From Special Relativity to Feynman Diagrams.pdf

Are eigenvectors of 3 u p r and v p r will be

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are eigenvectors of 3 , u ( p , r ) and v( p , r ) will be eigenvectors of 3 . In Sect.9.4.1 of last chapter, a general method was applied to the construction of the single-particle quantum states | p , r acted on by a unitary irreducible repre- sentation of the Lorentz group. The method consisted in first constructing the states of the particle | ¯ p , r in some special frame S 0 in which the momentum of the par- ticle is the standard one ¯ p , and on which an irreducible representation R of the little group G ( 0 ) of ¯ p acts ( ¯ p = ( mc , 0 ) and G ( 0 ) = SU ( 2 ) for massive particles, while ¯ p = ( E , E , 0 , 0 )/ c and G ( 0 ) is effectively SO ( 2 ) for massless particles). A generic state | p , r is then constructed by acting on | ¯ p , r by means of U ( p ) , see ( 9.111 ), that is the representative on the quantum states of the simple Lorentz boost p connecting ¯ p to p : p = p ¯ p . This suffices to define the representative U ( ) of a generic Lorentz transformation, see ( 9.112 ). In this section we have applied this prescription to the construction of both the positive and negative energy eigen- states of the momentum operators. The role of | p , r is now played by the spinors u ( p , r ), v( p , r ) , and that of U ( ) by the matrix S ( ) , as it follows by comparing ( 10.149 ) with ( 9.112 ). It is instructive at this point to show that the expressions for u ( p , r ), v( p , r ) given in ( 10.154 ) or, equivalently, ( 10.157 ), for massive fermions, could have been obtained from the corresponding spinors u ( 0 , r ), v( 0 , r ) in S 0 using the prescription ( 9.111 ), namely by acting on them through the Lorentz boost S ( p ) : u ( p , r ) = S ( p ) u ( 0 , r ) ; v( p , r ) = S ( p )v( 0 , r ). (10.163) This is readily proven using the matrix form ( 10.118 ) of S ( p ) derived in Sect.10.4.4 and the definition of u ( 0 , r ), v( 0 , r ) in ( 10.152 ). The matrix product on the right hand side of ( 10.163 ) should then be compared with the matrix form of u ( p , r ), v( p , r ) in ( 10.157 ).
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340 10 Relativistic Wave Equations 10.6.1 Useful Properties of the u ( p , r ) and v( p , r ) Spinors In the following we shall prove some properties of the spinors u ( p , r ) and v( p , r ) describing solutions with definite four-momentum. Let us compute the Dirac conjugates of u ( p , r ) e v( p , r ) : ¯ u ( p , r ) = u ( p , r 0 = u ( 0 , r ) p + mc 2 m ( E p + mc 2 ) γ 0 = u ( 0 , r 0 γ 0 p + mc 2 m ( E p + mc 2 ) γ 0 = ¯ u ( 0 , r ) p + mc 2 m ( E p + mc 2 ) . (10.164) In an analogous way one finds: ¯ v( p , r ) = ¯ v( 0 , r ) p + mc 2 m ( E p + mc 2 ) . (10.165) Recalling the property ( 10.156 ), from ( 10.164 ) and ( 10.165 ) we obtain the equations of motion obeyed by the Dirac spinors ¯ u ( p , r ) e ¯ v( p , r ) : ¯ u ( p , r )( p mc ) = 0 , ¯ v( p , r )( p + mc ) = 0 . (10.166) Next we use the relations: ( p + mc ) 2 = 2 mc ( p + mc ), ( p mc ) 2 = 2 mc ( p + mc ), (10.167) which follow from ( 10.142 ) and the mass-shell condition p 2 = m 2 c 2 , to compute ¯ u ( p , r ) u ( p , r ) : ¯ u ( p , r ) u ( p , r ) = 2 mc 2 m ( E p + mc 2 ) ¯ u ( 0 , r )( p + mc ) u ( 0 , r ) = c E p + mc 2 r , 0 , 0 )( p + mc ) 0 0 ϕ r = ϕ r · ϕ r = δ rr , (10.168)
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10.6 Plane Wave Solutions to the Dirac Equation 341 With analogous computations one also finds: ¯ v( p , r )v( p , r ) = c E p + mc 2 ¯ v( 0 , r )( p + mc )v( 0 , r ) = c E p + mc 2 ( 0 , 0 , ϕ r )( p + mc ) 0 0 ϕ r = − δ rr , (10.169) and moreover ¯ u ( p , r )v( p , r ) ¯ u ( 0 , r )( p + mc )( p + mc )v( 0 , r ) = 0 = ¯ v( p , r ) u ( p , r ). (10.170) Summarizing, we have obtained the relations ¯ u ( p , r ) u ( p , r ) = δ rr , = −¯ v( p , r )v( p , r ), ¯ u ( p , r )v( p , r ) = 0 . (10.171) Next we show that: u ( p , r ) u ( p , r ) = E p mc 2 δ rr 0
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