量子力学的困惑

温伯格

这是温伯格教授刚刚发表在纽约书评杂志上的文章，表达了他对目前量子力学的看法。去年温伯格教授答应为我主编的文集《Collapse of the Wave Function — Models, Origin and Ontology》（剑桥大学出版社，2018）写一篇文章。可最近通过电子邮件（weinberg@physics.utexas.edu）联系不上他。有那位老师知道温伯格教授的其它联系方式或能够联系上他，请告知。谢谢！

 (中译文在后面)

The Trouble with Quantum Mechanics

Steven Weinberg

JANUARY 19, 2017 ISSUE

Eric J. Heller

    The physicist Eric J. Heller’s Transport XIII (2003), inspired by electron flow experiments conducted at Harvard. According to Heller, the image ‘shows two kinds of chaos: a random quantum wave on the surface of a sphere, and chaotic classical electron paths in a semiconductor launched over a range of angles from a particular point. Even though one is quantum mechanical and the other classical, they are related: the chaotic classical paths cause random quantum waves to appear when the classical system is solved quantum mechanically.’

    The development of quantum mechanics in the first decades of the twentieth century came as a shock to many physicists. Today, despite the great successes of quantum mechanics, arguments continue about its meaning, and its future.

1.

The first shock came as a challenge to the clear categories to which physicists by 1900 had become accustomed. There were particles—atoms, and then electrons and atomic nuclei—and there were fields—conditions of space that pervade regions in which electric, magnetic, and gravitational forces are exerted. Light waves were clearly recognized as self-sustaining oscillations of electric and magnetic fields. But in order to understand the light emitted by heated bodies, Albert Einstein in 1905 found it necessary to describe light waves as streams of massless particles, later called photons.

    Then in the 1920s, according to theories of Louis de Broglie and Erwin Schrödinger, it appeared that electrons, which had always been recognized as particles, under some circumstances behaved as waves. In order to account for the energies of the stable states of atoms, physicists had to give up the notion that electrons in atoms are little Newtonian planets in orbit around the atomic nucleus. Electrons in atoms are better described as waves, fitting around the nucleus like sound waves fitting into an organ pipe.1 The world’s categories had become all muddled.

    Worse yet, the electron waves are not waves of electronic matter, in the way that ocean waves are waves of water. Rather, as Max Born came to realize, the electron waves are waves of probability. That is, when a free electron collides with an atom, we cannot in principle say in what direction it will bounce off. The electron wave, after encountering the atom, spreads out in all directions, like an ocean wave after striking a reef. As Born recognized, this does not mean that the electron itself spreads out. Instead, the undivided electron goes in some one direction, but not a precisely predictable direction. It is more likely to go in a direction where the wave is more intense, but any direction is possible.

    Probability was not unfamiliar to the physicists of the 1920s, but it had generally been thought to reflect an imperfect knowledge of whatever was under study, not an indeterminism in the underlying physical laws. Newton’s theories of motion and gravitation had set the standard of deterministic laws. When we have reasonably precise knowledge of the location and velocity of each body in the solar system at a given moment, Newton’s laws tell us with good accuracy where they will all be for a long time in the future. Probability enters Newtonian physics only when our knowledge is imperfect, as for example when we do not have precise knowledge of how a pair of dice is thrown. But with the new quantum mechanics, the moment-to-moment determinism of the laws of physics themselves seemed to be lost.

    All very strange. In a 1926 letter to Born, Einstein complained: Quantum mechanics is very impressive. But an inner voice tells me that it is not yet the real thing. The theory produces a good deal but hardly brings us closer to the secret of the Old One. I am at all events convinced thatHe does not play dice.2

    As late as 1964, in his Messenger lectures at Cornell, Richard Feynman lamented, “I think I can safely say that no one understands quantum mechanics.”3 With quantum mechanics, the break with the past was so sharp that all earlier physical theories became known as “classical.”

    The weirdness of quantum mechanics did not matter for most purposes. Physicists learned how to use it to do increasingly precise calculations of the energy levels of atoms, and of the probabilities that particles will scatter in one direction or another when they collide. Lawrence Krauss has labeled the quantum mechanical calculation of one effect in the spectrum of hydrogen “the best, most accurate prediction in all of science.”4 Beyond atomic physics, early applications of quantum mechanics listed by the physicist Gino Segrè included the binding of atoms in molecules, the radioactive decay of atomic nuclei, electrical conduction, magnetism, and electromagnetic radiation.5 Later applications spanned theories of semiconductivity and superconductivity, white dwarf stars and neutron stars, nuclear forces, and elementary particles. Even the most adventurous modern speculations, such as string theory, are based on the principles of quantum mechanics.

    Many physicists came to think that the reaction of Einstein and Feynman and others to the unfamiliar aspects of quantum mechanics had been overblown. This used to be my view. After all, Newton’s theories too had been unpalatable to many of his contemporaries. Newton had introduced what his critics saw as an occult force, gravity, which was unrelated to any sort of tangible pushing and pulling, and which could not be explained on the basis of philosophy or pure mathematics. Also, his theories had renounced a chief aim of Ptolemy and Kepler, to calculate the sizes of planetary orbits from first principles. But in the end the opposition to Newtonianism faded away. Newton and his followers succeeded in accounting not only for the motions of planets and falling apples, but also for the movements of comets and moons and the shape of the earth and the change in direction of its axis of rotation. By the end of the eighteenth century this success had established Newton’s theories of motion and gravitation as correct, or at least as a marvelously accurate approximation. Evidently it is a mistake to demand too strictly that new physical theories should fit some preconceived philosophical standard. 

    In quantum mechanics the state of a system is not described by giving the position and velocity of every particle and the values and rates of change of various fields, as in classical physics. Instead, the state of any system at any moment is described by a wave function, essentially a list of numbers, one number for every possible configuration of the system.6If the system is a single particle, then there is a number for every possible position in space that the particle may occupy. This is something like the description of a sound wave in classical physics, except that for a sound wave a number for each position in space gives the pressure of the air at that point, while for a particle in quantum mechanics the wave function’s number for a given position reflects the probability that the particle is at that position. What is so terrible about that? Certainly, it was a tragic mistake for Einstein and Schrödinger to step away from using quantum mechanics, isolating themselves in their later lives from the exciting progress made by others.

2.

Even so, I’m not as sure as I once was about the future of quantum mechanics. It is a bad sign that those physicists today who are most comfortable with quantum mechanics do not agree with one another about what it all means. The dispute arises chiefly regarding the nature of measurement in quantum mechanics. This issue can be illustrated by considering a simple example, measurement of the spin of an electron. (A particle’s spin in any direction is a measure of the amount of rotation of matter around a line pointing in that direction.)

    All theories agree, and experiment confirms, that when one measures the amount of spin of an electron in any arbitrarily chosen direction there are only two possible results. One possible result will be equal to a positive number, a universal constant of nature. (This is the constant that Max Planck originally introduced in his 1900 theory of heat radiation, denoted h, divided by 4π.) The other possible result is its opposite, the negative of the first. These positive or negative values of the spin correspond to an electron that is spinning either clockwise or counter-clockwise in the chosen direction.

    But it is only when a measurement is made that these are the sole two possibilities. An electron spin that has not been measured is like a musical chord, formed from a superposition of two notes that correspond to positive or negative spins, each note with its own amplitude. Just as a chord creates a sound distinct from each of its constituent notes, the state of an electron spin that has not yet been measured is a superposition of the two possible states of definite spin, the superposition differing qualitatively from either state. In this musical analogy, the act of measuring the spin somehow shifts all the intensity of the chord to one of the notes, which we then hear on its own.

    This can be put in terms of the wave function. If we disregard everything about an electron but its spin, there is not much that is wavelike about its wave function. It is just a pair of numbers, one number for each sign of the spin in some chosen direction, analogous to the amplitudes of each of the two notes in a chord.7 The wave function of an electron whose spin has not been measured generally has nonzero values for spins of both signs.

    There is a rule of quantum mechanics, known as the Born rule, that tells us how to use the wave function to calculate the probabilities of getting various possible results in experiments. For example, the Born rule tells us that the probabilities of finding either a positive or a negative result when the spin in some chosen direction is measured are proportional to the squares of the numbers in the wave function for those two states of the spin.8

    The introduction of probability into the principles of physics was disturbing to past physicists, but the trouble with quantum mechanics is not that it involves probabilities. We can live with that. The trouble is that in quantum mechanics the way that wave functions change with time is governed by an equation, the Schrödinger equation, that does not involve probabilities. It is just as deterministic as Newton’s equations of motion and gravitation. That is, given the wave function at any moment, the Schrödinger equation will tell you precisely what the wave function will be at any future time. There is not even the possibility of chaos, the extreme sensitivity to initial conditions that is possible in Newtonian mechanics. So if we regard the whole process of measurement as being governed by the equations of quantum mechanics, and these equations are perfectly deterministic, how do probabilities get into quantum mechanics?

    One common answer is that, in a measurement, the spin (or whatever else is measured) is put in an interaction with a macroscopic environment that jitters in an unpredictable way. For example, the environment might be the shower of photons in a beam of light that is used to observe the system, as unpredictable in practice as a shower of raindrops. Such an environment causes the superposition of different states in the wave function to break down, leading to an unpredictable result of the measurement. (This is called decoherence.) It is as if a noisy background somehow unpredictably left only one of the notes of a chord audible. But this begs the question. If the deterministic Schrödinger equation governs the changes through time not only of the spin but also of the measuring apparatus and the physicist using it, then the results of measurement should not in principle be unpredictable. So we still have to ask, how do probabilities get into quantum mechanics?

    One response to this puzzle was given in the 1920s by Niels Bohr, in what came to be called the Copenhagen interpretation of quantum mechanics. According to Bohr, in a measurement the state of a system such as a spin collapses to one result or another in a way that cannot itself be described by quantum mechanics, and is truly unpredictable. This answer is now widely felt to be unacceptable. There seems no way to locate the boundary between the realms in which, according to Bohr, quantum mechanics does or does not apply. As it happens, I was a graduate student at Bohr’s institute in Copenhagen, but he was very great and I was very young, and I never had a chance to ask him about this.

    Today there are two widely followed approaches to quantum mechanics, the “realist” and “instrumentalist” approaches, which view the origin of probability in measurement in two very different ways.9 For reasons I will explain, neither approach seems to me quite satisfactory.10

3.

The instrumentalist approach is a descendant of the Copenhagen interpretation, but instead of imagining a boundary beyond which reality is not described by quantum mechanics, it rejects quantum mechanics altogether as a description of reality. There is still a wave function, but it is not real like a particle or a field. Instead it is merely an instrument that provides predictions of the probabilities of various outcomes when measurements are made.

    It seems to me that the trouble with this approach is not only that it gives up on an ancient aim of science: to say what is really going on out there. It is a surrender of a particularly unfortunate kind. In the instrumentalist approach, we have to assume, as fundamental laws of nature, the rules (such as the Born rule I mentioned earlier) for using the wave function to calculate the probabilities of various results when humans make measurements. Thus humans are brought into the laws of nature at the most fundamental level. According to Eugene Wigner, a pioneer of quantum mechanics, “it was not possible to formulate the laws of quantum mechanics in a fully consistent way without reference to the consciousness.”11

    Thus the instrumentalist approach turns its back on a vision that became possible after Darwin, of a world governed by impersonal physical laws that control human behavior along with everything else. It is not that we object to thinking about humans. Rather, we want to understand the relation of humans to nature, not just assuming the character of this relation by incorporating it in what we suppose are nature’s fundamental laws, but rather by deduction from laws that make no explicit reference to humans. We may in the end have to give up this goal, but I think not yet.

    Some physicists who adopt an instrumentalist approach argue that the probabilities we infer from the wave function are objective probabilities, independent of whether humans are making a measurement. I don’t find this tenable. In quantum mechanics these probabilities do not exist until people choose what to measure, such as the spin in one or another direction. Unlike the case of classical physics, a choice must be made, because in quantum mechanics not everything can be simultaneously measured. As Werner Heisenberg realized, a particle cannot have, at the same time, both a definite position and a definite velocity. The measuring of one precludes the measuring of the other. Likewise, if we know the wave function that describes the spin of an electron we can calculate the probability that the electron would have a positive spin in the north direction if that were measured, or the probability that the electron would have a positive spin in the east direction if that were measured, but we cannot ask about the probability of the spins being found positive in both directions because there is no state in which an electron has a definite spin in two different directions.

4.

These problems are partly avoided in the realist—as opposed to the instrumentalist—approach to quantum mechanics. Here one takes the wave function and its deterministic evolution seriously as a description of reality. But this raises other problems.

                   in Schrödinger; drawing by David Levine

    The realist approach has a very strange implication, first worked out in the 1957 Princeton Ph.D. thesis of the late Hugh Everett. When a physicist measures the spin of an electron, say in the north direction, the wave function of the electron and the measuring apparatus and the physicist are supposed, in the realist approach, to evolve deterministically, as dictated by the Schrödinger equation; but in consequence of their interaction during the measurement, the wave function becomes a superposition of two terms, in one of which the electron spin is positive and everyone in the world who looks into it thinks it is positive, and in the other the spin is negative and everyone thinks it is negative. Since in each term of the wave function everyone shares a belief that the spin has one definite sign, the existence of the superposition is undetectable. In effect the history of the world has split into two streams, uncorrelated with each other.

    This is strange enough, but the fission of history would not only occur when someone measures a spin. In the realist approach the history of the world is endlessly splitting; it does so every time a macroscopic body becomes tied in with a choice of quantum states. This inconceivably huge variety of histories has provided material for science fiction,12 and it offers a rationale for a multiverse, in which the particular cosmic history in which we find ourselves is constrained by the requirement that it must be one of the histories in which conditions are sufficiently benign to allow conscious beings to exist. But the vista of all these parallel histories is deeply unsettling, and like many other physicists I would prefer a single history.

    There is another thing that is unsatisfactory about the realist approach, beyond our parochial preferences. In this approach the wave function of the multiverse evolves deterministically. We can still talk of probabilities as the fractions of the time that various possible results are found when measurements are performed many times in any one history; but the rules that govern what probabilities are observed would have to follow from the deterministic evolution of the whole multiverse. If this were not the case, to predict probabilities we would need to make some additional assumption about what happens when humans make measurements, and we would be back with the shortcomings of the instrumentalist approach. Several attempts following the realist approach have come close to deducing rules like the Born rule that we know work well experimentally, but I think without final success.

    The realist approach to quantum mechanics had already run into a different sort of trouble long before Everett wrote about multiple histories. It was emphasized in a 1935 paper by Einstein with his coworkers Boris Podolsky and Nathan Rosen, and arises in connection with the phenomenon of “entanglement.”13

    We naturally tend to think that reality can be described locally. I can say what is happening in my laboratory, and you can say what is happening in yours, but we don’t have to talk about both at the same time. But in quantum mechanics it is possible for a system to be in an entangled state that involves correlations between parts of the system that are arbitrarily far apart, like the two ends of a very long rigid stick.

    For instance, suppose we have a pair of electrons whose total spin in any direction is zero. In such a state, the wave function (ignoring everything but spin) is a sum of two terms: in one term, electron A has positive spin and electron B has negative spin in, say, the north direction, while in the other term in the wave function the positive and negative signs are reversed. The electron spins are said to be entangled. If nothing is done to interfere with these spins, this entangled state will persist even if the electrons fly apart to a great distance. However far apart they are, we can only talk about the wave function of the two electrons, not of each separately. Entanglement contributed to Einstein’s distrust of quantum mechanics as much or more than the appearance of probabilities.

    Strange as it is, the entanglement entailed by quantum mechanics is actually observed experimentally. But how can something so nonlocal represent reality?

5.

What then must be done about the shortcomings of quantum mechanics? One reasonable response is contained in the legendary advice to inquiring students: “Shut up and calculate!” There is no argument about how to use quantum mechanics, only how to describe what it means, so perhaps the problem is merely one of words.

    On the other hand, the problems of understanding measurement in the present form of quantum mechanics may be warning us that the theory needs modification. Quantum mechanics works so well for atoms that any new theory would have to be nearly indistinguishable from quantum mechanics when applied to such small things. But a new theory might be designed so that the superpositions of states of large things like physicists and their apparatus even in isolation suffer an actual rapid spontaneous collapse, in which probabilities evolve to give the results expected in quantum mechanics. The many histories of Everett would naturally collapse to a single history. The goal in inventing a new theory is to make this happen not by giving measurement any special status in the laws of physics, but as part of what in the post-quantum theory would be the ordinary processes of physics.

    One difficulty in developing such a new theory is that we get no direction from experiment—all data so far agree with ordinary quantum mechanics. We do get some help, however, from some general principles, which turn out to provide surprisingly strict constraints on any new theory.

    Obviously, probabilities must all be positive numbers, and add up to 100 percent. There is another requirement, satisfied in ordinary quantum mechanics, that in entangled states the evolution of probabilities during measurements cannot be used to send instantaneous signals, which would violate the theory of relativity. Special relativity requires that no signal can travel faster than the speed of light. When these requirements are put together, it turns out that the most general evolution of probabilities satisfies an equation of a class known as Lindblad equations.14 The class of Lindblad equations contains the Schrödinger equation of ordinary quantum mechanics as a special case, but in general these equations involve a variety of new quantities that represent a departure from quantum mechanics. These are quantities whose details of course we now don’t know. Though it has been scarcely noticed outside the theoretical community, there already is a line of interesting papers, going back to an influential 1986 article by Gian Carlo Ghirardi, Alberto Rimini, and Tullio Weber at Trieste, that use the Lindblad equations to generalize quantum mechanics in various ways.

    Lately I have been thinking about a possible experimental search for signs of departure from ordinary quantum mechanics in atomic clocks. At the heart of any atomic clock is a device invented by the late Norman Ramsey for tuning the frequency of microwave or visible radiation to the known natural frequency at which the wave function of an atom oscillates when it is in a superposition of two states of different energy. This natural frequency equals the difference in the energies of the two atomic states used in the clock, divided by Planck’s constant. It is the same under all external conditions, and therefore serves as a fixed reference for frequency, in the way that a platinum-iridium cylinder at Sèvres serves as a fixed reference for mass.

    Tuning the frequency of an electromagnetic wave to this reference frequency works a little like tuning the frequency of a metronome to match another metronome. If you start the two metronomes together and the beats still match after a thousand beats, you know that their frequencies are equal at least to about one part in a thousand. Quantum mechanical calculations show that in some atomic clocks the tuning should be precise to one part in a hundred million billion, and this precision is indeed realized. But if the corrections to quantum mechanics represented by the new terms in the Lindblad equations (expressed as energies) were as large as one part in a hundred million billion of the energy difference of the atomic states used in the clock, this precision would have been quite lost. The new terms must therefore be even smaller than this.

    How significant is this limit? Unfortunately, these ideas about modifications of quantum mechanics are not only speculative but also vague, and we have no idea how big we should expect the corrections to quantum mechanics to be. Regarding not only this issue, but more generally the future of quantum mechanics, I have to echo Viola inTwelfth Night: “O time, thou must untangle this, not I.”

 

Refrences

1. Conditions on sound waves at the closed or open ends of an organ pipe require that either an odd number of quarter wave lengths or an even or an odd number of half wave lengths must just fit into the pipe, which limits the possible notes that can be produced by the pipe. In an atom the wave function must satisfy conditions of continuity and finiteness close to and far from the nucleus, which similarly limit the possible energies of atomic states.  ↩

2. Quoted by Abraham Pais in ‘Subtle Is the Lord’: The Science and the Life of Albert Einstein (Oxford University Press, 1982), p. 443.  ↩

 

3. Richard Feynman, The Character of Physical Law (MIT Press, 1967), p. 129.  ↩

 

4. Lawrence M. Krauss, A Universe from Nothing (Free Press, 2012), p. 138.  ↩

 

5. Gino Segrè, Ordinary Geniuses (Viking, 2011).  ↩

 

6. These are complex numbers, that is, quantities of the general form a+ib, where a and b are ordinary real numbers and i is the square root of minus one.  ↩

 

7. Simple as it is, such a wave function incorporates much more information than just a choice between positive and negative spin. It is this extra information that makes quantum computers, which store information in this sort of wave function, so much more powerful than ordinary digital computers.  ↩

 

8. To be precise, these “squares” are squares of the absolute values of the complex numbers in the wave function. For a complex number of the form a+ib, the square of the absolute value is the square of a plus the square of b. ↩

 

9. The opposition between these two approaches is nicely described by Sean Carroll in The Big Picture (Dutton, 2016).  ↩

 

10. I go into this in mathematical detail in Section 3.7 of Lectures on Quantum Mechanics, second edition (Cambridge University Press, 2015).  ↩

 

11. Quoted by Marcelo Gleiser, The Island of Knowledge (Basic Books, 2014), p. 222.  ↩

 

12. For instance, Northern Lights by Philip Pullman (Scholastic, 1995), and the early “Mirror, Mirror” episode of Star Trek.  ↩

 

13.Entanglement was recently discussed by Jim Holt in these pages, November 10, 2016.  ↩

 

14. This equation is named for Göran Lindblad, but it was also independently discovered by Vittorio Gorini, Andrzej Kossakowski, and George Sudarshan. ↩

 

 

 

 

量子力学的困惑

温伯格 

 

 

    本文是著名理论物理学家Steven Weinberg为纽约书评所撰写，将于1月19日出版。温伯格因统一弱相互作用与电磁作用而荣获诺贝尔物理学奖，其对量子力学本质的思考和挣扎，尤其发人深省。

 

    20世纪头十年间量子力学的发展给许多物理学家带来冲击。时至今日，尽管量子力学已经取得巨大成功，关于它的意义与未来的争论却仍在继续。

 

一

    量子力学的第一个冲击是对物理学家在1900年以前早已习惯的范畴所带来的挑战。那时我们有粒子——原子、然后是电子和原子核——然后有场——这是电、磁力以及引力可以彰显的空间环境。光则被清晰的认作电磁场的自持振荡。然而为了理解受热物体的发光问题，1905年阿尔伯特·爱因斯坦发现需要把光波表述成无质量的粒子束，这些粒子后被称为光子。

    到了1920年代，根据路易斯·德布罗意和厄尔文·薛定谔的理论，被一直看作典型粒子的电子，似乎在某些情况下表现出了波动性。为了解释原子的稳定能级，物理学家们不得不放弃了电子如同牛顿行星一般在轨道内围绕原子核转动的见解。原子中的电子更像是围绕契合在原子核周围的波，如同琴管中的声波一样[1]。至此这个世界的范畴变得乱套了。

    更糟糕的是，电子波并非是电子物质的波，这和海浪是水波完全不同。的确，马克斯·波恩开始意识到电子波是概率波。也就是说，当一个自由电子同一个原子发生碰撞后，我们原则上不能预测它会弹射到哪个方向上去。电子波在与原子碰撞后会遍及所有方向，这和海浪撞到暗礁上类似。正如伯恩所意识到的，这并不意味电子本身被散播各处，不可分割的电子仍被散射至某个方向但不是一个被精确预测的方向上。电子更有可能在一个波分布更稠密的方向上但同时其他所有方向都有可能。

    1920年代的物理学家对概率并非陌生，但是概率通常被看作是那些还在研究中的不完美知识的反映，而不是反映了潜在物理学定律中的非决定性。牛顿的运动与引力理论确立了决定论规律的准则。当我们精准的知道某一给定时刻太阳系中物体的位置和速度时，牛顿定律就能很精确的告诉我们未来很长时间内它们都在什么地方。只有我们的所知并不完善时概率才会出现在牛顿物理学中，比如我们无法精确预测一对骰子将掷出几点。然而对于新的量子力学，这种物理学规律的即时确定性似乎消失了。

    一切都如此奇怪。1926年爱因斯坦在一封写给波恩的信中这样抱怨：量子力学令人印象深刻。但是我的内心中有个声音告诉我这并非真实。这个理论很好但却很难让我们更接近上帝的秘密，我十分确定他不玩骰子[2]。

到了1964年，理查德·费曼在康奈尔的先驱讲座中哀叹：“我想我可以肯定的说没有人能理解量子力学”[3]。量子力学与旧时代的决裂如此鲜明，以至于之前所有的物理学理论都被称为“经典”的。

    量子力学的古怪在大多数场合倒也没什么。物理学家学会了如何利用它来更精确的计算原子能级，以及粒子碰撞时沿某个方向的散射概率。劳伦斯·克劳斯给量子力学对氢原子能谱中某个效应的计算冠以“整个科学中最好的最准确的预测”[4]。原子物理之外，基诺·沙格瑞列出了量子力学的早期应用，包括分子中的原子束缚、原子核的放射性衰变、导电性、磁性以及电磁辐射[5]。接下来的应用涵盖了半导体和超导理论、白矮星和中子星、核力、以及基本粒子。即使是现代最具冒险精神的探索，譬如弦论，也筑基于量子力学原理。

    很多物理学家开始觉得爱因斯坦、费曼以及其他一些人对于量子力学的那些新奇之事的反应有些夸张。这也曾经是我的看法。毕竟，牛顿的理论也曾经让他的同辈们感到不舒服。牛顿引入过让批评者难以理解的力-引力。它跟任何接触式的推拉都无关，而且很难在哲学或者是纯数学的基础上加以解释。他的理论也放弃了托勒密和开普勒的主要目标，即通过第一原理来计算行星轨道。但是对牛顿理论的反对声终归烟消云散。牛顿和他的追随者不仅成功的解释了行星运动和苹果下落，也解释了彗星和卫星运动以及地球的形状和它转动轴方向的改变。到十八世纪结束前这些成就已经确立了牛顿的运动和引力理论是正确的，或至少是一种极为精准的近似。显然，过分要求新的物理学理论应该符合某些预想的哲学标准本身就是一个错误。

    不同于经典物理，量子力学中系统的状态不是由每个粒子的位置和速度、以及各种场的值与变化率来描述的。取而代之的，任意时刻的系统状态由波函数描述，它本质上就是一组数字，每个数字都对应着一个可能的系统构形[6]。对于单粒子系统，所有的粒子可能占据的空间位置对对应一个这样的数字。这类似于经典物理中声波的描述，不同在于代表声波每个空间位置的数字给出了那个点上的气压，而量子力学中代表某个特定位置粒子波函数的数字反映出那个点上的粒子存在概率。这有什么可怕的呢？显然，对于爱因斯坦和薛定谔来说，逃避使用量子力学是一个令人扼腕的失误，这将他们自己与其他人取得的那些令人激动的进展彻底隔绝。

二

    即便如此，我也不像以前那样确信量子力学的未来。一个不好的信号是即使那些最适应量子力学的物理学家们也无法就它的意义达成共识。这种分歧主要产生于量子力学中测量的本质。这个问题可以用一个简单的例子来说明：电子自旋的测量（一个粒子在任意方向的自旋是它围绕该方向轴的转动量）。所有的理论和实验都支持的结论是：测量一个电子沿某个选定方向的自旋只能得到两种可能的值。一个是正的普适常数 (这个常数是在1900年最先由马克斯·普朗克在他的热辐射理论中提出，大小为h/4π)。另外一个则是前面这个的相反数。这两个自旋值正好对应于电子沿选定方向上的顺时针或逆时针旋转。

    不过只有当测量完成它们才会成为唯二的可能。一个电子自旋在测量前就像一个音乐和弦一样，由两个音符叠加而成，这两个音符分别对应正负自旋，每个音符都有自己的大小。如同一个和弦奏出不同于组分音符的声音，电子自旋在测量前是由确定自旋的两个态叠加而成，这种叠加态在定性上完全不同于其中任意一个态。同奏乐类似，对自旋的测量行为就像是一下把和弦调到某个特定的音符上去，从而我们只能听到这单个音符。

    这些可以用波函数来说明。如果我们忽略其他关于电子的一切而只考虑自旋，那它的波函数跟波动性其实没什么关系。只有两个数，每个数代表自旋沿某个选定方向的正负，类似于和弦中每个音符的振幅[7]。在测量自旋前，电子波函数通常对于正负自旋都有非零值。

    量子力学中的波恩定则告诉我们如何计算实验中得到各种不同结果的概率。举例来说，波恩定则告诉我们测量发现特定方向自旋的正或负值的概率正比于这两个自旋态的波函数中数字的平方[8]。

    把概率引入物理学原理曾困扰物理学家，但是量子力学的真正困难不在于概率。这点我们可以承受。困难在于量子力学波函数随时演化的方程，薛定谔方程，本身并不涉及概率。它就像牛顿运动方程和引力方程一样具有确定性。也就是说，一旦给定某时刻的波函数，薛定谔方程就能够准确告诉你未来任意时刻的波函数。甚至不会出现混沌（一种牛顿力学中对初始条件极其敏感的现象）的可能性。所以如果我们认定整个测量过程都是由量子力学方程来确定，而这些方程又是确定性的，那量子力学中的概率究竟是怎么来的呢？

    一个普通的答案是，在测量中自旋（或其他被测量）被放置在一个与之相互作用的宏观环境中，这个环境则以一种无法预测的方式震动。举例来说，这个环境可能是一束用来观测系统的光束中的大量光子，在实际中它如同一阵倾盆大雨一样无法预测。这样的环境引起了波函数叠加态的坍缩，最终导致了测量的不可预测性（即所谓的退相干）。就像是一个嘈杂的背景不知怎么的就让一个和弦只能发出一个音符。但是这个答案回避了问题实质。如果确定性的薛定谔方程不仅决定了自旋而且连同测量仪器以及使用它的物理学家的随时演化，那么原则上测量结果不应是不可预测的。所以我们仍然要问，量子力学中的概率究竟是怎么来的？

    这个谜题的一个回答是1920年代尼尔斯·波尔给出的，后世称之为量子力学的哥本哈根表述。根据波尔的见解，测量过程中系统状态（比如自旋）会以一种量子力学无法描述的方式坍塌成一种结果或是另一种，它本质上就是无法预测的。这个答案今天普遍认为是不可接受的。按照波尔的意思，似乎根本就无法区分哪里量子力学是适用的哪里不是。碰巧那时我还是哥本哈根波尔研究所的一个研究生，但那时波尔声望正隆而我还很年轻，我从未有机会向他问起这个问题。

    今天存在有两种广泛采用的对待量子力学的方式，一种是“现实主义”，另一种是“工具主义”，这两种方式看待测量中概率的起源截然不同[9]。但是基于我下面要给出的理由，它们在我看来都不太令人满意[10]。

三

    工具主义其实与哥本哈根表述一脉相承的，但是它不再构想量子力学所无法描述的现实的边界，它直接否认了量子力学是对现实的一种描述。波函数仍然存在，不过它不代表现实的粒子或者场，取而代之的它仅仅作为提供测量结果预测的工具。

    对我来说似乎它的问题不仅仅在于放弃了自古以来科学的目标：寻求世界的终极奥义。它更是以一种令人遗憾的方式投降。在工具主义论中，我们必须假定，当人们开始测量时，应用波函数计算测量结果的概率的规则（例如前文提到的伯恩定则）是自然界的基本法则。于是乎人类本身就被带入自然界最基本的规律层次。正如一位量子力学的先驱，尤金·魏格纳所说，“永远无法用一种完全自洽的却又跟意识无关的方式构建起量子力学的定律”[11]。

    工具主义与达尔文之后变为可能的一个观点背道而驰，那就是这个世界被非人力的自然法则所统治，人类行为以及其他所有一切都要受其统御。这并非是我们要反对这样思考人类。我们其实更想要理解人类与自然的关系，不是简单的通过把它并入我们自以为的自然界的基本规律中来设想这个关系的本质，而是从不显含人类的基本规律中推导而出这个关系的本质。或许我们终将不得不放弃这个远大目标，但是我认为至少现在还不是时候。

    有些物理学家采用工具主义的方法，他们声称我们从波函数中得到的概率是客观存在的概率，不依赖于人们究竟有没有做测量。我则不认为这观点是站得住脚的。量子力学中这些概率只有当人们选择什么去测量时才存在，比如沿某个方向上的自旋。不同于经典物理，量子力学中必定存在一个选择，这是因为量子力学中不是所有量可以同时被测量。正如维尔纳·海森堡意识到的，一个粒子不能同时有一个确定的位置和速度。测量其中一个就无法测量另一个。同样的，如果我们知道一个电子自旋的波函数，我们就可以去计算我们测量得出这个电子朝北的方向上有正自旋的概率，或者是测量得到朝东方向上有正自旋的概率。但是我们不能问同时在两个方向上的正自旋的概率是多少，因为没有一个态可以表示电子在两个不同方向上都有确定自旋。

四

    与工具主义相反的另一种对待量子力学的方式——现实主义避免了部分上面提到的问题。现实主义者切实的把波函数及其确定性的演化当作对现实的描述。但是这也带来其他的问题。

    现实主义有一个非常奇怪的推论，最早是1957年在已故的休·艾弗雷特的普林斯顿的博士毕业论文中提出的。当一个物理学家测量一个电子自旋时，比如朝北方向上，电子、测量仪器连同实施测量的物理学家的波函数的演化都假定是确定性的，均由薛定谔方程给出。但是随着这几者在测量中发生相互作用，波函数变成两项的叠加，一个是电子自旋是正值，这个世界的每个人去观测都会看到它是正值，而另一个则是负值，同样世界每个人都认为它是负的。因为对于波函数的每一项每个人都坚信电子自旋只有一个确定符号，于是这种叠加态的存在根本无法探测。从而这个世界的历史便分裂为彼此完全不相关的两支。

    这就够奇怪了，然而历史的分裂不仅仅会发生在某人去测量自旋时。在现实主义者的观点中，这个世界的历史时时都在进行无穷无尽的分裂; 每当有宏观物体伴随量子状态的选择时历史就会分裂。这种不可思议的历史分裂为科幻小说提供了素材[12]，而且为多重宇宙提供了依据，众多宇宙之中某个特定宇宙历史中的我们发现自己被限定在条件优渥从而允许有意识生命存在的历史中的一个。但是展望这些平行历史令人深深不安，同其他很多物理学家一样，我倾向于单一存在的历史。

    在我们狭隘的各人喜好之外，现实主义论中还有件事让人不爽。这种观点中多重宇宙的波函数的确进行确定性的演化。我们仍然可以论及在不同时间段上在任意某个历史中测量多次得到多个可能结果的概率，但是决定这些观测概率的规则必须依从整个多重宇宙的决定性演化。若非如此，那预测概率时我们就得额外假设人们在测量时发生了什么，这样我们就回到了工具主义的缺点上。尽管一些现实主义的尝试已经得到类似于波恩定则这样和实验配合很好的推论，但我觉得他们都不会取得最终的成功。

    其实早在艾弗雷特提出多重历史很久之前，量子力学的现实主义论就已陷入另一个麻烦之中。这个麻烦是在1935年爱因斯坦与他的合作者鲍里斯·波尔多斯基和南森·罗斯一起撰写的文章中提出的，与所谓的“纠缠”现象有关[13]。

    我们一般都自然认为可以“局域”的描述现实。我可以告诉你我实验室发生了什么，你可以告诉我你实验室怎么样，不过我们没必要非得同时说两个。但是在量子力学中，系统可以处于距离很远但相互关联的两部分（像刚棒的两端）的纠缠态中。

    举例来说，假设我们有一对总自旋沿任意方向都为零的电子。这样一个态的波函数（只考虑自旋部分）是两项之和：一项中，沿北方向上电子A自旋为正，B自旋为负，另一项中正负号正好反过来。这时两个电子的自旋就可以说纠缠在一起了。只要不去干涉这对自旋，即使是两个电子分开很远距离，这样一个纠缠态仍会一直持续。无论分开多远，我们也只能讨论两个电子的波函数而不是单独一个的。纠缠带给爱因斯坦对量子力学的不信任感甚至超过概率的出现。

    虽然听起来很奇怪，但从量子力学那里继承来的纠缠事实上已经在实验上被观测到。但是这种如此“非局域”的东西如何能代表现实呢？

五

    针对量子力学的缺点又应该做些什么呢？一个合理的回应包含在了那句经典的给爱追究问题学生的建议中：“Shut up and calculate” 其实如何去用量子力学并无争议，有争议的是如何阐述它的意义，所以或许问题仅仅就是一个词而已。

    另一方面，如何在当前量子力学框架下理解测量的问题或许是在警告我们理论仍需要修正。量子力学对原子解释的如此完美，以至于任何应用到如此小的对象上的新理论都和量子力学近乎不可分辨。但是或许新理论可以仔细设计，使得大物体比如物理学家和他们的仪器即使在孤立的情况下也可以发生快速的自发式坍缩，从而由概率演化能给出量子力学的期待值。艾弗雷特的多重历史也自然的坍缩成一个。发明新理论的目标即是如此，但不是通过给测量在物理学规律中一个特殊地位而达成，而是使之作为那些成为正常物理进程的后量子力学理论的一部分。

    发展这样新理论有一个困难是实验没能给我们指明方向—目前所有的实验数据都符合通常的量子力学。我们倒是从一些普适原理中得到些许帮助，但是这些都最终令人惊讶的演变为对新理论的严苛限制。

    显然，概率必须为正数，其总和必须为100%。还有一个通常的量子力学已经满足的条件，就是纠缠态中测量过程中概率的演化不能用来发出瞬时信号，否则就违反相对论。狭义相对论要求不能有任何信号传递速度超过光速。当把这些条件合在一起，最一般的概率演化就满足一组方程（即所谓的林布莱德方程）中的一个[14]。这组林布莱德方程涵盖了通常量子力学中的薛定谔方程作为一个特例。但是这些方程同时涉及了一系列背离量子力学的量。关于这些量的细节我们目前无疑毫无了解。尽管几乎不为理论界之外所注意到，还是有了一些很有意思的文章，比如在利亚斯特的吉安·卡洛·吉拉尔地、阿尔贝托·里米尼以及图里奥·韦伯在1986年写的颇有影响力的文章，就用林布莱德方程以不同的方式来一般化量子力学。

    近来我一直在思考原子钟中一个可能寻找到背离通常量子力学迹象的实验。在每个原子钟的核心都有一个已故的诺曼·拉姆齐发明的装置，它是用来调节微波或是可见辐射的频率到一个已知的自然频率上，在这个频率上当一个原子的波函数正处于两个不同能级的叠加态时会发生振荡。这个自然频率就正好等于原子钟采用的两个原子能级之差再除以普朗克常数。同塞弗尔的铂铱合金圆柱体作为质量的固定基准一样，这个频率在任何外部条件下都保持不变，因此可以作为频率的固定基准。

    把一个电磁波的频率调节到这个基准频率上就有点像调节一个节拍器的频率和另一个节拍器匹配。如果你同时启动两个节拍器而且在敲了一千下后它们还是保持一致，那么你就就知道它们的频率至少在千分之一的精度上相同。量子力学计算表明在一些原子钟中调节精度可达10^-17，而且这种精度确实已实现。但是如果林布莱德方程中那些代表着对量子力学修正的项（以能量的形式）的量级到了原子钟中应用的两个原子能级差的1/10^17，那么这个精度也已经明显不够用了。如此说来新的修正项想必比这个量级还要小。

    这个极限究竟有多显著？可惜的是，这些对量子力学修正的想法不仅带有推测性质而且还很模糊，我们也不知道应该期待量子力学的修正究竟有多大。想到此处更是思及量子力学的未来，我唯有引用维奥拉在《第十二夜》中的话："O time, thou must untangle this, not I"。“啊时间！你必须解决此事，而不是我”。

 

参考文献

1. 开口或者闭合琴管中的声波条件要求1/4波长的奇数倍或是半波长的整数倍正好契合管子，这样就限制了琴管可以奏出的音符。原子中，波函数必须符合远离和靠近原子核的连续性和有限性条件，这也同样限制了可能的原子态能级。

2. 引自Abraham Pais ，‘Subtle Is the Lord’: The Science and the Life of Albert Einstein (Oxford University Press, 1982), p. 443.  

3. Richard Feynman, The Character of Physical Law (MIT Press, 1967), p. 129.  

4.Lawrence M. Krauss, A Universe from Nothing (Free Press, 2012), p. 138.  

5. Gino Segrè, Ordinary Geniuses (Viking, 2011).  

6. 这都是些复数，通常采用a+ib的形式，a，b均为实数，i为-1的平方根。             

7. 如此简单，这样一个波函数包含的信息，远远多于只是选一个正负自旋。正是这些额外信息造就了量子计算机，其信息都由波函数来存储，性能也远超传统数字电脑。

8. 更精确的说，是波函数中复数绝对值的平方。对于复数a+ib，这个值是a2+b2

9. Sean Carroll在The Big Picture (Dutton, 2016)中很好地阐述了两种论点的对立

10. 更多数学细节可以参见Lectures on Quantum Mechanics, second edition (Cambridge University Press, 2015)， 第3.7节  

11. 引自 Marcelo Gleiser, The Island of Knowledge (Basic Books, 2014), p. 222.  

12. 比如, Northern Lights by Philip Pullman (Scholastic, 1995), 以及早期星际迷航中的 “Mirror, Mirror”剧集 

13. Jim Holt 最近对纠缠在这些方面进行了讨论, November 10, 2016.  

14. 这个方程因戈兰·林布莱德得名, 但亦由维托里·奥戈里尼、安杰伊·科萨科夫斯基以及乔治·苏达山独立提出。