From Special Relativity to Feynman Diagrams.pdf

Ε 1 μ p i ε 2 μ p 21 if we denote by ε μ p r

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ε ( 1 ) μ ( p ) ± i ε ( 2 ) μ ( p ). 21 If we denote by ε μ ( p , r ) such complex vectors we can write the photon field operator in the following form: ˆ A μ ( x ) = c d 3 p ( 2 π ) 3 V 2 E p 2 r = 1 ε μ ( p , r ) a ( p , r ) e i p · x + ε μ ( p , r ) a ( p , r ) e i p · x . (11.278) where a ( p , r ) are the complex combinations a 1 ( p ) ia 2 ( p ). 11.7.2 Poincaré Transformations and Discrete Symmetries Let us recall the transformation property of the classical electromagnetic field under a Poincaré transformation: A μ ( x ) ( , x 0 ) −→ A μ ( x ) = μ ν A ν ( x ) = O ( , x 0 ) A μ ( x ), (11.279) where μ and ν indices are raised and lowered using the Lorentzian metric ( μ ν η μρ ρ σ η σν ) = ( T ) μ ν and, as usual, x = x x 0 . From the relation ( 11.269 ) we deduce, just as we did for the scalar and Dirac fields, the transformation property of the field operator ˆ A μ ( x ) : ˆ A μ ( x ) ( , x 0 ) −→ U ˆ A μ ( x ) U = μ ν ˆ A ν ( x ) = O ( , x 0 ) ˆ A μ ( x ), (11.280) where U = U ( , x 0 ) is the unitary operator implementing the Poincaré transfor- mation on the physical multi-photon states. The commutation relations between the 20 Note that this correspondence should take into account a normalization factor due to the fact that A μ ( x ) does not have the dimension of a wave-function: x | s 0 . 21 In Chap.6 the direction of motion was chosen along the X -axis so that the transverse directions were 2 and 3. Here the motion is chosen along the Z -axis.
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426 11 Quantization of Boson and Fermion Fields infinitesimal generators J μν , P μ of U ( , x 0 ) and the ˆ A ν ( x ), which characterize its transformation properties, are deduced just as we did for the lower spin fields, namely by writing ( 11.280 ) for an infinitesimal transformation and expanding it to first order in the parameters. Let us now evaluate the action of the discrete symmetries C , P , T on the photon field. Since a photon coincides with its own antiparticle, the action of C only amounts to a multiplication by a factor η C = ± 1 : U ( C ) ˆ A μ ( x ) U ( C ) = η C ˆ A μ ( x ). (11.281) We shall choose η C = − 1 for reasons we are going to illustrate below, so that the photon is odd under charge conjugation . As for P , T , the transformation properties read: U ( P ) ˆ A μ ( x ) U ( P ) = η P P μ ν ˆ A ν ( x P ) = η P η μμ ˆ A μ ( x P ), (11.282) U ( T ) ˆ A μ ( x ) U ( T ) = η T T μ ν ˆ A ν ( x T ) = − η T η μμ ˆ A μ ( x T ), (11.283) with no summation over μ. In the above formulas we have defined x P P x = ( ct , x ), and x T T x = ( ct , x ). In order to determine action of U ( C ), U ( P ) and U ( T ) on the a operators which reproduces ( 11.281 ), ( 11.282 ) and ( 11.283 ), one follows the same procedure illustrated for the scalar and Dirac field, though we shall refrain from doing it here. 11.8 Quantum Electrodynamics In Sect.10.7 of Chap.10 , we have studied the interaction of a charged Dirac field ψ( x ) (such as an electron) with the electromagnetic one A μ ( x ). The description of such interaction was obtained by applying to the free Dirac equation the minimal coupling prescription ( 10.210 ). The resulting equations of motion could be derived from the Lagrangian density ( 10.228 ). If we include the electromagnetic field in the description by adding to L in ( 10.228 ) the term L e . m .
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