Why The Bogus Argument Fails
To prove spin/statistics, it is necessary to use relativity (though there are a few nice methods which do not use field theoretic tools). In relativity, there are no local fields which are pure creation operators or annihilation operators. Every local field both creates particles and annihilates the corresponding antiparticle. This means that in relativity, the product of the free real spin-0 field has a nonzero vacuum expectation value, because in addition to creating particles and annihilating particles, it also includes a part which creates and then annihilates a particle:
And now the heuristic argument can be used to see that G(x) is equal to G(−x), which tells you that the fields cannot be anti-commuting.
Read more about this topic: Spin-statistics Theorem
Famous quotes containing the words argument and/or fails:
“Argument is conclusive ... but ... it does not remove doubt, so that the mind may rest in the sure knowledge of the truth, unless it finds it by the method of experiment.... For if any man who never saw fire proved by satisfactory arguments that fire burns ... his hearer’s mind would never be satisfied, nor would he avoid the fire until he put his hand in it ... that he might learn by experiment what argument taught.”
—Roger Bacon (c. 1214–1294)
“The host, the housekeeper, it is
who fails you. He had forgotten
to make room for you at the hearth
or set a place for you at the table
or leave the doors unlocked for you.”
—Denise Levertov (b. 1923)