aa r X i v : . [ h e p - t h ] A p r Quantizing Geometry or Geometrizing the Quantum?
Benjamin Koch
Pontificia Universidad Cat´olica de Chile,Av. Vicu˜na Mackenna 4860,Santiago, Chile (Dated: October 31, 2018)The unsatisfactory status of the search for a consistent and predictive quantization of gravity istaken as motivation to study the question whether geometrical laws could be more fundamentalthan quantization procedures. In such an approach the quantum mechanical laws should emergefrom the geometrical theory. A toy model that incorporates the idea is presented and its necessaryformulation in configuration space is emphasized.
PACS numbers: 04.62.+v, 03.65.Ta
I. QUANTIZING GEOMETRY
The dream of finding a unified description of all phys-ical phenomena is facing a profound problem: “
The deepincompatibility between the indefinite nature of quantummechanics and the rigid geometrical formulation of gen-eral relativity. ” A common assumption is that quantummechanics, as it is usually formulated, is a fact of “na-ture” and thus it is more fundamental than general rel-ativity. Consequently most approaches try to apply oneof the well defined quantization procedures to the phys-ical degrees of freedom of space-time (or to some deepertheory that gives rise to space-time). Some of the mostpopular approaches along this line are:
String theory is for many the most famous candidatefor a unified theory of nature [1]. It also lead to interest-ing conjectures about the relation between certain treelevel string theories and quantum field theory [2]. Butuntil today it could not live up to its promises concern-ing the uniqueness of what this theory actually predictsand explains.
Loop quantum gravity is a canonical approach for thequantization of space-time. In earlier stages of its devel-opment it lead to the development of geometrodynam-ics [3] and [4] supergravity that has very nice featuresat the Planck scale [5]. However, up to now it was notpossible to show that it really contains general relativityin some classical limit [6].
Causal dynamical triangulation and causal sets are dis-ciplines that earn more and more attention [7]. Theyshow the emergence of four dimensional space-time bystarting from a discrete causal structure. Until now thoseapproaches are limited to asking very basic questions onsuch as the dimensionality of space-time but they don’tallow to derive an effective gravitational action.
Induced gravity theories try to show the emergence ofcurved space-time in a mean field approximation of someunderlying microscopic degrees of freedom [8]. It is as-sumed that this mechanism is similar to the mechanismthat allows to get fluid dynamics from Bose-Einstein con-densation. Up to now those models manage to mimicsome possible features of (quantized) general relativity but a complete picture is still missing. An other alterna-tive for emergence of gravity is based on the idea thatthe existence of curved space-time emerges from non-geometric statistical laws [9, 10].
Renormalization group approaches are working in theimaginary time formalism. Given an ultra violet (UV)completion and the existence of a non-trivial fixed pointin the running couplings of the completed gravitationalaction this approach might present a renormalizable ver-sion of gravity [11, 12]. Until now the strict applicabilityof the imaginary time formalism and the form of the UVcompletion are open issues.
Anisotropic models postulate a different scaling behav-ior of space and time in the UV regime, which allows toconstruct a power counting renormalizable theory [13] inthe UV. However, recent studies claim that the infraredlimit of the theory is not identical to massless gravity [14].
Further research has been done on asymptotic quanti-zation [15], twistors [16], non-commutative [17, 18] anddiscretized [19] geometry.Despite of impressive progress in some directions, theoriginal task remains unsolved in all those approaches.
II. GEOMETRIZING THE QUANTUM
Given the problems in applying the laws of quantummechanics to the geometry of space-time we want to askthe following question:“
Could it be that (classical) geometry is more funda-mental than the rules of quantization? ” A. Conceptual problems
Necessarily, answering this question with “yes” wouldmean that the undeniable observable effects of quantiza-tion have to emerge from the deeper theory (in this casea classical geometrical theory). Such an approach facesimmediately two mayor problems • Determinism is, in contrast to quantum mechanics, part of mostgeometric theories (such as general relativity). Thismeans for example that in causal geometrical the-ories uncertainties are just a result of unknown ini-tial conditions, whereas in standard quantum me-chanics they are an irrenunciable concept. • Non-locality:
In principle it is possible to construct determinis-tic (hidden variable) theories that are in agreementwith the predictions of quantum mechanics. How-ever, those theories have to pay a price in order toevade “no go” theorems such as the Bell inequal-ities [20]. They have to contain non-local interac-tions.
B. A conceptual bridge
There exists a self consistent deterministic formulationof quantum mechanics, which also reproduces all typicalexperimental results [43]. It was first suggested by deBroglie, then shown to be consistent by Bohm [21, 22] andlater further developed by several authors [20, 23, 24]. Itwill be referred to as dBB (de Broglie-Bohm) theory. Inthis proposal, the dBB theory will be an essential piecewhen building the bridge from classical geometry to aquantum theory. We will now shortly present its formu-lation for the case of a relativistic system of n-bosonicparticles as given by [24]: Let | i be state vector of thevacuum and | n i be an arbitrary n-particle state. Thecorresponding n-particle wave function is [24] ψ ( x ; . . . ; x n ) = P s √ n ! h | ˆΦ( x ) . . . ˆΦ( x n ) | n i , (1)where the ˆΦ( x j ) are scalar Klein-Gordon field operators.The symbol P s denotes symmetrization over all positions x j which we will keep in mind but not write explicitly anymore. For free fields, the wave function (1) satisfies theequation n X j ∂ mj ∂ jm + n M ~ ψ ( x ; . . . ; x n ) = 0 . (2)The mass of a single particle is given by M . The index j indicates on which one of the n particle coordinates thedifferential operator has to act and the index m is thetypical space-time index in four flat dimensions. Sincequantum mechanics is formulated in terms of equal timecommutation relations, the relevant equation only con-tains one single time variable τ = t = t . . . t n . Doingthis, equation (2) contains n identical time derivativeswhich can be absorbed into a single ∂ t by redefining t = τ / √ n .The key step to the dBB interpretation comes from split-ting the wavefunction up ψ = P exp( iS/ ~ ) and postulat-ing the three-momentum of the particle j to be − ~∂ j S . Including this definition one has three coupled real differ-ential equations. For further convenience the coordinatesfor the n particles can be labeled as x L = ( t, ~x , ~x , . . . , ~x n ) , (3)which also implies the 1 + 3 n dimensional co- and con-travariant derivatives ∂ L and ∂ L . Now the three realequations of the dBB theory read2 M Q ≡ ( ∂ L S )( ∂ L S ) − nM with (4) Q ≡ ~ M ∂ L ∂ L PP , ≡ ∂ L (cid:0) P ( ∂ L S ) (cid:1) , (5) p L ≡ M dx L ds ≡ − ∂ L S . (6)Applying the total derivative d/ds ≡ dx L /ds∂ L to eq. (6)gives a Newtonian type of equation of motion d x L ds = ( ∂ N S )( ∂ L ∂ N S ) M . (7)It is crucial to note that this theory addresses the twopreviously mentioned conceptual issues and thus makesthe dBB theory a good framework for the program of ge-ometrizing the quantum.First, it is deterministic in the sense that given initialpositions and given initial field configurations for S and P determine the final state of the system.Second, it is deeply non-local, because the functions S and P simultaneously depend on the positions of all then-particles.A further remark: the dBB theory contextual and there-fore not affected by the Kochen-Specker theorem [25]. III. EMERGENT QUANTUM MECHANICS
The idea that quantum mechanics might not be fun-damental but rather emerge from an underlying classicalsystem has been proposed in various ways.
A. Various appearances of the idea
Although the focus of this paper is on the possiblegeometric origin of quantum mechanics it is instructive togive a list of proposals that point into a similar direction.
Statistical emergence of quantum mechanics:
In [26, 27] it was shown that quantum mechanical corre-lations arise when considering finite subsystems of clas-sical statistical systems with originally infinite degrees offreedom. An application of this observation to quantumgravity is perceivable but was not attempted yet.
Gauge emergence of quantum mechanics:
Based on a new kind of local gauge transformation anon-linear field theory has been proposed that containsquantum field theory as an infrared limit [28]. Also aspecial classical supersymmetric model was suggested togive rise to a quantum mechanical system [29]. A possibleunification with general relativity was not explored yet.
Dissipative emergence of both, quantum and gravity:
Dissipative deterministic systems can give rise to quan-tum operators and symmetries that are not present in theoriginal theory at the microscopic scale [30, 31]. Furtherconjecturing that those symmetries are the ones of dif-feomorphism invariance (general relativity) might give anidentikit picture of a future theory of quantum gravity.
Geometrical emergence of quantum mechanics:
The similarity between Weyl geometry and the struc-ture of quantum mechanical equations was first noticedin [32]. Other studies in this direction focused on theRicci flow [33, 34] or on a geometric reduction of thedimensionality of space-time [35, 36]. Using local confor-mal transformations (Weyl geometry) it was even pos-sible to formulate a geometrical theory that contains incertain limits both general relativity and the equationsof Bohmian mechanics [37–39]. The impressive successof those (Weyl geometry) models is limited to the singleparticle case because the dBB theory is only consistentif it also contains the non-local interactions due to multiparticle dynamics.
B. Geometry of configuration space
It was shown that existing models for the geometricalemergence of quantum mechanics are incomplete, sincethey can’t explain the non-local interactions in the multiparticle dBB theory. Continuing previous work in thisdirection [40, 41] a possible way to fill this gap will bepresented.The 1 + 3 n dimensional configuration space of n-particles with a common time cordinate will be consid-ered. Following the notation in eq. (3) the coordinates inthis (possibly curved) space-time will be denoted asˆ x Λ = (ˆ t, ˆ ~x , . . . , ˆ ~x n ) . (8)The toy model for the curvature of this space will bea single scalar equation which is a 1 + 3 n dimensionalanalog to the Nordstrom theory [42]ˆ R (cid:12)(cid:12)(cid:12) S = κ ˆ T M (cid:12)(cid:12)(cid:12) S . (9)The left hand side contains the Ricci scalar (correspond-ing to a metric ˆ g ΛΣ ). The right hand side contains somecoupling constant κ and the trace of the energy momen-tum tensor ˆ T . The symbol | S indicates complete sym-metrization of the terms with respect to the interchangeof two configuration coordinates ˆ x i ↔ ˆ x j , . . . . Just likein the case of the bosonic Klein-Gordon equation we willkeep this in mind without explicitly writing it into thefollowing equations. The symmetrization further fixesthe coordinate system for the four dimensional subspaces and forces all block diagonal submetrics to be identicalˆ g µνi ↔ ˆ g µνj . In order to describe the local conformal partof this theory separately and for simplification one as-sumes the metric ˆ g to split up into a conformal function φ ( x ) and a flat part η ˆ g ΛΓ = φ n − η LG . (10)The inverse of the metric (10) isˆ g ΛΓ = φ − n − η LG . (11)Indices with a lower Greek and a lower Roman index canbe identified ˆ ∂ Λ ≡ ∂ L . From this follows for example thatthe adjoint derivatives are not identical, in both notationsˆ ∂ Λ = ˆ g ΛΣ ˆ ∂ Σ = φ − n − η LS ∂ S = φ − n − ∂ L . (12) The geometrical dual to the first dBB equation:
An Extension of the Hamilton Jacobi stress energy tensorˆ T M can be defined by subtracting a mass term ˆ M forevery particleˆ T M = ˆ p Λ ˆ p Λ − n ˆ M G (13)= ( ˆ ∂ Λ S H )( ˆ ∂ Λ S H ) − n ˆ M G = φ − n − (cid:0) ( ∂ L S H )( ∂ L S H ) − nM G (cid:1) = φ − n − T M . The Hamilton principle function S H defines the local mo-mentum ˆ p Λ = ˆ M G d ˆ x Λ /d ˆ s = − ˆ ∂ Λ S H . Combining (13),(12), (10), and (9) gives12 nκ (1 − n ) ∂ L ∂ L φφ = ( ∂ L S H )( ∂ L S H ) − nM G . (14)This is exactly the first dBB equation (4) if one identifies φ ( x ) = P ( x ) , (15) S H ( x ) = S ( x ) ,κ = 12 n (1 − n ) / ~ ,M = M G . Note that the matching conditions demand a negativecoupling κ . The geometrical dual to the second dBB equation:
In order to find the dual to the second Bohmian equa-tion one can exploit that the stress-energy tensor (13) iscovariantly conservedˆ ∇ Λ ˆ T Λ∆ = 0 . (16)This is true if the following relations are fulfilledˆ ∇ Λ ( ˆ ∂ Λ S H ) = 0 , (17)( ˆ ∂ Λ S H ) ˆ ∇ ∆ ( ˆ ∂ Λ S H ) = 0 , (18)( ˆ ∂ Λ S H ) ˆ ∇ Λ ( ˆ ∂ ∆ S H ) = 0 . (19)In addition to the covariant conservation of momentum(17) and the conservation of squared momentum (18) thetensor nature of (13) also demands (19). In order tocalculate the covariant derivatives in (17-19), one needsto know the Levi Civita connectionΓ ΣΛ∆ = 12 g ΣΞ ( ∂ Λ g ∆Ξ + ∂ ∆ g ΞΛ − ∂ Ξ g Λ∆ ) (20)= 12 φ − n − h ( ∂ L φ n − ) δ SD + ( ∂ D φ n − ) δ SL − ( ∂ S φ n − ) η LD i . It is this form of the connection that gives rise to the non-metricity in Weyl geometry. Using eq. (20), the condition(17) readsˆ ∇ Λ ( ˆ ∂ Λ S H ) = φ − n n − ∂ L (cid:2) φ ( ∂ L S H ) (cid:3) = 0 . (21)With the matching conditions (15), the above equationis identical to the second Bohmian equation (5). The geometrical dual to the third dBB equation:
According to the Hamilton-Jacobi formalism the deriva-tives of the Hamilton principle function ( S H ) define themomenta ˆ p Λ ≡ − ( ˆ ∂ Λ S H ) . (22)Therefore, with the prescription (12) and the matchingcondition (15) one sees that the third Bohmian equation(6) is fulfilled. The geometrical dual to the dBB equation of motion:
From differential geometry it is known that the validity of the geodesics equation of motion results in the con-servation of the stress energy tensor. Nevertheless, it isa good consistency check [41] to explicitly calculate thegeodesic equation d ˆ x Λ d ˆ s + ˆΓ Λ∆Σ d ˆ x ∆ d ˆ s d ˆ x Σ d ˆ s = ˆ p Λ · f (ˆ x ) . (23)Inserting eq. (20) into eq. (23) and using the matchingconditions eq. (15) the dBB equation of motion (7) isobtained. IV. SUMMARY
It is advocated that “geometrizing the quantum” mightbe a viable alternative to the standard approaches toquantum gravity. The main conceptual problems of thenew approach are discussed. Using the example of ascalar geometrical toy model (incorporating gravity is be-yond the scope of this proposal) and mapping this modelto the dBB interpretation of the multi particle Klein-Gordon equation, it is shown how those problems can beevaded. It is argued that such a mechanism only canwork consistently if the geometrical theory is formulatedin the (1 + 3 n dimensional) configuration space of thesystem.The author wants to thank J.M. Isidro and D. Dolce fortheir remarks. [1] D. J. Gross, J. A. Harvey, E. J. Martinec and R. Rohm,Nucl. Phys. B , 253 (1985); E. Witten, Phys. Rev. D , 314 (1991).[2] J. M. Maldacena, Adv. Theor. Math. Phys. ,231 (1998) [Int. J. Theor. Phys. , 1113 (1999)][arXiv:hep-th/9711200].[3] J. A. Wheeler, Annals Phys. , 604 (1957).[4] S. Deser and B. Zumino, Phys. Lett. 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