Richard Hamming: "There are thoughts you can't think about"

"There are smells that you cannot sense, wavelengths of light that you cannot see, sounds that you cannot hear, ... there are thoughts that you cannot think about."

Hi, Habr.
Remember the awesome article "You and your work" (+219, 1928 bookmarks, 328k reads)?

So Hamming (yes, yes, self-checking and self-correcting Hamming codes ) has a whole book based on his lectures. Let's translate it, because the man is talking.

This book is not just about IT, it is a book about the thinking style of incredibly cool people. “This is not just a charge of positive thinking; it describes the conditions that increase the chances of doing a great job. ”
')
Who wants to help with the translation - write in a personal or mail magisterludi2016@yandex.ru

Chapter 24. Quantum Mechanics

(for the translation, thanks to Nate Blankinstein)

Most physicists now believe that they have a basic description of the Universe [although they currently recognize that 90-99% of the Universe is in the form of "dark matter", about which they know nothing except that it is experiencing gravity] . You must understand that in all science there are only descriptions of how something happens and nothing about why this happens. Newton gave us a formula expressing how gravity works, and he did not make any hypotheses about what it is, nor through what medium it works, let alone why it works. In fact, he did not even believe in "long-range".

Reasons to discuss quantum mechanics, KM, are:

1. this is fundamental physics,
2. she has many unexpected intellectual implications and
3. gives a number of models for work.

At the end of the 19th – beginning of the 20th century, physics encountered a number of problems. Among them were the following:

1. classical physics considered continuously changing objects, and atomic spectra, obviously, turned out to be discrete,
2. electric charges, when moving along a path other than a straight line, emit energy, therefore the atomic model present at that time with an electron rotating around the center would have to quickly radiate energy and collapse into the nucleus, but in fact the atoms are obviously stable,
3. the radiation spectrum of an absolutely black body, measured in laboratories, had a definite shape, but the theories, which well describe one or the other end, gave infinite energy for the opposite end, and
4. many other problems, often centered around the contradictions between discrete and continuous.

Max Planck (1858–1949) put experimental data on blackbody radiation on an empirical curve, and they lay down so well that he “knew” that this was “the right formula.” He was about to bring her out, but he ran into difficulties. In the end, he applied the standard method of decomposing energy into finite portions, and then he took the limit. In the course of mathematical analysis, we do the same; the integral is approximated by a finite number of small rectangles, these rectangles are summed, and then the limit is taken by striving their width to zero. Fortunately for Planck, the formula worked only until he went to the limit, and no matter how he took this limit, the formula disappeared. Finally, he, being a very good, honest physicist, decided that he needed to dwell on a certain size, and this was exactly what was Planck's constant!

The result was presented at a meeting (December 1900), and then published, but was ignored. Even Planck had little faith in him, until Einstein showed how such finite portions of energy, called quanta, also explain the photoelectric effect. This marked the beginning of quantum mechanics. But progress still went slowly, although Bohr developed a model of the atom in which electrons were enclosed in certain orbits and radiated energy only by switching between them. This model is derived from the theory of spectral lines, built on the basis of arithmetic formulas without a known physical basis.

Before I continue, let me explain how this part of the story influenced my behavior in science. Obviously, Planck led to the creation of a theory that the approximating curve was very close to the data and had the appropriate shape. Therefore, I reasoned that if I helped someone do something like this, I would rather strive to express everything in terms of functions that, in their opinion, would be correct for this area, and not in standard polynomials. Therefore, I abandoned the standard polynomial approach to approximation, which is most often used by numerical analytics and statistics, in favor of a more complex approach: finding which class of functions I should use. I usually look for a class of functions that need to be used, asking analysts about it, and then applying all the facts that they think are relevant - all in the hope that I, nevertheless, sometime have a significant insight ( from their side). Well, I never managed to make a contribution as big as a CM, but it often happened that by fitting the task to their knowledge, I made smaller insights.

In 1925, the new CM was founded by two people - Heisenberg and Schrodinger. Heisenberg took such a position that he turned only to measurable quantities, for example, to spectral lines, and thus came to matrix mechanics. Schrödinger adopted a wave approach based on de Broglie’s earlier work, and found an appropriate theory. Both mathematical structures, as you know, allow for discrete eigenvalues ​​that correspond to discrete energy levels of spectral lines. Schrödinger, Eckart and others it was shown that these two theories, although they looked very different, were in many ways equivalent to each other.

Moral: there is no need for a single theory to describe a set of observations, instead two theories, which look completely different, can converge in all the predicted details. It is impossible to move from a set of data to the only correct theory! This I noted in the last chapter.

Another story will illustrate this point more clearly. A few years ago, when I took over the thesis for a Ph.D. from another professor, I discovered that they take random input signals and measure the corresponding outputs. I also learned that it was “well known” - that is, it was known, but it was almost never mentioned - that the completely different internal structures of the “black boxes” that they studied could give exactly the same results, provided that the inputs were the same, of course . There was no way, using such measurements as they did, to distinguish between two rather different structures. Again, you cannot get a unique theory simply from a data set.

New CM was born around 1925 and was a great success. She assumes that energy and many other things in physics live in the form of discrete pieces, but these pieces are so small that we, relatively large objects in relation to them, simply cannot perceive them except with the help of subtle experiments or in special situations. .

Thus, the situation was as follows: classical Newtonian mechanics, which was very well tested in many ways and even successfully predicted the positions of unknown planets, were replaced by two theories: relativity at high speeds, large masses and high energies, and CM for small sizes. Both theories were initially recognized as non-intuitive, but over time they became widely accepted, especially the special theory of relativity. You may recall that in Newton's time, gravity (action at a distance) was not considered reasonable.

Newton assumed that light is in its nature particles, but it also “drove” some parts of the theory. Initially it was thought that light was made up of particles that move along straight lines, but later Jung’s wave theory of light, which you were probably taught in optics classes, began to dominate the corpuscular model. Now we have to come to terms with the fact that light, apparently, consists of quanta, and quanta are both particles and waves at the same time. Practically every teacher teaching a CM is forced, one way or another, to declare: “I cannot explain this duality, you will get used to it!”

Again, I stop and note the obvious lessons that need to be learned from this wave-particle duality. After almost 70 years, during which no worthy explanation of this duality has appeared, one has to ask oneself: “Maybe this is one of those things that it is impossible to think about?” Or, probably, it is simply impossible to express in words. There are smells that you cannot sense, wavelengths of light that you cannot see, sounds that you cannot hear, it all depends on the limits of your senses, so why do you mind observing that, given the “wiring” of your brain, there is thoughts you can't think about? CM gives a possible example. For almost 70 years, all these smart people who taught KM - and no one has found a widely recognized explanation of the fundamental fact of KM: wave-wave dualism. You just gotta get used to it, they said.

This, in turn, shows: in developing the theory, they went to the touch, not quite “knowing” what they were doing. When they found an effect in formulas that can be interpreted in the real world, they declared that they had taken a step forward. During the creation of the CM, Born noted that the wave function in the Schrodinger theory should have the squared amplitude interpreted as the probability of seeing something. Similarly for the matrix mechanics of Heisenberg. From the very beginning, complex numbers dominated the whole theory, hence the need to take the square of the absolute value in order to get a real probability. Dirac observed that the photon interacts only with itself, so the probability should be attributed to individual photons, therefore, the probability in QM is not the average of the set of all photons (or electrons from the Davison-Germer experiment), as many books on probability theory her.

Heisenberg derived the uncertainty principle, which concludes that the conjugate variables, in the sense of the Fourier transform, are subject to the condition that the product of their uncertainties must exceed a fixed number that includes the Planck constant. I commented earlier in Chapter 17 that this is a theorem in Fourier transforms: any linear theory must have a corresponding uncertainty principle, but among physicists this is still widely seen as a physical effect, an effect from Nature, and not from the Mathematics model.

The fact that the probability of events was all that the theory provided provided many people to wonder if it could be that at a lower level of this part of Nature there is a very definite structure, and we only observe its statistical mechanics (but do not forget the observation of Dirac, see above). Von Neumann, in his classic work on CM, proved that there are no hidden variables, which means the absence of a lower structure and the fact that Nature is essentially probabilistic - all this is a point of view that Einstein would never recognize. But the evidence turned out to be erroneous, then new evidence was found and, in turn, also proved to be erroneous - the current situation - believe what you want to believe.

Man is not a rational animal, he is a rationalizing animal.

Consequently, you will find that you often believe in what you want to believe, and not in carefully considered results.

This probabilistic basis of CM, which does not have anything definite under itself, attracted the attention of many philosophers, and they began to discuss the general subject of free will. The classic statement against free will is a remark: “You, being who you are, in that situation, what it is, you can act differently than you act?” Apparently, this question cannot be solved experimentally, therefore the controversy continues. Personally, I, and this is only my conviction, I see that no connection between the two — Nature can in principle be probabilistic, and this does not mean that we can influence it in any way, therefore we cannot “choose” - taking into account if you accept the existence of the forces of "official" physics. Even in ancient Greece, Democritus (about 460 BC) said: “All atoms and voids”. This is still the main position of most physicists - they believe that they know everything that is (in the sense of there are no unknown forces that they have not yet discovered).

This is a religious matter to a large extent - you can believe as you wish in this matter. If we do not have free will, then the widespread belief in the punishment of God (or gods) for our actions seems unfair - we must do so if you take a deterministic approach! On the other hand, if it is reasonable to believe in justice from our God (or gods), then some free will must be around. (For Calvinists, on the contrary). And, of course, “infinite mercy” implies forgiveness for everything you've ever done; see the Adamida Buddha sect in Japan around 1000 as an example of the extremes of such beliefs.

I do not think it is wise to argue about such issues based on CM. I doubt it is a secret that physicists know everything. By my years, I came to believe that there are such things as self-awareness, awareness that cannot be ignored, as they are ignored in the theories of "atoms and voids." But how such things, if they exist (and in what ways they exist), can interact with the real world of atoms, is not entirely clear to me. The theory of psychophysical parallelism (the psychic and physical worlds go along independent parallel paths without intersecting, but in complete agreement), which I was taught in the early course of psychology, seemed to me to be absolutely foolish even then. Therefore, I have nothing to offer you in these matters, except for not pinning high hopes on CM on my convictions.

But the worst in the CM was still to come. Alain Aspect, in Paris, conducted several experiments that, at least, caused problems. Two particles with opposite spins are sent in opposite directions. Their polarization is unknown, but it is believed that when one is measured in one, the other is detected exactly in the opposite polarization. It is also one of the main statements of the CM - that only the act of measurement puts the wave function in a certain state; before measurement, you only have a probability distribution. Thus, the orientation of one measuring instrument at one end of the experiment immediately gives — and we mean really immediately — what is measured at the other, remote end of the experiment — about 12 meters or so! And this may seem to be contradictory in both special and general theories of relativity! I said “appear” because theories predict that you cannot transmit any useful signals faster than at the speed of light. You can swing a bright beam of light, as from a beacon, and the furthest point will move faster than the speed of light; but you cannot transmit signals faster according to both theories. Aspect's experiments seem to force us to accept non-local effects — what happens in one place depends on things at a distance, and the transmitted effect in no sense passes through the local areas between them, but goes there immediately. But apparently, you can not use this effect to transmit useful signals.

Others conducted similar experiments, demonstrating the same effect. Apparently, there are non-local effects in CM. Two systems that once "entangled", as they say, can interact forever, there is no such thing as an isolated object, as we say about their use in classical experiments. Einstein could not accept non-local effects, and many other people could not. But experiments have been conducted for more than a decade, and many hypotheses have been developed to circumvent the conclusion about non-local effects, but few of them have been widely accepted among physicists.

Einstein did not like the idea of ​​non-local effects, and he created a famous article by Einstein-Podolsky-Rosen (EPR), which showed that there were limitations to which we could observe if non-local effects existed. Bell clarified this in the form of the famous "Bell inequalities" on ratios, apparently independent dimensions of probability, and this result is now widely accepted. Non-local effects seem to mean that something can happen instantaneously, without taking time to get from cause to action, like the polarization states of two particles in Aspect experiments.

Thus, once again, CM directly contradicts our beliefs and instincts, which, of course, are based on the human scale, and not on the microscopic scale of atoms. CM is stranger than we ever believed, and it seems to get stranger the longer we study it.

It is important to note that even though I indicated that we may never be able to understand QM in the classical sense of “understand”, we nonetheless created a formal mathematical structure that we can use very effectively. Thus, while we are moving into the future and, possibly, we encounter many other things that we cannot “understand”, we can still create formal mathematical structures that will allow us to work with them in any way. Unsatisfactory? Yes! But it's amazing how you get used to CM after you work with it for a long time. This is almost the same story as handling complex numbers — all the words of teachers about complex arithmetic equivalent to ordered pairs of real numbers with a special multiplication rule meant little to you; your belief in the "reality" of complex numbers was based on their long-term use and the vision that they often provide reasonable, useful predictions. Similarly, faith in Newton's gravity (action at a distance).

I do not pretend to know in any detail what the future will bring, but I believe that since at each stage of progress we tend to attack simpler problems, the future will include more and more things that our brains, being stitched like they are, they cannot “understand” in the classical sense. However, the future is not hopeless. I suspect that we will need many different mathematical models to help ourselves, and I do not think that this is just a prejudice of mathematics. Thus, the future should be filled with interesting opportunities for those who have the intellectual courage to think and use Mathematical models as a basis for “understanding” of Nature. Creating and using new and other types of mathematics seems to me one of the things that you can expect to be mandatory if you need to get an “understanding”. The mathematics of the past was designed to fit the obvious situations, and, as we have already mentioned, we sought to first consider them. When we explore new areas, we can expect new types of mathematics - and even just following the leading edge of science, you will have to study them as they arise!

I put the word "understand" in quotes, because I don’t even pretend to understand what I mean. We all know what we mean by “understanding,” until we try to directly say what it means - and then it disappears! Saint Augustine (died 604 AD) remarked that he knew what “time” is, until you asked him, and then he did not know! I leave you to the future to try to explain (better than I can) what you mean by "understand."



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Who wants to help with the translation - write in a personal or mail magisterludi2016@yandex.ru