aa r X i v : . [ g r- q c ] M a r February 2008
Is Quantum Gravity Necessary?
S. C arlip ∗ Department of PhysicsUniversity of CaliforniaDavis, CA 95616USA
Abstract
In view of the enormous difficulties we seem to face in quantizing generalrelativity, we should perhaps consider the possibility that gravity is afundamentally classical interaction. Theoretical arguments against suchmixed classical-quantum models are strong, but not conclusive, and thequestion is ultimately one for experiment. I review some work in progresson the possibility of experimental tests, exploiting the nonlinearity of theclassical-quantum coupling, that could help settle this question. ∗ email: [email protected] he first attempts to quantize general relativity date back to the early 1930s [1]. Inthe 75 years that have followed, we have learned an enormous amount: gauge-fixing andFaddeev-Popov ghosts, background field methods, the effective action formalism, thecanonical analysis of constrained systems, the investigation of gauge-invariant observ-ables, and much of what we know about topology in quantum field theory grew out ofattempts to quantize gravity. But despite the extraordinary work of a great many out-standing physicists, a complete, consistent, and compelling quantum theory of gravitystill seems distant [2].In view of this history, we should perhaps consider the possibility that we are askingthe wrong question. It could be that gravity is simply not quantum mechanical. Theprospect of a fundamentally classical theory of gravity is unpalatable; in Duff’s words [3],it “seems to be the very antithesis of the economy of thought which is surely the basisof theoretical physics.” But the matter is ultimately one for experiment. As Rosenfeldhas put it [4],It is nice to have at one’s disposal such exquisite mathematical tools as thepresent methods of quantum field theory, but one should not forget thatthese methods have been elaborated in order to describe definite empiricalsituations, in which they find their only justification. Any question as totheir range of application can only be answered by experience, not by for-mal argumentation. Even the legendary Chicago machine cannot deliver thesausages if it is not supplied with hogs.There are old arguments that fundamentally classical fields are incompatible withquantum mechanics, in the sense that they could be used to violate the uncertaintyprinciple [5]. Details depend on how the classical field interacts with a quantum system.Eppley and Hannah [6, 7] have considered two cases:1. A classical gravitational measurement collapses the quantum wave function: thenmomentum is not conserved. Consider a quantum object in a coherent state witha very small uncertainty in momentum and a correspondingly large uncertainty inposition. Measure its position by scattering a very short wavelength gravitationalwave, causing its state to change to one with a very small uncertainty in positionand a large uncertainty in momentum. If gravity is classical, the gravitationalwave can carry an arbitrarily small momentum, despite its short wavelength; yetby the uncertainty principle, the quantum system must sometimes experience alarge change in momentum.2. A classical gravitational measurement does not collapses the quantum wave func-tion: then signals can be sent faster than light. Place a proton in a box, in a statein which it has an equal probability of being in the left or right half. Split the boxin half and carry one half to a remote location. Monitor your half continuouslywith gravitational measurements, while a colleague performs a nongravitationalmeasurement of the other half. Your colleague’s measurement will collapse the1ave function, causing an instantaneous and detectable change in the half of thebox you are monitoring.Page and Geilker [8] add a third case:3. Neither classical nor quantum measurements collapse the wave function (Everettinterpretation): then gravitational fields will not be observed to have localizedsources. Consider a gravitating mass in a superposition of two widely separatedposition eigenstates. If its classical gravitational field depends on its quantumwave function, its gravitational attraction should point toward some intermediate“average” location [9, 10]. Page and Geilker tested this experimentally, but theoutcome is already apparent in, say, the observed gravitational field of the Moon.But while such arguments are certainly suggestive, they are not really conclusive[4, 7, 11, 12]. For instance, there are inherent non-quantum limits to gravitational mea-surements [12, 13], whose implications for an Eppley-Hannah-type argument have yet tobe fully explored. The general question of whether one can consistently couple classicaland quantum systems is a matter of ongoing research—see, for example, [14–20]—andis not yet resolved.The thought experiments of Eppley, Hannah, and others do, however, suggest thata fundamentally classical theory of gravity is likely to require changes to quantum me-chanics as well. As I shall argue below, once one allows a coupling between classical andquantum systems, quantum mechanics almost inevitably becomes nonlinear, suggestingthe possibility of sensitive new experimental tests.
1. Semiclassical gravity and the Schr¨odinger-Newton equation
If we wish to couple classical gravity and quantum matter, we need field equationsfor gravity. The standard Einstein equations, G ab = 8 π ˆ T ab , (1.1)no longer make sense, since they now equate a c-number with an operator. We might tryto interpret (1.1) as an eigenvalue equation, but this picture fails: the components of thestress-energy tensor do not commute, and cannot be simultaneously diagonalized [10].The obvious next step is to replace the right-hand side of (1.1) with an expectationvalue, G ab = 8 π h ψ | ˆ T ab | ψ i , (1.2)leading to the model of “semiclassical gravity” first proposed by Møller [21] and Rosen-feld [4], and derived from an action principle by Kibble and Randjbar-Daemi [22]. Seenmerely as a Hartree approximation to a full quantum theory of gravity, such a modelseems uncontroversial. But as Kibble and Randjbar-Daemi emphasized [22], seen asa fundamental theory, the model implies nonlinearities in quantum mechanics: the2chr¨odinger equation for the wave function | ψ i depends on the metric, which now de-pends in turn on the wave function. ∗ Adler has observed that semiclassical gravitycontains self-interaction terms that are not present in a Hartree approximation [23],further differentiating it from a mere approximation to a full quantum theory.Several technical problems with semiclassical gravity have been pointed out in theliterature. Field redefinition ambiguities can lead to inequivalent quantizations of thesame classical theory [3]; renormalization may either require classical curvature-squaredterms in the action that can lead to negative energies [24] or new matter vertices thatimply noncausal behavior at short distances [25]; and it is not obvious that an abruptchange in the right-hand side of (1.2) due to wave function collapse can be consistentwith conservation of the left-hand side [10]. Again, though, these objections do not seemconclusive. The nonlinearity of semiclassical gravity, on the other hand, suggests thatexperimental tests may be possible: gravity is very weak, but limits on nonlinearities inquantum mechanics are very strong [26].To address this question, it is useful to start with the Newtonian approximation to(1.2), the Schr¨odinger-Newton equation [27, 28] i ~ ∂ψ∂t = − ~ m ∇ ψ − m Φ ψ, ∇ Φ = 4 πGm | ψ | . (1.3)As in full semiclassical gravity, this model treats matter quantum mechanically, butdescribes gravity in terms of a classical Newtonian potential Φ sourced by the expectationvalue of the mass density. Despite the nonlinearities of the coupled system (1.3), thestandard probability interpretation of the wave function remains consistent; in particular,the probability current continuity equation ∂∂t | ψ | = ~ ∇ · (cid:20) i ~ m (cid:16) ψ ∗ ~ ∇ ψ − ψ ~ ∇ ψ ∗ (cid:17)(cid:21) (1.4)still holds, and total probability is conserved. A number of authors have studied thissystem [29–32], and we know a good bit about the stationary states with low energyeigenvalues, but time evolution has proven to be much more problematic [33–35].
2. Estimates and numerics
The question, then, is whether the nonlinearities in the Schr¨odinger-Newton equation(1.3) are large enough to lead to observable consequences. Let us begin with a roughestimate. Consider a particle of mass m with a localized initial wave function—forsimplicity, a Gaussian, ψ ( r,
0) = (cid:16) απ (cid:17) / e − αr / (2.1)with width α − / . The time evolution of ψ will depend on two competing effects, thequantum mechanical spreading of the wave function and its Newtonian “self-gravitation,” ∗ Dirac was also apparently aware of this; see [12], p. 1. r p ∼ α − / (cid:18) α ~ m t (cid:19) / , (2.2)which “accelerates” outward at a rate a out = ¨ r p ∼ ~ /m r p . This will balance theinward gravitational acceleration a in ∼ Gm/r p at t = 0 when m ∼ (cid:18) ~ √ αG (cid:19) / . (2.3)This is almost certainly an overestimate: as t increases, a out drops more quickly than a in , so even if wave packet spreading dominates initially, self-gravity may eventually win.For more precise results, one must solve (1.3) numerically. Note that although theinitial data (2.1) depend on two parameters, α and m , the Schr¨odinger-Newton equationis invariant under the rescalings m → µm, ~x → µ − ~x, t → µ t, ψ → µ / ψ, (2.4)so it is enough to consider a one-parameter set of solutions. Peter Salzman and I havenumerically simulated the evolution of an initial Gaussian wave function [36, 37]. Wefind the expected qualitative results:1. For small masses, the behavior is virtually identical to that of a free particle, whileas m increases, the wave packet spreads more slowly.2. In a transitional range of mass, the wave packet is unstable, fluctuating rapidlyand developing growing oscillations. (A similar instability is seen in [33–35].)3. For large masses, the wave packet undergoes “gravitational collapse.”Surprisingly, though, we find that the “collapse” behavior occurs at considerably lowermasses than the estimate (2.3) suggests. For the initial width of α = 5 × m − usedin the simulations, the mass (2.3) is on the order of 10 u, while collapse first appears inthe simulations for masses of about 10 u. † This result is somewhat unexpected, althoughnot implausible in view of the highly nonlinear nature of the problem. Fortunately, it isnow being tested by another group, using different, independently developed code.Assuming the validity of our simulations, we can use the scaling behavior (2.4) toobtain the parameters for gravitational collapse. We find that a wave packet of initialwidth w = α − / will shrink if its initial mass lies in a range m − ( w ) < m < m + ( w ), with m − / w/ µ m) − / , m + / . × ( w/ µ m) − / . (2.5) † Anticipating a discussion of molecular interferometry, I am giving masses in unified atomic mass units. m > m + , we have not been able to run the simulation long enough to reliablydetermine the outcome.) The numerically obtained collapse times, in nanoseconds, are t − / . × − ( w/ µ m) − / t + / . × − ( w/ µ m) − / . (2.6)
3. Experimental tests
Are nonlinearities at the level described above experimentally testable? To get ameasurable signal, one needs to use as large a mass as possible while still maintainingobservable quantum behavior. The best bet seems to be molecular interferometry, wherea “collapsing” wave packet would lead to suppression of interference. At this writing,the heaviest molecule that has experimentally exhibited interference is fluorofullerene,C F , with a mass of 1632 u [38]. The grating slits in the fluorofullerene experimenthave a width w ∼ . µ m. From (2.5), semiclassical gravity would predict a loss ofinterference for a wave packet of this width for masses greater than about m ∼ u [39–42]. If the next generation of molecularinterferometry experiments can come even close to this limit, a clean test of semiclassicalgravity should be well within reach. Acknowledgments
I would like to thank my colleagues at Peyresq 11, including Brandon Carter, LarryFord, Bei-Lok Hu, Seif Randjbar-Daemi, and Albert Roura, for a great many usefulcomments and suggestions. This work was supported in part by U.S. Department ofEnergy grant DE-FG02-91ER40674. 5 eferences [1] C. Rovelli, in
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