- #1
UnreliableObserver
- 4
- 4
In scattering theory, the quantity of interest is the amplitude for the system—initially prepared as a collection of (approximate) momentum eigenstates—to evolve into some other collection of momentum eigenstates. For example, for scattering, the amplitude we're interested in is where is the time at which we prepare the state, is the time at which we observe the result of the scattering process, and is the time evolution operator.
We can imagine finding operators capable of creating such states from the vacuum, in which case we could write as or, in the Heisenberg picture, where . I hope this is all rather uncontroversial so far.
Now, it can be shown (see e.g. Srednicki QFT) that the operators that create single-particle momentum eigenstates from the vacuum are where
However after this point things start to fall apart for me. According to everything I've read, these operators are used to create the "initial" and "final" states and , with which the corresponding S-matrix element is given as . But how can this be right? If these operators do indeed create single-particle momentum eigenstates, then isn't this inner product very trivial? If we refer back to the Heisenberg picture expression , we are free to write it as , but these states are clearly not products of momentum eigenstates, but rather whatever complicated mess results from the action of on the vacuum (as they must be to give a non-trivial amplitude). The products of eigenstates are those in , so shouldn't the operators instead assume the role of those in ?
We can imagine finding operators
Now, it can be shown (see e.g. Srednicki QFT) that the operators that create single-particle momentum eigenstates from the vacuum are
Last edited: