Quantum Theory - Particle Physics Group

CHAPTER 9. SYMMETRIES AND TRANSFORMATION GROUPS 176(without proof). For massless particles, however, W 2 = 0 so that s cannot be determinedby W! The physical reason for this problem is that massless particles can never be in theircenter of mass frame. In fact, the spin quantum number j refers to the rotation group SO(3)which cannot be used to classify massless particles exactly because of the non-existence of acenter of mass frame (indeed, if photons could be described by spin j = 1, then the magneticquantum number would have three allowed values; but we know that photons only have thetwo tranversal polarizations).The intrinsic angular momentum of massless particles therefore has to be described by adifferent conserved quantity. Equation (9.92) implies that ⃗p · ⃗S is a constant of motion[⃗p · ⃗S,H] = 0. (9.96)If p = |⃗p| ≠ 0, which is always the case for massless particles, we can define the helicitys p =⃗p · ⃗Sp , (9.97)which is the spin component in the direction of the velocity of the particle. For solutions ofthe Dirac equation its eigenvalues can be shown to be s p = ±/2. For a given momentum aDirac particle can have two different helicities for positive-energy and two different helicities fornegative energy solutions, so that the four degrees of freedom describe particles and antiparticlesof both helicities. For the massless neutrinos, however, only positive helicity (left-handed)particles and negative helicity (right-handed) anti-particles exist in the standard model ofparticle interactions. The massless photons with s p = ± and the gravitons with s p = ±2 aretheir own antiparticles and they exist with two rather than 2s p + 1 polarizations.9.4.3 Dirac conjugation and Lorentz tensorsIf we try to construct a conserved current j µ = (cρ,⃗j), which satisfies the continuity equation˙ρ + div⃗j = 0 and thus generalizes the probability density current of chapter 2, it is natural toconsider the quantity ψ † γ µ ψ, which however is not real(ψ † γ µ ψ) ∗ = (ψ † γ µ ψ) † = ψ † (γ µ ) † ψ ≠ ψ † γ µ ψ (9.98)because the γ µ is anti-Hermitian for µ ≠ 0. It can, hence, also not transform as a 4-vectorbecause Lorentz transformations would mix the real 0-component with the imaginary spacialcomponents.An appropriate real current can be constructed by replacing the Hermitian conjugate ψ † bythe Dirac conjugate spinorψ = ψ † γ 0 , j µ = ψγ µ ψ. (9.99)