Editing Spin–statistics theorem (section)

==Proofs==
An elementary explanation for the spin–statistics theorem cannot be given despite the fact that the theorem is so simple to state. In ''[[The Feynman Lectures on Physics]]'', [[Richard Feynman]] said that this probably means that we do not have a complete understanding of the fundamental principle involved.<ref name="FeynmanIII-4">{{cite book |last=Feynman |first=Richard P. |url=https://www.feynmanlectures.caltech.edu/III_04.html |title=The Feynman Lectures on Physics |author2=Robert B. Leighton |author3=Matthew Sands |publisher=Addison-Wesley |year=1965 |isbn=978-0-201-02118-9 |volume=3 |pages=4.1}}</ref>

Numerous notable proofs have been published, with different kinds of limitations and assumptions. They are all "negative proofs", meaning that they establish that integer spin fields cannot result in fermion statistics while half-integral spin fields cannot result in boson statistics.<ref name=DuckSudarshanBook>{{Cite book |last1=Duck |first1=Ian |title=Pauli and the spin-statistics theorem |last2=Sudarshan |first2=Ennackel Chandy George |last3=Sudarshan |first3=E. C. G. |date=1998 |publisher=World Scientific |isbn=978-981-02-3114-9 |edition=1. reprint |location=Singapore}}</ref>{{rp|487}} 

Proofs that avoid using any relativistic quantum field theory mechanism have defects. Many such proofs rely on a claim that 
<math display="block">
 |\psi(\alpha_1, \alpha_2, \alpha_3, \dots)|^2 = |\hat{P}\psi(\alpha_1, \alpha_2, \alpha_3, \dots)|^2,
</math>
where the operator <math>\hat{P}</math> permutes the coordinates. However, the value on the left-hand side represents the probability of particle&nbsp;1 at <math>r_1</math>, particle&nbsp;2 at <math>r_2</math>, and so on, and is thus quantum-mechanically invalid for indistinguishable particles.<ref name=CurceanuGillaspyHilborn>{{Cite journal |last1=Curceanu |first1=Catalina |last2=Gillaspy |first2=J. D. |last3=Hilborn |first3=Robert C. |date=2012-07-01 |title=Resource Letter SS–1: The Spin-Statistics Connection |url=https://pubs.aip.org/ajp/article/80/7/561/1039586/Resource-Letter-SS-1-The-Spin-Statistics |journal=American Journal of Physics |language=en |volume=80 |issue=7 |pages=561–577 |doi=10.1119/1.4704899 |issn=0002-9505|url-access=subscription }}</ref>{{rp|567}}

The first proof was formulated<ref>{{cite journal |author1=Markus Fierz |title=Über die relativistische Theorie kräftefreier Teilchen mit beliebigem Spin |lang=de |journal=Helvetica Physica Acta |volume=12 |issue=1 |pages=3–37 |year=1939 |doi=10.5169/seals-110930 |author1-link=Markus Fierz |bibcode=1939AcHPh..12....3F}}</ref> in 1939 by [[Markus Fierz]], a student of [[Wolfgang Pauli]], and was rederived in a more systematic way by Pauli the following year.<ref>{{cite journal |author1=Wolfgang Pauli |title=The Connection Between Spin and Statistics |journal=[[Physical Review]] |volume=58 |issue=8 |pages=716–722 |date=15 October 1940 |doi=10.1103/PhysRev.58.716 |bibcode = 1940PhRv...58..716P |author1-link=Wolfgang Pauli}}</ref> 
In a later summary, Pauli listed three postulates within relativistic quantum field theory as required for these versions of the theorem:
# Any state with particle occupation has higher energy than the [[vacuum state]].
# Spatially separated measurements do not disturb each other (they commute).
# Physical probabilities are positive (the metric of the Hilbert space is positive-definite).
Their analysis neglected particle interactions other than commutation/anti-commutation of the state.<ref name=PauliOnFeynman/><ref name=DuckSudarshanBook/>{{rp|374}}

In 1949 [[Richard Feynman]] gave a completely different type of proof<ref>{{Cite book |last=Richard Feynman |title=Quantum Electrodynamics |publisher=[[Basic Books]] |year=1961 |isbn=978-0-201-36075-2 |chapter=The theory of positrons |author-link=Richard Feynman}} A reprint of Feynman's 1949 paper in Physical Review.</ref> based on [[vacuum polarization]], which was later critiqued by Pauli.<ref name=PauliOnFeynman>{{cite journal |author1=Wolfgang Pauli |title=On the Connection Between Spin and Statistics |journal=[[Progress of Theoretical Physics]] |volume=5 |issue=4 |pages=526–543 |year=1950 |doi=10.1143/ptp/5.4.526 |bibcode=1950PThPh...5..526P |doi-access=free}}</ref><ref name=DuckSudarshanBook/>{{rp|368}} Pauli showed that Feynman's proof explicitly relied on the first two postulates he used and implicitly used the third one by first allowing negative probabilities but then rejecting field theory results with probabilities greater than one.

A proof by [[Julian Schwinger]] in 1950 based on time-reversal invariance<ref name=SchwingerI>{{cite journal |title=The Quantum Theory of Fields I |author1=Julian Schwinger |journal=Physical Review |volume=82 |issue=6 |pages=914–917 |date=June 15, 1951 |doi=10.1103/PhysRev.82.914 |bibcode=1951PhRv...82..914S |s2cid=121971249 }}</ref> followed a proof by [[Frederik Belinfante]] in 1940 based on charge-conjugation invariance, leading to a connection to the [[CPT theorem]] more fully developed by Pauli in 1955.<ref>{{Cite book |last=Pauli |first=Wolfgang |chapter-url=http://link.springer.com/10.1007/978-3-322-90270-2_41 |title=Wolfgang Pauli |date=1988 |publisher=Vieweg+Teubner Verlag |isbn=978-3-322-90271-9 |editor-last=Enz |editor-first=Charles P. |location=Wiesbaden |pages=459–479 |language=de |chapter=Exclusion Principle, Lorentz Group and Reflection of Space-Time and Charge |doi=10.1007/978-3-322-90270-2_41 |editor-last2=v. Meyenn |editor-first2=Karl}}</ref> These proofs were notably difficult to follow.<ref name=DuckSudarshanBook/>{{rp|393}}

Work on the [[Wightman axioms|axiomatization of quantum field theory]] by [[Arthur Wightman]] lead to a theorem that stated that the expectation value of the product of two fields, <math>\phi(x)\phi(y)</math>, could be [[analytical continuation| analytically continued]] to all separations <math>(x - y)</math>.<ref name=DuckSudarshanBook/>{{rp|425}} (The first two postulates of the Pauli-era proofs involve the vacuum state and fields at separate locations.) The new result allowed more rigorous proofs of the spin–statistics theorems by [[Gerhart Lüders]] and [[Bruno Zumino]]<ref>{{Cite journal |last1=Lüders |first1=Gerhart |last2=Zumino |first2=Bruno |date=1958-06-15 |title=Connection between Spin and Statistics |url=https://link.aps.org/doi/10.1103/PhysRev.110.1450 |journal=Physical Review |language=en |volume=110 |issue=6 |pages=1450–1453 |doi=10.1103/PhysRev.110.1450 |issn=0031-899X|url-access=subscription }}</ref> and by Peter Burgoyne.<ref name=DuckSudarshanBook/>{{rp|393}} In 1957 [[Res Jost]] derived the CPT theorem using the spin–statistics theorem, and Burgoyne's proof of the spin–statistics theorem in 1958 required no constraints on the interactions nor on the form of the field theories. These results are among the most rigorous practical theorems.<ref>{{Cite book |last=Pais |first=Abraham |title=Inward bound: of matter and forces in the physical world |date=2002 |publisher=Clarendon Press [u.a.] |isbn=978-0-19-851997-3 |edition=Reprint |location=Oxford}}</ref>{{rp|529}}

In spite of these successes, Feynman, in his 1963 undergraduate lecture that discussed the spin–statistics connection, says: "We apologize for the fact that we cannot give you an elementary explanation."<ref name="FeynmanIII-4"/> Neuenschwander echoed this in 1994, asking whether there was any progress,<ref>{{Cite journal |last=Neuenschwander |first=Dwight E. |date=1994-11-01 |title=Question #7. The spin–statistics theorem |url=https://pubs.aip.org/ajp/article/62/11/972/1040578/Question-7-The-spin-statistics-theorem |journal=American Journal of Physics |language=en |volume=62 |issue=11 |pages=972 |doi=10.1119/1.17652 |issn=0002-9505|url-access=subscription }}</ref> spurring additional proofs and books.<ref name=DuckSudarshanBook/> Neuenschwander's 2013 popularization of the spin–statistics connection suggested that simple explanations remain elusive.<ref>{{Cite magazine |last=Neuenschwander |first=Dwight E. |date=2015-07-28 |title=The Spin-Statistics Theorem and Identical Particle Distribution Functions |url=https://www.sigmapisigma.org/sigmapisigma/radiations/fall/2013/spin-statistics-theorem-and-identical-particle-distribution-functions |magazine=Radiations |page=27 |language=en}}</ref>