Here B is a 3 vector and \u03c3 \u03c8 also transforms like a 3 vector Also i \u03c3 i \u03c8 is

Here b is a 3 vector and σ ψ also transforms like a

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Here,~Bis a 3-vector, and·ψalso transforms like a 3-vector. Also,iσiψis rotation invariant, and then we can guess and check thattψ-iσiψ= 0is Lorentz-invariant. In fact, if we defineσμ= (1,~σ), thenσμμψ= 0is the Dirac equation form= 0. Maybe we could guess thatσμμψ=isthe massive case, but this is wrong. So enough guessing.This follows naturally from representations of the Lorentz group. There aretheseΛRz=R(θz) =1cosθzsinθz-sinθzcosθz1,Bβx=coshβxsinhβxsinhβxcoshβx11.Now we can look at the infinitesimal generators and extract theLie algebra.Then we getRz=R(θz) =0θz-θz0,βx=βxβx00.Then we getΔV0=βiVi,ΔVi=βiV0-ijkθjVk.If we write theitimes the rotation generators asJiand theitimes the boostgenerators asKi, hen we have[Ji, Jj] =iijkJk,[Ji, Kj] =iijkKk,[Ki, Kj] =-iijkJk.
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Physics 253a Notes4313October 16, 2018We were talking about representations of the Lorentz group. The Lorentz groupis defined asL={ΛF: (ΛF)gΛF=g}.Once we have these matrices, there is a group operation, so we get a groupO(1,3). There is the obvious 4-vector representationVμΛμνVν.To find a representation, we could instead look at the representations of the Liealgebra, which is generated byiJ1, iJ2, iJ3andiK1, iK2, iK3.13.1Representations of the Lorentz groupThe Lie algebra structure is given by[Ji, Jj] =iijkJk,[Ji, Ki] =iijkKk,[Ki, Kj] =-iijkJk,which is calledo(1,3). How do we find representations of this? We observe thatthere is a convenient linear combination,J+i=12(Ji+iKi),J-i=12(Ji-iKi).Then the relations become[J+i, J+j] =iijkJ+k,[J-i, J-j] =iijkJ-k,[J+i, J-j] = 0.This means that we really haveo(1,3) =su(2)×su(2).But we know the finite-dimensional representations ofsu(2): they are just(2j+1)-dimensional vector spaces. So irreducible representations ofso(1,3) arejust indexed by (A, B), which is a (2A+ 1)(2B+ 1)-dimensional representationobtained by taking the tensor (box) product.Now given any representation ofso(1,3), we will also get a representationsofso(3) by looking at~J=~J++~J-.Now we can think about which spinrepresentations appear in each (A, B). Here is a table of this for small values:(A, B)dimspin ofso(3)-representations(0,0)10(12,0)212(0,12)212(12,12)41, 0(1,0)31Table 1: Representations ofsu(2)×su(2) restricted toso(3)
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Physics 253a Notes4413.2Dirac spinorsSo what are these representations? If we look at (12,0) and (0,12), these differentrepresentations with spin12. These are calledright-handed spinorsψLJ+=σi2,J-= 0,~J=J++J-=2,~K=i(J--J+) =-i2,andleft-handed spinorsψRJ+= 0,J-=σi2,~J=2,~K= +i2.
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