation (4) isthe geodesic equation which reflects the universal coupling of gravity to matter. Incontrast to its Newtonian analogue, it corresponds to free motion in the geometrydescribed by g µν (‘equivalence principle’). So for the description of matter, the pair ( T µν , g αβ ) is needed, and one needs a rather involved initial value formulation todetermine the spacetime metric (see next section).A major feature of GR, and one that is particularly relevant for its quantization, is background independence . This must be carefuly distinguished from mere generalcovariance, which means form invariance of equations under an arbitrary change ofcoordinates. In contrast, background independence means that there are no absolute(non-dynamical) fields in the theory – this applies to GR, where the metric is adynamical quantity that acts on matter and is acted upon by it. As Jürgen Ehlershas remarked (Ehlers (2007), p. 91): “Conceptually, the background independencemust be seen as the principal achievement of general relativity theory; it is, however,at the same time the main obstacle to overcome if general relativity theory andquantum theory are to be united.” In GR, the law of motion (4) cannot be formulatedindependently from the field equations (1) – in fact, it follows from them by employing(2). This would not be possible in a theory with an absolute background, that is, withan absolute non-dynamical spacetime.In 1918, Weyl generalized the notion of the Levi-Civita connection that occurs in(4) to a symmetric linear connection, Weyl (1918), see also the extended discussion Recent doubts on this Λ -observation are expressed e.g. in Di Valentino et al. (2020). Test bodies in GR cannot be mass points. The mass-to-radius ratio of objects has an upper boundof c / G ; the concept of a mass point is replaced by a black hole. Claus Kiefer in Raum, Zeit, Materie , Weyl (1993). For this concept, a metrical structure on themanifold is not needed, only the notion of a parallel transport for vectors and tensors,which provides the means to connect different points on the manifold. In contrast tothe Levi-Civita connection, his more general connection need not be derivable froma metric. Weyl distinguishes, in fact, three levels of geometry: the first level is thetopological manifold (which he calls situs manifold or empty world ), the secondlevel the affinely connected manifold, and the third level the metric continuum (whichhe also calls “ether”); see also Schrödinger (1954) for a lucid presentation.The notion of a symmetric linear connection allowed Weyl to construct a gener-alization of Einstein’s theory. In his theory, the magnitude of vectors is not fixed,but the connection allows the comparison of magnitudes in different points. Thisintroduces a new freedom into the theory – the freedom to perform gauge transfor-mations. The metric is here determined only up to a (spacetime-dependent) factor.The exponent of this factor can be connected with a function that behaves as theelectromagnetic vector potential (here interpreted as a one-form). Weyl thought thathe has constructed in this way a unified theory of gravity and electromagnetism; fordetails, see Weyl (1993), p. 121 ff. In the above hierarchy, Weyl’s theory can belocated between the second and third level: in it, spacetime has a conformal struc-ture, which provides a more general framework than the structure of Riemanniangeometry.Weyl was convinced that fundamental geometric relations should only refer toinfinitesimally neighbouring points (
Nahgeometrie instead of
Ferngeometrie ). Thisprinciple plays a key role in both the 1918 and the 1929 versions of gauge theories.In Weyl (2000), p. 115, he writes (emphasis by Weyl): “
Only in the infinitely smallcan we expect to encounter everywhere the same elementary laws , thus the worldmust be understood from its behaviour in the infinitely small. In spite of its formal elegance, Weyl’s theory is empirically wrong, as was soonrealized by Einstein. The reason is that a non-integrable connection leads to path-dependent frequencies for atomic spectra, in contrast to observations. But his theorycan nevertheless be seen as the origin of our modern gauge theories. A decade later,in 1929, Weyl came up with a gauge theory of electromagnetism and the Dirac field.Instead of the real conformal factor multiplying the metric, there occurs now a phasefactor for which the exponent is a one-dimensional integral over the vector potential.A non-integrable connection is manifested there, for example, in the Aharonov–Bohm effect. In its non-Abelian generalization, gauge invariance is a key ingredientto the Standard Model of particle physics, see e.g. Dosch (2007) for a review. TheStandard Model (extended by massive neutrinos) is experimentally extremely well Analysis situs is an older name for topology. Here, the words ‘gauge’ (
Eichung ), ‘to gauge’ ( eichen ), and ‘gauge invariance’ (
Eich-Invarianz )enter. Their original meaning arises from providing standards for physical quantities (includingdistances), which is different from their later abstract use in the description of intrinsic symmetriesin gauge theories. The German original reads: “
Nur im Unendlichkleinen dürfen wir erwarten auf die elementaren,überall gleichen Gesetze zu stoßen , darum muß die Welt aus ihrem Verhalten im Unendlichkleinenverstanden werden.pace, Time, Matter in Quantum Gravity 5 tested, and no obvious deviation from it is seen so far in experiments at the LargeHadron Collider (LHC) and elsewhere.But what about gravity and spacetime? In its standard formulation, GR is not agauge theory. The reason is that the connection Γ µαβ is not independent there, but isderived from a metric. On thus has the chain g µν −→ Γ µαβ −→ R µαβγ , where R µαβγ denotes the Riemann curvature tensor. For gauge theories, the firststep in this chain is lacking. Gauge theories of gravity do, however, exist, and theyare needed for the consistent implementation of fermions, see Blagojević and Hehl(2013). Weyl’s original theory is a special case of this general class, but it isimportant to emphasize that the coupling of Weyl’s vector potential is not to theelectrodynamic current – as its creator believed – but to the dilaton current (becausethe one-parameter dilation group is gauged), see Hehl et al. (1988).One of the striking properties of GR is that it exhibits its own incompleteness.This is expressed in the singularity theorems which state that, under general condi-tions, singularities in spacetime are unavoidable, see Hawking and Penrose (1996).Singularities are here understood in the sense of geodesic incompleteness – timelikeor null geodesics as found from (4) terminate at finite proper time or finite affineparameter value. In most physically relevant cases, the occurrence of singularitiesis connected with regions of infinite curvature or energy density; notable examplesare the singularities characterizing the beginning of the Universe (“big bang”) andthe interior of black holes. One of the hopes connected with the construction of aquantum theory of gravity is that such a theory will avoid singularities. This hopemay be extended to a different type of singularities in our present physical theories –the infinities that arise in almost every local quantum field theory. One has learnt tocope with the latter singularities by employing sophisticated methods of regulariza-tion and renormalization. Nevertheless, one would expect that a truly fundamentaltheory will be finite from the onset. The reason is that the occurrence of singularitiesis connected with an unsufficient understanding of the microstructure of spacetime.True infinities should not occur in any sensible description of Nature, cf. Ellis et al. (2018).One possible solution to the singularity problem is to avoid a continuum for thespacetime structure and to assume instead that spacetime is built up from discreteentities. There are indications for such a discrete structure in some approaches toquantum gravity, but the last work has not yet been spoken. Interestingly, Riemannhimself envisaged the possibility of a continuous as well as a discrete manifold; thesmallest entities he calls quanta (Jost (2016), p. 32):
Definite portions of a manifoldness, distinguished by a mark or by a boundary, are calledQuanta. Their comparison with regard to quantity is accomplished in the case of discrete See also the contributions by Hehl and Obukhov and by Scholz to this volume. Claus Kiefermagnitudes by counting, in the case of continuous magnitudes by measuring.(Translated by William Kingdon Clifford 1873) Weyl, in his commentary to Riemann’s text, speculates that the final answer to theproblem of space may be found in its discrete nature. What happens to this when the quantum of action ~ comes into play? One ofthe early pioneers of attempts to quantizing gravity, Matvei Bronstein, through theapplication of thoughts experiments, arrived at the necessity of introducing minimaldistances in spacetime, thus abandoning the idea of a metric continuum. He writes The elimination of the logical inconsistencies connected with this [his thought experiments]requires a radical reconstruction of the theory, and in particular, the rejection of a Riemanniangeometry dealing, as we see here, with values unobservable in principle, and perhaps alsothe rejection of our ordinary concepts of space and time, modifying them by some muchdeeper and nonevident concepts.
Wer’s nicht glaubt, bezahlt einen Taler . In Bronstein’s analysis, quantities appear that can be found by combining G , c ,and ~ into units of length, time, and mass (or energy). They were first presented byMax Planck in 1899 (one year before the ‘official’ introduction of the quantum ofaction into physics!) and are called Planck units in his honour. They read l P = r ~ Gc ≈ . × − m (5) t P = l P c = r ~ Gc ≈ . × − s (6) m P = ~ l P c = r ~ cG ≈ . × − kg ≈ . × GeV / c . (7)At the end of his 1899 paper, Planck wrote the following prophetic sentences, seeKiefer (2012), p. 6: These quantities retain their natural meaning as long as the laws of gravitation, of lightpropagation in vacuum, and the two laws of the theory of heat remain valid; they musttherefore, if measured in various ways by all kinds of intelligent beings, always turn out tobe the same.
The German original reads (Jost (2013), p. 31): “Bestimmte, durch ein Merkmal oder eine Grenzeunterschiedene Theile einer Mannigfaltigkeit heissen Quanta. Ihre Vergleichung der Quantität nachgeschieht bei den discreten Grössen durch Zählung, bei den stetigen durch Messung.” “Sehen wir von der ersten Möglichkeit ab, es könnte ‘das dem Raum zugrunde liegende Wirklicheeine diskrete Mannigfaltigkeit bilden’ ( obschon in ihr vielleicht einmal die endgültige Antwort aufdas Raumproblem enthalten sein wird , my emphasis) . . . The quotation is from Kiefer (2012), p. 20. See e.g. Kiefer (2012), p. 5. The German original reads: “Diese Grössen behalten ihre natürliche Bedeutung so lange bei,als die Gesetze der Gravitation, der Lichtfortpflanzung im Vacuum und die beiden Hauptsätze derWärmetheorie in Gültigkeit bleiben, sie müssen also, von den verschiedensten Intelligenzen nachden verschiedensten Methoden gemessen, sich immer wieder als die nämlichen ergeben.”pace, Time, Matter in Quantum Gravity 7
One can form a dimensionless number out of these Planck units by bringing thecosmological constant Λ into play. Inserting the present observational value for Λ (see above), this gives l Λ ≡ G ~ Λ c ≈ . × − . (8)The smallness of this number is one of the biggest open puzzles in fundamentalphysics. Only a fundamental unified theory of all interactions is expected to providea satisfactory explanation.What are the general arguments that speak in favour of a quantum theory ofgravity? First, as mentioned above, there is the singularity problem of classicalgeneral relativity, which points to the incompleteness of Einstein’s theory. Second,the search for a unified theory of all interactions should include quantum gravity:gravity interacts universally to all fields of Nature, and all non-gravitational fieldsare successfully described by quantum (field) theory so far, so a quantum descriptionshould apply to gravity, too. Third, a very general argument was put forward byRichard Feynman in 1957, see Kiefer (2012), p. 18: if we generate a superposition oftwo masses at different locations, their gravitational fields should also be superposed,unless the superposition principle of quantum theory breaks down. A quantum theoryof gravity is needed to describe such superpositions. It is clear that such a statecan no longer correspond to a classical spacetime. There are at present interestingsuggestions for the possibility to observing the gravitational field generated by aquantum superposition in laboratory experiments, see Carlesso et al. (2019) andreferences therein.Several approaches to quantum gravity exist, but there is so far no consensusin the community, see Kiefer (2012). The ideal case would be to construct a finitequantum theory of all interactions from which present physical theories can bederived as approximations (or “effective field theories”) in appropriate limits. Theonly reasonable candidate is string theory. In this theory, the dimension of spacetimeassumes the number ten or eleven. Unfortunately, it is so far not clear how to recoverthe Standard Model from string theory and how to test it by experiments. Connectedwith this is the difficulty to proceed in a more or less unique way from the ten oreleven spacetime dimensions to the four dimensions of the observed world.The main alternatives to finding a unified theory are the more modest attempts toconstruct first a quantum theory of the gravitational field and to relegate unification toa later step. The usual starting point is GR, but quantization methods may be appliedto any other gravitational theory. Standard methods are path integral quantizationand canonical quantization. We shall focus below on the canonical quantization ofGR using metric variables, because conceptual issues dealing with space and timeare most transparent in this approach, see Kiefer (2009). See e.g. Kiefer (2012) for a comprehensive discussion. Claus Kiefer
Besides ordinary three-dimensional space (or four-dimensional spacetime), the con-cept of configuration space plays an eminent role in physics. In mechanics, thisis the N -dimensional space generated by all configurations, described by coordi-nates { q a } , a = , . . . N , that the system can assume. In field theory, it is infinite-dimensional of possible field configurations. In quantum theory, it will enter theargument of the wave function (functional) and lead to the central property of entan-glement.What is the configuration space in general relativity? As John Wheeler writes(Wheeler (1968), p. 245): “A decade and more of work by Dirac, Bergmann, Schild,Pirani, Anderson, Higgs, Arnowitt, Deser, Misner, DeWitt, and others has taughtus through many a hard knock that Einstein’s geometrodynamics deals with thedynamics of geometry: of 3-geometry, not 4-geometry.” Most of these developmentshappened after Weyl’s death in 1955. In fact, upon application of the canonical (orHamiltonian) formalism, Einstein’s theory can be written as a dynamical system forthe three-metric h ab and its canonical momentum π ab on a spacelike hypersurface Σ .The ten Einstein equations can be formulated as four constraints, that is, restrictionson initial data h ab and π ab on Σ , and six evolution equations. The four constraintsread (per spacepoint) H ⊥ = κ G ab cd π ab π cd − ( κ ) − √ h ( ( ) R − Λ ) + √ h ρ ≈ H a = − ∇ b π ab + √ h j a ≈ , (10)with the (inverse) DeWitt metric G ab cd = √ h ( h ac h bd + h ad h bc − h ab h cd ) (11)and κ given by (3). Here, ( ) R denotes the three-dimensional Ricci scalar and h the determinant of h ab ; ρ and j a denote matter density and current, respectively.The constraint H ⊥ ≈ H a ≈ ≈ π ab is related tothe extrinsic curvature K cd of Σ by π ab = G ab cd K cd κ , (12)where G ab cd denotes the DeWitt metric itself (the inverse of the expression in (11)).This quantity plays the role of a metric in the space of all Riemannian three-metrics h ab , a space called Riem Σ .Is Riem Σ the configuration space of GR? Not yet. The constraints H a ≈ geometries ,not the space of all three- metrics . This is what Wheeler called superspace , here pace, Time, Matter in Quantum Gravity 9 denoted by S( Σ ) , see Wheeler (1968). It is the arena for classical and quantumgeometrodynamics. One can formally write S( Σ ) : = Riem Σ / Diff Σ , where Diff Σ denotes the group of three-dimensional diffeomorphisms (“coordinatetransformations”). By going to superspace, the momentum constraints are automat-ically fulfilled. Whereas Riem Σ has a simple topological structure, the topologicalstructure of S( Σ ) is very complicated because it inherits (via Diff Σ ) some of thetopological information contained in Σ ; see Giulini (2009) for details.The DeWitt metric has pointwise a Lorentzian signature with one negative andfive positive directions, that is, it has negative, null, and positive directions. Due tothe minus sign, the kinetic term for the gravitational field is indefinite. It is importantto note that this minus sign is unrelated to the signature of spacetime; starting witha four-dimensional Euclidean space instead of a four-dimensional spacetime, thesame signature for the DeWitt metric is found. The presence of this minus sign isrelated to the attractive nature of gravity. It is also worth mentioning that the DeWittmetric reveals a surprising analogy with the elasticity tensor in three-dimensionalelasticity theory and the local and linear constitutive tensor in four-dimensionalelectrodynamics, see Hehl and Kiefer (2018). This analogy could be of importancefor theories of emergent gravity.Constraints and evolution equations have an intricate relationship; see e.g.Giulini and Kiefer (2007). Let me summarize the main features as well as point-ing out analogies with electrodynamics. First, there is an important connection withthe (covariant) conservation law of energy–momentum. The constraints are pre-served in time if and only if the energy–momentum tensor of matter has vanishingcovariant divergence. In electrodynamics, the Gauss constraint is preserved in timeif and only if electric charge is conserved.Second, Einstein’s equations represent the unique propagation law consistentwith the constraints. To be more concrete, if the constraints are valid on an “initial”hypersurface and if the dynamical evolution equations (the pure spatial componentsof the Einstein equations) hold, the constraints hold on every hypersurface. And ifthe constraints hold on every hypersurface, the dynamical evolution equations hold.Again, there is an analogy with electrodynamics: Maxwell’s equations are the uniquepropagation law consistent with the Gauss constraint.It must be emphasized that the picture of a spacetime foliated by a one-parameterfamily of hypersurfaces only emerges after the dynamical equations are solved. Then,spacetime can be interpreted as a “trajectory of spaces”. Before this is done, one onlyhas a three-dimensonal manifold Σ with given topology, equipped with the canonicalvariables satisfying the constraints (9) and (10).This fact that spacetime is not given from the outset but must be constructedthrough an initial value formulation, is an expression of the background independencediscussed in the previous section. In this sense, the analogy with electrodynamics ona given external spacetime breaks down. Background independence is related withthe classical version of what is called the problem of time : if we restrict ourselves to compact three-manifolds Σ , the total Hamiltonian of GR is a combination of theconstraints (9) and (10). Thus, no external time parameter exists; all physical timeparameters are to be constructed from within our system, that is, as functional of thecanonical variables. A priori, there is no preferred choice of such an intrinsic timeparameter. It is this absence of an external time and the non-preference of an intrinsicone that is known as the problem of time in (classical) canonical gravity. Still, afterthe solution of the dynamical equations, spacetime as a trajectory of spaces exists.This is different in the quantum theory where it leads to the far-reaching quantumversion of the problem of time (next section).The possibility of constructing spacetime in the way just described, is also re-flected in the closure of the Poisson algebra for the constraints (9) and (10): {H ⊥ ( x ) , H ⊥ ( y )} = − σδ , a ( x , y ) (cid:16) h ab ( x )H b ( x ) + h ab ( y )H b ( y ) (cid:17) (13) {H a ( x ) , H ⊥ ( y )} = H ⊥ ( x ) δ , a ( x , y ) (14) {H a ( x ) , H b ( y )} = H b ( x ) δ , a ( x , y ) + H a ( y ) δ , b ( x , y ) (15)It is not a Lie algebra, though, because the Poisson bracket between two Hamiltonianconstraints at different points also contains (the inverse of) the three-metric, h ab .We also remark that the signature of the spacetime metric enters here in the formof the parameter σ : in fact, σ = − σ = In the last section, we have reviewed the canonical (Hamiltonian) formulation of GR.Here, we discuss the quantum version of this, see e.g. Kiefer (2012) for a compre-hensive treatment. We follow Dirac’s heuristic approach and transform the classicalconstraints (9) and (10) into conditions on physically allowed wave functionals.These wave functionals are defined on the space of all three-metrics (the above spaceRiem Σ ) and matter fields on Σ . The quantum version of (9) readsˆ H ⊥ Ψ ≡ (cid:18) − π G ~ G abcd δ δ h ab δ h cd −( π G ) − √ h (cid:0) ( ) R − Λ (cid:1) + √ h ˆ ρ (cid:17) Ψ = In the asymptotically flat case, additional boundary terms are present.pace, Time, Matter in Quantum Gravity 11 and is called the
Wheeler–DeWitt equation . We note that the kinetic term in this equa-tion only has formal meaning before the issues of factor ordering and regularizationare successfully addressed. The quantum implementation of (10) readsˆ H a Ψ ≡ − ∇ b ~ i δ Ψ δ h ab + √ h ˆ j a Ψ = x a ¯ x a = x a + δ N a ( x ) , the three-metric transforms as h ab ( x ) 7→ ¯ h ab ( x ) = h ab ( x ) − D a δ N b ( x ) − D b δ N a ( x ) . The wave functional then transforms according to Ψ [ h ab ] 7→ Ψ [ h ab ] − ∫ d x δ Ψ δ h ab ( x ) D a δ N b ( x ) . Assuming the invariance of the wave functional under this transformation, one is ledto D a δ Ψ δ h ab = . This is exactly (17) (restricted here to the vacuum case).A simple analogy to (17) is Gauss’s law in quantum electrodynamics (or itsgeneralization to the non-Abelian case). The quantized version of the constraint ∇ E ≈ ~ i ∇ δ Ψ [ A ] δ A = , where A is the vector potential. This equation reflects the invariance of Ψ underspatial gauge transformations of the form A → A + ∇ λ .The constraints can only be implemented in the form (16) and (17) if the quantumversion of the constraint algebra (13) – (15) holds without extra c-number terms onthe right-hand side. Otherwise, only a part of the quantum constraints (or even none)holds in this form. The situation is reminiscent of string theory where the Virasoroalgebra displays such extra (central or Schwinger) terms. More general quantumconstraints hold there provided the number of spacetime dimensions is restricted toa specific number (ten in the case of superstrings). It is imaginable that a restrictionin the number of spacetime dimensions arises also here from a consistent treatmentof the quantum constraint algebra. But so far, this is not clear at all. For a recent attempt into this direction, see Feng (2018). This problem was already known to Dirac and was the reason why he abandoned working onquantum gravity. In his last contribution to this field, he remarked, Dirac (1968), p. 543: “Theproblem of the quantization of the gravitational field is thus left in a rather uncertain state. Ifone accepts Schwinger’s plausible methods, the problem is solved. [Dirac refers to a heuristic2 Claus Kiefer
In the last section, we have seen that we can interpret spacetime as a generalizedtrajectory of spaces. In its construction, the four constraint equations and the sixdynamical equations are inextricably interwoven. What happens in the quantumtheory? There, the trajecory of spaces has disappeared, in the same way as theordinary mechanical trajectory of a particle has disappeared in quantum mechanics.The three-metric h ab and its momentum π cd play the role of the q i and p j inmechanics, so it is clear that in quantum gravity h ab and π cd cannot be “determinedsimultaneously”, which means that spacetime is absent at the most fundamental level,and only the configuration space of all three-metrics respective three-geometriesremains. This is clearly displayed in Table 1 on p. 248 in Wheeler (1968).From this point of view it is clear that in the quantum theory only the constraintssurvive. The evolution equations lose their meaning in the absence of a spacetime.In a certain sense, this is anticipated in the classical theory by the strong connectionbetween constraints and evolution equations as discussed in the previous section.The absence of spacetime, and in particular of time, is usually understood as thequantum version of the problem of time . It means that the quantum world at thefundamental level is timeless – it just is . Weyl has attributed such a static picturealready to the classical spacetime of GR. In Weyl (2000) p. 150, he writes: The objective world just is , it does not happen . Only from the view of the consciousnesscrawling upwards in the worldline of my life a sector of this world “lives up” and passes byat him as a spatial picture in temporal transformation. In the quantum theory, there is not even a spacetime and a worldline with a consciousobserver, at least not at the most fundamental level. So how can we relate this pictureof timelessness, forced upon us by a straightforward extrapolation of establishedphysical theories, with the standard concept of time in physics? There are two pointsto be discussed here.First, as already mentioned, the DeWitt metric (12) has an indefinite signature: one minus and five plus . This means that the Wheeler–DeWitt equation has a local hyper-bolic structure through which part of the three-metric is distinguished as an intrinsictimelike variable . One can show that this role is played by the “local scale” √ h . Insimple cosmological models of homogeneous and isotropic (Friedmann–Lemaître)universes, this is directly related to the scale factor , a . Using units with 2 G / π = (cid:18) ~ a ∂∂ a (cid:18) a ∂∂ a (cid:19) − ~ a ∂ ∂φ − a + Λ a + m a φ (cid:19) ψ ( a , φ ) = . (18) regularization proposed by Schwinger in 1962, C.K.] But one cannot be happy with such methodswithout having a reliable procedure for handling quadratic expressions in the δ -function.” Such areliable procedure is still missing. But see Feng (2018). The German original reads: “Die objektive Welt ist schlechthin, sie geschieht nicht. Nur vor demBlick des in der Weltlinie meines Lebens emporkriechenden Bewußtseins “lebt” ein Ausschnittdieser Welt “auf” und zieht an ihm vorüber als räumliches, in zeitlicher Wandlung begriffenesBild.” [emphasis by Weyl]pace, Time, Matter in Quantum Gravity 13
Additional gravitational and matter degrees of freedom come with kinetic termsthat differ in sign from the kinetic term with respect to a . For equations such as (18),one can thus formulate an initial value problem with respect to intrinsic time a . Theconfiguration space is here two-dimensional and spanned by the two variables a and φ . Standard quantum theory employs the mathematical structure of a Hilbert space inorder to implement the probability interpretation for the quantum state. An importantproperty is the unitary evolution of this state; it guarantees the conservation of thetotal probability with respect to the external time t . But what happens when there isno external time, as we have seen is the case in quantum gravity? There is no commonopinion on this, but it is at least far from clear whether a Hilbert-space structure isneeded at all, and if yes, which one. This is also known as the Hilbert-space problem and is evidently related to the problem of time. The second point concerns the recovery of the standard (general relativistic)notion of time from the fundamentally timeless theory of gravity. The standardway proceeds via a Born–Oppenheimer type of approximation scheme, similarlyto molecular physics. For this to work, the quantum state, which is a solution of(16) and (17), must be of a special form. For such a state one can recover anapproximate notion of semiclassical (WKB) time. One can show that this WKBtime (which, in fact, is a “many-fingered time”) corresponds to the notion of timein Einstein’s theory. Equations (16) and (17) then lead to a functional Schrödingerequation describing the limit of quantum field theory in curved spacetime, the lattergiven by Einstein’s equations. It is in this limit that one can apply the standardHilbert-space structure and the associated probability interpretation. Higher ordersof this approximation allow the derivation of quantum-gravitational correctionsterms, which, for example, give corrections to the Cosmic Microwave Background(CMB) anisotropy spectrum proportional to the inverse Planck-mass squared. Suchterms follow from a straightforward expansion of (16) and (17) and could in principlegive a first observational test of quantum geometrodynamics, Brizuela et al. (2016).Quantum geometrodynamics, like practically all approaches to quantum gravity,is a linear theory in the quantum states and thus obeys the superposition principle.This means that most states do not correspond to any classical three-geometry. Thesituation resembles, of course, Schrödinger’s cat. Like there, one can employ theprocess of decoherence to understand why such weird superpositons are not observed,see Joos et al. (2003). Decoherence is the irreversible and unavoidable interaction ofa quantum system with the irrelevant degrees of freedom of its “environment”. Inquantum cosmology, one can consider, for example, the variables a and φ in (18) asdescribing the (relevant) quantum system, while small density perturbations and tinygravitational waves can play the role of the environment. The entanglement between Except phantom fields , which play a role in connection with discussions about dark energy, cf.Bouhmadi-López et al. (2019), Di Valentino et al. (2020). See e.g. Kiefer (2012, 2013) for a detailed discussion of this and the other conceptual issuesdiscussed below. “Environment” is a metaphor here. It stands for other degrees of freedom in configuration spacewhich become entangled with the quantum system, but which cannot be observed themselves.4 Claus Kiefer system and environment leads to the suppression of interferences between different a and different φ (within some limits); in this sense, classical geometry and classicaluniverse emerge. The same holds for the emergence of structure in the universe fromprimordial quantum fluctuations, see Kiefer and Polarski (2009).It is evident from the above that the question about the correct interpretation ofquantum theory enters here with its full power. Since by definition the Universeas a whole is a strictly closed quantum system, one cannot invoke any classicalmeasurement agent as acting from the outside. Following DeWitt(1967), the standardinterpretation used, at least implicitly, is the Everett interpretation, which states thatall components in the linear superposition are real. It is obvious that at the level of (18) there is no intrinsic difference between bigbang and big crunch; both correspond to the region a approaching zero in config-uration space. This has important consequences for cosmological models in whichclassically the universe expands and recollapses, see Zeh (2007). In the quantumversion, there is no trajectory describing the expansion and the recollapse. The onlystructure available is an equation of the form (18) in which only the scale factor a (and other variables) enter. The natural way to solve such an equation is to specifyinitial values on constant- a hypersurfaces in configuration space and to evolve themfrom smaller a to larger a . In more complicated models, one can evolve also theentanglement entropy between degrees of freedom in this way. If the entropy is lowat small a (as is suggested by observations), it will increase all along from small a to large a . There is then a formal reversal of the arrow of time at the classicalturning point, although this cannot be noticed by any observer, because the classicalevolution comes to an end before the region of the classical turning point is reached.We have limited the discussion here to quantum geometrodynamics. The mainconclusions also hold for the path-integral approach and to loop quantum gravity. In loop quantum gravity, there are analogies with gauge theories, for example withFaradays’s lines of forces, see Frittelli et al. (1994). Still, it is not a gauge theory byitself, and many conceptual issues such as the semiclassical limit are much less clearthan in quantum geometrodynamics.
Very early on, Einstein was concerned with a fundamental duality oberved in thephysical description of Nature: the duality between fields and matter. This dualityis the prime motivation for introducing the concept of light quanta in his importantpaper on the photoelectric effect from 1905. At that time, the only known dynamical Alternatives are the de Broglie–Bohm approach and collapse models, which both are more newtheories than new interpretations. The situation in string theory so far is less clear; there are indications that not only the concept ofspacetime, but also the concept of space is modified, as is discussed in the context of the AdS/CFTconjecture.pace, Time, Matter in Quantum Gravity 15 field was the electromagnetic field; ten years later, with GR, the gravitational fieldjoined in.In
Raum, Zeit, Materie , Weyl writes at the end of the main text, see Weyl (1993),p. 317:
In the darkness, which still wraps up the problem of matter, perhaps quantum theory is thefirst dawning light. Here, the hope is expressed that quantum theory, which in 1918 was still in itsinfancy, may provide a solution for this duality. This is certainly along the lines ofEinstein’s 1905 light quan ta hypothesis. But, ten years later, the final quantum theorygave a totally different picture: central notions of the theory are wave functions andthe probability interpretation. Einstein was repelled by this, especially by the featureof entanglement, which seems to provide a “spooky” action at a distance. This iswhy he focused on a unified theory of gravity and electrodynamics. He hoped tounderstand “particles” as solitonic solutions of field equations. His project did notsucceed.A somewhat different direction to understand ‘matter from space’ was pursued byJohn Wheeler in the 1950s, see Wheeler (1962). The idea is that mass, charge, andother particle properties originate from a non-trivial topological structure of space,the most famous example being Wheeler’s wormhole. This is most interesting, buthas not led to anything close to a fundamental theory. Weyl’s 1929 idea of understanding the interaction of electrons with the electro-magnetic field by the gauge principle turned out to be more promising. The StandardModel of particle physics is an extremely successful gauge theory, and virtually allof its extensions make use of this principle, too. Gauge fields can also be describedin a geometric way by adopting the mathematical structure of fibre bundles. Still, thisis relatively far from the geometric concepts of GR, which deal with spacetime andnot with the internal degrees of freedom of gauge theories. Perhaps gauge theoriesof gravity may help in finding a unified field theory, see Blagojević and Hehl (2013).Our physical theories all employ a metric to represent matter fields and theirinteractions, so GR is always relevant, even in situations where its effects are small.As Jürgen Ehlers writes in Ehlers (2007), p. 91: “Since inertial mass is separable fromactive, gravity-producing mass, an ultimate understanding of mass can be expectedonly from a theory comprising inertia and gravity.” This should also apply for theorigin of the masses in the Standard Model. The Higgs mechanism provides onlya partial answer; the masses of elementary (non-composite) particles are given bythe coupling to the Higgs, but the masses of composite particles such as proton andneutron cannot be explained. In fact, it seems that the mass of the proton mostly arisesfrom the binding energy of its constituents – quarks and gluons – and not from theirmasses, which to first order are negligible. Invoking the inverse of Einstein’s famousformula, m = E / c , one can speculate that mass ultimately originates from energy,see Wilczek (1999/2000). It is hard to imagine that this origin can be understood The German original reads: “In dem Dunkel, welches das Problem der Materie annoch umhüllt,ist vielleicht die Quantentheorie das erste anbrechende Licht.” For a recent account of matter from (the topology of) space, see e.g. Giulini (2018).6 Claus Kiefer without gravity. Perhaps a unified theory at the fundamental level is conformallyinvariant, similar to Weyl’s 1918 theory, expressing the irrelevance of masses at highenergies (small scales); masses would then only emerge as an effective, low-energyconcept.Unfortunately, despite many attempts, the duality of matter and fields remainsunresolved, even in present approaches to quantum gravity. An exception may bestring theory, but this approach has its own problems and it is far from clear whetherit can be tested empirically. Perhaps the solution to the problem of matter may arrivefrom a completely unexpected direction. Space, time, and matter continue to becentral concepts for research in the 21st century. The question posed in the title ofEinstein (1919), “Do gravitational fields play an essential role in the constitution ofmaterial elementary particles?” will most likely have to be answered by a definite yes . Acknowledgements
I am grateful to Silvia De Bianchi and Friedrich Hehl for their comments onmy manuscript.
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