Box 3.1 Compton Scattering Transition Amplitude Derivation

Two ways (same particles in and out) for Compton scattering

.

                                                                                                          

All terms in S ⁽ⁿ⁾ where n ≠ 2 go to zero below since the initial state (the ket) will be acted on by operators which will result in bras and kets which are unequal (non-orthogonal eignenstates) and whose inner products are therefore zero.  Similarly, only the  term of all the n = 2 terms will result in the same particle states in the bra and ket.  The S matrix transition amplitude for Compton scattering is thus (with incoming particles unprimed, outgoing primed)

                                          (B3.1-1)

                                                      (B3.1-2)

                    .                                    (B3.1-3)

After the operators raise and lower the ket, only two terms in (B3.1-3) remain (i.e., have identical bra and ket).  They are

                                                         (B3.1-4)

Continuing with only the first term in (B3.1-4), we have (with Dirac indices explicitly shown)

      .                          (B3.1-5)

Re-arranging factors in the above, we have

                  (B3.1-6)

 

Noting that

        ,                     (B3.1-7)

we find thus,

                                                      q = p + k = p^/ + k^/                                                           (B3.1-8)           

and

              .                                                        (B3.1-9)

Then

    ,               (B3.1-10)

where the Feynman amplitude is

     .                                                 (B3.1-11)

Similarly, for the second term in (B3.1-4), one gets the same relation for  as (B3.1-10) except that M_B1 is replaced with

     .                                                  (B3.1-12)

Thus,

                    (B3.1-13)

 

Return to Unifying Chart 3 Part 2.