# Spin Statistics Theorem

Under the spin statistics theorem of quantum physics is defined as the theoretical basis for the empirical finding that all elementary particles with half-integer spin of the Fermi-Dirac statistics follow, d. H. are so-called fermions , whereas all particles with integer spin follow the Bose-Einstein statistics , i.e. H. are so-called bosons .

## Explanation of the terms

Spin is the intrinsic angular momentum of the particles. All currently detected particles have either an integer (0, 1, 2, ...) or half-integer (1/2, 3/2, 5/2, ...) spin, each in units of the reduced Planck constant . ${\ displaystyle \ textstyle \ hbar = {\ frac {h} {2 \ pi}}}$

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 (of the same kind): only one 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. when the parameters of two fermions are interchanged, their sign changes, while the wave function of a group of indistinguishable bosons is symmetrical , i.e. H. does not change its sign when the parameters of two bosons are interchanged .

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

The Fermi-Dirac statistics yield u. a. the basis for the explanation of the periodic table of the elements and the stability of atoms and macroscopic matter, the Bose-Einstein statistics and the like. a. the explanation for the superfluidity of 4 He at low temperatures.

## Discovery of the rationale

Although the spin and the two statistics 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 spin and statistics. Relativistic quantum field theory plays a decisive role in both of the explanations and in the numerous generalizations and refinements that followed . According to Pauli (and Fierz), for particles with an integer spin, the Bose statistics follow from the fact that observables for space-like distances must be commensurable (relativistic causality). For fields with half-integer spin, on the other hand, the Fermi statistics follow from the requirement of the existence of a state of lowest energy ( ground state ). In both cases, relativistic arguments are decisive. These efforts found a certain initial conclusion with the work of Gerhart Lüders and Bruno Zumino and N. Burgoyne and illustrated in the book by Arthur Wightman and Ray Streater . Lüders, Zumino and Burgoyne in particular gave evidence in the case of interactive fields (Pauli treated non-interactive fields). However, some additional assumptions were also made (positive-definite metric in Hilbert space , vacuum as the state of lowest energy, either commutator or anti-commutator for space-like distances in the same field). Feynman criticized these justifications for their complexity and concluded that the basic principle was not fully understood. An article by Dwight E. Neuenschwander in the American Journal of Physics also called for a simpler explanation, which produced a number of answers, but most of them concluded that even twenty years after Feynman's challenge, no really simple evidence is in sight. Even Ian Duck and George Sudarshan , who wrote a book in response with reprints of the most important works and an attempt to explain their own (based on the treatment by Julian Schwinger ), had to admit that although they suppressed relativistic arguments, the derivation was not complete could perform non-relativistically. Attempts to derive the spin statistics theorem non-relativistically are considered insufficient or not convincing. There have been various attempts at this by Michael Berry and others. Carl R. Hagen showed that in Galileo covariant field theories (i.e. the non-relativistic limit case of the Galileo group ) there is no connection between spin and statistics (both statistics are possible with any spin). and Arthur Wightman also argued that in the non-relativistic case there is no connection between spin and statistics.

Arguments that use the topological properties of the turntable to explain, such as by Feynman himself in one of his most recent publications, are not considered convincing. An older demonstration experiment used by Feynman (rotation of a coffee cup held in a hand by arm movements, which does not find its way back to the starting position with a simple 360 ​​degree rotation, but only at 720 degrees) shows the different behavior of spinors and vectors during 360 degree rotations. According to Robert C. Hilborn, these arguments provide an analogy at best, but no evidence.

Oscar Wallace Greenberg and Rabindra Mohapatra proposed in 1989 to search for minor violations of the spin statistics theorem. Generalized statistics ( anyon ) between those of fermions and bosons have been discussed for quasiparticles in solid state physics.

## literature

• Ray F. Streater and Arthur S. Wightman: PCT, Spin, Statistics and All That , Bibliographisches Institut, Mannheim 1964. Title of the English original: PCT, Spin & Statistics, and All That , Benjamin 1964
• Ian Duck and Ennackel Chandy George Sudarshan: Pauli and the Spin-Statistics Theorem , World Scientific, Singapore 1997 (with reprints)
• Curceanu, Gillasby, Hilborn, Resource Letter SS-1: The Spin-Statistics Connection, American Journal of Physics, Volume 80, 2012, pp. 561-577

## Individual evidence

1. Markus Fierz: About the relativistic theory of force-free particles with any spin. In: Helvetica Physica Acta . tape 12 , 1939, pp. 3–17 ( e-periodica.ch ).
2. ^ Wolfgang Pauli: The Connection Between Spin and Statistics . In: The Physical Review . tape 58 , 1940, p. 716-722 , doi : 10.1103 / PhysRev.58.716 .
3. ^ Lüders, Zumino: Connection between spin and statistics . In: Phys. Rev. Band 110 , 1958, pp. 1450 .
4. ^ Burgoyne: On the connection between spin and statistics . In: Il Nuovo Cimento . tape 8 , 1958, pp. 657 .
5. ^ R. Streater, A. Wightman: PCT, Spin, Statistics and All That . BI university paperback, 1964.
6. a b A. Wightman: The spin-statistics connection: some pedagogical remarks in response to Neuenschwander's question . In: Electronic Journal of Differential Equations . Conf. 04, 2000, pp. 207-213 ( txstate.edu [PDF]).
7. ^ RP Feynman, RB Leighton, M. Sands: The Feynman Lectures on Physics . Addison-Wesley, Reading, Mass., USA 1965, p. 4.3 .
8. DE Neuenschwander: Question ♯7. The spin statistics theorem . In: American Journal of Physics . tape 62 , 1994, pp. 972 , doi : 10.1119 / 1.17652 .
9. ^ Duck, Sudarshan: Pauli and the Spin Statistics Theorem . World Scientific, 1997, ISBN 978-981-02-3114-9 .
10. ^ RE Allen, AR Mondragon: No spin-statistics connection in nonrelativistic quantum mechanics . arxiv : quant-ph / 0304088 .
11. C. Curceanu, JD Gillasby, RC Hilborn: Resource Letter SS-1: The Spin-Statistics Connection . In: American Journal of Physics . tape 80 , 2012, p. 561-577 , doi : 10.1119 / 1.4704899 .
12. MV Berry, JM Robbins: Indistinguishability of quantum particles: spin, statistics and the geometric phase . In: Proc. Roy. Soc. A . tape 453 , 1997, pp. 1771-1790 , doi : 10.1098 / rspa.1997.0096 , JSTOR : 53019 .
13. Murray Peshkin: Spin and Statistics in Nonrelativistic Quantum Mechanics: The Zero Spin Case . In: Phys. Rev. A . tape 67 , 2003, p. 042102 , doi : 10.1103 / PhysRevA.67.042102 , arxiv : quant-ph / 0207017 .
14. ^ CR Hagen: Spin and statistics in Galilean covariant field theory . In: Phys. Rev. A . tape 70 , 2004, pp. 012101 , doi : 10.1103 / PhysRevA.70.012101 .
15. ^ RP Feynman: The reason for antiparticles . In: RP Feynman, S. Weinberg (Ed.): Elementary Particles and the Laws of Physics . Cambridge University Press, Cambridge, England 1987, pp. 1-59 .
16. It comes from Paul Dirac and can be found, for example, in HJ Bernstein, AV Phillips: Fiber bundles and quantum theory . In: Scientific American . tape 245 , no. 1 , July 1981, p. 122 , doi : 10.1038 / scientificamerican0781-122 (see also: Spektrum der Wissenschaft (1981) issue 9, pp. 89-105). , and corresponding sign changes at 360 degree rotations have been proven in neutron experiments.
17. Hilborn: Answer to Question 7 . In: American Journal of Physics . tape 63 , 1995, pp. 298 , doi : 10.1119 / 1.17953 .
18. Greenberg, Mohapatra: Phenomenology of small variations of Bose and Fermi statistics . In: Phys. Rev. D . tape 39 , 1989, pp. 2032-2038 , doi : 10.1103 / PhysRevD.39.2032 .
19. Review by Arthur S. Wightman: Pauli and the Spin-Statistics Theorem (book review), Am. J. Phys., Volume 67, 1999, pp. 742-746 (1999)