Quantum Paradoxes

Quantum Theory for the Perplexed
By Yakir Aharonov Daniel Rohrlich

John Wiley & Sons

Copyright © 2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
All right reserved.

ISBN: 978-3-527-40391-2

Chapter One

The Uses of Paradox

On November 9, 1919, The New York Times reported solar eclipse observations confirming a prediction of Einstein's general theory of relativity: rays of starlight bend near the sun. It also reported that when Einstein sent his theory to the publishers, "he warned them that there were not more than twelve persons in the world who would understand it...." Was there a time when only "twelve wise men" understood the general theory of relativity? "I do not believe there ever was such a time," commented Feynman. "There might have been a time when only one man did, because he was the only guy who caught on, before he wrote his paper. But after people read the paper a lot of people understood the theory of relativity in some way or other, certainly more than twelve. On the other hand, I think I can safely say that nobody understands quantum mechanics."

What is the problem with quantum mechanics? It is a spectacularly successful theory. It governs the structure of all matter. Measurements of Planck's constant are accurate to better than a part in a million, and still more accurate measurements confirm predictions of quantum electrodynamics. But along with the spectacular successes of quantum mechanics come spectacular difficulties of interpretation. "Do not keep saying to yourself, if you can possibly avoid it, 'But how can it be like that?"' Feynman continued, "because you will get 'down the drain', into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that."

We can stop asking ourselves, "But how can it be like that?" We may indeed despair of asking a question that Einstein, Schrödinger and Feynman could not answer. But we cannot stop using quantum mechanics. So the problem is that everybody uses quantum mechanics and nobody knows how it can be like that. Our relationship with quantum mechanics recalls a Woody Allen joke:

This guy goes to a psychiatrist and says, "Doc, my brother's crazy - he thinks he's a chicken! And, uh, the doctor says, "Well, why don't you turn him in?" And the guy says, "I would, but I need the eggs!"

We say, "Quantum mechanics is crazy - but we need the eggs!"

Such a relationship with quantum mechanics is paradoxical. In this book, we will not be satisfied to have a paradoxical relationship with quantum mechanics. We will not stop asking, "How can it be like that?" But we will use paradox repeatedly in order to understand quantum mechanics better.

1.1 Paradox in Physics

We will use paradox to probe quantum mechanics. Can paradox be useful? The history of physics shows how useful. As Wheeler put it, "No progress without a paradox!" In this section, we define and classify physics paradoxes; the next sections present examples of each class.

A paradox is an argument that starts with apparently acceptable assumptions and leads by apparently valid deductions to an apparent contradiction. Since logic admits no contradictions, either the apparently acceptable assumptions are not acceptable, or the apparently valid deductions are not valid, or the apparent contradiction is not a contradiction. A paradox is useful because it can show that something is wrong even when everything appears to be right. It does not show what is wrong. But something is wrong - something we thought we understood - and a paradox moves us to reexamine the argument until we find out what is wrong.

We can classify physics paradoxes according to what is wrong. There are three broad classes: "errors", "gaps" and "contradictions".

Many paradoxes arise from errors. An error in logic or in our understanding of the laws of physics easily leads us to an apparent contradiction. Our error may be simple or it may be subtle, but it is just an error; once we recognize it, we have resolved the paradox. What distinguishes the first class is that these paradoxes do not arise from any flaw in the theory. In the special theory of relativity, for example, erroneous assumptions about simultaneity lead us to paradox. (See Sect. 1.2.) Resolving the paradox, we improve our understanding of special relativity, but we do not improve the theory. Another example of a paradox arising from an error is Einstein's clock-in-the-box paradox. (See Sect. 2.4.) Einstein made an error and arrived at an apparent contradiction in quantum theory. The resolution of the paradox came as a surprise, but it did not show quantum theory to be flawed in any way.

Other paradoxes do show a physical theory to be flawed. A gap in physical theory is a flaw. As an example of a gap, consider Wheeler's paradox of black hole entropy. According to the general theory of relativity, nothing can escape a black hole. We, as outside observers, can measure the electric and gravitational fields of a black hole, and hence its charge and mass (and angular momentum); but we have no other access to a black hole. So a black hole at rest has only three properties: charge, mass and angular momentum. Such a simple physical system can hardly have much entropy. Now suppose a complicated physical system, containing a lot of entropy, falls into a black hole. What happens to the entropy? Apparently it vanishes. But vanishing entropy violates the second law of thermodynamics. Wheeler told his student Bekenstein about this paradox:

The idea that a black hole had no entropy troubled me, but I didn't see any escape from this conclusion. In a joking mood one day in my office, I remarked to Jacob Bekenstein that I always feel like a criminal when I put a cup of hot tea next to a glass of iced tea and then let the two come to a common temperature, conserving the world's energy but increasing the world's entropy. My crime, I said to Jacob, echoes down to the end of time, for there is no way to erase or undo it. But let a black hole swim by and let me drop the hot tea and the cold tea into it. Then is not all evidence of my crime erased forever? This remark was all that Jacob needed. Bekenstein proposed that a black hole has entropy proportional to the square of its mass. If any physical system falls into a black hole, the mass of the black hole increases - and hence the entropy. He demonstrated that the increase in entropy is at least as great as the entropy of the infalling system, thus corroborating the second law and resolving Wheeler's paradox.

Wheeler's paradox indicated a flaw - but not a fatal flaw - in general relativity and thermodynamics. The resolution of the paradox did not invalidate either theory. The apparent contradiction between the theories arose from a gap in thermodynamics - we didn't know how to extend the concept of entropy to black holes - and Bekenstein's proposal filled the conceptual gap. Another paradox in the second class came, in turn, from Bekenstein's proposal: if thermodynamics extends to black holes, then black holes must emit as well as absorb heat. But nothing can escape a black hole! This paradox, too, arose from a conceptual gap, as Hawking discovered: one consequence of the uncertainty principle is that black holes radiate. Many such paradoxes appear in this book.

A contradiction in physical theory is a fatal flaw. Paradoxes in the third class are associated with revolutions in physics, because they indicate that the physical theory behind the paradox is wrong. Bohr faced such a paradox in 1911. In that year, Rutherford reported experiments on neutral atoms, showing that the positive charges in atoms - but not the negative charges (electrons) - are concentrated in nuclei. According to classical theory, such atoms should be unstable: like all accelerating charges, the electrons should radiate energy, and fall into the nuclei. Matter should collapse in a split second. So why is matter stable? Bohr realized that this paradox had no resolution in classical physics. Only a new physical theory - quantum theory - could resolve it. The only resolution was a revolution.

The paradox arose for Bohr as a contradiction between physical theory and experiment. Especially useful are paradoxes that arise as contradictions within physical theory. Such a paradox can show that a physical theory is wrong even when no experiment contradicts it. The paradox then starts us searching for a new theory. (See Sects. 1.4 and 2.2.)

1.2 Errors

Every student of special relativity encounters the Twin Paradox. Here is a Triplet Paradox. Dumpy, Grumpy and Jump - identical triplets wearing synchronized wristwatches - once lived together happily at home. But Grumpy got mad at Dumpy and decided to move to another city. When he arrived, his watch was still synchronized with his brothers' watches, because he travelled very slowly compared to the speed of light. (In this paradox we set the speed of light to 1000 m/s.)

A month later, Jump decided to visit Grumpy. Dumpy accompanied Jump to the train station, and Jump took a seat in the train. Then the train accelerated, within a second, to 100 m/s. At the end of this second, Jump's cabin passed Dumpy on the platform. Jump and Dumpy glanced at each other through a cabin window and noticed that their watches still showed the same time (to within a second). Hence Jump did not age appreciably during the acceleration. For the rest of the trip, the train's speed and direction were constant. When it arrived, it stopped within a second.

Dumpy and Grumpy expected that Jump would be slightly younger than them when he arrived, and that his watch would lag behind their watches, for Jump had been moving fast relative to them. But Jump expected the opposite: Dumpy and Grumpy would be slightly younger, and their watches would lag behind his. He told himself, "After one second of acceleration, Dumpy's watch and mine showed the same time (to within a second); and Dumpy's and Grumpy's watches were still synchronized. Afterwards, the inertial reference frame of Dumpy and Grumpy moved fast relative to mine; so time passed more slowly for them, and their watches now lag behind mine." When he arrived, he discovered that his watch lagged Grumpy's by about half a minute! On the one hand, Jump's expectation should be just as correct as that of his brothers; there can be no preferred frame in special relativity. On the other hand, Jump and his brothers cannot all be correct. So special relativity contradicts itself!

The Triplet Paradox belongs to the class of errors in that it does not arise from any flaw or misconception in the special theory of relativity. It arises, rather, from incorrect intuition. We can often use paradoxes in this class to improve our intuition.

1.3 Gaps

In 1856, Clausius stated the second law of thermodynamics as follows: heat cannot flow from a colder body to a hotter body without an accompanying process (i.e. work). Clausius regarded the second law as exact, and tried to derive it from the laws of mechanics. In 1871 he published a paper in which he offered a mechanical explanation of the second law. He did not know that Boltzmann had published much the same explanation five years earlier. Boltzmann (who, like Clausius, regarded the second law as exact) was quick to claim priority. Yet Clausius did not wholly concede. Maxwell was amused. "But it is rare sport to see those learned Germans contending for the priority of the discovery that the 2nd law of [theta][DELTA]cs is the Hamiltonische Princip...." he wrote. "The Hamiltonische Princip, the while, soars along in a region unvexed by statistical considerations....". Boltzmann and Clausius were both wrong. The second law has no mechanical explanation; it is statistical.

What made Maxwell so sure that the second law is statistical? In 1859 he had calculated that the distribution of molecular speeds in any gas, hot or cold, would range from zero to infinity. (Molecules were still an untested hypothesis at the time.) In 1867 he had considered the following thought experiment. Gas fills a sealed, insulated box, divided by a diaphragm. The gas is hot on one side of the diaphragm and cold on the other side; yet there are fast molecules in the cold gas and slow molecules in the hot gas. "Now conceive a finite being who knows the paths and velocities of all the molecules by simple inspection but who can do no work except open and close a hole in the diaphragm by means of a slide without mass." The being opens and closes the hole in such a way that fast molecules in the cold gas enter the hot gas and slow molecules in the hot gas enter the cold gas. Energy gradually flows from the cold gas to the hot gas. After many molecules have crossed through the hole, "the hot system has got hotter and the cold colder and yet no work has been done, only the intelligence of a very observant and neat-fingered being has been employed". The "neat-fingered being" soon had a name: "Maxwell's demon".

Maxwell's demon violates the second law of thermodynamics, as formulated by Clausius: it does no work, yet it causes heat to flow from a cold gas to a hot gas. It does not, however, violate the laws of mechanics. Hence the second law cannot be a mechanical law. Maxwell's thought experiment was a paradox for Clausius's formulation; it does not disprove the second law, but it shows that the second law can only be a statistical law.

Another formulation of the second law states that the entropy of a closed system always tends to increase to thermal equilibrium. But this formulation, too, leads to a paradox. It assumes an arrow of time, relative to which entropy tends to increase. But what if there is no arrow of time? What if the "arrow of time" is no more intrinsic than the "arrow of space" defined by gravity? (See Fig. 1.1.) Suppose that two sealed, insulated boxes are filled with gas, e.g. helium in one box and neon in the other, and at time t = 0, neither gas is at thermal equilibrium. Now on the one hand, if the boxes are perfectly insulated, they could contain two opposite arrows of time. Assume that the gases have contrary evolutions: the entropy of the neon increases in time while the entropy of the helium decreases in (the same) time. Such an assumption is plausible since the laws of mechanics are invariant under time reversal and the boxes do not interact. On the other hand, suppose the boxes do interact, with an interaction that is independent of time; assume that the position and momentum of each atom at t = 0 is the same as before. According to the second law, the combined entropy of the two gases always tends to increase; that is, any perturbation of the helium atoms, however small, will destroy the precise coordination of their positions and momenta that allows their entropy to decrease. So in the evolution of the two gases after t = 0, their total entropy increases. But the same reasoning applies in reverse to the evolution of the gases before t = 0: their total entropy must decrease until t = 0. Extrapolation forwards from t = 0 implies that the neon (with its increasing entropy) overwhelms the helium; extrapolation backwards from t = 0 implies that the helium overwhelms the neon. This paradox shows that the second law contains no arrow of time. (See also Chap. 10.)

The second law is almost exact, i.e. the probability of a significant violation is very small. Maxwell's demon can violate the second law, yet the probability of a significant violation is very small. Still, after Maxwell, the demon turned up in new paradoxes. The demon kept turning up, because it is easier to imagine a demon that can violate the second law significantly, than to prove that it can't. For example, in Fig. 1.2 the demon is a trapdoor that apparently allows only fast molecules of the cold gas to enter the hot gas. In 1914, Smoluchowski showed that this demon fails to violate the second law significantly because the trapdoor itself thermalizes, eventually opening and closing in random fluctuations. More recent paradoxes allow Maxwell's demon to measure and compute. Their resolution involves an application of information theory to thermodynamics.

All the paradoxes in this section belong to the class of gaps; they show up flaws or gaps in how we understand the second law, but do not invalidate it. The resolutions of these paradoxes correct our formulation of the second law and extend the concepts we use to apply it, but do not contradict the formalism of thermodynamics.


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