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I'm trying to follow chapter 4 about interacting fields in Peskin and Schröder. They define the S-matrix by $$_{\mathrm{out}}\langle p_1 p_2 | k_a k_b\rangle_{\mathrm {in}} = \langle p_1 p_2 | S | k_a k_b\rangle,$$ where $S = \lim_{T\rightarrow \infty}e^{-i2HT}$. The states on the right hand side of the equation are eigenstate of the momentum operator. Furthermore, they are said to be eigenstates of $H$ as well (in 4.6 below eq. 4.87).

But if the states are eigenstates of $H$, the above scalar product becomes very trivial right? So what's going on here?

Charlie
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user2224350
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2 Answers2

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This answer is essentially a citation of this answer, given by Arnold Neumaier.

The resolution is essentially that while the asymptotic single particle states in the full hamiltonian $|p \rangle $ (which is what is what I'm guessing is what Peskin meant) are eigenstates of the full hamiltonian, the product states of those asymptotic states (the only ones that have non-trivial scattering) $|p_1, p_2 \rangle$ are not eigenstates of the full hamiltonian $H$, and so we would expect

$$_{in}\langle p_1, p_2 \cdots | k_A, k_B \rangle_{out} = \lim_{t\to \infty} \langle p_1, p_2 \cdots| e^{-iH2t} | k_A, k_B \rangle $$

to have a nontrivial overlap.

InertialObserver
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So I figured that the in and out states are indeed eigenstates of the full Hamiltonian. However, they can still have a non-trivial overlap since the energy-eigenstates of the full Hamiltonian obviously have a high degeneracy (eg with multiple particles you have a lot of possible momenta combinations that have the same total momentum and energy).

More precisely, the in- and out states are scattering states of the full Hamiltonian that fulfill certain boundary conditions such that they "match" free multi-particle states (but with physical mass due to self-interaction) long before and long after the scattering respectively, see Lippmann-Schwinger eq.

A nice way to gain some intuition is to look at potential scattering in non-relativistic quantum mechanics. For example chapter 32.1 Potentialstreuung illustrates nicely how in-states can be interpreted in terms of the scattering of an incoming wavepacket (the text only considers in-states so they're never explicitly called like that). Unfortunately, the text book is in German, but maybe anyone knows a similar text derivation in english..?

user2224350
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