Lorentz-Invariant Localization for Elementary Systems

Philips has axiomatically defined sets of localized states which are Lorentz-invariant (while Newton-Wigner sets of localized states are not) and has divided these sets into three classes. Philips's conjecture concerning spin-zero localized states, namely, that the postulates define the sets of localized states in a unique way, is proved to be incorrect by finding a class-III set that satisfies Philips's postulates. Philips's work is discussed and his calculations (spin-zero case) are repeated without using some (explicit and implicit) unnecessary hypotheses. These calculations are also extended to the spin-\textonehalf{} case, for which it is proved that there are only class-III sets of localized states. The results are discussed. Incidentally, an explicit form of the effects induced by a Lorentz transformation on representation space is found for both the spin-zero and spin-\textonehalf{} cases.