# Is there a “time-independent” formalism for scattering theory of time-dependent interactions?

The title may look idiot but, please, let me try to explain.

Motivation. By "time-independent" formalism I refer to that part of the formalism which makes use of concepts like asymptotic in/out states $\Psi ^\pm$, Möller wave matrices $\Omega ^\pm$ etc. For example, consider the interaction of a scalar field $\phi$ with an external source $j$: $$\mathscr L = \frac{1}{2}(\partial \phi)^2-\frac{m^2}{2}\phi ^2 -j\phi,$$ where $j=j(x)$ is a smooth function of compact support. In ref. [1] this problem is solved (actually for the photon field interacting with an external current) by introducing the asymptotic fields: $$\phi_{\text{in}} (x)=\phi (x) -\intop \text d ^4 y \Delta _R(x-y)j(y),\\ \phi_{\text{out}} (x)=\phi (x) -\intop \text d ^4 y \Delta _A(x-y)j(y).$$ Using these fields, one may obtain, for example, the one-meson to one-meson scattering amplitude as: $$S(\mathbf p' \leftarrow \mathbf p )=\langle \mathbf p ' \text { out}\vert \mathbf p \text { in}\rangle = \langle 0 \text{ out}\vert a_{\text {out}}(\mathbf p')a^\dagger _{\text {in}}(\mathbf p )\vert 0\text{ in}\rangle.$$
If the source $j$ doesn't depend on time (and so, BTW, doesn't have compact support), I perfectly understand the meaning of $S(\mathbf p ' \leftarrow \mathbf p)$. On the other hand, for a time dependent source, I don't understand how to relate this amplitude with an actual experimental situation. I mean, it is quite obvious that the way in which the meson scatters must necessarily depend on the time the experiment is performed. If it is performed long before or long after the source has been turned on, then the meson won't scatter at all.

Simpler model. Note that I'm not concerned with the somewhat trivial case in which the "time dependence" amounts to an adiabatic turn-off. For concreteness, let's consider the following toy model of "potential scattering": $$H(t)=H_0 +f(t)V,$$ where $H_0 = -\frac{1}{2m}\nabla ^2$ is the kinetic energy, and $V$ is a separable potential, of the form: $$V=g(\psi _0,\cdot),$$ with $\psi _0$ a fixed vector in $L^2(\mathbb R ^3)$. $f$ is an arbitrary function of time, like $f(t)=\cos (\omega (t))$ or $f(t)=e^{-bt^2}$.

Question. Is it possible to describe the scattering of the particle off $V(t)=f(t)V$ in terms of asymptotic "in/out" states and/or in terms of a time independent operator $S$? What is the physical meaning of these states, i.e. how would one calculate, say, a (time dependent) scattering cross section starting from them?

Reference request. I've found some time ago a few papers describing scattering theory from a time dependent electric field (if I manage to find the link, I'll post it here), however they were a bit too specialized. Can you provide a reference, textbook or paper, where the formalism is explained from scratch?

[1] C. Itzykson, J. Zuber, "Quantum Field Theory".

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• I've tried my best to make the question clear, if I failed please let me know. – pppqqq Jan 19 '17 at 14:02
• What is your mathematical background? There are suitable mathematical tools that allow to deal with such type of problems. A good reference textbook would be the third volume of Reed and Simon's series on mathematical methods of modern physics. In particular, the Haag-Ruelle scattering theory may be closely related to what you are interested in. – yuggib Jan 19 '17 at 14:18
• I've only seen some basic rigorous results in potential scattering, like the strong convergence of $\Omega ^\pm$ and the generalized eigenfunction expansion of Ikebe. I'd honestly content myself with some non-mathematical exposition, such as those of formal scattering theory which are found in the physics '60s literature, or Weinberg's "QFT Vol. I". In any case, I'll give a look at Reed and Simon, thank you! – pppqqq Jan 19 '17 at 14:34