Chapter 11 Chapter 10 Inequalities

one This trip in Wonderland of the Many Worlds may confuse everyone, but like the long obscure poem Jabberwocky that Alice reads in the mirror, it should undoubtedly leave a strong impression.Indeed, imagine ourselves splitting over time into projections in multiple worlds, and these copies grow geometrically to infinity.Such a wonderful scene really adds a bit of irony to the universe in which we live.Some people may think that there is nothing more lovely than consciousness in such a model. What do Everett and those scientists who support many worlds value in it? However, the benefits of MWI are also obvious. Its greatest feat is to kick the observer out of physics.Now the entire universe just evolves strictly according to the wave function, and there is no need to humbly ask for help from the observer or the choice of intelligent creatures.Physicists now no longer have to worry about the miraculous collapse and put ugly patches on the beautiful theoretical framework to explain the mechanism of the R process.Our poor Schrödinger cat finally got rid of the torment of being dead and alive, and instead lived happily (one dead and one alive) in two different worlds.

The important thing is that nature can make its own decisions again, and it doesn’t have to linger in the shadow of the observer, until some conscious master appreciates an observation before it becomes a reality, otherwise it has to bury its life in the superposition of probability waves .In MWI, the universe itself becomes the sole master again, and any observer is a part of it, which is split and projected into various worlds as it evolves.The splitting of the universe only depends on the introduction of the environment and the irreversible amplification process. Such an objective picture is still in line with the traditional taste of most scientists. At least it will not drive people crazy like the Copenhagen School, so that they can't sleep well.

A by-product of MWI is that it falls back on the determinism of classical theory.Because as far as the Schrödinger equation itself is concerned, it is decisive, that is to say, given the state at a certain time t, we can deduce it from both positive and negative directions to obtain the state of the system at any time.In this sense, the passage of time is nothing but an illusion!In addition, since there is no collapse or R process, only deterministic U process, randomness will no longer be messed with differently from person to person.In this sense, God does not play dice anymore. He stands at a high angle and has a bird’s-eye view of the wave function of the entire universe, so everything is still under control: the universe as a whole is still strictly evolving according to the definite Schrödinger equation.Nor does the electron have to roll a dice and make a random choice to go through a slit: it just goes through a slit in each of the two worlds at the same time.However, for us ordinary people, because we are entangled in the world of mortals, our inherent limitations confuse our eyes, so that we can only see the shadow of a certain world.And in this projection, reality is random, jumpy, surprising.

(*By the way, here is a clarification of the wording problem. For MWI, people generally like to call multiple branches World (World), and their sum is called Universe (Universe). In this way, there is only one universe, which follows the Schrödinger equation development, and there are many worlds, which keep splitting over time. But some people like to call each branch a universe, and their sum is called a multiverse (Multiverse), such as the famous multi-universe physicist David Deutsch .This is just a matter of name, multi-world or multi-universe, they refer to the same meaning.) However, although MWI can also be justified, in any case, there are many universes in reality, which sounds too weird to ordinary people.Even for philosophically elegant reasons (especially Occam's razor), people feel that MWI should be taken with caution: it's not very admirable to drag the entire universe down for the sake of a small electron.But among cosmologists, the MWI is a popular and well-received idea.In particular, it does not require the special status of the observer, but attributes the history and evolution of the universe to itself, which makes cosmologists who have suffered from the Copenhagen interpretation and the curse of participatory models feel very comfortable.Generally speaking, physicists who are engaged in quantum gravity (such as superstring) and cosmology prefer MWI, and if the scope is expanded to general scientists, the proportion of people who think it is weird and unacceptable will increase greatly.Among the many-worlds supporters, there are Feynman, Weinberg, and Hawking that we are familiar with, and some people put the founder of the quark model, Murray Gell-Mann, who won the Nobel Prize in Physics in 1969, also Included in it, but as one of the founders of the consistent history interpretation of quantum theory, we still leave him to the corresponding chapter of the history, although this interpretation can actually be regarded as an enhanced version of MWI.

Those who expressed direct opposition to MWI include Bell, Stein, Kent, and Penrose.Some of them, such as Penrose, are also engaged in gravity, which can be regarded as very unique. However, for the readers of our history, maybe you don’t need to pay attention to the different philosophical tastes of cosmologists or other scientists. The important thing is that now we have a Copenhagen explanation and a multi-universe explanation. Only then can we know, which one should we believe?It is normal for people to have different aesthetic viewpoints in life. For example, you like Beethoven and I like Mozart, you like Li Bai and I like Du Fu. It is not surprising, but science, especially natural science, is different.Isn't science great because it can become the unique law of the universe without being influenced by subjective will?Economists may fight each other for various models, but the ultimate goal of physics is not to be practical, but to explore the deep hidden mysteries of nature.It must treat all kinds of assumptions with the strictest attitude, pick out those unqualified and remove them from its own system, so as to maintain its immortal vitality forever.The history of science should be a process of constantly examining oneself, constantly taking practice as the only criterion, and constantly climbing towards that Platonic ideal.For this, it must provide a screening mechanism to kick out those theories that look beautiful but do not conform to the facts. This has become an important sign that it is different from philosophy or religion.

Perhaps we can accept that famous and controversial philosopher of science, Carl.According to Karl Popper's opinion, the dividing line between science and metaphysics is drawn here on falsifiability.That is, a scientific claim must be falsified.For example, I said: There are no white crows in the world.This is a claim that conforms to the scientific method, because as long as you really find a white crow, you can prove me wrong and overthrow my theory.But, as in the example we gave earlier, suppose I claim to have an invisible dragon in my garage. , this is not a scientific assertion, because you can't prove me wrong anyway.If we accept these unprovable assertions as science, then there will be many funny things in science: in addition to flying dragons, there will be three-headed dogs, eight-legged donkeys, and monkeys who speak Chinese.Anyway, you can't prove that dogs with three heads don't exist, can you?

If Hertz hadn't discovered sparks caused by electromagnetic waves in his experiments in 1887, then Maxwell's theory would have been falsified.If Eddington had not detected the displacement of those stars during the 1919 eclipse, then Einstein's theory of relativity would have been disproved (although the experiment is not without problems today).If Wu Jianxiong and others did not find the expected effect in the experiment in 1956︱1957, then Yang and Li's assumption of parity violation under weak action would be falsified.No matter at that time or in the future, you can design some experiments. If the result is so-and-so, you can prove that the theory is incorrect. This is the falsifiability of science.Of course, some concepts have been falsified, such as the flat earth theory, phlogiston, and light ether, but no matter what, we can at least say that the expressions they adopt are in line with the scientific method.

Others, such as God, are hard to say. There is no experiment that can prove that God does not exist (it is not necessary to prove that it does not exist, but there is no such possibility).So we might as well kick it out of the realm of science and leave it to religious buffs to ponder. Going back to the history, in order to make our two explanations conform to Popper's principle, can we design an experiment to identify which one is credible and which one is false?Copenhagen explained that the observer caused the wave function to collapse, and MWI said that the universe split, but, for us in reality, there is no observable difference!In any case, the fact must be that the electron appears somewhere on the screen, seemingly randomly, according to the probability of the wave function, doesn't it?Even if we observe a million times, we can't tell which one is wrong, Copenhagen or many worlds!

Since the 1970s, the decoherence theory (decoherence) proposed, developed and popularized by Dieter Zeh, Wojciech H Zurek, Gell-Mann and others seems to have a certain effect on Everett’s multiverse explanation. huge help.As we have discussed a little earlier, this theory explains how an object transitions from a microscopic superposition state to a macroscopic definite state: it mainly involves the macroscopic nature of objects such as detectors or cats, that is, compared to electrons, The number of degrees of freedom is much greater, and their interaction with the environment.This theory is a duck to water in MWI. It explains why the world does not show superposition on a large scale, and explains how the world splits up. These are things that MWI could not explain before.Broadly speaking, when an instrument observes a system, it also becomes entangled with its environment, causing the instrument's superposition state to rapidly degenerate into a classical correlation.We say this very roughly, and it can in fact be proved mathematically.If we use the system's so-called density matrix (Desity Matrix) to represent it, then the elements on the diagonal of this matrix represent the classical probability states, and other places represent the correlation between these states.We will see that when decoherence occurs, the density matrix of the instrument or cat rapidly diagonalizes, making the quantum superposition property gone forever (see attached picture).This process is so fast that we don't notice it at all.

However, although decoherence theory is a powerful supplement to MWI, it cannot prove that MWI is the only explanation.Decoherence can explain why in a universe full of quantum superposition and uncertainty, the world still seems classical and objective when we look at the world on a daily large scale, but it cannot answer whether the wave function always develops normally, or whether it will occasionally ground jump.In fact, we can also use decoherence in the Copenhagen interpretation to determine the boundary between observers and non-observers according to their respective size, or number of degrees of freedom!Those that are prone to decoherence may be more qualified to appear as observers, and the so-called observation may be just an irreversible amplification process.But in the final analysis, we are still not sure whether it is Copenhagen or multiverse!

When Popper was in his later years (he died in 1994), I think his mood would be more complicated.On the one hand, some of his conclusions back then were correct. For example, quantum mechanics itself did not exclude deterministic factors (nor did it exclude indeterminism).Regarding the principle of complementarity, he was almost completely convinced by Bohr in Copenhagen, but now he can still reconsider other alternatives.On the other hand, we will also be very interested to know what Popper thinks about the status quo in which various interpretations in the field of quantum theory are side by side and can hardly be distinguished by practice. But let's describe some interesting experiments that strongly support MWI, including the crazy quantum suicide, and the currently hot quantum computer that claims to use multiple parallel worlds to work together. After-dinner gossip: falsification and confirmation Regarding the definition of science, the two factions of verification and falsification have been arguing endlessly. This topic is too big, and we are not interested in participating. Here we just talk about the issue of verification and falsification. How can a proposition be stated as scientific?According to the falsificationists, it must be possible to be proven false.For example, all crows are black, then you can prove the error of this proposition as long as you find a crow that is not black, so there is no problem with this proposition.On the contrary, if you have to prove it before accepting this assertion, it will be difficult, and in fact it is impossible!Unless you have caught all the crows, how can you know that you have caught all the crows in the world? For scientific theories, verification is almost impossible.For example, if we say that the law of the universe is F=ma, what we are talking about here is a kind of universality, but how do you prove it?Only when you have observed the phenomena in every corner of the universe since ancient times and found no exceptions can you prove this point.Even so, there is no guarantee that this rule will still work in the future.In fact, almost no scientific theory can be proven. As long as it can be proven wrong but has not been proved wrong (according to Popper, with a positive attitude towards falsification), we temporarily accept it as reliable theory.It has been admitted since Hume that a limited number of instances (no matter how many) alone can form a basis for verification. However, according to empiricists like Locke, the basis of all our knowledge comes from our experience, and the establishment of science is a kind of inductivism on experience.For example, we see the sun rising from the east every day, and it has been like this every day for thousands of years, so we should be able to reasonably deduce a rule from it: the sun rises from the east every day.And use it to predict that the sun will still rise from the east tomorrow.If we fall into Hume's agnosticism, then we can't talk about any knowledge at all, because everything tomorrow is uncertain anyway. According to inductivism, we induce a law from past phenomena, and when this phenomenon is repeated, it becomes a new confirmation of this law each time.For example, every time the sun rises again, the certainty of the proposition that the sun rises from the east every day is slightly confirmed again.Every time we see a black crow, the truth of the proposition that all crows are black increases slightly again, until we encounter a crow that is not black. Most of us probably think so, but this kind of empiricism has very interesting results.Let's do such a reasoning. Everyone knows that the inverse of a proposition is equivalent to itself.For example, all crows are black, which can be changed to an equivalent proposition. Anything that is not black is not a crow.Now if we meet a white cat, this phenomenon definitely confirms that nothing that is black is not a crow (white cats are not black, and white cats are not crows), so likewise, it also slightly confirms that all crows are black This original proposition. All in all, meeting a white cat slightly increases the probability that all crows are black.Interesting, right? This paradox was proposed by the famous German logical positivist Carl G Hempel, who also studied mathematics with Hilbert in his youth.If you accept this conclusion, then the next time your instructor asks you to go to the field to prove a proposition such as insects with six legs, you don’t have to go out in the wind and rain.You can contribute just as much to the proposition by sitting at home and observing numerous instances of things that don't have six legs that aren't insects (like tables, chairs, lamps, yourself) as you can by actually observing insects in the wild! Our understanding of the theory of knowledge is still very superficial. two The creepy and ironic quantum suicide experiment was proposed by Hans Moravec, Bruno Marchal et al. in the late 1980s and developed in 1998 by cosmologist Max Tegmark in his well-known paper promoting MWI and rehash.This is actually a live-action version of Schrödinger's cat.Everyone knows that in the cat experiment, if the atom decays, the cat will be poisoned to death, otherwise it will survive.In this regard, the explanation of the Copenhagen school is: before we observe it, the cat is both dead and alive, but after the observation, the wave function of the cat collapses, and the cat is either dead or alive. MWI claims that each experiment must produce a live cat and a dead cat at the same time, but they exist in two parallel worlds. What is the real difference between the two?The key point is that the Copenhagen School believes that there is always only one cat, and it starts to be in a superposition state. After the collapse, there is a 50% possibility of death and a 50% possibility of life.However, the multiverse believes that cats are not superimposed, but split into two, one dead and one alive, there must be a live cat! Now if there is a benevolent man who has the courage to dedicate himself to science, he volunteers to replace that hapless cat.Out of humanitarianism, in order to save him from suffering, we changed the gas bottle into a gun.If the atom decays (or utilizes some other quantum mechanism, such as a photon passing through the half-silvering), the gun bangs and sends our friend on his way.On the contrary, the gun only made a hollow sound. Now comes the key question, when a photon reaches the half-coated mirror, according to the Copenhagen school, half of you may hear a squeak and be safe, and the other half is not so wonderful, you hear a slam and don’t know anything.And according to the multiverse, there must be one of you who hears the bang, and another one who hears the bang in another world.But the problem is, the person who heard the bang died immediately, and he lost all feeling, and the world has no meaning for you.The only world that makes sense to you is the one you live in. So, in terms of the anthropic principle (which we discussed earlier), the only existences that make sense to you are those worlds in which you live.You'll just keep on living forever just hearing 哢!Because the multiverse is different from Copenhagen, there will always be one of you living in a certain world! Let's fire a photon every second to the half-coated mirror to activate the mechanism.At this time, Copenhagen predicts that even if you are very lucky, you will hear a few crackles at most and die eventually.But the prophecy of the multiverse is: there will always be one you alive, and his world is the only meaningful existence for you.As long as you sit in front of the gun, from your own perspective, you will only ever hear the crack every second, and you will never die (although in a surprising number of other worlds, you have died. all over the world, but those worlds mean nothing to you)! But as soon as you move away from the gun, you'll hear the bang again, as these worlds regain meaning to you, and you live to witness them.All in all, the multiverse prophecy is this: as long as you're in front of the gun, (for you) it never fires, and as soon as you move away, it starts bangping randomly again. So, for the tester himself, if he keeps hearing the Q and lives well, he can be confident to a large extent that the multiverse explanation is correct.If he is dead, then the Copenhagen interpretation is correct.But this is meaningless to him, everyone is dead. You may be confused by the anthropic principle here.In any case, isn't it a very, very small probability that the gun will always fail (if n times, the probability is one/two^n)?How can you say that the gun must behave this way for you?But the problem is, the premise for you is that you must exist! Let us take this example: If you are a male, you will definitely find such an interesting fact: your father has a son, your grandfather has a son, and your great-grandfather has a son. , wars and horses, hunger and barrenness, not only can they survive, but they will continue to have offspring, and there will always be a son. This is a very small probability (if you are a woman, you can push it to the way of your natal family).But if you say with emotion that your existence is a once-in-a-century miracle, it would be ridiculous.Obviously, the prerequisite for you to be able to feel emotion is your existence itself!In fact, objectively speaking, the probability of a family having sons in n generations is extremely small, but for you and me, it is necessary, the probability is 100%!In the same way, some people lament the ingenuity of the universe, and the probability of its creation is so low, but according to the anthropic principle, the universe must be like this!In Quantum Suicide, as long as you are always present, it is 100% for you that the gun does not fire! But it is a pity: even if you find that the multiverse explanation is correct, it is only knowledge for you alone.The facts are always the same as far as we, the bystanders, are concerned: you are shot dead after a number of shots.All we can do is argue around your dead body, whether according to Copenhagen, you have disappeared from the universe forever, or according to MWI, you are still alive and well in some world.The probability that we outsiders are projected into the world you live in is extremely low and can almost be ignored, but for you, it is 100% necessary for you to exist in that world!And, since the worlds can't interfere with each other, you can never come to us from that world and tell us that the multiverse is true! In fact, Tegmark and others don’t need to bother to design any quantum suicide experiments. According to their thinking, if the multiverse explanation is correct, then for someone, no matter how he tries to commit suicide, he will not die!If he wiped his neck with a knife, then because the knife is made up of a group of particles conforming to the Schrödinger wave equation, there is always a very, very small, but unrealistic possibility that these particles have quantum tunneling at that moment. This way penetrated the person's neck without any damage, thus keeping the person alive!Of course, this probability is extremely small, but according to MWI, everything that could happen has actually happened, so this phenomenon will always happen in a certain world!Objectively speaking, this person is dead in 99.99999999% of the worlds, but from his subjective point of view, he is still alive!It doesn't matter what the method is, whether it's jumping off a building, lying on a rail, or hanging himself, there are always some worlds that keep him alive.From the person's own point of view, no matter how he dies, he won't die! This is the strange theory derived from the quantum suicide thought experiment, euphemistically called quantum immortality.As far as the subjective point of view is concerned, not only can a person never complete suicide, but in fact once he begins to exist, he will never disappear!There is always some quantum effect that keeps a person from aging, and according to the MWI, these very low probabilities always correspond to some actual world!If the multiverse theory is correct, then we have the corollary that once a consciousness begins to exist, it must be immortal from its own perspective! (Gosh, how did we get to consciousness again!) This is the strongest version of the anthropic principle, also known as the ultimate anthropic principle. It is conceivable that supporters of the multiverse theory such as Tegmark felt dumbfounded when they saw that their proposal was deduced into such a strange theory.The University of Pennsylvania cosmologist had to come out and say that immortality is not an orthodox inference from MWI.He said that before a person dies, he also goes through some kind of non-quantization process, so that the so-called consciousness cannot continue to last forever.Unfortunately, few people believed his defense. This question will no doubt be of interest to scientists and philosophers alike.Proponents of MWI would also criticize that the deaths of humans in a large cosmic sample cannot simply be ignored, because we still know next to nothing about consciousness, and how it exists continuously has not been examined at all.Some biased opinions will hold that if consciousness must exist continuously in some branches of the universe, then we should conclude that it not only exists all the time, but also continues forever, that is, we should not lose consciousness (such as sleep). or coma).But maybe there are some worlds where we never sleep, who knows?Besides, falling asleep for a while and then waking up again does not seem to be insignificant to consciousness.And more importantly, perhaps it is the question of how to define what you are in the many worlds.In short, there are endless logical circles here, and there is almost nothing that can be tested in practice, all empty-to-empty.I don't think Popper would be happy with that! Regarding the suicide experiment itself, I don't think anyone would actually try it just to test Copenhagen and MWI!Because no matter what, only you know the result of the experiment, and you cannot tell it to the masses.And if the Copenhagen Interpretation is unfortunately correct, you're out of luck.Although it is said that if you hear the Tao in the morning, you can die in the evening, but generally speaking, after hearing the Tao, it is better to use it to do something more meaningful.Also, even if you really don't die at gunpoint, you can't really tell if that's the result of a multi-world prophecy, or if it's just your luck being very, very, very good.The most you can say is: I declare with 99.99999999% certainty that many worlds are correct.that is it. According to Shikhovtsev's latest biography, Everett himself somehow believed that his consciousness would continue along some branch of the universe that didn't lead to death (of course he didn't know about suicide experiments).But the tragedy and irony is that his family believed in parallel universes so much that his daughter Liz (Liz) said in her suicide note that she went to another parallel world to meet him ( Of course, she didn't kill herself to test this theory).Maybe the Everett family actually met in a world, but at least in this world we are in now (and most other worlds), we see that people cannot be resurrected after death.So, at least in consideration of the feelings of family and friends in most worlds, I strongly advise readers not to attempt this with scientific zeal. We've been on the many-worlds road long enough, and, as we did with the Copenhagen School, our quests have gotten weirder and weirder in the end, rough and weedy and exhausting, and in the end There will be consciousness again, immortality and other metaphysical things (what the hell)!We still retreat in spite of the difficulties, return to the original fork in the road, and see if there are other different options.But before we leave this path, there is one more thing worth mentioning, and that is the so-called quantum computer.In 1977, Everett accepted the invitation of Wheeler and DeWitt to give lectures at the University of Texas. During lunch, DeWitt specially arranged for one of Wheeler's students to sit next to Everett. Asked him about Hilbert spaces.This student is David.David Deutsch. three The invention of the computer is one of the most important events in the 20th century. The emergence of this new thing has fundamentally changed the human society, making our ability break through the limit and reach an unimaginable level.Today, computers have penetrated into every corner of our lives, and we can hardly move without it.Among other things, the history you are reading is typed and edited on my laptop, although using a modern PC for word processing would be overkill, or using Ian.In Stewart's words, it's like driving a Rolls-Royce.Royce delivered milk, but thanks to the progress of the times, this luxury has entered thousands of households after all.Moreover, in today's information business society, its replacement is so fast that people have to worry about upgrading their old computers every two or three years, and it is not without heartache to take out those good ones to the capitalists. Money that is easy to accumulate. Looking back at the history of computer development, people often lament the rapid development of science and technology.Usually we regard the ENIAC of the University of Pennsylvania in 1946 as the world's first electronic computer, but of course, with different definitions of the concept of computer, people often refer to the German Konrad Zuse The Z3 built in 1941, the ABC (Atanasoff-Berry Computer) built by Iowa State University during World War II, or the Collosus built by the Turing team to break the German code.In any case, these were big, unwieldy things that could fill an entire room, some crammed with unsightly tubes, some trailing long wires, input and output by punched paper or tape, and modern It's like the difference between beauty and the beast compared to a light and delicate home computer.However, if we mathematically idealize these two seemingly very different, Beauty and the Beast are essentially the same!Regardless of whether it is a huge early computer or the PC we use today, they can actually be simplified into such a machine: it reads an input each time, and depending on its internal state at the time, it follows a pre-compiled rule table. Output the corresponding operation: this operation can be to write the output, or change the internal state, or simply do nothing or even shut down.The key here is that the input and output of our machine can be infinite, but its internal state and rule table must be finite.This model is actually the prototype of all computers. It was created by Alan, one of the founders of modern computers.Turing (Alan Turing), also known as Turing Machine (The Turing Machine).In Turing's original paper, it was described as a sort of box with an infinitely long tape running through it, with input at one end and output at the other.Information is recorded on the tape, generally a sequence of 0s and 1s.The machine moves the tape as needed, reads data from one end, performs operations according to a compiled rule table, and finally outputs the results of calculations on the other end. The computers we use today, no matter how sophisticated and complex they may appear to be, are essentially Turing machines.It reads in the data stream, processes it according to a specific algorithm, and outputs the result at the other end.In this sense, the difference between the Pentium 4 and the 286 is that the former is faster and more efficient, but they are also Turing machines, and they can actually do the same amount of things!I mean, the 286 can do what the Pentium can do, given enough time and output space (to record temporary storage). 286 is already too advanced. Even if it degenerates into the most primitive form of Turing machine, that is, the machine that can only move the tape left or right and act accordingly, they can solve as many things, but It's just a question of speed and efficiency. The information that computers process is at the most basic level binary code, in other words, a sequential stream of sums and ones.Friends who are a little familiar with computers know that we call each bit of information a bit (bit, which is actually an abbreviation for binary digit). For example, information 1010 contains four bits.Eight bits are equal to one byte, 1024 bytes are 1K, 1024K=1M, 1024M=1G, everyone must be very clear. For traditional computers, a bit is the smallest unit of information.It's either one, or one, corresponding to the circuit being on or off.If a computer reads ten bits of information, it is equivalent to saying that it reads a ten-bit binary number (for example, 1010101010), and each bit of this number is a definite 0 or 1.This seems to be taken for granted in people's eyes. But let's move on to the amazing quantum world.A bit is the smallest unit in the flow of information, which looks like a quantum!Let's recall the strange sights we saw on the way. What is the most puzzling thing about quantum theory?is uncertainty.We can't pinpoint exactly where an electron is, we don't know whether it passed through the left or right slit, we don't know whether Schrödinger's cat is dead or alive.According to the fundamental equations of quantum theory, all possibilities add up linearly!Electrons pass through the left and right slits at the same time, and Schrödinger's cat is alive and dead at the same time.只有當實際觀測它的時候，上帝才隨機地擲一下骰子，告訴我們一個確定的結果，或者他老人家不擲骰子，而是把我們投影到兩個不同的宇宙中去。 大家不要忘記，我們的電腦也是由微觀的原子組成的，它當然也服從量子定律（事實上所有的機器肯定都是服從量子論的，只不過對於傳統的機器來說，它們的工作原理並不主要建立在量子效應上）。假如我們的資訊由一個個電子來傳輸，我們規定，當一個電子是左旋的時候，它代表了0，當它是右旋的時候，則代表1（通常我們會以上和下來表示自旋方向，不過可能有讀者會對上旋感到困惑，我們換個稱呼，這無所謂）。現在問題來了，當我們的電子到達時，它是處於量子疊加態的。這豈不是說，它同時代表了0和1？ 這就對了，在我們的量子電腦裡，一個bit不僅只有0或者1的可能性，它更可以表示一個0和1的疊加！一個比特可以同時記錄0和1，我們把它稱作一個量子比特（qubit）。假如我們的量子電腦讀入了一個10bits的資訊，所得到的就不僅僅是一個十位的二進位數字了，事實上，因為每個bit都處在0和1的疊加態，我們的電腦所處理的是2^10個十位數的疊加！ 換句話說，同樣是讀入10bits的資訊，傳統的電腦只能處理一個十位的二進位數字，而如果是量子電腦，則可以同時處理2^10個這樣的數！ 利用量子演化來進行某種圖靈機式的計算早在七十年代和八十年代初便由Bennett，Benioff等人進行了初步的討論。到了一九八二年，那位極富傳奇色彩的美國物理學家理查．費因曼（Richard Feynman）注意到，當我們試圖使用電腦來類比某些物理過程，例如量子疊加的時候，計算量會隨著類比物件的增加而指數式地增長，以致使得傳統的模擬很快變得不可能。費因曼並未因此感到氣餒，相反，他敏銳地想到，也許我們的電腦可以使用實際的量子過程來類比物理現象！如果說類比一個疊加需要很大的計算量的話，為什麼不用疊加本身去模擬它呢？每一個疊加都是一個不同的計算，當所有這些計算都最終完成之後，我們再對它進行某種么正運算，把一個最終我們需要的答案投影到輸出中去。費因曼猜想，這在理論上是可行的，而他的確猜對了！ 一九八五年，我們那位在埃弗萊特的諄諄教導和多宇宙論的薰陶下成長起來的大衛．德義奇閃亮登場了。他仿照圖靈當年走的老路子，成功地證明了，一台普適的量子電腦是可能的。所謂普適機（universal machine）的概念可能對大家有點陌生以及令人困惑，它可以回到圖靈那裡，其基本思想是，存在某種圖靈機，把一段指令編成合適的編碼對其輸入，可以令這台機器模擬任何圖靈機的行為。我無意在這裡過於深入細節，因為那是相當費腦筋的事情，雖然其中的數學一點也不複雜。如果各位有興趣深入探索的話可以參閱一些介紹圖靈工作的文章（我個人還是比較推薦彭羅斯的《皇帝新腦》），在這裡各位所需要瞭解的無非是：我們聰明睿智的德義奇先生證明了一件事，那就是我們理論上可以建造一種機器，它可以類比任何特殊量子電腦的過程，從而使得一切形式的量子計算成為可能。傳統的電腦處理資訊流的時候用到的是所謂的布林邏輯門（Boolean Logic Gate），比如AND，OR，NOT，XOR等等。在量子電腦中只需把它們換成相應的量子邏輯門即可。 說了那麼多，一台量子電腦有什麼好處呢？ 德義奇證明，量子電腦無法實現超越演算法的任務，也就是說，它無法比普通的圖靈機做得更多。從某種確定的意義上來說，量子電腦也是一種圖靈機。但和傳統的機器不同，它的內態是不確定的，它同時可以執行多個指向下一階段的操作。如果把傳統的電腦稱為決定性的圖靈機（Deterministic Turing Machine,DTM），量子電腦則是非決定性的圖靈機（NDTM）。德義奇同時證明，它將具有比傳統的電腦大得多的效率。用術語來講，執行同一任務時它所要求的複雜性（complexity）要低得多。理由是顯而易見的，量子電腦執行的是一種平行計算，正如我們前面舉的例子，當一個10bits的資訊被處理時，量子電腦實際上操作了2^10個態！ 在如今這個資訊時代，網上交易和電子商務的浪潮正席捲全球，從政府至平民百姓，都越來越依賴於電腦和網路系統。與此同時，電子安全的問題也顯得越來越嚴峻，誰都不想駭客們大搖大擺地破解你的密碼，侵入你的系統篡改你的資料，然後把你銀行裡的存款提得精光，這就需要我們對私隱資料執行嚴格的加密保護。目前流行的加密演算法不少，很多都是依賴於這樣一個靠山，也即所謂的大數不可分解性。大家中學裡都苦練過因式分解，也做過質因數分解的練習，比如把15這個數字分解成它的質因數的乘積，我們就會得到15＝5×3這樣一個唯一的答案。 問題是，分解15看起來很簡單，但如果要分解一個很大很大的數，我們所遭遇到的困難就變得幾乎不可克服了。比如，把10949769651859分解成它的質因數的乘積，我們該怎麼做呢？糟糕的是，在解決這種問題上，我們還沒有發現一種有效的演算法。一種笨辦法就是用所有已知的質數去一個一個地試，最後我們會發現10949769651859＝4220851×2594209（數字取自德義奇的著作The Fabric of Reality），但這是異常低效的。更遺憾的是，隨著數字的加大，這種方法所費的時間呈現出幾何式的增長！每當它增加一位數，我們就要多費三倍多的時間來分解它，很快我們就會發現，就算計算時間超過宇宙的年齡，我們也無法完成這個任務。當然我們可以改進我們的演算法，但目前所知最好的演算法（我想應該是GNFS）所需的複雜性也只不過比指數性的增長稍好，仍未達到多項式的要求（所謂多項式，指的是當處理數位的位元數n增大時，演算法所費時間按照多項式的形式，也就是n^k的速度增長）。 所以，如果我們用一個大數來保護我們的秘密，只有當這個大數被成功分解時才會洩密，我們應當是可以感覺非常安全的。因為從上面的分析可以看出，想使用暴力方法，也就是窮舉法來破解這樣的密碼幾乎是不可能的。雖然我們的處理器速度每隔十八個月就翻倍，但也遠遠追不上安全性的增長：只要給我們的大數增加一兩位數，就可以保好幾十年的平安。目前最流行的一些加密術，比如公開金鑰的RSA演算法正是建築在這個基礎之上。 但量子電腦實現的可能使得所有的這些演算法在瞬間人人自危。量子電腦的並行機制使得它可以同時處理多個計算，這使得大數不再成為障礙！一九九四年，貝爾實驗室的彼得．肖（Peter Shor）創造了一種利用量子電腦的演算法，可以有效地分解大數（複雜性符合多項式！）。比如我們要分解一個二百五十位元的數位，如果用傳統電腦的話，就算我們利用最有效的演算法，把全世界所有的電腦都聯網到一起聯合工作，也要花上幾百萬年的漫長時間。但如果用量子電腦的話，只需幾分鐘！一台量子電腦在分解二百五十位元數的時候，同時處理了10^500個不同的計算！ 更糟的事情接踵而來。在肖發明了他的演算法之後，一九九六年貝爾實驗室的另一位科學家洛弗．格魯弗（Lov Grover）很快發現了另一種演算法，可以有效地搜索未排序的資料庫。如果我們想從一個有n個記錄但未排序的資料庫中找出一個特定的記錄的話，大概只好靠隨機地碰運氣，平均試n/2次才會得到結果，但如果用格魯弗的演算法，複雜性則下降到根號n次。這使得另一種著名的非公開金鑰系統加密演算法，DES面臨崩潰。現在幾乎所有的人都開始關注量子計算，更多的量子演算法肯定會接連不斷地被創造出來，如果真的能夠造出量子電腦，那麼對於現在所有的加密演算法，不管是RSA，DES，或者別的什麼橢圓曲線，都可以看成是末日的來臨。最可怕的是，因為量子並行運算內在的機制，即使我們不斷增加密碼的位元數，也只不過給破解者增加很小的代價罷了，這些加密術實際上都破產了！ 二○○一年，IBM的一個小組演示了肖的演算法，他們利用七個量子比特把15分解成了3和5的乘積。當然，這只是非常初步的進展，我們還不知道，是否真的可以造出有實際價值的量子電腦，量子態的糾纏非常容易退相干，這使得我們面臨著技術上的嚴重困難。雖然二○○二年，斯坦弗和日本的科學家聲稱，一台矽量子電腦是可以利用現在的技術實現的，二○○三年，馬里蘭大學的科學家們成功地實現了相距0.7毫米的兩個量子比特的互相糾纏，一切都在向好的方向發展，但也許量子電腦真正的運用還要過好幾十年才會實現。這個專案是目前最為熱門的話題之一，讓我們且拭目以待。 就算強大的量子電腦真的問世了，電子安全的前景也並非一片黯淡，俗話說得好，上帝在這裡關上了門，但又在別處開了一扇窗。量子論不但給我們提供了威力無比的計算破解能力，也讓我們看到了另一種可能性：一種永無可能破解的加密方法。這是另一個炙手可熱的話題：量子加密術（quantum cryptography）。如果篇幅允許，我們在史話的最後會簡單描述一下這方面的情況。這種加密術之所以能夠實現，是因為神奇的量子可以突破愛因斯坦的上帝所安排下的束縛那個宿命般神秘的不等式。而這，也就是我們馬上要去討論的內容。 但是，在本節的最後，我們還是回到多宇宙解釋上來。我們如何去解釋量子電腦那神奇的計算能力呢？德義奇聲稱，唯一的可能是它利用了多個宇宙，把計算放在多個平行宇宙中同時進行，最後匯總那個結果。拿肖的演算法來說，我們已經提到，當它分解一個二百五十位數的時候，同時進行著10^500個計算。德義奇憤憤不平地請求那些不相信MWI的人解釋這個事實：如果不是把計算同時放到10^500個宇宙中進行的話，它哪來的資源可以進行如此驚人的運算？他特別指出，整個宇宙也只不過包含大約10^80個粒子而已。但是，雖然把計算放在多個平行宇宙中進行是一種可能的說法（雖然聽上去仍然古怪），其實MWI並不是唯一的解釋。基本上，量子電腦所依賴的只是量子論的基本方程，而不是某個解釋。它的模型是從數學上建築起來的，和你如何去解釋它無干。你可以把它想像成10^500個宇宙中的每一台電腦在進行著計算，但也完全可以按照哥本哈根解釋，想像成未觀測（輸出結果）前，在這個宇宙中存在著10^500台疊加的電腦在同時幹活！至於這是如何實現的，我們是沒有權利去討論的，正如我們不知道電子如何同時穿過了雙縫，貓如何同時又死又活一樣。這聽起來不可思議，但在許多人看來，比起瞬間突然分裂出了10^500個宇宙，其古怪程度也半斤八兩。正如柯文尼在《時間之箭》中說的那樣，即使這樣一種電腦造出來，也未必能證明多世界一定就比其它解釋優越。關鍵是，我們還沒有得到實實在在可以去判斷的證據，也許我們還是應該去看看還有沒有別的道路，它們都通向哪些更為奇特的方向。 Four 我們終於可以從多世界這條道路上抽身而退，再好好反思一下量子論的意義。前面我們留下的那塊意識怪獸的牌子還歷歷在目，而在多宇宙這裡我們的境遇也不見得好多少，也許可以用德威特的原話，立一塊精神分裂的牌子來警醒世人注意。在哥本哈根那裡，我們時刻擔心的是如何才能使波函數坍縮，而在多宇宙那裡，問題變成了我在宇宙中究竟算是個什麼東西。假如我們每時每刻都不停地被投影到無數的世界，那麼究竟哪一個才算是真正的我呢？或者，我這個概念乾脆就應該定義成由此刻開始，同時包含了將來那n條宇宙岔路裡的所有我的一個集合？如果是這樣的話，那麼量子永生聽起來就不那麼荒誕了：在這個集合中我總在某條分支上活著嘛。假如你不認同，認為我只不過是某時某刻的一個存在，隨著每一次量子測量而分裂成無數個新的不同的我，那麼難道我們的精神只不過是一種暫態的概念，它完全不具有連續性？生活在一個無時無刻不在分裂的宇宙中，無時無刻都有無窮個新的我的分身被製造出來，天知道我們為什麼還會覺得時間是平滑而且連續的，天知道為什麼我們的自我意識的連續性沒有遭到割裂。 不管是哥本哈根還是多宇宙，其實都是在努力地試圖解釋量子世界中的這樣一個奇妙性質：疊加性。正如我們已經在史話中反覆為大家所揭示的那樣，當沒有觀測前，古怪的量子精靈始終處在不確定的狀態，必須描述為所有的可能性的疊加。電子既在這裡又在那裡，在實際觀測之前並不像以前經典世界中我們不言而喻地假定的那樣，有一個唯一確定的位置。當一個光子從A點運動到B點，它並不具有經典力學所默認的一條確定的軌跡。相反，它的軌跡是一團模糊，是所有可能的軌跡的總和！而且不單單是所有可能的空間軌跡，事實上，它是全部空間以及全部時間的路徑的總和！換句話說，光子從A到B，是一個過去、現在、未來所有可能的路線的疊加。在此基礎之上費因曼建立了他的路徑積分（path integral）方法，用以計算量子體系在四維空間中的機率振幅。我們在史話的前面已經看到了海森堡的矩陣和薛定諤的波，費因曼的路徑積分是第三種描述量子體系的手段。但同樣可以證明，它和前兩者是完全等價的，只不過是又一種不同的數學表達形式罷了。配合費因曼圖，這種方法簡單實用，而且非常巧妙。把它運用到原子體系中，我們會驚奇地發現在絕大部分路徑上，作用量都互相抵消，只留下少數可能的軌道，而這正和觀測相符！ 我們必須承認，量子論在現實中是成功的，它能夠完美地解釋和說明觀測到的現象。可是要承認疊加，不管是哥本哈根式的疊加還是多宇宙式的疊加，這和我們對於現實世界的常識始終有著巨大的衝突。我們還是不由地懷念那流金的古典時代，那時候現實世界仍然保留著高貴的客觀性血統，它簡單明確，符合常識，一個電子始終有著確定的位置和動量，不以我們的意志或者觀測行為而轉移，也不會莫名其妙地分裂，而只是一絲不苟地在一個優美的宇宙規則的統治下按照嚴格的因果律而運行。哦，這樣的場景溫馨而暖人心扉，簡直就是物理學家們夢中的桃花源，難道我們真的無法再現這樣的理想，回到那個令人懷念的時代了嗎？ 且慢，這裡就有一條道路，打著一個大看板：回到經典。它甚至把愛因斯坦拉出來作為它的代言人：這條道路通向愛因斯坦的夢想。天哪，愛因斯坦的夢想，不就是那個古典客觀，簡潔明確，一切都由嚴格的因果性來主宰的世界嗎？那裡面既沒有擲骰子的上帝，也沒有多如牛毛的宇宙拷貝，這是多麼教人心動的情景。我們還猶豫什麼呢，趕快去看看吧！ 時空倒轉，我們先要回到一九二七年，回到布魯塞爾的第五屆索爾維會議，再回味一下那場決定了量子論興起的大辯論。我們在史話的第八章已經描寫了這次名留青史的會議的一些情景，我們還記得法國的那位貴族德布羅意在會上講述了他的導波理論，但遭到了泡利的質疑。在第五屆索爾維會議上，玻爾的互補原理還剛剛出臺，粒子和波動還正打得不亦樂乎，德布羅意的導波正是試圖解決這一矛盾的一個嘗試。我們都還記得，德布羅意發現，每當一個粒子前進時，都伴隨著一個波，這深刻地揭示了波粒二象性的難題。但德布羅意並不相信玻爾的互補原理，亦即電子同時又是粒子又是波的解釋。德布羅意想像，電子始終是一個實實在在的粒子，但它的確受到時時伴隨著它的那個波的影響，這個波就像盲人的導航犬，為它探測周圍的道路的情況，指引它如何運動，也就是我們為什麼把它稱作導波的原因。德布羅意的理論裡沒有波恩統計解釋的地位，它完全是確定和實在論的。量子效應表面上的隨機性完全是由一些我們不可知的變數所造成的，換句話說，量子論是一個不完全的理論，它沒有考慮到一些不可見的變數，所以才顯得不可預測。假如把那些額外的變數考慮進去，整個系統是確定和可預測的，符合嚴格因果關係的。這樣的理論稱為隱變數理論（Hidden Variable Theory）。 德布羅意理論生不逢時，正遇上偉大的互補原理出臺的那一刻，加上它本身的不成熟，於是遭到了眾多的批評，而最終判處它死刑的是一九三二年的馮諾伊曼。我們也許還記得，馮諾伊曼在那一年為量子論打下了嚴密的數學基礎，他證明了量子體系的一些奇特性質比如無限後退。然而在這些之外，他還順便證明了一件事，那就是：任何隱變數理論都不可能對測量行為給出確定的預測。換句話說，隱變數理論試圖把隨機性從量子論中趕走的努力是不可能實現的，任何隱變數理論不管它是什麼樣的註定都要失敗。 馮諾伊曼那華麗的天才傾倒每一個人，沒有人對這位二十世紀最偉大的數學家之一產生懷疑。隱變數理論那無助的努力似乎已經逃脫不了悲慘的下場，而愛因斯坦對於嚴格的因果性的信念似乎也註定要化為泡影。德布羅意接受這一現實，他在內心深處不像玻爾那樣頑強而充滿鬥志，而是以一種貴族式的風度放棄了他的觀點。整個三、四十年代，哥本哈根解釋一統天下，量子的不確定性精神深植在物理學的血液之中，眾多的電子和光子化身為波函數神秘地在宇宙中彌漫，眾星拱月般地烘托出那位偉大的智者尼爾斯．玻爾的魔力來。 一九六九年諾貝爾物理獎得主蓋爾曼後來調侃地說：玻爾給整整一代的物理學家洗了腦，使他們相信，事情已經最終解決了。 約翰．貝爾則氣忿忿地說：德布羅意在一九二七年就提出了他的理論。當時，以我現在看來是丟臉的一種方式，被物理學界一笑置之，因為他的論據沒有被駁倒，只是被簡單地踐踏了。 誰能想到，就連像馮諾伊曼這樣的天才，也有陰溝裡翻船的時候。他的證明不成立！馮諾伊曼關於隱函數理論無法對觀測給出唯一確定的解的證明建立在五個前提假設上，在這五個假設中，前四個都是沒有什麼問題的，關鍵就在第五個那裡。我們都知道，在量子力學裡，對一個確定的系統進行觀測，我們是無法得到一個確定的結果的，它按照隨機性輸出，每次的結果可能都不一樣。但是我們可以按照公式計算出它的期望（平均）值。假如對於一個確定的態向量Φ我們進行觀測X，那麼我們可以把它坍縮後的期望值寫成。正如我們一再強調的那樣，量子論是線性的，它可以疊加。如果我們進行了兩次觀測X，Y，它們的期望值也是線性的，即應該有關係： ＝＋ 但是在隱函數理論中，我們認為系統光由態向量Φ來描述是不完全的，它還具有不可見的隱藏函數，或者隱藏的態向量H。把H考慮進去後，每次觀測的結果就不再隨機，而是唯一確定的。現在，馮諾伊曼假設：對於確定的系統來說，即使包含了隱函數H之後，它們也是可以疊加的。即有： ＝＋ 這裡的問題大大地有。對於前一個式子來說，我們討論的是平均情況。也就是說，假如真的有隱函數H的話，那麼我們單單考慮Φ時，它其實包含了所有的H的可能分佈，得到的是關於H的平均值。但把具體的H考慮進去後，我們所說的就不是平均情況了！相反，考慮了H後，按照隱函數理論的精神，就無所謂期望值，而是每次都得到唯一的確定的結果。關鍵是，平均值可以相加，並不代表一個個單獨的情況都能夠相加！ 我們這樣打比方：假設我們扔骰子，骰子可以擲出1-6點，那麼我們每扔一個骰子，平均得到的點數是3.5。這是一個平均數，能夠按線性疊加，也就是說，假如我們同時扔兩粒骰子，得到的平均點數可以看成是兩次扔一粒骰子所得到的平均數的和，也就是3.5＋3.5＝7點。再通俗一點，假設ABC三個人同時扔骰子，A一次扔兩粒，B和C都一次扔一粒，那麼從長遠的平均情況來看，A得到的平均點數等於B和C之和。 但馮諾伊曼的假設就變味了。他其實是假定，任何一次我們同時扔兩粒骰子，它必定等於兩個人各扔一粒骰子的點數之和！也就是說只要三個人同時扔骰子，不管是哪一次，A得到的點數必定等於B加C。這可大大未必，當A擲出12點的時候，B和C很可能各只擲出一點。雖然從平均情況來看A的確等於B加C，但這並非意味著每回合都必須如此！ 馮諾伊曼的證明建立在這樣一個不牢靠的基礎上，自然最終轟然崩潰。終結他的人是大衛．玻姆（David Bohm），當代最著名的量子力學專家之一。玻姆出生於賓夕法尼亞，他曾在愛因斯坦和奧本海默的手下學習（事實上，他是奧本海默在伯克利所收的最後一個研究生），愛因斯坦的理想也深深打動著玻姆，使他決意去追尋一個回到嚴格的因果律，恢復宇宙原有秩序的理論。一九五二年，玻姆復活了德布羅意的導波，成功地創立了一個完整的隱函數體系。全世界的物理學家都吃驚得說不出話來：馮諾伊曼不是已經把這種可能性徹底排除掉了嗎？現在居然有人舉出了一個反例！ 奇怪的是，發現馮諾伊曼的錯誤並不需要太高的數學技巧和洞察能力，但它硬是在二十年的時間裡沒有引起值得一提的注意。David Mermin揶揄道，真不知道它自發表以來是否有過任何專家或者學生真正研究過它。貝爾在訪談裡毫不客氣地說：你可以這樣引用我的話：馮諾伊曼的證明不僅是錯誤的，更是愚蠢的！ 看來我們在前進的路上仍然需要保持十二分的小心。 飯後閒話：第五公設 馮諾伊曼栽在了他的第五個假設上，這似乎是冥冥中的天道迴圈，二○○○年前，偉大的歐幾里德也曾經在他的第五個公設上小小地絆過一下。 無論怎樣形容《幾何原本》的偉大也不會顯得過分誇張，它所奠定的公理化思想和演繹體系，直接孕育了現代科學，給它提供了最強大的力量。《幾何原本》把幾何學的所有命題推理都建築在一開頭給出的五個公理和五個公設上，用這些最基本的磚石建築起了一幢高不可攀的大廈。 對於歐氏所給出的那五個公理和前四個公設（適用於幾何學的他稱為公設），人們都可以接受。但對於第五個公設，人們覺得有一些不太滿意。這個假設原來的形式比較冗長，人們常把它改成一個等價的表述方式：過已知直線外的一個特定的點，能夠且只能夠作一條直線與已知直線平行。長期以來，人們對這個公設的正確性是不懷疑的，但覺得它似乎太複雜了，也許不應該把它當作一個公理，而能夠從別的公理中把它推導出來。但二○○○年過去了，竟然沒有一個數學家做到這一點（許多時候有人聲稱他證明了，但他們的證明都是錯的）！ 歐幾里德本人顯然也對這個公設感到不安，相比其他四個公設，第五公設簡直複雜到家了（其他四個公設是：一，可以在任意兩點間劃一直線。二，可以延長一線段做一直線。三，圓心和半徑決定一個圓。四，所有的直角都相等）。在《幾何原本》中，他小心翼翼地儘量避免使用這一公設，直到沒有辦法的時候才不得不用它，比如在要證明任意三角形的內角和為一百八十度的時候。 長期的失敗使得人們不由地想，難道第五公設是不可證明的？如果我們用反證法，假設它不成立，那麼假如我們匯出矛盾，自然就可以反過來證明第五公設本身的正確性。但如果假設第五公設不成立，結果卻導致不出矛盾呢？ 俄國數學家羅巴切夫斯基（N. Lobatchevsky）正是這樣做的。他假設第五公設不成立，也就是說，過直線外一點，可以作一條以上的直線與已知直線平行，並以此為基礎進行推演。結果他得到了一系列稀奇古怪的結果，可是它們卻是一個自成體系的系統，它們沒有矛盾，在邏輯上是自洽的！一種不同於歐幾里得的幾何非歐幾何誕生了！ 從不同於第五公設的其他假設出發，我們可以得到和歐幾里得原來的版本稍有不同的一些定理。比如三角形內角和等於一百八十度是從第五公設推出來的，假如過一點可以作一條以上的平行線，那麼三角形的內角和便小於一百八十度了。反之，要是過一點無法作已知直線的平行線，結果就是三角形的內角和大於一百八十度。對於後者來說容易想像的就是球面，任何看上去平行的直線最終必定交匯。比方說在地球的赤道上所有的經線似乎都互相平行，但它們最終都在兩極點相交。如果你在地球表面畫一個三角形，它的內角和會超出一百八十度，當然，你得畫得足夠大才測量得到。傳說高斯曾經把三座山峰當作三角形的三個頂點來測量它們的內角和，但似乎沒有發現什麼，不過他要是在星系間做這樣的測量，其結果就會很明顯了：星系的品質造成了空間的明顯彎曲。 羅巴切夫斯基假設過一點可以做一條以上的直線與已知直線平行，另一位數學家黎曼則假設無法作這樣的平行線，創立了黎曼非歐幾何。他把情況推廣到n維中去，徹底奠定了非歐幾何的基礎。更重要的是，他的體系被運用到物理中去，並最終孕育了二十世紀最傑出的科學巨構廣義相對論。 five 玻姆的隱變數理論是德布羅意導波的一個增強版，只不過他把所謂的導波換成了量子勢（quantum potential）的概念。在他的描述中，電子或者光子始終是一個實實在在的粒子，不論我們是否觀察它，它都具有確定的位置和動量。但是，一個電子除了具有通常的一些性質，比如電磁勢之外，還具有所謂的量子勢。這其實就是一種類似波動的東西，它按照薛定諤方程發展，在電子的周圍擴散開去。但是，量子勢所產生的效應和它的強度無關，而只和它的形狀有關，這使它可以一直延伸到宇宙的盡頭，而不發生衰減。 在玻姆理論裡，我們必須把電子想像成這樣一種東西：它本質上是一個經典的粒子，但以它為中心發散出一種勢場，這種勢彌漫在整個宇宙中，使它每時每刻都對周圍的環境瞭若指掌。當一個電子向一個雙縫進發時，它的量子勢會在它到達之前便感應到雙縫的存在，從而指導它按照標準的干涉模式行動。如果我們試圖關閉一條狹縫，無處不在的量子勢便會感應到這一變化，從而引導電子改變它的行為模式。特別地，如果你試圖去測量一個電子的具體位置的話，你的測量儀器將首先與它的量子勢發生作用，這將使電子本身發生微妙的變化，這種變化是不可預測的，因為主宰它們的是一些隱變數，你無法直接探測到它們。 玻姆用的數學手法十分高超，他的體系的確基本做到了傳統的量子力學所能做到的一切！但是，讓我們感到不舒服的是，這樣一個隱變數理論始終似乎顯得有些多餘。量子力學從世紀初一路走來，諸位物理大師為它打造了金光閃閃的基本數學形式。它是如此漂亮而簡潔，在實際中又是如此管用，以致於我們覺