By Peskin and Schroeder
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Extra resources for An Introduction to Quantum Field Theory
Ty L#+ = T L"+ P! L" = P L" \orthochronous" ; + x? yT ! L#; = PT L"+ \nonorthochronous" P \proper" \improper" At the same time that we discuss P and T , it will be convenient to discuss a third (non-spacetime) discrete operation: charge conjugation, denoted by C . Under this operation, particles and antiparticles are interchanged.
To do this we need to know one more property of the matrices. With a short computation you can verify that S ] = (J ) or equivalently, ;1 + i ! S 2 ;1 ; i ! S 2 ; = 1 ; 2i ! J : This equation is just the in nitesimal form of 1 1 2 where 1 2 = (3:29) ; (3:30) = exp ; 2i ! 19)). 29) says that the matrices are invariant under simultaneous rotations of their vector and spinor indices (just like the i under spatial rotations). In other words, we can \take the vector index on seriously," and dot into @ to form a Lorentz-invariant di erential operator.
The sum over r is accomplished most easily by; choosing the spinors r to be eigenstates of 3 . We then nd that for s = 10 , the ;one-particle state is an eigenstate of Jz with eigenvalue +1=2, while for s = 01 , it is an eigenstate of Jz with eigenvalue ;1=2. This result is exactly what we expect for electrons. An analogous calculation determines the spin of a zero-momentum antifermion. But in this case, since the order of the b and by terms in Jz is reversed, we get an extra minus sign from evaluating bp byp by0] = ; bypbp by0].
An Introduction to Quantum Field Theory by Peskin and Schroeder