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:
“Any authentic work of art must start an argument between the artist and his audience.”
—Rebecca West (1892–1983)
“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)