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:
“No one thinks anything silly is suitable when they are an adolescent. Such an enormous share of their own behavior is silly that they lose all proper perspective on silliness, like a baker who is nauseated by the sight of his own eclairs. This provides another good argument for the emerging theory that the best use of cryogenics is to freeze all human beings when they are between the ages of twelve and nineteen.”
—Anna Quindlen (20th century)
“If fancy then
Unequal fails beneath the pleasing task,
Ah, what shall language do?”
—James Thomson (1700–1748)