QFT S-matrix explanations are incomprehensible

[jostpuur]

TL;DR Summary

I've failed to understand S-matrix explanations. Does anyone feel like understanding them?

The first look at a scattering process is something like this: We define an initial state

$$|\textrm{in}\rangle = \int dp_1dp_2 f_{\textrm{in,1}}(p_1) f_{\textrm{in,2}}(p_2) a_{p_1}^{\dagger} a_{p_2}^{\dagger} |0\rangle$$

Here $f_{\textrm{in,1}}$ and $f_{\textrm{in,2}}$ are wavefunctions that define some wavepackets that are about collide. Schrodinger equation will determine what happens, so we define an out state as

$$|\textrm{out}\rangle = e^{-\frac{it}{\hbar}H} |\textrm{in}\rangle$$

If it turns out at that N particles fly out from the collision as some wavepackets, then something like

$$|\textrm{out}\rangle \approx \int dq_1 dq_2 \cdots dq_N f_{\textrm{out,1}}(q_1) f_{\textrm{out,2}}(q_2)\cdots f_{\textrm{out,N}}(q_N) a_{q_1}^{\dagger} a_{q_2}^{\dagger}\cdots a_{q_N}^{\dagger}|0\rangle$$

is true. So far I feel like I understand what this all means. However, in the fully developed QFT the scattering is not handled like above. Instead we define an initial state as

$$|\textrm{in}\rangle = |p_1,p_2\rangle = a_{p_1}^{\dagger} a_{p_2}^{\dagger} |0\rangle$$

So instead of wavepackets we wave planewaves that extend to infinities. Then we have an S-matrix operator that works so that it will give an amplitude for N particles flying out as

$$\langle q_1, q_2,\cdots, q_N|S|p_1,p_2\rangle$$

What confuses me about this is that planewaves cannot really collide, can they? Wavepackets are something can actually collide, but planewaves are somekind of artificial tool? So how do you make the S-matrix operator work so that it makes the planewaves collide?

One formula for S-matrix is

$$S = \lim_{t_{\textrm{B}}\to\infty} \lim_{t_{\textrm{A}}\to -\infty} e^{\frac{i}{\hbar}t_{\textrm{B}}H_0} e^{-\frac{i}{\hbar}(t_{\textrm{B}}-t_{\textrm{A}})H}e^{-\frac{i}{\hbar}t_{\textrm{A}}H_0}$$

It is nice that a formula exists, but I still don't understand that what calculations turn the wavepackets into planewaves.

 

[strangerep]

What confuses me about this is that planewaves cannot really collide, can they?

Sure. Why not? Whether this has any nontrivial outcome depends on whether there's an interaction term in the total Hamiltonian.

Wavepackets are something can actually collide, but planewaves are somekind of artificial tool?

Wavepackets can be expressed as a linear combination of planewaves (as in Fourier theory).

I still don't understand that what calculations turn the wavepackets into planewaves.

If we can compute a general formula for the scattering of any (combination of) incoming planewaves into any combination of outgoing planewaves then we have a theory that can be compared with experiment.

One of the early chapters in Peskin& Schroeder has a section on how physically realistic beams (like in an accelerator) are modeled in these terms, and then how to compute scattering cross sections therefrom.

 

[Demystifier]

What confuses me about this is that planewaves cannot really collide, can they?

This is not necessarily related to QFT, the same issue arises also in nonrelativistic QM. I would suggest you to first try to understand it in this simpler context.

The crucial insight is the fact that QM is linear. So to understand how a wave packet scatters, you can expand the packet into plane waves and consider each plane wave separately. A nice analysis can be found e.g. in the book Bohm and Hiley, The Undivided Universe, Sec. 5.2.