The following post is based on some notes I prepared for a journal club talk I’m going to give on quantum gravity in a few hours. A postscript equivalent is here, with a few modifications.

Disclaimer: The whole point of our journal club talks is to give talks on interesting topics about which we are not experts! For me, quantum gravity fits this description in spades. Caveat emptor.

#### Introduction

Every physicist learns as an undergraduate (if not before) that we don’t yet have a single theory unifying quantum mechanics and general relativity, i.e., a theory of quantum gravity. What is often not explained is why it is difficult to come up with such a theory. In this journal club I want to ask and partially answer two questions: (1) what makes it so difficult to put quantum mechanics and general relativity together; and (2) what approaches might one take to developing a theory of quantum gravity?

You might wonder if this is an appropriate topic for a forum such as this. After all, none of us here, including myself, are experts on string theory, loop quantum gravity, twistors, or any of the other approaches to quantum gravity that have been proposed and are currently being pursued.

However, we don’t yet know that any of these approaches is correct, and so there’s no harm in going back and thinking through some of the basic aspects of the problem, from asn elementary point of view. This can be done by anyone who knows the rudiments of quantum mechanics and of the general theory of relativity.

If you like, you can view it as trying to solve the problem of quantum gravity without first “looking in the back of the book” to see the best attempted answers that other people have come up with. This procedure of first thinking things through for yourself has the advantage that it is likely to greatly increase the depth of your understanding of other people’s work if you later do investigate topics such as string theory, etc.

A related disclaimer is that I personally know only a miniscule fraction of all the modern thinking on quantum gravity. I prepared this lecture to force myself to think through in a naive way some of the problems involved in constructing a quantum theory of gravity, only pausing occasionally to peek in the back of the book. I won’t try to acknowledge my sources, which were many, but suffice to say that I doubt there’s anything here that hasn’t been thought before. Furthermore, people who’ve thought hard about quantum gravity over and extended period are likely to find much of what I say obvious, naive, absurd, or some combination thereof. Frankly, I don’t recommend that such people look through these notes — they’ll likely find it rather frustrating! For those less expert even than myself, perhaps you’ll find these notes a useful entertainment, and maybe they’ll stimulate you to think further on the subject.

#### Standard formulations of quantum mechanics and general relativity

Let’s start off by reminding ourselves of the standard formulations used for quantum mechanics and general relativity. I expect that most attendees at this journal club are extremely familiar with the basic principles of quantum mechanics, and, indeed, use them every day of their working lives. You may be rather less familiar with general relativity. I’ve tried to construct the lecture so you can follow the overall gist, even so.

Recall that the standard formulation of quantum mechanics contains the following elements:

• The postulate that for every physical system there is a state vector in a Hilbert space which provides the most complete possible description of that system.
• The postulate that the dynamics of a closed quantum system are described by a Hamiltonian and Schroedinger’s equation.
• The postulate that a measurement on a system is described using an observable, a Hermitian operator acting on state space, which is used to describe measurement according to some rule for: (1) calculating measurement probabilities; and (2) describing the relationship between prior and posterior states.
• The postulate that the state space for a composite quantum system is built up by taking the tensor product of individual state spaces. In the special case when those systems are indistinguishable, the postulate is modified so that the state space is either the symmetric or antisymmetric subspace of the total tensor product, depending on whether the systems are bosons or fermions.

It’s worth pointing out that this is merely the most common formulation of quantum mechanics. Other formulations are possible, and may be extremely valuable. It’s certainly possible that the right way of constructing a quantum theory of gravity is to start from some different formulation of quantum mechanics. My reason for describing this formulation of quantum mechanics — probably the most commonly used formulation — is so that we’re all working off the same page.

Let’s switch now to discuss general relativity. Recall that the standard formulation of general relativity contains the following elements:

• The postulate that spacetime is a four-dimensional pseudo-Riemannian manifold, with metric signature (+1,-1,-1,-1).
• The postulate that material in spacetime is described by a two-index tensor T known as the stress-energy tensor. The stress-energy tensor describes not only thinks like mass and energy, but also describes the transport of mass and energy, so it has aspects that are both static and dynamic.
• The postulate known as the Einstein field equations: $$G = 8\pi T$$. This postulate connects the stress-energy tensor T to the Einstein tensor, G. In its mathematical definition G is fundamentally a geometric object, i.e., it is determined by the “shape” of spacetime. The physical content of the Einstein field equations is therefore that the shape of spacetime is determined by the matter distribution, and vice versa.An interesting point is that because the stress-energy tensor contains components describing the transport of matter, the transport properties of matter are actually determined by the geometry. For example, it can easily be shown that, as a consequence of the Einstein field equations, test particles follow geodesics of spacetime.
• Since 1998 it has been thought that the Einstein equations need to be modifed, becoming $$G+\Lambda g = 8 \pi T$$, where g is the metric tensor, and $$\Lambda$$ is a non-zero constant known as the cosmological constant. Rather remarkably, it turns out that, once again, test particles follow geodesics of spacetime. However, for a given stress-energy tensor, the shape of spacetime will itself be different, and so the geodesics will be different.

In an ideal world, of course, we wouldn’t just unify quantum mechanics and general relativity. We’d actually construct a single theory which incorporates both general relativity and the entire standard model of particle physics. So it’s arguable that we shouldn’t just be thinking about the standard formulation of quantum mechanics, but rather about the entire edifice of the standard model. I’m not going to do that here, because: (1) talking about vanilla quantum mechanics is plenty enough for one lecture; (2) it illustrates many of the problems that arise in the standard model, anyway; and (3) I’m a lot more comfortable with elementary quantum mechanics than I am with the standard model, and I expect much of my audience is, too.

#### Comparing the elements of general relativity and quantum mechanics

Let’s go through and look at each element in the standard formulations of general relativity and quantum mechanics, attempting as we do to understand some of the problems which arise when we try to unify the two theories.

Before getting started with the comparisons, let me make an aside on my presentation style. Conventionally, a good lecture is much like a good movie or a good book, in that a problem or situation is set up, preferably one involving high drama, the tension mounts, and then the problem is partially or fully resolved. Unfortunately, today is going to be a litany of problems, with markedly little success in resolution, and so the lecture may feel a little unsatisfying for those hoping, consciously or unconsciously, for a resolution.

Spacetime: In standard quantum mechanics, we usually work with respect to a fixed background spacetime of allowed configurations. By contrast, in general relativity, the metric tensor specifying the structure of spacetime is one of the physical variables of the theory. If we follow the usual prescriptions of quantum mechanics, we conclude that the metric tensor itself ought to be replaced by some suitable quantum mechanical observable, or set of observables. If one does this, it is no longer so clear that space and time can be treated as background parameters in quantum mechanics. How, for example, are we supposed to treat Schroedinger’s equation, when the physical structure of time itself is variable? Perhaps we ought to aim for an effective equation of the form

$$i \frac{d|\psi\rangle}{d\langle t \rangle} = H |\psi\rangle$$

derived from some deeper underlying theory?

Stress-energy tensor: In general relativity T is used to describe the configuration of material bodies. Standard quantum mechanics tells us that T needs to be replaced by a suitable set of observables. In and of itself this is not obviously a major problem. However, a problem arises (again) in connection with the possible quantization of space and time. As usually understood in general relativity, T is a function of location p on the underlying four-dimensional manifold. The natural analogue in a quantized version is an observable $$\hat T(p)$$ which is again a function of position on the manifold. However, as described above, it seems likely that p itself should be replaced by some quantum equivalent, and it is not so clear how $$\hat T$$ ought to be constructed then. One possibility is that $$\hat T$$ becomes a function of some suitable $$\hat p$$. A related problem is that the standard definition of the components of T often involve tangent vectors (essentially, velocity 4-vectors) to the underlying manifold. As for the position, p, perhaps such tangent vectors should be replaced by quantized equivalents.

Einstein field equations (with and without the cosmological constant): Consider the usual general relativistic formulation of the field equations: $$G+\Lambda g = 8\pi T$$. The problem with constructing a quantum version ought by now to be obvious: quantum mechanics tells us that the quantities on the left — geometric quantities, to do with the shape of spacetime — are all associated with some notion of a background configuration, ordinarily left unquantized, while the quantities on the right are physical variables that ought to be quantized.

One natural speculation in this vein is that in any quantum theory of gravity we ought to have

$$G+\Lambda g = 8 \pi \langle T \rangle$$

as an effective equation of the theory.

Hilbert space and quantum states: There is no obvious incompatability with general relativity, perhaps because it is so unclear which Hilbert space or quantum state one might use in a description of gravitation.

The Hamiltonian and Schroedinger’s equation: As already mentioned, this presents a challenge because it is not so clear how to describe time in quantum gravity. Something else which is of concern is that for many standard physical forms Schroedinger’s equation often gives rise to faster than light effects. In order to alleviate this problem we must move to a relativistic wave equation, or to a quantum field theory.

In this vein, let me mention one natural candidate description for the dynamics of a free (quantum) test particle moving in the background of a fixed (classical) spacetime. First, start with a relativistically invariant wave equation such as the Klein-Gordon equation, which can be used to describe a free spin zero particle,

$$-\hbar^2 \frac{\partial^2 \psi}{\partial^2 t} = -\hbar^2 c^2 \nabla^2 \psi + m^2 c^4 \psi,$$

or the Dirac wave equation, which can be used to describe a free spin 1/2 particle,

$$i \hbar \frac{\partial \psi}{\partial t} = \left(i \hbar c \alpha \cdot \nabla – \beta mc^2 \right) \psi,$$

where $$\alpha_x,\alpha_y,\alpha_z$$ and $$\beta$$ are the four Dirac matrices. In the case of the Klein-Gordon equation there is a clear prescription for how to take this over to a curved spacetime: simply replace derivatives by appropriate covariant derivatives, giving:

$$-\hbar^2 \nabla^2_; \psi = m^2 c^2 \psi.$$

In flat spacetime this will have the same behaviour as the Klein-Gordon equation. In a fixed background curved spacetime we would expect this equation to describe a free spin zero test particle.

The same basic procedure can be followed in the case of the Dirac equation, replacing derivatives wherever necessary by covariant derivatives. I have not explicitly checked that the resulting equation is invariantly defined, but expect that it is (exercise!), and can be used to describe a free spin 1/2 test particle in a fixed background curved spacetime. It would be interesting to study the solutions of such equations for some simple nontrivial geometries, such as the Schwarzschild geometry. For metrics with sufficient symmetry, it may be possible to obtain analytic (or at least perturbative) solutions; in any case, it should be possible to investigate these problems numerically.

Of course, although it would be interesting to study this prescription, we should expect it to be inadequate in various ways. We have described a means of studying a quantum test particle moving against a fixed classical background spacetime. In reality: (1) the background may not be classical; (2) the particle itself modifies the background; and (3) because of quantum indeterminancy, the particle may modify the background in different ways. In the language of the many-worlds interpretation, it seems reasonable to expect that the which branch of the wavefunction we are in (representing different particle positions) may have some bearing on the structure of spacetime itself: in particular, different branches will correspond to different spacetimes.

This discussion highlights another significant incompatibility between general relativity and quantum mechanics. In general relativity, we know that test particles follow well-defined trajectories — geodesics of spacetime. This is a simple consequence of the field equations themselves. In quantum mechanics, no particle can follow a well-defined trajectory: the only way this could happen is if the Hamiltonian commuted with the position variables, in which case the particle would be stationary. In any case, this commutation condition can not occur when the momentum contributes to the Hamiltonian, as is typically the case.

Observables: One striking difference between quantum mechanics and general relativity is that the description of measurement is much more complex in the former. Several questions that might arise include:

• Should wave function collapse occur instantaneously? This depends on how one interprest the wave function.
• Should measurements be a purely local phenomena, or can we make a measurement across an entire slice of spacetime? Across all of spacetime?
• Should we worry that in the usual description of measurement, time and space are treated in a manifestly unsymmetric manner?
• What observables would one expect to have in a quantum theory of gravity?

The tensor product structure and indistinguishable particles:One cause for concern here is that the notion of distinguishability itself is often framed in terms of the spatial separation of particles. If the structure of space itself really ought to be thought of in quantum terms, it is perhaps not so clear that the concepts of distinguishable, indistinguishable, and spatially separated particles even make sense. This may be a hint that in a quantum theory of gravity such concepts may be absent at the foundation, though they would need to emerge as consequences of the theory.

Quantum field theory: So far, we’ve concentrated on identifying incompatabilities between general relativity and quantum mechanics. Of course, fundamental modern physics is cast in terms of an extension of quantum mechanics known as quantum field theory, and it is worth investigating what problems arise when one attempts to unify general relativity with the entire edifice of quantum field theory. We won’t do this in any kind of fullness here, but will make one comment in relation to the canonical quantization procedure usually used to construct quantum field theories. The standard procedure is to start from some classical field equation, such as the wave equation, $$(\nabla^2 – 1/c^2 \partial^2 / \partial t^2 ) \phi = 0$$, to expand the solution as a linear combination of solutions for individual field modes, to regard the different mode coefficients as dynamical variables, and to then quantize by imposing canonical commutation relationships on those variables. This procedure can be carried out for many of the standard field equations, such as the wave equation, the Dirac equation, and the Klein-Gordon equation, because in each case the equation is a linear equation, and thus the solution space has a linear structure. In the case of general relativity, the field equations are nonlinear in what seems like the natural field variables — the metric tensor — and it is not possible to even get started with this procedure. One could, of course, try linearizing the field equations, and starting from there. My understanding is that when this is done the resulting quantum field theory is nonrenormalizable (?), and thus unsatisfactory.

#### Discussion

Perhaps the most striking feature of the above discussion is an asymmetry between general relativity and quantum mechanics. Quantum mechanics, like Newton’s laws of motion, is not so much a physical theory as a framework for constructing physical theories, with many important quantities (the state, the state space, the Hamiltonian, the relevant observables) left unspecified. General relativity is much more prescriptive, specifying as it does an equation relating the distribution of material entities to the shape of spacetime, and, as a consequence, controlling the matter-energy dynamics. Once we’ve set up the initial matter-energy distribution and structure of spacetime, general relativity gives us no further control. In the analogous quantum mechanical situation we still have to specify the dynamics, and the measurements to be performed.

There is therefore a sense in which quantum mechanics is a more wideranging and flexible framework than general relativity. This is arguably a bug, not a feature, since one of general relativity’s most appealing points is its prescriptiveness; once we have the Einstein equations, we get everything else for free, in some sense. However, it also suggests that while the right approach may be to extend the quantum mechanical framework to incorporate general relativity, it is exceedingly unlikely that the right approach is to extend general relativity to incorporate quantum mechanics. On the other hand, it may also be that some extension or reformulation of quantum mechanics is necessary to incorporate gravity. Such an extension would have to be carried out rather carefully: results such as Gleason’s theorem show that quantum mechanics is surprisingly sensitive to small changes.

As an aside, let me also take this opportunity to point out something which often bugs me: the widely-made assertion that quantum gravity effects will become important at the Planck length — about $$10^{-35}$$ meters — and the notion of spacetime will break down at that length. Anyone claiming this, in my opinion, ought to be asked why the notion of mass doesn’t break down at the Planck mass, which has the rather hefty value of about $$10^{-8}$$ kilograms.

#### A toy model

Just for fun, let me propose a simple toy model for quantum gravity, inspired by the Klein-Gordon equation. I’m sure this is wrong or inadequate somehow, but after an hour or so’s thought, I can’t yet see why. I include it here primarily as a stimulant to further thought.

The idea is to look for a four-dimensional pseudo-Riemannian manifold M, with metric signature (-,+,+,+), and a function $$\psi : M \rightarrow C$$, such that the following equations have a solution:

$$G + \Lambda g = 8 \pi T$$

$$T^{\mu \nu} = v^\mu v^\nu$$

$$v^0 = \frac{i\hbar}{2mc^2}( \psi^* \psi^{;0}- \psi \psi^{;0 *})$$

$$v^j = \frac{-i\hbar}{2m}( \psi^* \psi^{;j}- \psi \psi^{;j *}),$$

where m, c, $$\Lambda$$ are all constants with their usual meanings, j = 1,2,3, and the expression for $$T^{\mu \nu}$$ may need a proportionality constant, probably related to m, out the front. The expressions for $$v^0$$ and $$v^j$$ are covariant versions of the corresponding expressions for the charge and current densities associated to the Klein-Gordon equation — see Chapter~13 of Schiff’s well-known text on quantum mechanics (3rd ed., Mc-Graw Hill, 1968); note that Schiff calls this equation the “relativistic Schroedinger equation”. A subtlety is that the covariant derivative itself depends on the metric g, and so these equations are potentially extremely restrictive; it is by no means obvious that a solution ever exists. However, if we take seriously the idea that $$T^{\mu \nu}$$ needs a proportionality constant related to m, then we can see that in the test particle limit, $$m \rightarrow 0$$, these equations have as a solution any $$\psi$$, and flat spacetime, which is not unreasonable.

#### Conclusion

The picture I have painted is somewhat bleak, which is perhaps not surprising: finding a quantum theory of gravity is not a trivial problem! However, the good news is that many further steps naturally suggest themselves:

• At many points, my analysis has been incomplete, in that I haven’t thoroughly mapped out a catalogue of all the possible alternatives. A more thorough analysis of the possibilities should be done.
• The analysis needs to be extended to incorporate modern relativistic quantum field theory.
• Computer pioneer Alan Kay has said “A change of perspective is worth 80 IQ points”. It would be fruitful to repeat this exercise from the point of view of some of the other formulations people have of general relativity and quantum mechanics. I’d particularly like to do this for the initial value and action formulations of general relativity, and for the quasidistribution and nonlocal hidden variable formulations of quantum mechanics. It may also be useful to attempt to construct modifications of either or both theories in order to solve some of the problems that we’ve described here.
• Read up on some of the work that other people have done on quantum gravity, from a variety of points of view. Things to learn might include: supersymmetry, string theory, loop quantum gravity, twistors, Euclidean quantum gravity, Hawking radiation, the Unruh effect, the Wheeler-de Witt equation, Penrose’s gravitational collapse, 1- and 2-dimensional quantum gravity, gravitational wave astronomy, work on the cosmological constant, …

#### Acknolwedgments

Thanks to David Poulin for comments and encouragement.

“Itâ€™s worth pointing out that this is merely the most common formulation of quantum mechanics. Other formulations are possible, and may be extremely valuable.”

It is probably largely incidental, but I thought I would mention in this context a new preprint (quant-ph/0504102) by a colleague of yours at U. of Q., Tony Bracken.

2. Re the Dirac equation in a fixed curved space-time: I believe Robert Wald treats this case in his book on _Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics_. But it’s been, dear Lord, seven years since I read that.

I hope you won’t neglect the relevance of causal sets and related notions. I believe you know about Seth Lloyd’s recent (Jan 2005) preprint. He gave two talks at Perimeter Institute this week. Also see:

Causal Sets: Discrete Gravity

Topology, order and causal structure

A domain of spacetime intervals in general relativity

4. Thanks for the pointer, Cosma.

5. “As an aside, let me also take this opportunity to point out something which often bugs me: the widely-made assertion that quantum gravity effects will become important at the Planck length â€” about 10^{-35} meters â€” and the notion of spacetime will break down at that length. Anyone claiming this, in my opinion, ought to be asked why the notion of mass doesnâ€™t break down at the Planck mass, which has the rather hefty value of about 10^{-8} kilograms.”

Physics doofus that I am, let me give this a go. Aren’t quantum effects believed to become important at the Planck length because if you tried to localize a particle inside a sphere of radius less than l_p, it would acquire so much energy that it would collapse to a black hole? I don’t know what the analogous interpretation is for the Planck mass — maybe that if you converted that mass into energy, it would be the maximum energy that you could impart to a single particle? Anyone?

Since Scott Aaronson has offered a comment, I thought I would quote some remarks of his that have substantial relevance to the topic of this post (with apologies for butchering the typography):

Before going further, we should contrast our approach with previous approaches to hidden variables, the most famous of which is Bohmian mechanics [2]. Our main criticism of Bohmian mechanics is that it commits itself to a Hilbert space of particle positions and momenta. Furthermore, it is crucial that the positions and momenta be continuous, in order for particles to evolve deterministically. To see this, let |Li and |Ri be discrete positions, and suppose a particle is in state |Li at time t0, and state (|Li + |Ri) /p2 at a later time t1. Then a hidden variable representing the position would have entropy 0 at t1, since it is always |Li then; but entropy 1 at t1, since it is |Li or |Ri both with 1/2 probability. Therefore the earlier value cannot determine the later one.[44] It follows that Bohmian mechanics is incompatible with the belief that all physical observables are discrete. But in our view, there are strong reasons to hold that belief, which include black hole entropy bounds; the existence of a natural minimum length scale (10âˆ’33 cm); results on area quantization in quantum gravity [18]; the fact that many physical quantities once thought to be continuous have turned out to be discrete; the infinities of quantum field theory; the implausibility of analog â€œhypercomputersâ€; and conceptual problems raised by the independence of the continuum hypothesis.

[From Quantum Computing and Hidden Variables, p. 5 (posted on arXiv.org as quant-ph/0408035 and quant-ph/0408119)]

7. Michael just a note on your postscript notes: If I recall correctly, you don’t need the time derivatives of the stress-energy tensor in the ADM formalism, but you do need hydrodynamic equations to describe the evolution of the stress-energy tensor. Also note that specifying the initial conditions for the metric and its “time” derivative (often the external curvature) is often a very very difficult problem because there are all sorts of constraints that must be statisfied.

8. Thanks for that, Dave. It looks like Wald has a chapter on this stuff, which I really ought to read. I did think a bit about the constraints on the initial metric, and am glad / not surprised to hear it’s not trivial!