From Special Relativity to Feynman Diagrams.pdf

One can prove on general grounds that that there

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solutions, and viceversa. One can prove on general grounds that that there exists a matrix in spinor space, called the charge-conjugation matrix with the following properties C 1 γ μ C = − γ T μ ; C T = − C ; C = C 1 . (10.184) In the standard representation we may identify the C matrix as C = i γ 2 γ 0 = 0 i σ 2 i σ 2 0 . (10.185) Given a Dirac field ψ( x ) , we define its charge conjugate spinor ψ c ( x ) as follows: ψ c ( x ) C ¯ ψ T ( x ). (10.186) The operation which maps ψ( x ) into its charge conjugate ψ c ( x ) is called charge- conjugation . Let us show that charge conjugation is a correspondence between pos- itive and negative energy solutions.

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344 10 Relativistic Wave Equations To this end let us consider the positive energy plane wave described by the spinor u ( p , r ). Its Dirac conjugate ¯ u will satisfy the following equation: ¯ u ( p ) ( p mc ) = 0 . By transposition we have γ T μ p μ mc ¯ u T ( p ) = 0 If we now multiply the above equation to the left by the C matrix and use the property ( 10.184 ) we obtain ( p + mc ) C u T ( p ) = 0 , (10.187) which shows that charge-conjugate spinor u c ( p ) = C u T ( p ) satisfies the second of ( 10.148 ) and should therefore coincide with a spinor v( p ) defining the negative energy solution ψ ( ) p with opposite momentum p . Besides changing the value of the momentum, charge-conjugation also reverses the spin orientation. Going, for the sake of simplicity, to the rest frame, where a positive energy solution with spin projection / 2 along a given direction, is described by u ( 0 , 1 ) = ( 1 , 0 , 0 , 0 ) T , see ( 10.152 ), we find for the charge conjugate spinor u c C γ 0 u (note that γ 0 T = γ 0 ) u c ( 0 , r ) = C γ 0 u ( 0 , r = 1 ) = ( 0 , 0 , 0 , 1 ) T = v( 0 , r = 2 ), that is a negative energy spinor with spin projection / 2 . In general the reader can verify that u c ( 0 , r ) = rs v( 0 , s ), (10.188) where summation over s = 1 , 2 is understood, and ( rs ) is the matrix i σ 2 : 11 = 22 = 0 , 12 = − 21 = 1. Let us now evaluate u c ( p , r ) using the explicit form of u ( p , r ) given in ( 10.154 ): u c ( p , r ) = C γ 0 u ( p , r ) = C γ 0 p + mc 2 m ( mc 2 + E p ) u ( 0 , r ) = C p T + mc 2 m ( mc 2 + E p ) γ 0 u ( 0 , r ) = p + mc 2 m ( mc 2 + E p ) u c ( 0 , r ) = rs p + mc 2 m ( mc 2 + E p ) v( 0 , s ) = rs v( p , s ). (10.189)
10.6 Plane Wave Solutions to the Dirac Equation 345 In the above derivation we have used the properties C p T C 1 = − p and γ 0 p = p T γ 0 . We shall see in the next chapter that, upon quantizing the Dirac field, negative energysolutions ψ ( ) p , r withmomentum p andacertainspincomponent(upordown relative to a given direction) are reinterpreted as creation operators of antiparticles with positive energy, momentum p and opposite spin component. Thus the charge conjugation operation can be viewed as the operation which interchanges particles with antiparticles with the same momentum and spin. As far as the electric charge is concerned we need to describe the coupling of a charge conjugate spinor to an external electromagnetic field as it was done for the scalar field. This will be discussed in Sect.10.7 . We anticipate that the electric charge of a charge conjugate spinor describing an antiparticle is opposite to that of the corresponding particle.

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