Regularization and Renormalization
(See, for instance, Mandl and Shaw [17, Chap. 9] and Peskin and Schroeder [23, Chap. 7].)
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Regularization and Renormalization
(See, for instance, Mandl and Shaw [17, Chap. 9] and Peskin and Schroeder [23, Chap. 7].) In previous chapters, we have evaluated some radiative effects in the S-matrix (Chap. 4) and covariant-evolution operator formulations (Chap. 8). In this chapter, we discuss the important processes of renormalization and regularization in some detail. Many integrals appearing in QED are divergent, and these divergences can be removed by replacing the bare electron mass and charge by the corresponding physical quantities. Since infinities are involved, this process of renormalization is a delicate matter. In order to do this properly, the integrals first have to be regularized, which implies that the integrals are modified so that they become finite. This has to be done so that the process is gauge-independent. After renormalization, the regularization modification is removed. Several regularization schemes have been developed, and we shall consider some of them in this chapter. If the procedure is performed properly, the way of regularization should have no effect on the final result.
12.1 The Free-Electron QED 12.1.1 The Free-Electron Propagator The wave functions for free electrons are given by (D.29) in Appendix D pC .x/ D .2/3=2 uC .p/ eip x eiEp t ; p .x/ D .2/3=2 u .p/ eip x eiEp t
(12.1)
where p is the momentum vector and pC represents positive-energy states (r D 1; 2) p and p negative-energy states (r D 3; 4). Ep D cp0 D c 2 p2 C m2 c 4 . The coordinate representation of the free-electron propagator (4.10) then becomes hx1 jSOFfreejx2 i D
Z
d! X p;r .x 1 / p;r .x 2 / i!.t1 t2 / e ; 2 p;r ! "free p .1 i/
I. Lindgren, Relativistic Many-Body Theory: A New Field-Theoretical Approach, Springer Series on Atomic, Optical, and Plasma Physics 63, c Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-8309-1 12,
(12.2)
237
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12 Regularization and Renormalization
free where "free p is the energy eigenvalue of the free-electron function (Ep D j"p j). The Fourier transform with respect to time then becomes
hx1 jSOFfreejx2 i D
X p;r .x 1 / p;r .x 2 / p;r
Z
! "free p .1 i/
)
d3 p X eip.x1 x2 / u .p/ u .p/ r r .2/3 r ! "free p .1 i/ Z 3 1 d p uC .p/ uC .p/ D 3 .2/ ! Ep .1 i/ 1 eip.x1 x2 / : C u .p/ u .p/ ! C Ep .1 i/
D
The square bracket above is the Fourier transform of the propagator, and using the relations (D.35, D.36), this becomes1 SFfree .!; p/ D
1 1 1 C 2 ! Ep .1 i/ ! C Ep .1 i/ 2 1 c ˛ p C ˇmc 1 C 2p0 ! Ep .1 i/ ! C Ep .1 i/
! C c˛ p C ˇmc 2 ! C c ˛ p C ˇmc 2 D ! 2 Ep2 C i ! 2 .c 2 p2 m2 c 4 / .1 i/ 1 D ! .c ˛ p C ˇmc 2 / .1 i/ D
(12.3)
with Ep2 D c 2 p02 D c 2 p2 C m2 c 4 and ˛ˇ D ˇ˛. This can also be expressed SFfree .!; p/ D
1 !
hfree D .p/ .1
i/
;
(12.4)
where hfree D .p/ is the momentum representation of the free-electron Dirac Hamiltonian operator (D.21), hO free O D .p/. Formally, we can write (12.3) inp covariant four-com
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