The Hilbert space of the theory $\cal H$ can be viewed **simultaneously** as a Fock space in **two different manners**:
$${\cal H} = {\cal F}_{symm}(K_{in}) = {\cal F}_{symm}(K_{out})\:.$$ Where $K_{in}$ and $K_{out}$ are the one-particle space of ingoing and outgoing free particles and I am assuming that the particles are Bosons for the sake of simplicity. Above:
$${\cal F}_{symm}(K) = {\mathbb C}\oplus K \oplus (K\otimes K)_{symm} \oplus (K\otimes K\otimes K)_{symm}\oplus \cdots $$

The point is now that, in general $K_{in}$ include not only vectors of $K_{out}$, but even of $(K_{out}\otimes K_{out})_{symm}$, $(K_{out}\otimes K_{out}\otimes K_{out})_{symm}$, and so on. In other words it is **false** that, for instance $K_{in} \not\perp (K_{out}\otimes \cdots (k \:times)\cdots \otimes K_{out})_{symm}$ for $k>1$.

This is the mathematical translation of the fact that an ingoing free particle, in the asymptotic past (when the interactions are switched off), due to interactions at finite time, may give rise to many free particles in the asymptotic future (when the interactions are again switched off). In general the number of particles is not preserved due to the interactions passing from $t=-\infty$ to $t=+\infty$.

In general, for every value of $n$ and $m$:

$(K_{in}\otimes \cdots (n \:times)\cdots \otimes K_{in})_{symm} \not\perp (K_{out}\otimes \cdots (m \:times)\cdots \otimes K_{out})_{symm}$.

These relations can be written using vectors:

$$\langle\psi^{(out)}_{1}\cdots \psi^{(out)}_{m}| \psi^{(in)}_{1}\cdots \psi^{(in)}_{n}\rangle \neq 0 \quad \mbox{for generic $n\neq m$,}$$

where, for instance

$$ |\psi^{(out)}_{1}\cdots \psi^{(out)}_{m}\rangle \in (K_{out}\otimes \cdots (m \:times)\cdots \otimes K_{out})_{symm}$$

is the generic symmetrised state made of $m$ outgoing particles with single states
$\psi^{(out)}_{1},\cdots, \psi^{(out)}_{m}\in K_{out}$, not necessarily pairwise different.

This post imported from StackExchange Physics at 2014-04-24 02:32 (UCT), posted by SE-user V. Moretti