lördag 25 juli 2015

Frank Wilczek: Ugly Answer to Ugly Question

In his new book A Beautiful Question: Finding Nature's Deep Design, Frank Wilczek (Nobel Prize in Physics 2004) starts out stating the questions (or paradoxes) which motivated the development of modern physics:

In the quantum world of atoms and light, Nature treats us to a show of strange and seemingly impossible feats. Two of these feats seemed, when discovered, particularly impossible:
  • Light comes in lumps. This is demonstrated in the photoelectric effect, as we’ll discuss momentarily. It came as a shock to physicists. After Maxwell’s electromagnetic theory was confirmed in Hertz’s experiments (and later many others), physicists had thought they understood what light is. Namely, light is electromagnetic waves. But electromagnetic waves are continuous.
  • Atoms have parts, but are perfectly rigid. Electrons were first clearly identified in 1897, by J. J. Thomson. The most basic facts about atoms were elucidated over the following fifteen years or so. In particular: atoms consist of tiny nuclei containing almost all of their mass and all of their positive electric charge, surrounded by enough negatively charged electrons to make a neutral whole. Atoms come in different sizes, depending on the chemical element, but they’re generally in the ballpark of $10^{-8}$ centimeters, a unit of length called an angstrom. Atomic nuclei, however, are a hundred thousand times smaller. The paradox: How can such a structure be stable? Why don’t the electrons simply succumb to the attractive force from the nucleus, and dive in.
  • These paradoxical facts led Einstein and Bohr, respectively, to propose some outrageous, half-right hypotheses that served as footholds on the steep ascent to modern quantum theory. 
  • After epic struggles, played out over more than a decade of effort and debate, an answer emerged. It has held up to this day, and its roots have grown so deep that it seems unlikely ever to topple.
Wilczek then proceeds to prepare us to accept the answers offered by the modern physics of quantum mechanics as the result of epic struggles:
  • The framework known as quantum theory, or quantum mechanics, was mostly in place by the late 1930s. 
  • Quantum theory is not a specific hypothesis, but a web of closely intertwined ideas. I do not mean to suggest quantum theory is vague—it is not. 
  • With rare and usually temporary exceptions, when faced with any concrete physical problem, all competent practitioners of quantum mechanics will agree about what it means to address that problem using quantum theory. 
  • But few, if any, would be able to say precisely what assumptions they have made to get there. Coming to terms with quantum theory is a process, through which the work will teach you how to do it.
We learn that quantum mechanics is not built on specific hypotheses or assumptions, but nevertheless is not vague, and instead rather is a process monitored by competent practitioners. In any case, Wilczek proceeds to give us a glimpse of the basic hypothesis:
  • In quantum theory’s description of the world, the fundamental objects are ....wave functions.
  • Any valid physical question about a physical system can be answered by consulting its wave function.
  • But the relation between question and answer is not straightforward. Both the way that wave functions answer questions and the answers they give have surprising—not to say weird—features.
OK, so we are now enlightened by understanding that the answers that come out are weird. Wilczek continues:
  • I will focus on the specific sorts of wave functions we need to describe the hydrogen atom: 
  • We are interested, then, in the wave function that describes a single electron bound by electric forces to a tiny, much heavier proton.
  • Before discussing the electron’s wave function, we’ll do well to describe its probability cloud. The probability cloud is closely related to the wave function. The probability cloud is easier to understand than the wave function, and its physical meaning is more obvious, but it is less fundamental. (Those oracular statements will be fleshed out momentarily).
  • Quantum mechanics does not give simple equations for probability clouds. Rather, probability clouds are calculated from wave functions.
  • The wave function of a single particle, like its probability cloud, assigns an amplitude to all possible positions of the particle. In other words, it assigns a number to every point in space. 
  • To pose questions, we must perform specific experiments that probe the wave function in different ways.
  • You get probabilities, not definite answers.
  • You don’t get access to the wave function itself, but only a peek at processed versions of it.
  • Answering different questions may require processing the wave function in different ways.
  • Each of those three points raises big issues.
Wilczek then tackles these issues by posing new questions, or lacking question by retreating to an admirable attitude of humility in a lesson of wisdom
  • The first raises the issue of determinism. Is calculating probabilities really the best we can do?
  • The second raises the issue of many worlds. What does the full wavefunction describe, when we’re not peeking? Does it represent a gigantic expansion of reality, or is it just a mind tool, no more real than a dream?
  • The third raises the issue of complementarity....It is a lesson in humility that quantum theory forces to our attention. To probe is to interact, and to interact is potentially to disturb.
  • Complementarity is both a feature of physical reality and a lesson in wisdom.
We see that Wilczek sells the usual broth of strange and seemingly impossible feats, weird features, and outrageous half-right hypotheses, all raising big issues. Wilczek sums up by the following quote of Walt Whitman under the headline COMPLEMENTARITY AS WISDOM:

             Do I contradict myself?
             Very well, then, I contradict myself,
              I am large, I contain multitudes.

But physics is not poetry, and contradictory poetry does not justify contradictory physics. Contradictory mathematical physics cannot be true real physics, not even meaningful poetry. To get big by contradiction is a trade of politics, which is ugly and not beautiful.

Nevertheless, Wilczek started his Nobel lecture as follows:
  • In theoretical physics, paradoxes are good. That’s paradoxical, since a paradox appears to be a contradiction, and contradictions imply serious error. But Nature cannot realize contradictions. When our physical theories lead to paradox we must find a way out. Paradoxes focus our attention, and we think harder.
We understand that to Wilczek/modern physicists, contradictions are good rather than catastrophical and the more paradox the better, since it makes physicists focus attention to think harder.  Beautiful. For more excuses, see What Is Quantum Theory. Wilczek here retells the story of the Father (or Dictator) of Quantum Mechanics, Niels Bohr:
  • How wonderful that we have met with a paradox. Now we have some hope of making progress.
The paradox presented itself in 1925, but what happened to the hope of progress? Is paradoxical physics the physics of our time? Does light come in lumps? Why are atoms stable? Despite paradoxes, no real progress for 90 years!!??

PS1 Here is the question killing the probability interpretation of the wave function: Since the wave function for the ground state of Hydrogen is non-zero even far away from the kernel, does it mean that there is a non-zero chance of experimentally detecting a Hydrogen ground state electron far away from the kernel it is associated with? Or the other way around, since the wave function is maximal at zero distance from the kernel, does it mean that one will mostly find the electron hiding inside the kernel?

PS2 Beauty is an expression of order and deep design, not of disorder and lack of design. An atomistic world ruled by chance can be beautiful only to a professional statistician obsessed by computing mean values.

PS3 Not Even Wrong presents the book as follows: Frank Wilczek’s new book, A Beautiful Question, is now out and if you’re at all interested in issues about beauty and the deep structure of reality, you should find a copy and spend some time with it. As he explains at the very beginning:
  • This book is a long meditation on a single question:
  • Does the world embody beautiful ideas?
To me (and I think to Wilczek), the answer to the question has always been an unambiguous “Yes”. The more difficult question is “what does such a claim about beauty and the world mean?” and that’s the central concern of the book.

PS4 Wilczek expresses a tendency shared by many modern physicists of pretending to know all of chemistry "in principle", simply by writing down a Schrödinger equation on a piece of paper, however without actually being able to predict anything specific because solutions of the equation cannot by computed: 
  • Wave functions that fully describe the physical state of several electrons occupy spaces of very high dimension. The wave function for two electrons lives in a six-dimensional space, the wave function for three electrons lives in a nine-dimensional space, and so forth. The equations for these wave functions rapidly become quite challenging to solve, even approximately, and even using the most powerful computers. This is why chemistry remains a thriving experimental enterprise, even though in principle we know the equations that govern it, and that should enable us to calculate the results of experiments in chemistry without having to perform them.
In this illusion game, the uncomputability of the Schrödinger's many-dimensional equation relieves the physicist from the real task of explaining the actual physics of chemistry, while the physicist can still safely take the role of being in charge of principal theoretical chemistry underlying a "thriving experimental enterprise", which "in principle" is superfluous. Beautiful? 

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