From Special Relativity to Feynman Diagrams.pdf

The above commutators completely define the

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The above commutators completely define the transformation properties of ˆ ψ under the action of the Poincaré group. Of course U , as well as its generators J μν , P μ , act on the c and d operators in the expansion of ˆ ψ. Let us define what such an action should be in order to reproduce the correct transformation property ( 11.191 ). To this end let usrecallthatthe u ( p , r ) and v( p , r ) spinorstransformunderaLorentztransformation as in ( 10.149 ) of Chap.10 , where the matrix R ( , p ) r s is a rotation in the spin- group, namely a SU ( 2 ) (for massive particles) or an SO ( 2 ) (for massless particles) transformation depending on the momentum p and the Lorentz transformation itself. Let us show that the transformation law ( 11.191 ) is correctly reproduced if: U ( , x 0 ) c ( p , s ) U ( , x 0 ) = e i p · x 0 R ( , 1 p ) s r c ( 1 p , r ), U ( , x 0 ) d ( p , s ) U ( , x 0 ) = e i p · x 0 [ R ( , 1 p ) s r ] d ( 1 p , r ). (11.194) Computing the hermitian conjugate of last equation we find: U ( , x 0 ) d ( p , s ) U ( , x 0 ) = e i p · x 0 R ( , 1 p ) s r d ( 1 p , r ). (11.195) Applying the above properties, the transformation rule for the spinor field operator reads:
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408 11 Quantization of Boson and Fermion Fields U ( , x 0 ) ˆ ψ( x ) U ( , x 0 ) = p , r mc 2 E p V U c ( p , r ) Uu ( p , r ) e i p · x + U d ( p , r ) U v( p , r ) e i p · x = p , r , s mc 2 E p V R ( , 1 p ) r s × c ( 1 p , s ) u ( p , r ) e i p · ( x + x 0 ) + d ( 1 p , s )v( p , r ) e i p · ( x + x 0 ) = p , r , s mc 2 E p V R ( , p ) r s × c ( p , s ) u ( p , r ) e i ( p ) · ( x + x 0 ) + d ( p , s )v( p , r ) e i ( p ) · ( x + x 0 ) = p , s mc 2 E p V c ( p , s ) S ( ) u ( p , s ) e i p · x + d ( p , s ) S ( )v( p , s ) e i p · x = S ( ) p , s mc 2 E p V c ( p , s ) u ( p , s ) e i p · x + d ( p , s )v( p , s ) e i p · x = S ( ) ˆ ψ( x ), (11.196) where we have changed summation variable from p to p = 1 p and, as usual, wrote x = 1 ( x + x 0 ). We have moreover used the transformation properties ( 10.149 ). 11.6.3 Discrete Transformations Let us now consider the three discrete transformations corresponding to parity P , charge conjugation C and time-reversal T for the Dirac quantum field. In the previous Chapterwehaveseenthatforthe classical Diracfieldthespacereflectioncorresponds to the active transformation (see (10.242)): ψ( x , t ) ψ ( x , t ) = η P γ 0 ψ( x , t ), (11.197) with respect to which it is easily verified that the Dirac equation is invariant. For the quantized field we must seek a unitary operator U ( P ) such that 16 U ( P ) ψ( x , t ) U ( P ) = η P γ 0 ψ( x , t ). (11.198) We can define through its action on the operators c ( p , r ), d ( p , r ), which should reproduce ( 11.198 ). Using the properties 16 We suppress here and in the following the “hat” symbol for the field operator.
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11.6 Invariant Commutation Rules for the Dirac Field 409 γ 0 u ( p , r ) = u ( p , r ), γ 0 v( p , r ) = − v( p , r ) which can be easily derived from the explicit form of the spinors u ( p , r ) and v( p , r ) given in ( 10.154 ) and ( 10.154 ), we find the operators c and d should transform under parity as follows U ( P ) c ( p , r ) U ( P ) = η P c ( p , r ), (11.199) U ( P ) d ( p , r ) U ( P ) = − η P d ( p , r ). (11.200) The explicit form of U ( P ) can be obtained following the same procedure as in the scalar field case. The result is U ( P ) = e i π 2 p , r c ( p , r ) c ( p , r ) d ( p , r ) d ( p , r ) η P c ( p , r ) c ( p , r ) η P d ( p , r ) d (
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