2.1. Non-commutative Spacetime Operators

The most popular argument for the non-commutativity of the spacetime coordinates and for the discreteness of spacetime dates back to Snyder Reference Snyder1947 and has found a famous expression in Doplicher et al. Reference Doplicher, Fredenhagen and Roberts1995. This discreteness encapsulates the quantum nature of spacetime or, equivalently, of the gravitational field. Doplicher and collaborators have argued that the quantum uncertainty relations of the spacetime coordinates emerge from the combination of Heisenberg's principle with general relativity. The argument, as I interpret it, should contain two main parts: the first shows that continuous spacetime loses its operational meaning at small scales already at the semi-classical level; the second would extend the argument to full quantum gravity by defending that this operational limit is due to the truly discrete nature of spacetime at Planck scale.

The first part of the argument encodes a rather common belief that the combination of the uncertainty relations and classical general relativity imposes restrictions on the operational meaning of classical spacetime. The reasoning runs along the following lines. If

1. 1. the greater the accuracy (or, equivalently, the smaller the uncertainty) in the measurement of spatio-temporal coordinates, the stronger the gravitational field generated by the measurement, and

2. 2. an increasing gravitational field eventually becomes strong enough as to collapse to a black hole, creating a closed trapped surface, and thus prevents any signal from leaving the region at stake, and

3. 3. an operational meaning can only be attached to a spacetime localization in case signals can leave the region measured,

then a localization measurement is operationally restricted to a certain maximum accuracy, or minimum uncertainty. The third premise is merely a definition of operational meaning and does not seem to be either problematic or substantial. The first two premises, on the other hand, are those that procure the mileage. The second premise epitomizes how gravity determines the causal structure of spacetime. This is an immediate consequence of classical general relativity and deserves all the credit that general relativity does. The first premise is a rather direct consequence of combining Heisenberg's uncertainty principle with the principles of general relativity. According to the uncertainty principle, measuring a spacetime coordinate with accuracy $1/ E$ generates an uncertainty E in the associated momentum. Thus, an uncontrollable energy of the order E is concentrated in the region to be localized. According to general relativity, the energy-momentum density $T_{\mu \nu }$ associated with this energy deforms the geometry of the spacetime. Note that the first two premises are implied by basic principles of well-trusted physical theories. While this certainly does not make them unassailable, our reasons for rejecting them would have to be substantial enough to challenge first principles of two of the most successful achievements of twentieth-century physics: quantum mechanics and general relativity.

One may argue, however, that these first principles will naturally be challenged at the level of quantum gravity. The fact that attempts to combine quantum mechanics with general relativity unveil their conceptual incommensurability can be taken to imply that some of their first principles must be violated in quantum gravity. In a sense, then, quantum mechanics and general relativity when combined already contain the seeds of their own destruction. Thus, one cannot expect that all principles that underpin the above premises will still be valid in full quantum gravity. But if not all premises hold, then the argument will of course collapse. Because the source energy associated with $T_{\mu \nu }$ is quantum and the argument as given is thus strictly semi-classical, the objection could conclude, it may very well turn out that the discreteness emerges only as an artefact of the manner in which quantum mechanics and general relativity were combined at the semi-classical level. Therefore, the argument as given so far must be complemented by a second part asserting that the operationally discrete spacetime at the semi-classical level results from an underlying discreteness at the fundamental Planck level.

Unfortunately, Doplicher and his collaborators do not provide such an addition. But neglecting to take this second leg seriously amounts to begging the question. It is undoubtedly true that if spacetime is discrete at the Planck level, it is reasonable to expect some signatures of this discreteness to surface at the semi-classical level. But the converse is not true, exactly because some or all of the premises made above may no longer obtain in full quantum gravity. The argument as it stands will hence not make any converts. Despite its appealing reliance on deeply entrenched physical principles, the argument thus falls short of proving that spacetime must be discrete (or, similarly, that gravity must be quantized) from the resources of trusted physical theories alone.

But the question then becomes how any argument drawing solely on accepted physical theories can possibly establish that gravity must be quantized. If a quantum theory of gravity would be part of the established corpus of theories, the proof would be easy. But alas, it is not! The failure of current physics to offer a straightforward and unique path to a quantum theory of gravity strongly suggests that the formulation of such a theory will require new physics. In this case, however, one cannot accept an argument from the resources of old physics alone to the effect that gravity must be quantized.

2.2. Inconsistency of Quantum-Classical Dynamics

Despite this limitation of arguments relying on principles of accepted physics, some ways of constructing a quantum theory of gravity without quantization can still be excluded. Peres and Terno (Reference Peres and Terno2001) and Terno (Reference Terno2006) have brought forth an argument which seems to preclude the possibility of a consistent semi-classical quantum theory of gravity which includes quantum backreactions. Consider a system consisting of two sectors, one governed by quantum field theory (the quantum sector), and a classical sector described by so-called Koopmanian dynamics. Koopmanian dynamics offers a generalized formalism that allows to cast a classical system in the mathematical formalism of a quantum theory, i.e., as a Hamiltonian system living in a Hilbert space. The conjugate variables of the classical sector will be represented by commuting multiplication operators, as opposed to the non-commuting operators of the quantum sector. Furthermore, assume that the hybrid dynamics of the combined classical-quantum system is described by a unitary evolution on the joint Hilbert space $\mathstrut{\cal H} =\mathstrut{\cal H} _{q}\otimes \mathstrut{\cal H} _{c}$.

Peres and Terno (Reference Peres and Terno2001) show that if we include terms with unobservable classical operators in the interaction part of the unitary evolution operator of the joint system, then the equations of motion of the combined system will no longer be the same as for the purely classical or the purely quantum equations of motion. But the failure to exhibit such formal equivalence, they argue, amounts to a violation of the correspondence principle, which is, of course, unacceptable. Such a formal correspondence between the equations of motion is required, Terno (Reference Terno2006) elaborates, in order to obtain the correct classical limit that must hold for the combined system. The reason for not obtaining the correct classical limit for hybrid systems with full interaction is that the evolution of the motion of observables differs if this formal correspondence does not apply. Since, to repeat, the presence of non-observable classical operators in the interaction term was responsible for the differences between the equations of motion, the interaction part can only contain observable classical operators. If this is the case, the quantum sector cannot influence the classical sector, thus precluding quantum backreactions. Even under these relatively weak assumptions it thus turns out to be impossible to define a mixed system with a hybrid dynamics that consistently includes the full interaction between the two sectors.

Although Peres and Terno's result also inhabits the semi-classical realm, I maintain that this does not render it vulnerable to the above charges as it did the preceding argument. The reason is simple: it does not compound two theories on whose principles it relies, but only offers a no-go result using a general formalism and invoking the unitarity of the combined evolution as well as the requirement for a quantum theory of gravity to possess the correct classical limit. These principles, it seems, constitute general desiderata for constructing theories rather than axioms of a specific theory that may become redundant as the theory is superseded. A full quantum theory of gravity is surely expected to comply at least with the correspondence principle. Insofar as Peres and Terno's assumptions transcend particular theories, they are immune to the objections aired above.

3.1. Sakharov's Induced Gravity

Most prominently, perhaps, among these alternative approaches is Sakharov's induced gravity theory. As it has recently been resurrected, it deserves particular attention.Footnote ² It claims to ingest Lorentz's (Reference Lorentz[1899] 1900) vision of the possibility of gravity merely being an effective force induced by residual electromagnetic forces. For Sakharov, gravity is thus not a fundamental physical field, but “induced,” i.e., emergent from quantum field theory like hydrodynamics emerges from molecular physics. Nota bene, since the interaction part of the action contains both classical and quantum terms, Sakharov's account leads to a type of semi-classical quantum gravity.

The general framework for an induced gravity theory in Sakharov's vein is set up by first assuming a Lorentzian manifold as a background on which to do quantum field theory. This background is a continuous, classical, unquantized spacetime. It is left free “to flap in the breeze,” i.e., no assumptions regarding its dynamical evolution are made. In particular, no Einstein equations—modified or not—enter the picture. When we do quantum field theory on this background spacetime, it turns out that the effective action at the one-loop level automatically contains terms proportional to the cosmological constant and to the Einstein-Hilbert action of general relativity, as well as higher order terms. Thus, it looks as if Einstein gravity is generated at the one-loop level from the interaction of quantum fields.

It may be instructive to note that gravity was not really created ex nihilo. At the bare minimum, a Lorentzian background manifold was assumed. Furthermore, the geometry of the background manifold acted as an external field for the quantum fields living on the background manifold. Terms encoding the geometry of the classical background thus cohabited in the Lagrangian of the effective action with the terms of the quantum fields. To the extent to which the background geometry was presupposed, arguably, the gravitational degree of freedom, but not its dynamics, was put into the theory from the start. Next, as the gravitational field is considered a degree of freedom, the variation of the effective action with respect to its variables automatically leads to the (semi-classical) Einstein equations. While the fact that the Einstein-Hilbert action is mimicked at the one-loop level is very suggestive, it is unclear whether this framework is sufficient to fully recover gravity. But it is important to note that gravity, although arguably not fully “induced,” was not quantized, as the geometry was assumed to be classical. Furthermore, the approach avoids Peres and Terno's prohibition of a semi-classical theory of gravity with quantum backreactions. Since no dynamical assumptions were made regarding the classical background (the classical sector), i.e., no classical dynamics imposed on the classical sector, the gravitational action is induced from the mutual interaction of the underlying quantum fields. The classical sector acts as a mere background and does not interact with the quantum sector. Therefore, no hybrid dynamics is necessitated in this approach.

Interestingly, there exist reverse efforts of constructing a fundamental quantum theory of gravity, subsumed under the heading “non-commutative geometry,” which assume only gravity to be a fundamental force and derive the standard model of electromagnetic, weak, and strong forces from gravity. However widespread and persuasive the belief in a fundamental theory containing gravity as a fundamental force may be, currently available observations and experiments only license inferences regarding the same semi-classical realm that Sakharov's approach describes. Not unlike the belief in the unity of nature, the belief in a fundamental theory including gravity is exposed by Sakharov as an additional commitment not warranted by the resources of empirical physics alone.

3.2. Jacobson's Gravitational Thermodynamics

More recently, Jacobson (Reference Jacobson1995) has offered a perspective that also cautioned against quantizing the Einstein equations. Rather than deriving the four laws of black hole thermodynamics from the classical Einstein equations, as did Bardeen, Carter, and Hawking (Reference Bardeen, Carter and Hawking1973), Jacobson inverts the derivation by recovering the Einstein equations from the proportionality of entropy and horizon surface area of a black hole together with the fundamental thermodynamical relation $\delta Q=TdS$, connecting heat Q, temperature T, and entropy S. The heat is interpreted as the energy flux across a causal horizon and the temperature as the Unruh temperature relative to an accelerated observer just inside a local Rindler horizon. This heat manifests itself via the gravitational field it generates. Like in conventional thermodynamics, where heat is interpreted as energy flux between unobservable degrees of freedom, the underlying mechanics of the energy flux is irrelevant. Assuming cosmic censorship, Jacobson formulates local gravitational thermodynamics for an observer by means of the boundary of her past (her “causal horizon”), associating this boundary with entropy. The system that radiates heat is identified with the degrees of freedom behind the horizon, separated from the observer's past by a causality barrier and is therefore unobservable.

As Jacobson shows, this interpretation imposes conditions on the curvature of spacetime such that the classical Einstein equations are implied. Therefore, he suggests that the Einstein equations can be more adequately analogized with the wave equation for sound in a medium, rather than interpreted as the dynamical equations for a fundamental field. These equations, he urges, as higher-level equations of state, should then not be quantized as if the gravitational field were fundamental, despite the fact that they may describe what is ultimately a quantum reality.

Like in Sakharov's approach, gravity does not represent a fundamental force. Rather, it emerges as a phenomenon supervenient on the energy flux from causally inaccessible degrees of freedom. Again, Peres and Terno's result does not apply. However, Jacobson's claim that gravity should not be quantized in this scheme because it represents a collective, higher-order degree of freedom is simply false. Physicists routinely quantize collective degrees of freedom such as sound (with “phonons” as quanta of sound). Whether or not a degree of freedom must be quantized or not does not depend on whether it is collective or individual, but on altogether different considerations. Hence, quantization cannot necessarily be escaped by Jacobsonian gravity; but it is not forced on it either.

Jacobson's incipient program, unfortunately, has so far not been worked out in any detail. It remains to be seen, therefore, whether it will be able to offer a full picture of the workings of gravity. If it will be successful in this undertaking, then the quantizers of gravity may have met a challenge.