# Protons follow Fermi Dirac's statistics

## Spin Statistics Theorem

Under the **Spin Statistics Theorem** Quantum theory is understood to be the theoretical justification for the empirical finding that all elementary particles with half-integer spin follow the Fermi-Dirac statistics, i.e. H. So-called fermions are, however, all particles with an integer spin of the Bose-Einstein statistics, i. H. are so-called bosons.

### Explanation of the terms

Spin is the particles' intrinsic angular momentum. A classic idea, such as a small rotating body, is not possible here. All particles have either integer (0, 1, 2, ...) or half-integer (1/2, 3/2, 5/2, ...) spin, each in units of the reduced Planck constant $ \ hbar = \ frac h {2 \ pi} $.

On the other hand, all particles follow either the Fermi-Dirac or the Bose-Einstein statistics. These statistics describe the collective behavior of indistinguishable particles: only a single fermion (Pauli principle), but any number of bosons can be in a certain quantum state. In the formalism of quantum mechanics this is expressed by the fact that the wave function of a group of indistinguishable fermions is antisymmetric, i.e. H. if the parameters of any two fermions are interchanged, their sign changes; on the other hand, the wave function of a group of indistinguishable bosons is symmetric, i. H. if the parameters of any two bosons are interchanged, their sign changes *Not*.

Examples of fermions are electrons, protons and neutrons, for bosons the photons, ^{4}He atoms and their nuclei, the alpha particles.

The Fermi-Dirac-Statistic provides inter alia the basis for the explanation of the periodic table of the elements, the Bose-Einstein statistics and others. the superfluidity of the ^{4}He at low temperatures.

### Discovery of the rationale

Although the spin and the two statistics mentioned above were already known in 1926, it was not until Markus Fierz in 1939 and Wolfgang Pauli in 1940 that theoretical reasons were found for the connection between the phenomena. Both justifications were based on methods of relativistic quantum field theory, and the opinion was formed that the connection could not be proven without (special) relativity theory. Pauli's proof was generalized and refined in the years that followed.

### literature

- Markus Fierz:
*About the relativistic theory of force-free particles with any spin*, Helv. Phys. Acta**12**, 3-17 (1939) - Wolfgang Pauli:
*The Connection Between Spin and Statistics*, Phys. Rev.**58**, 716-722 (1940). doi: 10.1103 / PhysRev.58.716 (PDF; 136 kB) - Ray F. Streater and Arthur S. Wightman:
*The principles of quantum field theory*, Bibliographisches Institut, Mannheim (1964). Title of the English original:*PCT, Spin & Statistics, and All That* - Ian Duck and Ennackel Chandy George Sudarshan:
*Pauli and the Spin-Statistics Theorem*, World Scientific, Singapore (1997) - Arthur S. Wightman:
*Pauli and the Spin-Statistics Theorem*(Book review), Am. J. Phys.**67**(8), 742-746 (1999) - Arthur Jabs:
*Connecting spin and statistics in quantum mechanics*, http://arXiv.org/abs/0810.2399 (Found. Phys.**40**, 776–792, 793–794 (2010), doi: 10.1007 / s10701-009-9351-4, doi: 10.1007 / s10701-009-9377-7)

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