- #1

jostpuur

- 2,116

- 19

- 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

[tex]

|\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

[/tex]

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

[tex]

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

[/tex]

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

[tex]

|\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

[/tex]

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

[tex]

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

[/tex]

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

[tex]

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

[/tex]

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

[tex]

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}

[/tex]

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

[tex]

|\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

[/tex]

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

[tex]

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

[/tex]

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

[tex]

|\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

[/tex]

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

[tex]

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

[/tex]

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

[tex]

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

[/tex]

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

[tex]

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}

[/tex]

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