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:
“Our argument ... will result, not upon logic by itself—though without logic we should never have got to this point—but upon the fortunate contingent fact that people who would take this logically possible view, after they had really imagined themselves in the other man’s position, are extremely rare.”
—Richard M. Hare (b. 1919)
“Someone is always at my elbow reminding me that I am the grand-daughter of slaves. It fails to register depression with me. Slavery is sixty years in the past. The operation was successful and the patient is doing well, thank you. The terrible struggle that made me an American out of a potential slave said “On the line!” The Reconstruction said “Go!” I am off to a flying start and I must not halt in the stretch to look behind and weep.”
—Zora Neale Hurston (1891–1960)