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Usually one assumes that long before and long after collision particles almost don't interact

because they're spatially separated. This statement is based on the assumption that

particles might be represented as wavepackets essentially localized in finite region of

space and particles are stable objects (i.e. - there is no self-interaction). Wherase

the first argument is well justified, the second according to me is flawed - the number

of particles is usually not conserved if the evolution is described by the hamiltonians

which are local, Poincare-invariant and not-free. It means that there should exist no limit

states as time goes to plus/minus infinity (it's imposible to turn off the self interaction).

On the other hand if one defines supspace [tex]H^{(1)}[/tex] of one-particles states as

an union of eignespace of mass operator with eignevalue m>0 (the one particles states

exist in the theory iff [tex] m=\sqrt{P_\mu P^\mu} [/tex] has some discrete eigenvalues

with m>0) then the subspace [tex]H^{(1)}[/tex] should be conserved in time (Hamiltonian

commute with the mass operator). Scheme based on this assumption is presented in

"Local quantum physics" by R. Haag (p. 88). He constructs asymptotic states without using

Moller operators (which don't exist in this context) and spliting hamiltonian into

interaction and free part. This approach seems to be very promising however at the first

sight it's not consistent with the picture of particle which arises from perturbative approach.

Feynmann diagrams indicates that something like asymptotic particle can exist only

effectivelly (electron emits some virtual photons, absorbs some but if one adds all

particles emitted and absorbed in the proces of self-interaction one would obtain

something like a free particle in the asymptotic future or past). On the other hand Haag

claims according to me that the notion of asymptotic particle might be more fundamental.

The "persisten self-interaction" of particles seems to me to be the main origin of the need

of renormalization in the theory. Beside this the well understanding of asymptotic states it's quite

important form conceptual point of view.