Fermion fields must satisfy anticommutation relation. But why? I know that unless they anti-commute the Pauli exclusion principle cannot be satisfied. But is there some other deeper/fundamental principle (such as locality or something) which dictates that fermions should anti-commute while bosons commute?
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1$\begingroup$ Related: physics.stackexchange.com/q/17893/2451, physics.stackexchange.com/q/65819/2451 , physics.stackexchange.com/q/134577/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Mar 11, 2018 at 4:40
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$\begingroup$ The unitary irreducible representations of the Poincaré group are indexed by two numbers, mass (a positive number) and spin (an half-integer number). Since every object in relativistic quantum mechanics should carry an irreducible unitary representation of the Poincaré group, it should then be characterized either by an half-integer or integer spin number. The spin-statistics theorem then yields that it should either be commuting or anti-commuting. It is therefore the relativistic nature of fields that yields their bosonic or fermionic nature. $\endgroup$– yuggibCommented Mar 11, 2018 at 9:38
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1 Answer
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You got it in the opposite direction.
You can always argue that there must be two kinds of fields - commuting and anti commuting.
Commuting fields are defined (rather named) as fermionic fields. Similarly for bosonic fields.
Later, through the Spin Statistics Theorem, Pauli showed that anti commuting fields should have half integer spin and commuting fields integer spins.
Thus it's said that fermions have half integer spins, and bosons have integer spins.
https://physics.stackexchange.com/a/390288/150769
Look up my answer in the above link for the details.
Cheers!
answered Mar 11, 2018 at 4:45
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$\begingroup$ Wasn't the Spin Statistics Theorem proved by Wolfgang Pauli in 1940? $\endgroup$ Commented Mar 11, 2018 at 5:00
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$\begingroup$ My bad, I wrote the wrong name. Thanks for the heads up. Edited.😊 $\endgroup$ Commented Mar 11, 2018 at 5:00
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$\begingroup$ "You can always argue that there must be two kinds of fields - commuting and anti commuting." This is very clearly false: there are fields that neither commute nor anti-commute. $\endgroup$ Commented Mar 11, 2018 at 14:57
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$\begingroup$ Do they pop up in theories with positive energy and positive norm fields? I don't think so. $\endgroup$ Commented Mar 11, 2018 at 15:00
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$\begingroup$ you can decompose any field to commutative and anti-commutative parts $\endgroup$ Commented Sep 24, 2022 at 1:31