R. Feynman "The Nature of Physical Laws" (fourth lecture)

Translation of the fourth lecture from the course “The Character of Physical Laws”, the brightest twentieth-century scholar Richard Feynman.
The lecture is called “Symmetry of physical laws” .
The course does not require special knowledge in physics and you can start watching from any lecture. And this lecture is probably the most interesting in the whole course.
In general, there will be two lectures today: one video lecture by Feynman, another small note by the professor of the Radboud University (Holland) Mikhail Katsnelson, explaining some principle about which Feynman kept silent.

Approximately 24 '28 "Feynman tries to talk about Pauli’s ban, this is a fundamental principle in physics that underlies a mass of things, for example, why does matter exist in the form we see it? Why does matter, solid matter, has the property not to pass through another substance (why do we not fall through the earth?). Because the nuclei are very tiny, and there is a huge distance between them. I tried to explain this to different people many times, but this is not so trivial as it seems and I have never succeeded.
If you know mathematics, you will like an overview article that explains this physical principle for mathematicians (you need to save the PDF there, otherwise it will not open)
The article uses rather complicated mathematics, and in some places I lack knowledge to keep track of the details in the reasoning and I lose the thread, but in general the article remains clear and very, very interesting.

So, Feynman thought that he couldn’t explain this topic to an unprepared audience, he says right there: sorry, I can’t help you with anything, learn math. He doesn’t say anything more about this principle.
However, this is a very interesting topic, and very important, which, alas, very few people can explain popularly, but Mikhail Katsnelson kindly allowed me to quote here my note on the Pauli prohibition principle, in my opinion this is one of the most sensible attempts to popularly explain this topic mathematically unprepared audience. I bring it here, maybe someone will be interested, so:

Pauli ban principle

About one of the basic laws of quantum physics and the entire universe.

In chess, there is such a rule - if the position is repeated three times, at the request of one of the players, a draw is fixed. Once I watched a scene in a chess club - P., disgruntled, strongly beckoning to the judge: “Yes, but I used it” ... I meant - one with a gnawed ear, the other with a whole. But, in terms of the chess code, the horses are indistinguishable. The position does not change when they are rearranged.
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So - with electrons the same garbage. And with photons. Microparticles are indistinguishable, and not nearly, but exactly. None of them has a gnawed ear or a sore back, which the others would not have.

Why am I so sure? In the end, the characteristics of the electron - mass, charge ... - are measured with finite accuracy, well, let's say, twelve significant digits. How do I know that the charges of two electrons cannot differ in the seventeenth decimal place?

The fact is that indistinguishability is not a quantitative, but a qualitative characteristic. One of the basic laws of physics is the Boltzmann principle, which relates a macroscopic quantity — entropy — with a microscopic — statistical weight, that is, the number of ways in which a given state can be realized. According to Boltzmann, entropy is proportional to the logarithm of statistical weight. By counting the latter, entropy can be found, and through it, in the end, all the other thermodynamic characteristics of the system.

Suppose we have a state in which two knights on a chessboard are in the fields a1 and c2, and there is nothing more. How many ways can this state be realized? The answer depends on whether we consider horses to be distinguishable or not. If they are distinguishable - by two (the horses can be interchanged), if they are indistinguishable - by one (the permutation of the horses does not change the state).

In classical physics, particles indistinguishable in their internal characteristics (mass, charge ...) differ, however, in history. You can always mark particles by their position in the phase space and say that this is the same electron that had such coordinates and such speed at zero GMT. But in quantum mechanics, due to the uncertainty relation, the position in the phase space is unknown - if the coordinates are known, the velocity is unknown, and vice versa. Therefore, permutation of two electrons in places should not be considered as a change in the state of the system — it should, in a sense, translate the wave function (aka state vector) containing all the information about the state into itself. Formally speaking, the wave function must be transformed by some irreducible representation of a group of permutations.

The fundamental experimental fact is that in our world only two possibilities are realized - symmetric (turning into themselves when rearranging any two particles) and antisymmetric (changing the sign with such a permutation) wave functions. In the first case, the particles are called bosons, in the second - fermions. The history of the names is as follows.

In 1924, an unknown Indian physicist, Bose, was sent to Einstein's manuscript of his article to anyone at that time, asking, if E. finds it possible, to submit it to the works of the Prussian Academy of Sciences. The article did not contain new physical results; it contained a new conclusion of the then already known result — the Planck law of radiation of a solid body — based on (introduced by E. in 1905) the concept of light as a combination of particles — photons. Bose showed that the Planck distribution means nothing more than the equation of state of an ideal photon gas - if photons are considered indistinguishable! The article was immediately published with the enthusiastic recommendation of Einstein. He immediately applied the Bose ideas to ordinary gas and showed that instead of the well-known Maxwell-Boltzmann distribution law, the law of classical statistics, other statistics arise (now it is called Bose-Einstein statistics). The difference is very significant at low temperatures, when, according to Einstein, quantum particles must accumulate in macroscopic quantities in the same quantum state. This phenomenon was called Bose-Einstein condensation (BEC) and was experimentally discovered relatively recently, after the advent of laser cooling methods for gases to ultra-low temperatures. However, the superfluidity of liquid helium is associated with the BEC phenomenon, but it is not a gas, but a liquid, and the connection there is rather indirect.

In the same 1925, when Einstein's article appeared, Pauli, analyzing the spectra of many-electron atoms, discovered the prohibition principle (the Pauli principle) - there can be no more than one electron in one quantum state. Immediately, Fermi proposed statistics for identical particles obeying the Pauli principle. Dirac realized the connection of this statistics with quantum mechanics, showing that it follows from the antisymmetry of the wave function during permutations (the new statistics began to be called Fermi-Dirac). Symmetry implies Bose-Einstein statistics.

The properties of particles obeying two statisticians (they are called fermions and bosons; to understand which ones, specifically, I give to readers as an exercise) differ like heaven and earth. At low temperatures, bosons accumulate in one state, and fermions fill one state with the lowest energy, up to the so-called Fermi energy. The thermal properties of the system of identical particles are radically different from the classical ones. For example, if electrons were fundamentally distinguishable, even in a millionth decimal place in one of the properties, this would change the type of statistics, and the heat capacity of metals at room temperature would be several times greater than it is! Similarly, from the very fact that lasers exist and work, it follows that the fundamental indistinguishability of photons.

In 1940, Pauli proved the theorem on the relationship of spin to statistics: particles with half-integer spin (in units of Planck’s constant) are fermions, with whole (in particular, zero) bosons. The Pauli theorem essentially uses the non-trivial geometric properties of three-dimensional space; in two dimensions, it is incorrect, and there may exist particles with intermediate statistics — the so-called anions.

Consider a system of noninteracting electrons. For distinguishable particles, the state space of the system is a direct product of the state spaces of independent particles, and the wave function is the product of wave functions. According to the principle of identity, the real state is an antisymmetrized work, that is, a determinant (it is called Slater in honor of the American physicist John Slater). Thus, even in the absence of interaction, the state of identical particles is “entangled” (entangled).

The main mathematical apparatus in quantum field theory is the so-called “path integrals”, a kind of summation over all possible states of the system. If for bosons these integrals are carried out using ordinary numerical functions, then for fermions (as suggested by the Soviet mathematician Berezin), it is necessary to use the so-called Grassmann variables, for which multiplication is not commutative, but anticommutative. The square of any Grassmann number is zero, which is the optimal mathematical expression for the Pauli principle: the zero probability of finding two fermions in the same state.

List of lectures:

Lecture 1 - EXAMPLE OF THE PHYSICAL LAW - THE LAW OF BRAIN
Lecture 2 - CONNECTION OF MATHEMATICS WITH PHYSICS
Lecture 3 - GREAT STANDARDS OF PRESERVATION
Lecture 4 - SYMMETRY OF PHYSICAL LAWS

Lecture 5 - DIFFERENCE OF THE PAST AND THE FUTURE
Lecture 6 - PROBABILITY AND UNCERTAINTY - QUANTUM MECHANICAL VIEW ON NATURE
Lecture 7 - SEARCHING FOR NEW LAWS

On Yandex video
All lectures on rutrekera with best quality and individual sabs
All lectures are in English (carefully silverlight! Only works in IE 8.0 and here I am told what else in FF)

UPD 01 If you have any questions, ask in the comments I will try to answer.
UPD 02 Mikhail Katsnelson has his own popular video lecture on a related topic. You can find it here.