is a vector particle, as opposed to the scalar we used in toy QED. However,the agreement up to a constant factor confirms that our picture of electron spinconservation at each vertex in the nonrelativistic limit is correct. In problem 2b,we’ll prove why this is true.(c) In the ultra-relativistic limit, we finddσdΩ=e464π2s4(t2+ut+u2)2t2u2=α2E2cm((1 + cosθ)2+ (1 + cosθ)(1-cosθ) + (1-cosθ)2)2(1-cos2θ)2=α2E2cm(3 + cos2θ)2sin4θ2.(a) The operatorsH±=12(1±ˆp·σ)are called helicity projectors, since they project onto plus and minus eigenstatesof the helicity operator12ˆp·σ(which measures spin in the direction of motion).For a massless particle, note that√p·σ=pE(1-ˆp·σ) =√2EH-(sinceH±are projection operators, with only 0 and 1 eigenvalues,√H±=H±). Meanwhile,√p·σ=pE(1 + ˆp·σ) =√2EH+.As we saw in class, for a massless particle, the Dirac equation decouples into twoWeyl equations. If we set:ψL= (1-γ52)ψψR= (1 +γ52)ψ(1)Thenpp·σψL= 0=⇒H+ψL= 0√p·σψR= 0=⇒H-ψR= 0In other words, for a massless fermion, the solutions to the Dirac equation arehelicity eigenstates.Now, letP±=12(1±γ5).Note thatγμγ5=-γ5γμ, soγμP±=P∓γμ. In particular,Phγ0γμ=γ0P-hγμ=γ0γμPh. Further,γ†5=γ5,
Phys 253a4soψhγμψh0=ψ†hγ0γμψh0=(Phψ)†γ0γμPh0ψ=ψ†Phγ0γμPh0ψ=ψ†γ0γμPhPh0ψWe note thatPhPh0= 0 unlessh=h0.(b) In the non-relativistic limit,p·σ=p·σ=m, so the positive-energy spinors areus≈√mξsξs!Here, we’re working in the Weyl basis, sousγμus=mξ†s(σμ+σμ)ξs0So forμ6= 0 the expression vanishes, and forμ= 0 we haveusγ0us= 2mξ†sξs0= 2mδss0,so in particularusγμusvanishes identically fors6=s0.(c) In the last problem set, we derived the non-relativistic hamiltonian for an electronH=mc2-ecA0+(p-eA)22m-e~2mΣ·B+O1c2whereΣis a 4×4 matrix withσon the 2×2 diagonal blocks. If we turn offthe magnetic field, thenΣcommutes withH, so that spin becomes a conservedquantum number.In the nonrelativistic limit, we can often ignore the spatial part of electromagneticsourcesJμrelative to the time componentJ0=ρ, the charge density. ButJisources magnetic fields, so without them there is nothing to cause electrons tospin-flip. Of course, magnetic fields DO exist in classical nonrelativistic systems,for instance from ferromagnets where a large number of small magnetic momentsare approximately aligned. This is a case where a large number offsets the smallratiov2c2.
Phys 253a 5
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