From Special Relativity to Feynman Diagrams.pdf

Are divergent constants given by δ m p p 2 m 2 b 1 4

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are divergent constants given by δ m = ( p ) | p 2 = m 2 , B = 1 4 γ μ p μ p 2 = m 2 . while ( c ) ( p ) is convergent and satisfies ( c ) ( p ) = γ μ ( c ) p μ = 0 for p 2 = m 2 . (12.270) We see that the entire divergence of ( p ) is contained in the infinite constants A and B . We can now insert the result ( 12.269 ) into the expression on the right hand side of ( 12.265 ), obtaining i p m 0 ( p ) = i p m 0 δ m B ( p m ) ( c ) ( p ) . (12.271) We see that the pole of S F ( p ) , which defines the mass of the particle, is no longer at p 2 = m 2 0 . If we choose the arbitrary parameter m to satisfy: m 0 + ( m ) = m , (12.272) where ( m ) ( p ) | p 2 = m 2 , ( 12.271 ) yields S F ( p ) = i ( p m )( 1 B ) ( c ) ( p ) . (12.273) Recalling that ( c ) vanishes for p 2 = m 2 , m becomes the mass of the particle, which is shifted from its original value m 0 , the shift being proportional to the diver- gent quantity δ m ( m ). Since δ m is divergent we conclude that the bare mass m 0 present in the original Lagrangian must be divergent as well, in order for the physical mass m to be finite. 32 The mass renormalization given by the mass shift ( 12.272 ) provides the removal of the divergent term δ m = ( m ) from the corrected propagator, but it still depends on the infinite constant B . 33 As it is apparent from 32 Naively one could think that the separation of the physical mass into the bare mass m 0 and the mass-shift δ m = ( m ) would correspond to the separation of the electron mass into a “mechanical” and a “electromagnetic” mass. However such separation is devoid of physical meaning since it cannot be observed. We also note that the process of mass renormalization is not a peculiarity of field theory. For example when an electron moves inside a solid it has a renormalized mass m , also called effective mass, which is different from the mass measured in the absence of the solid, i.e. the bare mass m 0 . However, differently from our case, the effective and bare mass can be measured separately, while in field theoretical case m 0 cannot be measured. 33 We observe that this term would give a vanishing contribution if we had an external on-shell state instead of the propagator in (12.264) since the term B ( p m ) in ( 12.269 ) is zero on the free electron wave function.
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520 12 Fields in Interaction the ( 12.273 ), this infinite constant changes the residue at the pole from its original value i to i ( 1 B ) 1 . To dispose of the divergent constant B we observe that neglecting higher order terms in e 2 0 we may write ( c ) ( p ) ( c ) ( p )( 1 B ), that is ( c ) ( p ) ( c ) ( p )( 1 B ) = ( c ) ( p ) Z 1 2 , where Z 2 ( 1 B ) 1 . (12.274) Equation ( 12.273 ) can be recast in the following form: S F ( p ) = i Z 2 ( p m ) ( c ) ( p ) . We see that the expression multiplying Z 2 is completely finite. On the other hand, the multiplicative constant Z 2 can be reabsorbed in a redefinition of the electron field, namely by defining a renormalized physical field ψ( x ) in terms of a bare unphysical one ψ 0 ( x ) as follows: ψ 0 = Z 1 2 2 ψ. (12.275) Recalling indeed the definition ( 12.110 ) of the Feynman propagator and its Fourier transform, we have S F = d 4 ξ e ip · ξ 0 | T (ψ( y + ξ) ¯ ψ( y )) | 0 = Z 1 2 d 4 ξ e ip · ξ 0 | T 0 ( y + ξ) ¯ ψ 0 ( y )) | 0 = i 1 ( p m ) ( c ) . (12.276)
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