Relativity and Charge Quantization

April 17, 2007

In textbook derivations of the Lorentz transformations for electric and magnetic fields from relativistic kinematics and “rest-frame” E-M laws, either one of two assumptions is used: that charge is conserved in all frames of reference, or that charge is quantized. I see no intuitive reason to favour the former, and most texts I’ve flipped through have used the latter. As it turns out, the latter implies the former, because if charge is quantized, then the total charge of any system must be an integer multiple of the smallest unit of charge, and integers are scalar invariants under Lorentz transformations, so the total charge must also be a Lorentz invariant. In other words, quantization of charge + Lorentz invariance => conservation of charge. Using conservation of charge, you get \rho_0 dV_0 = \rho dV, which, together with Lorentz contraction of space together and \mathbf{J} = \rho \mathbf{v}, implies that J_\mu is the product of a scalar invariant and the four-velocity, and hence is a four-vector itself. That allows one to go on to prove that A_\mu is a four-vector, blah blah.

What bothered me about this was that it seems counterintuitive that special relativistic electrodynamics is somehow dependent on the quantization of charge. Other than this mathematical requirement for justifying how the Lorentz transformations are a ‘derivation’ from ‘normal’ electrodynamics + relativistic kinematics, there doesn’t seem to be anything in electrodynamics itself that suggests it would work better with discrete rather than continuous charges. If anything, those damn differential equations cry out for lovely continuous flowing ribbons of charge rather than discrete blobs. A more plausible dependency scenario to me would be that of electrodynamics being directly dependent on the Lorentz invariance of charge. In which case, why don’t they just derive the Lorentz transformations using that assumption, rather than go through the more onerous* and (in my eyes) somewhat physically artificial route of “quantization, therefore charge conservation”? Could it be that like me, said textbook authors think that taking charge as a Lorentz invariant quantity as a fundamental assumption is not a sufficiently justifed move?

Another problem with that could, I suppose be dismissed as a “mere” historical problem. The quantization of electric charge was demonstrated only in 1913, eight years after Einstein had put forward special relativity. Presumably, special relativity had been integrated into electrodynamics before that (or, to be precise, people had worked out the details of how electromagnetism was inherently a relativistic theory). I suppose they used the (now judged to be unsatisfactory) Lorentz invariance of total charge as an assumption for that? And this new reason they give in modern textbooks is one of those things massaged in for pedagogical purposes, even though it was not the original reasoning through which the Lorentz transformations were derived?

I know nothing about particle theory, so I expect that someone is going to come along anytime now and say something like “you dumb shit, the relationship between charge quantization and special relativity is a simple consequence of electroweak unification”.

*Wangsness points out in his text that the inference from quantization to charge conservation depends not just on integers being Lorentz invariants, but also on the basic unit of charge being a Lorentz invariant.

Update: I am told that the textbook derivations are not, in fact, derivations, but justifications on the side for something we think should be true for other reasons. The reply M. O. gave was somewhat unclear, but he agreed when I paraphrased it as follows: we must make an assumption of some sort (in this case, charge invariance), in order to get into the logical circle of showing that the relevant quantities are four-vectors. My vague idea is that we can’t even empirically prove that charge is Lorentz invariant unless we already have the formulae for the Lorentz transformations of E and/or B. But we need the invariance of charge to ‘derive’ those transformations. So we have to assume something, and invariance of charge, I suppose, is as good a choice as any, and certainly better (more intuitively immediate) than assuming the specific four-vectors or the E and B tensor. (But, I maintain, quantization of charge is not — why do so many textbook authors use that? M. O. suggested that perhaps someone famous had used it.)

I wish there were “physics for philosophers” textbooks along the lines of “physics for poets” books. But nevermind the probable lack of market demand, there is a serious lack of people who are qualified to write those books. Most physicists certainly aren’t.

Monty Python: The German Programmes

April 14, 2007

No, they’re not just translated. I do not remember a running joke on Albrecht Duerer in the English version. And even the parts they reused (like the Lumberjack Song) were done with the actors clearly mouthing German.

They are here, found via Tyler Cowen.

Brendel on Comic Music

April 14, 2007

My depressive browsing through the incomparable Powell’s last night yielded this gem of a collection of essays by Alfred Brendel. From his essay on the ‘seriousness’ of classical music:

For most performers and virtually all concert audiences of our time, music is an entirely serious business. Performers are meant to function as heroes, dictators, poets, seducers, magicians, or helpless vessels of inspiration. The projection of comical music needs a performer who dares to be less than awe-inspiring, and does not take him- or herself too seriously. Comic music can be ruined, and made completely meaningless, by ‘serious’ performance. It is much more dependent on a performer’s understanding than an Allegro di bravura, a nocturne, or a funeral march. To manage to play a piece humorously is a special gift, yet, I am afraid, it is not enough: the public, expecting the celebration of religious rites, may not notice that something amusing is going on unless it is visibly encouraged to be amused.

Which explains Brendel’s facial contortions the last and only time I saw him in concert. More:

I admit that to expect a player to radiate amusement while performing is a tall order. The trouble is that many performers, on account of their concentration and nervous tension, look unduly grave or grim, no matter what they play. The first bars of a classical piece sets its mood. To sit down and start Haydn’s last C major Sonata with a tortured look is even worse than to embark on the so-called ‘Moonlight’ Sonata with a cheerful smile. Nobody will mistake the first movement of the ‘Moonlight’ for a cheerful piece,whereas the hilarious beginning of Haydn’s C major Sonata can easily sound wooden, and pointless. Before the first note, a discreet signal has to pass from the performer to the audience: ‘Caution! We are out for mischief.’

[Here he quotes an excerpt from the C major sonata]

When the English notion of ‘humour’ arrived in Germany, Lessing translated it as ‘Laune’. Laune, according to Kant, means, in its best sense, ‘the talent voluntarily to put oneself into a certain mental disposition, in which everything in judged quite differently from the ordinary method (reversed, in fact), and yet in accordance with certain rational principles in such a frame of mind’. This sounds to me like an apt description of the quality that a performer of comical music should be able to summon up. ‘But this manner,’ as Kant further says, ‘belongs rather to pleasant than to beautiful art, because the object of the latter must always show a certain dignity in itself…’*

For my part, I am perfectly happy to enjoy the ‘sublime in reverse’, and leave Kant’s dignity behind where Haydn and Beethoven took such obvious pleasure in doing so.

*Immanuel Kant: Critique of Judgment (1790) 54

Further proof that Kant is a humourless bastard. Proof that Brendel is not:

Update: After trying to be charitable towards them, I’ve sadly had to conclude that Pletnev and Richter don’t come anywhere close to Brendel’s humour for that piece. I resorted to buying Brendel’s recording of it off iTunes, from this recital. I was rather surprised that such accomplished pianists couldn’t master that movement, especially since Haydn marks clearly where the pauses, sforzandos and ritardandos should be. Yet Pletnev and Richter, at many points, just seem to ignore those.

Eliminating Initial Conditions — Or Not

April 11, 2007

Max Tegmark has a provocative paper up on the arXiv claiming that if there exists an external physical reality independent of humans, then that external physical reality is a mathematical structure. There’s too much in it to analyse in one blog post, so for now I’ll just comment on one section. Tegmark argues that if external physical reality is a mathematical structure (this statement he labels the Mathematical Universe Hypothesis — MUH), then physics no longer needs to explain initial conditions.

Just why initial conditions are undesirable is an assertion that itself deserves explanation. Tegmark explains the usual stance of physicists:

The traditional view of these matters is eloquently summarized by e.g. [3, 35] as splitting our quantitative description of the world into two domains, “laws of physics” and “initial conditions”. The former we understand and hail as the purview of physics, the latter we lack understanding for and merely take as an input to our calculations.

As Houtappel, van Dam and Wigner write in [35], initial conditions “are complicated and no accurate regularity has been discovered in them”. Physical laws explain the regular, so initial conditions are the things that fall outside the purview of physics. Since physicists like to think they can explain everything, this is a problem.

Taking a typical reductionist tack, Tegmark then explains how our increasing knowledge of the so-called fundamental laws of physics has led to the borderline between initial conditions and physical laws shifting at the expense of initial conditions. He claims that the MUH would complete this historical trend: “The MUH leaves no room for ‘initial conditions’, eliminating them altogether.” What he means by eliminating physical conditions, however, is somewhat confusing. He illustrates his point with the following diagram:
Tegmark’s inital conditions
As examples of the shifting boundary between initial conditions and physical laws, he describes various scientific revolutions in which what had been accepted as fundamental laws were reclassified as initial conditions. Thus Ptolemy thought that the circularity of orbits was a fundamental laws, but Newton reclassified that as an initial condition — gravitational orbits can be non-circular, so circularity is not a regularity that can be accounted for by a general physical law. Thus too, he claims, the string theory landscape, together with inflation, reclassifies what we now think of as fundamental physical laws (the weak force, the electromagnetic force, etc.) as initial conditions.

I say that this is a confusing formulation of his point, because this way, it would seem that we are conceding that more aspects of our universe are initial conditions rather than physical laws. In what sense, then, can the borderline between intial conditions and physical laws be shifting “at the expense of initial conditions”? In what sense can we be eliminating physical conditions when we move to the ultimate Theory of Everything of (say) the string theory landscape? If, as originally formulated, initial conditions are the part of the universe that physics cannot explain, then wouldn’t the physical-laws-as-initial-conditions theory mean that we are essentially wringing our hands and conceding that physics cannot go any further?

Let’s look more closely at why Tegmark thinks that a landscape theory eliminates initial conditions. It does so in a somewhat linguistically paradoxical fashion, so that one suspects that the tension between eliminating initial conditions and turning all laws into intial conditions stems from that inherent paradox. A landscape theory eliminates initial conditions by postulating that a range of different physical laws persists in different parts of the universe. Each part is communicationally isolated from the others via inflation, so we would never encounter signs of another part of the universe where, say, the fine structure constant has a different value from the one we love and know to 12 significant figures. The impossibility of detecting them notwithstanding, a landscape theory would claim that variants of the physical laws we know hold in other parts of the universe. Naively, one would think that this seems to only add to the problem: instead of eliminating initial conditions, we are now saying that many possible initial conditions exist! But that’s exactly the point, and it’s where the issue is badly formulated: The problem was not that any initial conditions at all exist, but why one set of initial conditions, as opposed to other sets, happens to exist in our universe. By saying that they all exist, we don’t have to tackle that problem, since (we then claim, somewhat controversially) ascribing an existent status to all of them means we aren’t discriminating between any of the possible sets of initial conditions. Thus the landscape theory would eliminate the need to explain the existence of a particular set of initial conditions as opposed to other sets. This is what Tegmark means by the MUH “leaving no room for ‘initial conditions'”.

This ties into Tegmark’s description of physics as progressively reclassifying fundamental physical laws as initial conditions. He is essentially saying that physics is progressively removing the need for an explanation for why certain physical laws pertain as opposed to others. In this view of physics, we can think of theoretical physics as an exercise in answering questions of the form “Why does law X pertain as opposed to some other law?” As we move to more “general” physical laws, we see that law X is only a special consequence of law Y, and law Y is only a special consequence of law Z, and so on. Before a given law is described as a consequence of some other more general laws, that law cannot be explained within physics — it must be accepted, as it were, as something analogous to a physical initial condition. This is how I think Tegmark ends up saying that the MUH eliminates initial conditions, rather than that it eliminates the need to explain why one particular set of initial conditions pertains. It’s because he is treating the fundamental laws as initial conditions of a sort — as things that cannot, at this time, be explained within physics. And it is true (maybe not, actually, but I don’t want to get into that argument) that the MUH eliminates laws-acting-as-initial-conditions by showing that those laws are actually consequences of some more encompassing set of laws.

Perhaps I’m cooking up a trivial linguistic error out of nothing, but it struck me as exceedingly sloppy prose to speak of progress as simultaneously:
1) eliminating initial conditions, and
2) reclassifying more laws as initial conditions.
Not to mention first making a firm distinction between laws and initial conditions, and then going on to treat laws as though they were initial conditions to be eliminated.

Further Confusion in Gell-Mann and Hartle’s Olbers’ Paradox

April 10, 2007

I finally found out where I misunderstood Huw Price’s argument against Gell-Mann and Hartle’s argument against the Gold universe. In short, I mistook his diagram for a time diagram: x in the diagram below is position, not time, as I should have guessed.
Price Olbers
I will now restate his argument. Gell-Mann and Hartle* claim that in a Gold universe, by time-symmetry, any radiation that makes it into the final singularity should be emitted at the big bang. Since the universe appears to be mostly transparent, a lot of radiation should make it to the final singularity, so we should see a corresponding amount of radiation emitted from the big bang. Since we do not, we do not live in a Gold universe. Price counters as follows: Suppose we are looking in the spatial direction -x at time -t, and we observe radiation emitted from the big bang due to the Gell-Mann-Hartle hypothesis. The time-symmetric counterpart of that radiation is radiation being emitted from the back of our eye at time t and subsequently travelling all the way to the final singularity without being absorbed. So the only radiation that could contribute to Gell-Mann and Hartle’s paradox is that which is emitted from the back of our eye (or telescope), which, as far as we know, is very little. In other words, if we are to observe Gell-Mann-Hartle radiation, then that radiation cannot be from the usual stars we observe, so we have no reason to believe that a large quantity of that radiation reaches the final singularity. Our act of observation gets in the way of the observable radiation’s time-symmetric counterpart reaching the final singularity. Therefore, although there might be plenty of extra radiation in a Gold universe, we would not be able to observe it.

A close reading of the Gell-Mann-Hartle paper, though, reveals an inconsistency in their argument — an inconsistency that Price seems to have missed as well. They write:

Suppose the universe to have initial and final classical distributions that are time-symmetric in the sense of (22.13).** Suppose further that these boundary conditions imply with high probability an initial epoch with stars in galaxies distributed approximately homogeneously and a similar final epoch of stars in galaxies at the symmetric time. Consider the radiation emitted from a particular star in the present epoch. If the universe is transparent, it is likely to reach the final epoch without being absorbed or scattered. There it may either be absorbed in the stars or proceed past them towards the final singularity. If a significant fraction of the radiation proceeds past, then by time-symmetry we should expect a corresponding amount of radiation to have been emitted from the big bang.

But if the initial and final epochs are just time-reversed versions of each other, then shouldn’t radiation emitted from a star in the present epoch necessarily be radiation absorbed by the star in the final epoch? How can radiation emitted from a star in the present epoch proceed to the final singularity in a truly symmetric Gold universe?

The answer, I’m afraid, doesn’t absolve Gell-Mann and Hartle from confusion, although it does show that Price, too, swallows the point that confuses them. It turns out that (22.13) is a statement about the statistical distribution of systems in an ensemble (a la statistical mechanics) in phase space: \rho^{cl} is the distribution function of systems over phase space, and \mathscr{T}^{-1} is the time-reversal operator. The time symmetry of the Hamiltonian entails that the initial and final distribution functions must be time reversals of each other — hence \rho^{cl}_f \left(q_t, p_t\right) = \mathscr{T}^{-1} \rho^{cl}_i \left(q_t, p_t\right). In Gell-Mann and Hartle’s words:

The entropy of the final distribution must be the same as the initial one. The thermodynamic arrow of time will run backwards on one side of the moment of time symmetry as compared to the other side. This does not mean, of course, that the histories of the ensemble need be individually time-symmetric… There would be appear to be no principle, for example, forbidding us to live on into the recontracting phase of the universe and see it as recontracting.

(They have a similar quantum mechanical argument that allows for a CPT-symmetric universe that has sets of histories in which individual members are CPT-asymmetric). Herein is where I think Price has misinterpreted Gell-Mann and Hartle (for real this time!). When Gell-Mann and Hartle speak of time reversals, they mean time reversals of the distributions of possible configurations of the universe over phase space. They do not mean that everything that occurs in any given universe will occur in reverse as the universe recontracts. This allows them to postulate radiation from present stars reaching the final singularity without the exact temporal reversal of that happening: without radiation from the big bang converging on the time-reversed analogues of present stars. But it also disallows them from arguing that a convergence of radiation at the final singularity implies a corresponding emission of radiation from the initial singularity. Similarly, Price cannot argue that any radiation we observe as being emitted from the big bang must be due to radiation emitted from the back of our eyes, since the radiation we observe as emitted from the big bang need not have an exact time-reversed analogue in our universe (or, quantum mechanically, in the history of the universe that we happen to be observing). Ultimately, Price isn’t wrong in taking them to be assuming that the history of our universe is time-symmetric, since they indeed do so implicitly. But that assumption does not follow from their statistical or quantum mechanical arguments.

To summarise:
1. Gell-Mann and Hartle’s “paradox” is constructed on the basis of an assumption that does not follow from their arguments in the quantum cosmology paper.
2. Price’s answer takes that assumption for granted as well, and shows that their argument is not compatible even with that. Which, on hindsight, is unsurprising, since Gell-Mann and Hartle don’t even apply their assumption consistently: they allow for radiated starlight to be a time-asymmetric feature of the universe, but do not allow for radiation from the big bang and the big crunch to be time-asymmetric.

*Gell-Mann, M. and Hartle, J. 1994: “Time Symmetry and Asymmetry in Quantum Mechanics and Quantum Cosmology,” in Halliwell, Perez-Mercader, and Zurek (1994), pp. 311-45.
**(22.13) reads \rho^{cl}_f \left(q_t, p_t\right) = \mathscr{T}^{-1} \rho^{cl}_i \left(q_t, p_t\right)

To Textbook Authors

April 9, 2007

For heaven’s sake, do not introduce a new, physically significant equation by saying “From Eqns 1-98, 5-24 and 11-38, we get [new equation]“. I am quite unable to follow Wangsness’ introductory E&M text because he does this so frequently. Even if one reads it sequentially, so that one has indeed encountered equations 1-98, 5-25 and 11-38 before getting to the new equation, one does not remember what they are. So one has to flip back to several previous chapters, and when one looks up equation 11-38, one finds that it in turn refers back to equations 10-56, 2-11 and whatnot. For this reason Wangsness’ book is absolutely useless if one wants a quick physical explanation of anything.

Compare this with Griffiths, who prefaces the presentation of each new equation with a lucid description of its physical meaning. I got through about half of the entire Griffiths text in the time it took me to get through one chapter of Wangsness, simply because Griffiths doesn’t require me to go sleuthing about for the justification of each equation. Also, I am much more willing to swallow the correctness of an equation if you offer me a physical justification for it there and then, instead of telling me that if you shift the quantities in some other equation about this way and take their curl, blah blah, you will get this equation.

This ties in with my general frustration about the mechanical computation that characterises most of my physics courses. I cannot watch someone calculate things for more than twenty minutes without falling asleep. More concepts, please, and less of this mechanical drilling on how we should shift variables about according to certain rules of thumb. There is no physical reasoning involved in all that. Nor any mathematical reasoning, even.

Joshua Bell Busks

April 7, 2007

The Washington Post gets Joshua Bell to busk in the D. C. metro. They asked Leonard Slatkin for his opinion on the idea:

“Let’s assume,” Slatkin said, “that he is not recognized and just taken for granted as a street musician . . . Still, I don’t think that if he’s really good, he’s going to go unnoticed. He’d get a larger audience in Europe . . . but, okay, out of 1,000 people, my guess is there might be 35 or 40 who will recognize the quality for what it is. Maybe 75 to 100 will stop and spend some time listening.”

So, a crowd would gather?

“Oh, yes.”

And how much will he make?

“About $150.”

Thanks, Maestro. As it happens, this is not hypothetical. It really happened.

“How’d I do?”

We’ll tell you in a minute.

“Well, who was the musician?”

Joshua Bell.


It would spoil the story to reveal now how much attention Bell got. So I am putting my opinions on the result of that experiment under the fold.

Read the rest of this entry »


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