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## Main Question or Discussion Point

Hi,

I'm working through Section 4-3 of Itzykzon and Zuber's QFT textbook, but I am a bit stuck while trying to understand some of the quantities and equations.

First of all, what is this "one-body scattering operator [itex]\mathcal{F}(A)[/itex]"? It is defined (eqn 4-89, page 188) as

[tex]\mathcal{F}(A) = e\gamma^{\mu}A_{\mu}(x) + e\gamma^{\mu}A_{\mu}(x)\frac{1}{\gamma^{\mu}P_{\mu}-m+i\epsilon}\mathcal{F}(A)[/tex]

What does this operator physically represent? For most of this section, it seems to be a trick to go through with the calculation. But the pair production probability in eqn 4-99 (page 191) is expressed in terms of an operator [itex]t[/itex] which is in turn defined in terms of the Hermitian adjoint of [itex]\mathcal{F}[/itex], in equation 4-98 (page 191).

Secondly, given a particular [itex]A^{\mu}(x)[/itex], I suppose I have to find [itex]\mathcal{F}[/itex] somehow. How do I do this? The defining expression above is recursive, and the only two situations for which the authors have actually gone through with the calculation are the lowest order emission rate and the constant field emission rate.

Thirdly, the 'lowest order' emission probability is linear in [itex]\alpha[/itex] (the coupling constant) [equation 4-105, page 192]. But so is the 'exact' emission probability for a constant electric field (equation 4-118, page 195). So, if I understand correctly, the higher order contributions vanish for a constant electric field. Without actually computing the rate, can this be justified physically?

I want to compute higher order corrections to 4-105. So, I need to figure out what [itex]\mathcal{F}(A)[/itex] really is, and how I can obtain it for an arbitrary [itex]A^{\mu}(x)[/itex].

Would appreciate your thoughts...

Thanks in advance.

I'm working through Section 4-3 of Itzykzon and Zuber's QFT textbook, but I am a bit stuck while trying to understand some of the quantities and equations.

First of all, what is this "one-body scattering operator [itex]\mathcal{F}(A)[/itex]"? It is defined (eqn 4-89, page 188) as

[tex]\mathcal{F}(A) = e\gamma^{\mu}A_{\mu}(x) + e\gamma^{\mu}A_{\mu}(x)\frac{1}{\gamma^{\mu}P_{\mu}-m+i\epsilon}\mathcal{F}(A)[/tex]

What does this operator physically represent? For most of this section, it seems to be a trick to go through with the calculation. But the pair production probability in eqn 4-99 (page 191) is expressed in terms of an operator [itex]t[/itex] which is in turn defined in terms of the Hermitian adjoint of [itex]\mathcal{F}[/itex], in equation 4-98 (page 191).

Secondly, given a particular [itex]A^{\mu}(x)[/itex], I suppose I have to find [itex]\mathcal{F}[/itex] somehow. How do I do this? The defining expression above is recursive, and the only two situations for which the authors have actually gone through with the calculation are the lowest order emission rate and the constant field emission rate.

Thirdly, the 'lowest order' emission probability is linear in [itex]\alpha[/itex] (the coupling constant) [equation 4-105, page 192]. But so is the 'exact' emission probability for a constant electric field (equation 4-118, page 195). So, if I understand correctly, the higher order contributions vanish for a constant electric field. Without actually computing the rate, can this be justified physically?

I want to compute higher order corrections to 4-105. So, I need to figure out what [itex]\mathcal{F}(A)[/itex] really is, and how I can obtain it for an arbitrary [itex]A^{\mu}(x)[/itex].

Would appreciate your thoughts...

Thanks in advance.