From Special Relativity to Feynman Diagrams.pdf

Where for the sake of clarity we have used a discrete

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where, for the sake of clarity, we have used a discrete notation: a p a ( p ), b p b ( p ). Now use the identity (see for instance [2] for a general proof) e i λ S Oe i λ S = O + i λ [ S , O ] + i 2 λ 2 2 ! [ S , [ S , O ]] + · · · . (11.86) Since S , a p = − a p S , S , a p = a p , we find e i λ S a p e i λ S = a p cos λ ia p sin λ, and the same relation for b p . Setting λ = η P π/ 2 , we get rid of the term in a ( p ), obtaining e i π 2 η P S a p e i π 2 η P S = − i η P a p , (11.87) e i π 2 η P S b p e i π 2 η P S = − i η P b p . (11.88) This is close to ( 11.85 ), but not yet correct. To get the exact result we multiply e i λ S by the further operator e i λ S , defined in such a way that 7 The intrinsic parity can only be fixed by experiment involving interactions, so that it is meaningful only when specified relative to other particles.
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11.3 Transformation Under the Poincaré Group 381 e i λ S a p e i λ S = ia p , (11.89) e i λ S b p e i λ S = ib p . (11.90) This is achieved by taking S = p a p a p + b p b p , and λ = − π/ 2 . The reader can show that [ S , S ] = 0 . Combining these results and defining U ( P ) e i π 2 S e i η P π 2 S = exp i π 2 ( S η P S ) = exp i π 2 p a p a p + b p b p η P a p a p η P b p b p . (11.91) Note that U ( P ) is indeed a unitary operator satisfying U ( P ) | 0 = | 0 , as can be easily seen expanding the exponentials. Thus the vacuum state has even parity . Moreover considering the momentum operator ( 11.45 ) we see that U ( P ) ˆ P U ( P ) = − ˆ P , consistently with the fact that the eigenvalues of the physical momentum are ordinary vectors under a space reflection. On the other hand U ( P ) commutes with the Hamiltonian , implying the conserva- tion of the parity operator: [ U ( P ), ˆ H ] = 0 . 8 On the quantum field ˆ φ( x , t ) we can also define a transformation with no analogue in the non-relativistic quantum theory: the charge conjugation C . It corresponds to exchanging particles for antiparticles, that is a p η C b p ; b p η C a p , (11.92) or, in terms of the field operator, ˆ φ( x ) η C ˆ φ ( x ), where η C is a constant which, defining C as an involutive transformation, can be chosen to be ± 1 . This operation is clearly a symmetry of the charged scalar theory. The construction of the unitary operator U ( C ) implementing such transformation on the Hilbert space of the states, namely 8 Since the parity transformation is involutive, U ( P ) 2 = ˆ I , its eigenvalues can only be ± 1 .
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382 11 Quantization of Boson and Fermion Fields U ( C ) a p U ( C ) = η C b p , U ( C ) b p U ( C ) = η C a p , U ( C ) ˆ φ( x ) U ( C ) = η C ˆ φ ( x ), (11.93) can be done by the same procedure used for the parity transformation. The result is U ( C ) = exp i π 2 η C p a p a p + b p b p η C a p b p η C b p a p . (11.94) It is easily verified that U ( C ) is unitary and satisfies U ( C ) | 0 = | 0 . Moreover, from ( 11.63 ) and ( 11.64 ) it follows U ( C ) ˆ J μ U ( C ) = − ˆ J μ ; U ( C ) ˆ QU ( C ) = − ˆ Q . (11.95) That means that, under charge conjugation the sign of the charge is flipped, according to our previous discussion in Sect.11.2.1.
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