RSS feed

One unanticipated effect of my move from Movable Type to WordPress was that the old RSS feed stopped working. There is a new feed at:

that seems to function correctly, at least so far as bloglines is concerned.

Comments or technical assistance from anyone who uses RSS would be very greatly appreciated! It’d be good to know if other people see the feed working properly, or not.

Categorized as General

The Planck length

Thanks to all those people who commented on the significance of the Planck length. Putting it all together, one plausible argument for the significance of the Planck length seems to be something like this.

First, suppose we have a particle of mass [tex]m[/tex] whose wavefunction is localized within a length [tex]L[/tex], in each co-ordinate. For each co-ordinate the standard deviation in position satisfies:

[tex]\Delta x^2 \leq L^2/4.[/tex]

The uncertainty principle tells us that the uncertainty in each momentum co-ordinate satisfies:

[tex]\Delta p^2 \geq \hbar^2/ L^2.[/tex]

It follows that:

[tex]\langle P^2 \rangle \geq 3 \hbar^2 / L^2,[/tex]

where now [tex]\langle P^2 \rangle[/tex] is the average of the square of the total momentum, i.e., not a single co-ordinate. Using the usual formula connecting energy and momentum for a free particle in special relativity, we obtain [tex]E^2 = P^2 c^2 + m^2 c^4[/tex]. If we assume that [tex]P^2[/tex] and [tex]\langle P^2 \rangle[/tex] can be identified then we obtain:

[tex]E^2 \geq 3 \hbar^2 c^2 / L^2.[/tex]

In short, strong localization in position forces a large momentum, which creates a large energy density.

It’s plausible that if we make the energy density large enough, we’ll create a black hole. Equating [tex]E[/tex] to [tex]M c^2[/tex] for some notional black hole mass M and seeing at what radius that is, we get the Planck length, up to a small constant. I haven’t put the details in because this stage of this argument is even more hokey than the rest: energy density depends on what frame you’re in. However, it is at least plausible.

If you take the “no hair” theorem of general relativity seriously, then you’d believe that such a black hole should have no internal structure. But the detailed wavefunction would seem to be such an internal structure, and so we have a problem, which seems likely to signify a breakdown in one or both of general relativity or quantum mechanics. How that breakdown would manifest itself, I don’t know.

There’s lots wrong with this argument: the sign ambiguity for E, the use of the free particle formula for energy, and so on. No doubt many of these difficulties disappear in some more sophisticated approaches to quantum gravity; I’ve just been using undergrad quantum and GR.

Nonetheless, this argument does seem suggestive that particle having wavefunctions with structure on the Planck scale would, indeed, be very interesting objects, and that it is at least somewhat likely that general relativity, quantum mechanics, or both, would break down at that level.

Categorized as General

Journal club on quantum gravity

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.


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: [tex]G = 8\pi T[/tex]. 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 [tex]G+\Lambda g = 8 \pi T[/tex], where g is the metric tensor, and [tex]\Lambda[/tex] 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

[tex]i \frac{d|\psi\rangle}{d\langle t \rangle} = H |\psi\rangle [/tex]

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 [tex]\hat T(p)[/tex] 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 [tex]\hat T[/tex] ought to be constructed then. One possibility is that [tex]\hat T[/tex] becomes a function of some suitable [tex]\hat p[/tex]. 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: [tex]G+\Lambda g = 8\pi T[/tex]. 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

[tex] G+\Lambda g = 8 \pi \langle T \rangle[/tex]

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,

[tex] -\hbar^2 \frac{\partial^2 \psi}{\partial^2 t} = -\hbar^2 c^2 \nabla^2 \psi + m^2 c^4 \psi,[/tex]

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

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

where [tex]\alpha_x,\alpha_y,\alpha_z[/tex] and [tex]\beta[/tex] 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:

[tex] -\hbar^2 \nabla^2_; \psi = m^2 c^2 \psi.[/tex]

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, [tex](\nabla^2 – 1/c^2 \partial^2 / \partial t^2 ) \phi = 0[/tex], 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.


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 [tex]10^{-35}[/tex] 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 [tex]10^{-8}[/tex] 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 [tex]\psi : M \rightarrow C[/tex], such that the following equations have a solution:

[tex]G + \Lambda g = 8 \pi T [/tex]

[tex] T^{\mu \nu} = v^\mu v^\nu [/tex]

[tex] v^0 = \frac{i\hbar}{2mc^2}( \psi^* \psi^{;0}- \psi \psi^{;0 *})[/tex]

[tex] v^j = \frac{-i\hbar}{2m}( \psi^* \psi^{;j}- \psi \psi^{;j *}),[/tex]

where m, c, [tex]\Lambda[/tex] are all constants with their usual meanings, j = 1,2,3, and the expression for [tex]T^{\mu \nu}[/tex] may need a proportionality constant, probably related to m, out the front. The expressions for [tex]v^0[/tex] and [tex]v^j[/tex] 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 [tex]T^{\mu \nu}[/tex] needs a proportionality constant related to m, then we can see that in the test particle limit, [tex]m \rightarrow 0[/tex], these equations have as a solution any [tex]\psi[/tex], and flat spacetime, which is not unreasonable.


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, …


Thanks to David Poulin for comments and encouragement.

Cluster-state quantum computation

One of my main areas of research interest for the past few years has been measurement-based models of quantum computation. In the standard accounts of quantum computing, a quantum computer is presented as a device that gets its power by performing coherent manipulations of superpositions of computational states, before a final measurement step destroys the superposition, singling out a single computational state to be read out.

Measurement-based quantum computing turns this picture on its head. In such a model, there are no coherent manipulations of superpositions. Instead, it’s just all measurements, all the time. In Debbie Leung’s memorable phrase, we compute by “pinging Nature”.

I’ve just written a short pedagogical review of one of these models, the so-called “one-way quantum computer”, or “cluster-state model” of quantum computation. A simple version of this model was recently implemented in the lab, as reported in Nature by the Zeilinger group.

The review is written to be accessible to anyone with a thorough grounding in basic quantum mechanics. The main part of the review is spent explaining what a quantum computer is, what the cluster state model is, and how clusters can be used to simulate an ordinary quantum computer. At the end, I also explain two new results: (1) a no-go theorem which, subject to some caveats (see paper), forbids us from obtaining the cluster experimentally by cooling a physical system to the ground state; and (2) a proof that clusters must be prepared in two or more dimensions if they are to be useful for quantum computing.

The review was written for a festschrift in honour of Tony Bracken and Angas Hurst, two well-known Australian mathematical physicists. Tony was my fourth-year honours thesis advisor, for which he suggested a wonderful topic: whether there is any connection between sometimes-negative “probability” distribution functions, like the Wigner function, and the Bell inequalities. It was a great topic for a student, combining a fundamental physical aspect with beautiful mathematics, and requiring only a little background to get started.

Categorized as General

The so-called breakdown of spacetime

An assertion which is often made is that quantum gravity effects will become important at the Planck length — about [tex]10^{-35}[/tex] meters — and the notion of spacetime will break down at that length. People like to wax lyrical about spacetime turning into some kind of “quantum foam” at that level.

This bugs me. If it really is the case, then why doesn’t the notion of mass doesn’t break down at the Planck mass, which has the rather hefty value of about [tex]10^{-8}[/tex] kilograms? What’s the critical difference between mass and length?

Categorized as General

Finite quantum de Finetti theorems

Attention conservation notice: This post needs knowledge of some elementary quantum mechanics to make sense.

David Poulin gave a seminar today, in which he described some beautiful results in quant-ph/0410229, by Koenig and Renner.

The main result is, I think, quite incredible.

Suppose we have N quantum systems, all with identical state spaces. Suppose rho is some fixed but completely arbitrary quantum state of those N systems.

Now suppose we pick out M of those systems. M is fixed, but we choose the subset of systems completely at random. Let’s call the resulting quantum state when we disregard the remaining systems rho’.

What can we say about rho’? Konig and Renner prove that, to a very good approximation, the state can be written in the form:

(*) int P(tau) tau^{\otimes M} d tau,

where the integral is over single-copy density matrices tau, and P(tau) is a normalized probability distribution.

How good an approximation? They prove that the trace distance between rho’ and the form (*) is at most c M / sqrt{N-M}, where c is some constant. As N goes to infinity, this goes to zero. (This gives rise to the result known as the quantum de Finetti theorem.)

Why is this representation theorem incredible?

From a mathematical point of view, the reason is that states of the form (*) are very special. The way we constructed rho’, it’s not difficult to see that the resulting state ought to be symmetric, but that is only enough to ensure that rho’ has a representation like (*), but possibly with P(tau) taking negative values, or with the tau operators not being density matrices. States of the form (*) are far more special.

There is also a reason that is more physical, almost philosophical, in nature. Imagine we have some very large number of systems, all with identical state spaces, but otherwise arbitrary. For example, consider the set of all electrons in the Universe.

Now pick out a small number of those systems, at random. The representation (*) tells us that to a very good approximation we can imagine that the state of the system was prepared by first sampling from the distribution P(tau), and then prepared M identical copies of tau. Furthermore, this is true, no matter what the original state was! The only way the original state enters the picture is that it controls what the distribution P(.) is.

In the N -> infinity limit all of this has been known for years; in recent years it’s received a lot of attention, due to the work of Caves, Fuchs, Schack and collaborators. I must admit, though, that until David’s talk I hadn’t appreciated how remarkable the results were, perhaps because I am always suspicious of any result involving infinite tensor products.

Categorized as General

Geodesics and the cosmological constant

Attention conservation notice: This post requires a little general relativity to understand the early bits, and quite a bit of general relativity later on. Later in the post I’ve also used some of the munged LaTeX beloved of theorists collaborating by email; hopefully this won’t obscure the main point.

One of the most exciting developments in physics over the past twenty years was the discovery that the cosmological constant (“Einstein’s greatest mistake”) was not, in fact, a mistake, but appears to be very real.

In standard general relativity – the type where the cosmological constant is zero – it can be shown that test particles follow geodesics of spacetime, i.e., locally, they simply fall freely, with relative acceleration between different particles due to spacetime curvature.

(Note that this geodesic behaviour is sometimes presented as though it’s a fundamental assumption of Einstein’s theory. Actually, it’s not – it follows as a consequence of the Einstein field equations, as we’ll see below.)

I got to wondering recently what paths test particles take when the cosmological constant is non-zero. Do they still follow geodesics, or is their behaviour modified?

The answer, which I’m sure is well-known to relativists, is that they still follow geodesics. So even when the cosmological constant is non-zero, particles still fall freely. What does change is the geometry, since the modification of the field equations means that the same distribution of matter will give rise to a different geometry, and thus to different geodesics.

Here’s an outline of the argument deriving geodesic paths, which is a simple variation on the argument given in Dirac’s text on general relativity for the case where the cosmological constant \Lambda = 0.

Start with the field equations, G+ \Lambda g = 8 \pi T. Suppose we evaluate the divergence of both sides. In index notation we obtain:

G^{\mu \nu}_{; \nu} + \Lambda g^{\mu \nu}_{;\nu} = 8 \pi T^{\mu \nu}_{;\nu}

The first term on the left vanishes because of the Bianchi identies, while the second term vanishes because of the metric compatability condition imposed on the connection. This tells us that the divergence of the stress-energy tensor vanishes:

T^{\mu \nu}_{;\nu} = 0

Now, suppose the stress-energy tensor has the form T^{\mu \nu} = \rho v^{\mu} v^{\nu}, where \rho is matter density, and v^\mu is the four-velocity. Using the product rule to evaluate the divergence, and conservation of mass-energy (\rho v^{\mu})_{;\mu} = 0) we end up with the equation v^{\nu} v ^{\mu}_{;\,nu} = 0, which is the geodesic equation.

Categorized as General

Academic ennui

Brad DeLong has a wonderful post on the perils of academic ennui, and how to overcome it: real ideas, hard fought, dearly (though provisionally) held, and people who care. Go read.

Categorized as General

21st century redux

Chad Orzel has made some comments on my post about the relative importance of physics and biology in the 21st century. Chad writes:

The crucial thing about physics in the middle of the last century was not the intellectual revolution that went on in the field, with Relativity and Quantum Mechanics supplanting the classical theories, but rather the material consequences of that revolution. Quantum Mechanics is important not because it forced scientists to re-think our relation to the universe, but because an understanding of quantum theory makes it possible to build devices like transistors and lasers. Relativity is important not because it transformed our understanding of space and time, but because understanding the theory makes it possible to build atom bombs and nuclear power plants. Everything that happened in the latter half of the twentieth century, from the Cold War to the Internet, is in some sense a result of the revolution in physics that took place in the first few decades of that century.

Seen in that way, none of the problems Michael mentions look like they stack up. Yes, a working theory of quantum gravity would be a major revolution in physics. But it doesn’t seem likely to have material consequences for the average person, unless some quirk of the theory makes either free energy or levitation possible. Quantum measurement and cosmology are fascinating topics, and the people who nail them down will richly deserve their Nobel Prizes, but I don’t think either is likely to have results that will re-make the world in the way that the transistor and the atomic bomb did.

The rub is in that phrase: “it doesn’t seem likely to have material consequences”. Neither did problems like resolving the ultraviolet catastrophe (which set in train the events leading to quantum mechanics), or the fact that the symmetry properties of Maxwell’s equations are different from those of ordinary Newtonian mechanics (which helped lead to relativity). However, resolving these problems caused the fundamental changes to our view of the world that enabled all the new technologies that Chad mentions.

It is, of course, an article of faith on my part that understanding quantum gravity, say, or quantum measurement, would have similar unexpected consequences. At present, I can no more predict those consequences than I could have predicted the laser before Planck discovered the first hints of quantum mechanics.

Categorized as General