It is often stated that the transition amplitude between eigenstates of the free-field Hamiltonian [itex]H_0[/itex] is encoded by the S-matrix, defined by(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\langle \mathrm{f} | U{_\mathrm{I}} (\infty,-\infty)|\mathrm{i} \rangle[/itex].

where [itex]U_{\mathrm{I}}[/itex] is the time-evolution operator in the interaction picture (e.g., [1]).

I would argue that this is not correct, and that the correct expression should be

[itex]S_{\mathrm{fi}}=\langle \mathrm{f} | U_0(\infty,-\infty) U{_\mathrm{I}} (\infty,-\infty)|\mathrm{i} \rangle[/itex]

where [itex]U_0(t,t') = \mathrm{e}^{-\mathrm{i} H_0 (t-t')}[/itex] is the time-evolution operator for the free-field.

My argument is quite simple, the transition amplitude in the Schrodinger picture is

[itex]\langle \mathrm{f} | U_{\mathrm{S}} (\infty,-\infty)|\mathrm{i} \rangle[/itex].

It is easy to show, however, that the Schrodinger evolution operator is related to the interaction evolution operator by [itex]U_{\mathrm{S}} = U_0 U_{\mathrm{I}}[/itex]. My argument seems to be supported by [2]. However, it concerns me that this is not addressed in mainstream QFT books.

Does anyone have any word on this apparent discrepancy?

[1] Mandl and Shaw, Quantum Field Theory.

[2] http://www.nyu.edu/classes/tuckerman/stat.mechII/lectures/lecture_21/node3.html

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# Interaction picture (Is there a mistake in QFT books?)

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