- #1
paweld
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Could anyone give me a good reference for construction of asymptotic states in QFT.
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.
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.