4n 2nm2 V J
The term in brackets looks like the transition rate, except that it is not on the "energy shell" defined by ui = Ea - Eb. Introducing a "virtual" transition rate,
where Wba(Ea-Eb) = Wba, permits us to write the energy shift in the following convenient form:
The integral has a singularity at u = Ea - Eb, which is defined using the "it prescription." With this prescription, the energy denominator Ea - Eb - u> is replaced by Ea - Eb -ui + ie, where the limit t —> 0 is understood. The sign of it is determined by causality. To see this, note that the denominator can be written
Ea - Eb - u + it Ea - Eb - to where P is the principal value integral. Hence the energy shift is now complex, with
27T ^ Jo Ea - Eb - Ui + it where Wba(Ea-Eb) = Wba, permits us to write the energy shift in the following convenient form:
ReAEa + ilmAEa
Hence, choosing +ie gives AEa a negative imaginary part, which equals \ of the total decay rate of the state o,
However, this is just what we should expect from the energy-time evolution factor
corresponding to exponential decay of the state a with a half-life equal to the reciprocal of the total decay rate ra 1
1 total lira
The +ie prescription therefore gives a decay in the probability \ipa(t)\ . If we had chosen a -ie prescription, we would have obtained an exponentially growing probability, contrary to causality.
The imaginary part of the energy shift makes the Hamiltonian appear to be non-Hermitian and the norm of ipa not conserved. However, when the entire Fock space is considered, it can be shown that the norm of the total system is conserved. A decrease in norm of ipa is accompanied by an increase in the norm of states with Eb < Ea and with one photon. In detail, the total state is
and the total norm
(3.83) is conserved.
Now, the real part of A Ea gives the shift in energy of the bound state, but it diverges. To see this, insert the expression for the decay rate, Eq. (3.67), into (3.77). Since |/>fca| is independent of u>, we obtain uidui
The integral diverges linearly, and we must introduce a high energy cutoff (upper limit) in order to define it. There are physical processes which we have ignored — one is the breakdown of the dipole approximation which is certainly unreliable for uj ~m — which naturally damp out the integral at high energies and help to define such a cutoff. But the sensitivity of the integral (3.84) to the precise choice of the cutoff makes the final result too sensitive to be useful for any reliable estimates. An even greater problem is that the result (3.84) is not physically observable. This leads us to the issue of mass renormalization.
If x < 0 we have directly
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