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02. FERMIONS AND BOSONS

Fermions and Bosons

Quantum physics divides the microscopic world into bosons and fermions.

The elementary constituents of matter are fermions: the structure and impenetrability of bodies is due to the electrons of the atoms that refuse to give space to their neighbors.

The mediating particles of the fundamental forces are bosons.

Between 1923 and 1925 Fermi published important contributions to quantum theory which, at the beginning of 1926, led to the formulation of the statistics that bears his name. In this fundamental work Fermi, starting from ideas on the statistical mechanics of a system of identical particles – ideas that he had already begun to develop at the institute directed by Paul Ehrenfest in Leiden – introduced Pauli’s ‘exclusion principle’ into the description, that is the selection rule hypothesized at the beginning of 1925, which allowed him to found an exhaustive theory of the behavior of those particles which, from that moment on, will take the name of “fermions“.

A few months later, in a completely autonomous way, the English physicist Paul Adrien Maurice Dirac will come to the same conclusions.

Today the theory bears the name of ‘Fermi-Dirac statistics’.

Anything wandering around in the atomic or subatomic world is either a “fermion” or a “boson“.

Our perception of reality, which appears primarily made up of matter and light, is only an impression, a distant reflection of the substrate of particles of matter and force – fermions and bosons respectively – described by the Standard Model. The very history of our Universe is a history of fermions and bosons. At the large collider LHC (Large Hadron Collider) of CERN, a primordial stage of matter is studied, made up of only quarks (fermions) and gluons (bosons), which is believed to have been crossed by the Universe during its evolution, a few moments later the Big Bang, before the atomic constituents, then the atoms and finally the stars were formed.

On 25 October 1926 Enrico Fermi wrote a letter to the English physicist Paul Adrien Maurice Dirac:

“Dear Sir! In your interesting work “On The Theory of Quantum Mechanics” you have advanced a theory of Ideal Gas based on the Pauli Exclusion Principle which is practically identical to the one I published at the beginning of 1926 … “.

Dirac promptly sent an apology letter that marked the birth of the Fermi-Dirac theory.

Subatomic particles and those that act as mediators of their own interactions behave according to the Fermi – Dirac theory or that of Bose – Einstein: they are either fermions or bosons, as Dirac called them.

Fermions refuse to be together, bosons instead prefer to be together.

Particles of matter are fermions: the solidity of a piece of iron is in the refusal of atomic electrons to share space with neighbors. George Gamow, father of the Big Bang theory and brilliant writer of works for the dissemination of scientific culture, as an example of fermion cites the charming Greta Garbo who in a film says “I want to be alone”.

Particles of light, photons, as well as other particles of the subatomic world, including the famous Higgs particle, are bosons.

Renzo Arbore would say, identifying himself: “The more we are, the better we are”.

The tendency of photons to stick together and all do the same thing underlies the power and accuracy of a laser light beam.

Let’s consider a “Fermi gas”, that is, a gas made of fermions that do not interact with each other.

Suppose this gas is composed of only two identical particles which can each occupy two states. Since the famous principle formulated by Wolfgang Ernst Pauli excludes that both particles can occupy the same state, our system has only one possible configuration.

If we consider an analogous system, but apply Bose-Einstein’s theory, there are three possible configurations.

Finally, if we consider the classical theory of the late nineteenth century, a theory linked to the names of James Clerk Maxwell and Ludwig Boltzmann, we count the following four possible configurations:

  • both particles in one state
  • both in the other
  • particle 1 in one state, 2 in the other
  • the other way around

What’s underneath?

Before the so-called “quantum revolution” of the early decades of the twentieth century, it was thought that two particles, however small, could always be distinguished, for example by continuously following the motion of each, and labeling them as particles 1 and 2. The study of atoms at the beginning of the twentieth century forced to abandon this idea. The exchange of two identical particles does not correspond to anything observable in the atomic world, not even in principle: the two exchanged particle states are a single state.

The two-particle minisystem shows the essential difference between fermions and bosons, which becomes dramatic when the numbers involved are high. Let us take the case of five identical particles which have a large number of states of increasing energy at their disposal.

The differences emerge at temperatures close to absolute zero (ie T = 0 K = – 273.15 ° C): the fermions precipitate in the five lowest levels, while the bosons all condense in the single lowest level. At high temperatures, the two quantum statistics, that of FermiDirac and that of BoseEinstein, and the classical one of MaxwellBoltzmann, all give approximately equal predictions.

Pauli discovered that the fermionboson dualism is connected to “spin”, a kind of intrinsic rotation of quantum particles. Using the quantum unit of measurement, the spin of bosons has an integer value (zero, one, two, etc.), that of fermions has a semi-integer value (one half, three means, etc.).

The angular momentum of a rotating top is represented by a vector, which can be simply displayed by an arrow, which indicates the direction, direction and intensity of the rotation.

The whole spin of a boson is also represented by a vector.

The description of the semi-integer spin of a fermion looks like a vector, it is drawn with a dart, but a surprising property makes it different.

If we imagine to rotate any object on itself, until it completes a complete revolution, it reappears equal to itself.

However, if we rotate the mathematical object that describes the state of a fermion by one complete revolution, an inversion of sign occurs; to regain the original object, two complete turns are necessary: ​​fermions are described by “spinors”, as Paul Ehrenfest baptized them in 1929, also observing that “it is really strange that no one before Pauli and Dirac has communicated the disconcerting news that a mysterious tribe known as the spinors inhabits our three-dimensional space ”.

Those strange mathematical objects were discovered by the mathematician Élie Joseph Cartan in 1913: they conquered other fields besides quantum physics. In 2007, another famous mathematician, Michael Francis Atiyah, wrote: “Nobody fully understands spinors”.

Their algebra is formally understood but their geometric meaning is mysterious. In a sense, they describe the “square root” of geometry, and just as it took ages to understand the concept of the “square root of -1″, the same could happen for spinors. ” Let’s take a paper ring – green outside and red inside – made by closing on itself a strip of green and red colored paper on both sides. By closing on itself a similar strip but with inverted faces at the junction, a two-color Moebius strip is obtained, a geometric figure whose invention is due to the mathematician August Ferdinand Möbius (Moebius). If an ant runs along one of the two faces of the ring, after a lap it finds itself at the starting point. In the other case, however, the ant, after one lap, finds the inverted color: he must make two laps to find himself at the starting point and color. The second case gives an idea of ​​the singular property of spinors.

In the case of composite particles, the resulting spin determines their properties and, since adding an even number of semi-integers gives an integer, an even number of fermions can combine and behave like a boson, with spectacular effects such as those of the Superfluidity of ‘Liquid Helium or Superconductivity.

The most abundant isotope of Helium (He) in the atmosphere is the type with mass number A = 4, consisting of an even number of fermions: two protons and two neutrons in the nucleus, plus two orbital electrons. It turns out to have integer spin and the behavior is boson-like. At temperatures close to absolute zero, the 4He isotope in the liquid state forms a Bose-Einstein condensate and its viscosity vanishes: it becomes a “superfluid“.