Timeline for Explicit form of S-matrix on the line
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11 events
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| Nov 13, 2019 at 18:32 | comment | added | Michael Engelhardt | The physical scattering information is all in the $S$ I gave, and you're just putting transformations around it to more properly describe asymptotic states and time evolution. | |
| Nov 13, 2019 at 18:26 | comment | added | Michael Engelhardt | ... Using the Moller machinery, you force yourself to use a different basis, consisting of superpositions of energy eigenstates, and you have to modify your notion of completeness. But let's say that all works out (otherwise, you're stuck anyway) - then, all you've really done is combine the $S$ I gave with changes of basis. You can decompose your bona fide Moller ingoing state into energy eigenstates, use the $S$ I gave, supply the proper time evolution phases, and reassemble your Moller outgoing state. (continued ...) | |
| Nov 13, 2019 at 18:10 | comment | added | Michael Engelhardt | Now, about using the Moller machinery (which I'm sure you've thought about much more than I, so I'm only taking a bird's eye view): I don't see how using the energy eigenstates that you invoke in your question is compatible with that. By virtue of being energy eigenstates, time evolution will never make the interacting and free states congruent, regardless of how far you evolve back into the past. So I don't see how it makes sense to ask, "what is $S(e^{ipx} ) $ ?". (continued in next comment ...) | |
| Nov 13, 2019 at 17:57 | comment | added | Michael Engelhardt | The less important comment first: The ordering of the rows in $S(p^2)$ of course simply corresponds to a choice of ordering the basis of outgoing states. The choice I specify in the last sentence of my answer is often adopted because it leads to $S^T =S$. You're completely free to reorder the basis, swapping the two rows in my $S(p^2 )$. Given the ordering I chose, I simply read off the first column of $S$ from your wave function; $\tilde{B} $ comes with the outgoing state $e^{-ipx} $ for $x\rightarrow -\infty $, $\tilde{A} $ comes with the outgoing state $e^{ipx} $ for $x\rightarrow \infty $. | |
| Nov 13, 2019 at 6:55 | comment | added | asv | Many thanks. Do you have an explanation why the first column of $S(p^2)$ looks as you wrote? I am looking not for an intuitive explanation like in Schiff's book, but rather more formal one based on the definition of $S$-matrix as composition of Moller operators etc. (And actually I think $\tilde A$ and $\tilde B$ should be interchanged, but this is less important.) | |
| Nov 12, 2019 at 23:45 | history | edited | Michael Engelhardt | CC BY-SA 4.0 |
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| Nov 12, 2019 at 22:04 | history | edited | Michael Engelhardt | CC BY-SA 4.0 |
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| Nov 12, 2019 at 21:55 | history | edited | Michael Engelhardt | CC BY-SA 4.0 |
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| Nov 12, 2019 at 21:01 | history | edited | Michael Engelhardt | CC BY-SA 4.0 |
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| Nov 12, 2019 at 18:59 | history | edited | Michael Engelhardt | CC BY-SA 4.0 |
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| Nov 12, 2019 at 18:32 | history | answered | Michael Engelhardt | CC BY-SA 4.0 |