Questions from Srednicki's Introduction to Interacting Field Theory using the LSZ Formula

Question

I have been reading through the chapter on the LSZ Reduction Formula from Srednicki's Quantum Field Theory, and I have a few questions about which I'm sort of confused. The questions are referenced from the following copy of the textbook: http://web.physics.ucsb.edu/~mark/qft.html

1. On page 49, it is mentioned that if we time evolve a wave packet, it spreads out in time; thus, the wave packet (particle) is localized far from the origin (where it was initially prepared) as $t \longrightarrow \pm \; \infty$. I'm not sure how to interpret the $t = - \; \infty$ bit. As time progress from 0, doesn't it move to $\infty$? Does the other argument have something to do with time-reversal. I'm quite confused on this front. I probably don't get something elementary, I suppose,

2. As defined in $(5.8)$ and $(5.9)$, how does $\langle f \rvert i \rangle$ describe the scattering amplitude in the interacting theory? If I'm correct, in a free field theory for spin-0 bosons the excitations of the field describe particles with well-defined three-momentum (by dirac delta functions). The particles don't interact with other particles. So my question is, how does the notion of scattering come about in an interacting theory; what is the physical process governing it? As it stands, Srednicki's analysis simply seems to imply the annihilation and creation of particles (at $\pm \; \infty$), which technically is allowed for in a free field theory?

3. pp. 51-52 state: "the next (second) excited state is that of two particles. These two particles could form a bound state with energy less than $2m$. but, to keep things simple, let us assume that there are no such bound states. Then the lowest possible energy of a two-particle state is $2m$. However, a two-particle state with zero total three-momentum can have any energy above $2m$, because the two particles could have some relative momentum that contributes to their total energy. How and why can a two-particle state have an energy of less than or greater than $2m$ (in an interacting theory). Again, if I'm correct, in a free field theory, the energy is exactly $2m$ for a two-particle state.

In summary, I seem to be confused about some aspects of Srednicki's introduction of an interacting field theory in the context of the LSZ Reduction Formula. It'd be great if someone could help alleviate some of these concerns.

Answer 1

1. Your confusion is elementary. It is specified that the wavepackets have a well-defined momentum, and that at $t=0$ they are located at the origin. Now since the wave packets have a well-defined momentum, they move at a constant velocity. An object which is at the origin at $t=0$ and moves at a constant velocity will be very far from the origin as $t \to \infty$. Going backwards in time, on object at the origin at $t=0$ must have come from very far away.

2. You are correct that in the non-interacting theory, the momentum states do not interact with each other. However in the interacting theory, they do interact. An analogy would be to think of billiard balls on a very large table. We start out at some very early time $t_i \to -\infty$ and place ("create") two billiard balls far apart from each other and roll them towards the origin so that they will be in the same place at $t=0$.

   Now if we imagine non-interacting billiard balls (think of "ghost" billiard balls), then they balls go right through each other and continue on to $t\to \infty$ with the same momentum. However, if we now think of real, physical billiard balls, they will collide and hte final momentum will not be equal to the initial momentum.

   Classicaly the two billiard balls will have a well-defined final momentum. In Quantum, the final state at $t_f \to -\infty$ will be a super-position of many momenta states, and you can find the probability of measuring a given final momentum state $|f\rangle$ by calculating the overlap of the momentum state $|f\rangle$ with the time evolved initial state $|i\rangle$.

3. Let's start with the free field case. Here the energy is not necessarily $2m$. For example, consider two billiard balls moving towards each other. Their energy is $2m+T$, where $T$ is the kinetic energy associated with the billiard balls' motion. This energy is greater than the mass energy $2m$. But how can the total energy be less than $2m$? This requires interactions. Consider the earth-sun system. The state where the earth and sun are infinitely separated and stationary corresponds to the state where the energy is equal to the mass energy. However to go from our current orbiting configuration to that inifinitely separated configuration, you would need to add energy to the earth to get it to escape the suns gravitational attraction. Thus the current energy of the earth sun system must be less than the mass energy.