ABSTRACT
CPΨ(r, t)(CP)−1 = η∗CPγ4γ2Ψ∗(−r, t), where ηCP is a phase factor. Using this definition and assuming the Dirac neutrino nature, prove the validity of the relation:
CP|ar(p) >= ηCP |br(−p) >,
where |ar(p) > (|br(p) >) designates the state to involve a particle (antiparticle) with the momentum p and polarization r. 3.24. If the vector field causal function is defined by the relation:
Dc′kl(x) = θ(x0)D (−) kl (x)− θ(−x0)D(+)kl (x) (A)
where
D (±) kl (x) =
±1 (2pi)3i
∫ eikxθ(±k0)δ(k2 −m2)
( gkl − klkn
m2
) dk,
the direct calculations of the expression (A) using the integral representation of the θfunction
θ(±x0) = 1 2pii
τ ∓ idτ,
leads to the expression different from
Dckl(x) =
( gkl +
m2 ∂2
∂xk∂xk
) Dc(x)
in neighborhood of the point x = 0. Find D (±) kl (x) and point out the reason of this contra-
diction. 3.25. In the photon case find the vector functions (spherical vectors) to be eigenfunctions both of the square angular momentum operator and of the angular momentum projection operator onto the given axis. [NOTE: Make use of the relation known from nonrelativistic quantum mechanics
[Lˆk, Aˆk] = iklmAˆm,
where Aˆm is some vector physical quantity to characterize a system.] 3.26. Show that the theory to be described by the Lagrangian of the problem 1.16 is quantized with the use of the indefinite metric.
