# Construction of asymptotic states in QFT

#### paweld

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 $$H^{(1)}$$ 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 $$m=\sqrt{P_\mu P^\mu}$$ has some discrete eigenvalues
with m>0) then the subspace $$H^{(1)}$$ 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.

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#### strangerep

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).
For theories without massless particles such as QED, the interaction does not
"turn off" at infinity. The modern way to see this is to examine the behaviour of
the interaction part of the Hamiltonian at timelike infinity (for massive particles)
or lightlike infinity (for massless particles). Kulish and Faddeev gave this method,
but their paper can be a bit hard to follow, so you might prefer the summarized
explanation in this paper:

Horan, Lavelle, McMullan,
"Asymptotic Dynamics in Quantum Field Theory
When does the coupling switch off?"
(Available as arXiv: hep-th/0002206)

(The Kulish-Faddeev reference is given therein.)

If you consult Google Scholar for these authors, you'll find lots of other

The method basically involves a stationary-phase approximation for the
case of asymptotic times. For massive particles, one finds that interactions
do usually disappear in this limit, but not when massless particles are
involved. One finds that (eg) the asymptotic electrons must be constructed
by dressing the bare fields with coherent photon operators (from which it
turns out that physical asymptotic electrons are accompanied by their
Coulomb fields. This is far more satisfactory physically, though perhaps
less easy to work with.

This method is different from the usual textbook approach of only
calculating IR-safe quantities such as the integrated cross-section.
But earlier authors such as Chung and Kibble showed that correct
choice of physical asymptotic states removes IR divergences to all orders
of perturbation theory.

I vaguely remember reading that the usual LSZ method doesn't work properly
with such massless particles, but I can't put my finger on the reference right now.
Maybe in one of the papers by the above authors, if you're interested enough
to search for it.

The "persistent self-interaction" of particles seems to me to be the main origin
of the need of renormalization in the theory.
It's the origin of IR divergences, but UV divergences have other causes.

Beside this the well understanding of asymptotic states it's quite important
from conceptual point of view.
I agree. Check out those references extensively, follow the citations, and
you'll find that vast amounts of work have been done on this subject
by many authors.

#### meopemuk

strangerep,

your recommendations are very good for an advanced student. My feeling is that paweld is just starting to learn the subject. So, I wouldn't recommend to begin with massless particles, for which ultraviolet and infrared divergences mix up and things become very complicated very fast. I would suggest to deal with massive particles first. Then, at least, one half of the problem (IR) is absent.

Eugene.

#### strangerep

My feeling is that paweld is just starting to learn the subject.
Re-reading the original post, I get the feeling paweld at least knows
enough to formulate some nontrivial questions.

So, I wouldn't recommend to begin with massless particles, for which ultraviolet and infrared divergences mix up and things become very complicated very fast. I would suggest to deal with massive particles first. Then, at least, one half of the problem (IR) is absent.
Certainly one should begin with massive particles -- but it's still useful to know
that the "no interaction at asymptotic times" assumption can be backed up by some
not-too-difficult maths applied to the Hamiltonian.

BTW, for any readers wondering why the QED (Coulomb) interaction has a problem,
Ballentine gives a brief explanation of what goes wrong in nonrelativistic
scattering for a Coulomb potential. See the end of section 16.2 in his textbook.

#### A. Neumaier

Could anyone give me a good reference for construction of asymptotic states in QFT.
Besides the LSZ reference given in post #3, the other (more rigorous) source is Haag-Ruelle theory, (look it up in http://scholar.google.com/scholar?q=Haag+Ruelle ).

"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.
Asymptotic particles are an idealization since they ''exist'' only for t-->+-inf, but this is an excellent approximation already at fairly short times before and after a scattering event.

Except asymptotically, the particle concept doesn't make sense in an interacting QFT.
By Haag's theorem, the Hilbert space carrying the physical representation of the Poincare group is not equivalent to the Hilbert space carrying the asymptotic representation.

This means that particles are identifiable as such only in a semiclassical approximation, where they are local lumps of energy with definite quantum numbers.

#### paweld

Thanks for all replies.

I'll present briefly Haag's approach to the problem:

1) Haag "Quantum Field Theories with Composite Particles and Asymptotic Conditions"
http://prola.aps.org/abstract/PR/v112/i2/p669_1
2) Haag "Local quantum physics" (book)

Haag makes the follwing assumptions concerning the one-partticle states in
the theory (the theory is modeled on some Hilbert space $$H$$ ):
i) The mass operator has some discrete eigenvalues. In
other words, the single particle subspace $$H^{(1)} \subset H$$ is not empty.
ii) In the subspace orthogonal to the vacuum the spectrum of M has a lower
bound m > 0 (technical assumption).
iii) For any single particle state with smooth momentum space wave
function $$\phi(p)$$ there exists an almost local operator Q generating it from the
vacuum (operator is local when it's integral of product some filed operators and functions
which decrese faster then any polynomial)

The one-particels states obtained in this way are stable as is the vacuum is (if I understand it corectly).

Using the above assumptions Haag (with the help of Ruell who proved some useful lemmas) managed
to construct asymptotic (many) particles states which transform according to some irreduciable Poincare
group representation (vectors in the Hilbert space $$H$$ which supposedly look like n-particle
states in asymptotic future or past). One-particle states in the asymptotic past and future are the same.

This assumption should be somehow reflected in perturbation approach but unfortunatelly
according to me it's controvesial matter. Why the created virtual particle don't appear in the asymptotic
states. Are there any situation when the S matrix element between one-particle state and many-particle
state is nonzero? Probably decays of particles - are they described by QFT?

The assumption that one-particle states are stable is quite usual one:
At the top of p. 67 in http://www.thphys.uni-heidelberg.de/...hmidt_QFT1.pdf [Broken] there is written:
"Note that states containing less than two particles do not interact, as there is nothing to interact with"
It's also assumed in the book "Relativistic Quantum Fields" by Bjorken and Drell.

The paper "Asymptotic Dynamics in Quantum Field Theory" shed some light on the problem of self-interaction
in asymptotic future or past. Although it tells very littel about states itself (vectors in Hilber space), it
proves that the expectaction value of interaction term of Hamiltonian goes to 0 for asymptotic times for theories
without massless particles and not very complicated interaction Hamiltonian. It means that effectively coupling between
particles long before/after collision might be considered as free. But perturbative approach to QFT still predicts that
the are virtual particle created which are absent in the asymptotic states.

The solution to my problem seems to be the "dressed approach". It gives me what I want
from the theory:

Postulate 10.1 (stability of vacuum and one-particle states) There is
no (self-)interaction in the vacuum and one-particle states, i.e., the time evo-
lution of these states is not aﬀected by interaction and is governed by the non-
interacting Hamiltonian H0 . Mathematically, this means that the interaction
Hamiltonian V is phys. (part of Eugene's book)

I wonder how the interaction hamiltonian look like, at least a few first terms.
What is the current progress in this approach?

Paweł

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#### meopemuk

Postulate 10.1 (stability of vacuum and one-particle states) There is
no (self-)interaction in the vacuum and one-particle states, i.e., the time evo-
lution of these states is not aﬀected by interaction and is governed by the non-
interacting Hamiltonian H0 . Mathematically, this means that the interaction
Hamiltonian V is phys. (part of Eugene's book)

I wonder how the interaction hamiltonian look like, at least a few first terms.
In the case of QED, few low-order interaction terms are shown in subsection 10.2.9 and sections 10.3 and 14.2.

What is the current progress in this approach?
Not many people are working in this field. You can google names like M. I. Shirokov, A. V. Shebeko, V. Yu. Korda for recent papers.

Eugene.

#### DrDu

I just want to remark that the asymptotic one-particle states of an interacting theory are in no way the same as the one-particle states of a non-interacting theory. That's why you still need virtual particles to describe the propagation of the incoming particles in a perturbational treatment.
Another example are bound states. E.g. hydrogen atoms or even molecules are stable asymptotic states of an interacting electron proton system. Nevertheless they have little resemblance with free protons or electrons which are asymptotic states in a non-interacting electron proton system.

#### meopemuk

I just want to remark that the asymptotic one-particle states of an interacting theory are in no way the same as the one-particle states of a non-interacting theory. That's why you still need virtual particles to describe the propagation of the incoming particles in a perturbational treatment.
This might be true in the traditional approach to QFT, but in the "dressed particle" approach both non-interacting and interacting theories have the same zero-particle and one-particle states. The idea is that single particle has nothing to interact with, so interaction can have no effect on its properties.

Eugene.

#### DrDu

How does this work in the "dressed particle" approach?
I'd thought that the definition of a single particle solutions (as a discrete eigenstate of the energy momentum operator) of which the asymptotic state space is constructed is unambiguous in both interacting and non-interacting theories and should be reproduced in any theory.

#### A. Neumaier

Haag makes the following assumptions concerning the one-particle states [...]

The one-particle states obtained in this way are stable as is the vacuum is (if I understand it correctly).

Using the above assumptions Haag (with the help of Ruelle who proved some useful lemmas) managed
to construct asymptotic (many) particles states which transform according to some irreducible Poincare
group representation (vectors in the Hilbert space $$H$$ which supposedly look like n-particle
states in asymptotic future or past). One-particle states in the asymptotic past and future are the same.
Yes; correctly understood.
This assumption should be somehow reflected in perturbation approach but unfortunatelly
according to me it's controversial matter. Why the created virtual particle don't appear in the asymptotic
states.
Nothing is controversial here. This is essentially the same as one is accustomed to see in a nonrelativistic theory. The physical (elementary or bound) particle states are represented in an asymptotic Fock space; while virtual particles are not representable as states in the physical Hilbert space.
decays of particles - are they described by QFT?
Unstable particles are very different from virtual particles. Unstable particles are states with a continuum spectrum highly peaked around their mass, with a width related to their lifetime. In QFT, they appear as poles of the analytically continued S-matrix.

In contrast, virtual particles have no associated states.
But perturbative approach to QFT still predicts that
the are virtual particle created which are absent in the asymptotic states.
No. perturbative QFT predicts S-matrix elements between states of real particles.
The virtual particles appear only as lines drawn on paper, representing integral kernels for the propagator of the appropriate field.
The solution to my problem seems to be the "dressed approach". It gives me what I want
from the theory:

Postulate 10.1 (stability of vacuum and one-particle states) There is
no (self-)interaction in the vacuum and one-particle states, i.e., the time evo-
lution of these states is not aﬀected by interaction and is governed by the non-
interacting Hamiltonian H0 . Mathematically, this means that the interaction
Hamiltonian V is phys. (part of Eugene's book)
Well, it gives the mess described in his book, with a completely unresolved infrared problem....

The Wightman form of the dynamics, described in https://www.physicsforums.com/showthread.php?t=476412 is much neater, and actually used a lot by people deriving on kinetic equations from QFT.

#### A. Neumaier

I just want to remark that the asymptotic one-particle states of an interacting theory are in no way the same as the one-particle states of a non-interacting theory. That's why you still need virtual particles to describe the propagation of the incoming particles in a perturbational treatment.
Another example are bound states. E.g. hydrogen atoms or even molecules are stable asymptotic states of an interacting electron proton system. Nevertheless they have little resemblance with free protons or electrons which are asymptotic states in a non-interacting electron proton system.
The asymptotic space of every nonrelativistic many-particle system, and of every relativistic QFT without asymptotic massless states consists of a Fock space that has creation operators for each bound state of the system. For example, for QCD, you have all mesons and baryons. For QED with protons, due to the infrared problem, the asymptotic space is not a Fock space, but the asymptotic particles are the electron, the proton, and the hydrogen atom, each dressed by their e/m field.

This has nothing to do with virtual particles. The asymptotic space consists of real particles, and in fact _defines_ the observable particles. But at finite times (which in practice means during collisions), the particle picture (being asymptotic) is inadequate.

#### A. Neumaier

How does this work in the "dressed particle" approach?
The essence is that one performs a formally unitary transform (with infinite matrix elements) to match the space at t=0 with that at t=+-inf, and reexpresses the transformed Hamiltonian in terms of c/a operators on the asymptotic Fock space. The computation of the coefficients involves the cancellation of the divergences and constitutes the renormalization of the theory.

Things break down for QED since photons are massless and the asymptotic space is not Fock, causing an infrared problem. This means that the resulting Hamiltonian is formally Hermitian but not self-adjoint, and the resulting domain problems cause IR divergences in higher order perturbation theory.

Things would also break down for QCD, but here because of confinement: The asymptotic space has no quark degrees of freedom, so it cannot match the space at t=0.

#### DrDu

This has nothing to do with virtual particles. The asymptotic space consists of real particles, and in fact _defines_ the observable particles. But at finite times (which in practice means during collisions), the particle picture (being asymptotic) is inadequate.
I meant that the virtual particles (as internal lines in Feynman diagrams) come into play when trying to describe formally the propagators for the asymtotic particles of an interacting field in terms of the particles of a free field.

#### DrDu

For QED with protons, due to the infrared problem, the asymptotic space is not a Fock space, but the asymptotic particles are the electron, the proton, and the hydrogen atom, each dressed by their e/m field.
I don't quite see why the appearance of bound states like hydrogen depends on the photon being massless. Wouldn't e.g. a Yukawa type interaction not also lead to bound states in the asymptotic region?

#### A. Neumaier

I meant that the virtual particles (as internal lines in Feynman diagrams) come into play when trying to describe formally the propagators for the asymptotic particles of an interacting field in terms of the particles of a free field.
The propagators for the asymptotic particles are themselves generalized free fields.

Assuming the asymptotic particle is related to an elementary field, one may formally expand the S-matrix elements in terms of the propagators of the bare fields appearing in the defining Lagrangian density. One gets an infinite series, in which each term is the sum of a number of ill-defined integrals represented by Feynman diagrams involving virtual particles. Thus the relationship to virtual particles is quite remote -- it is in the ill-defined part that must be renormalized away by infinite resummations.

The renormalization turns the bare propagators into the propagators for the asymptotic particles. The renormalized rules for the S-matrix only contain such renormalized propagators.

If the asymptotic particle is a hydrogen atom, this textbook way of proceeding no longer works.

#### A. Neumaier

I don't quite see why the appearance of bound states like hydrogen depends on the photon being massless. Wouldn't e.g. a Yukawa type interaction not also lead to bound states in the asymptotic region?
In massive QED (or in a massive Yukawa theory), one indeed has an asymptotic Fock space, with creators for electrons, protons, and hydrogen atoms.

It is not the bound state that depends on masslessness but the lack of Fockness.
If there are massless particles the long-range force generated by their fields does not permit to separate the massless fields from the asymptotic particle states. The particles degenerate to infraparticles, and these are not described by the usual Fock space.

#### A. Neumaier

I don't quite see why the appearance of bound states like hydrogen depends on the photon being massless. Wouldn't e.g. a Yukawa type interaction not also lead to bound states in the asymptotic region?
In massive QED (or in a massive Yukawa theory), one indeed has an asymptotic Fock space, with creators for electrons, protons, and hydrogen atoms.

It is not the bound state that depends on masslessness but the lack of Fock-ness. If there are massless particles the long-range force generated by their fields does not permit to separate the massless fields from the asymptotic particle states. The particles degenerate to infraparticles, and these are not described by the usual Fock space.

#### meopemuk

How does this work in the "dressed particle" approach?
The dressed particle approach forbids interaction operators that have non-trivial action on the vacuum and 1-particle states. If this restriction is satisfied, then there should be no differences between particles with interaction turned on and off. A reasonable interaction operator has a non-trivial action only if there are two or more particles present.

As Pawel mentioned correctly, all quantum field theories violate the above restriction. This is the reason why "physical" particles are different from "bare" particles in QFT. For the same reason "physical" vacuum is different from the "bare" vacuum. One can also trace the occurence of ultraviolet divergences to the same source. So, strictly speaking, QFT is not a "dressed particle" theory.

Luckily, it is possible to bring any QFT theory (I leave aside theories with massless particles for which some technical problems remain unresolved) to the "dressed particle" form by making a unitary "dressing" transformation. This transformation absorbs all ultraviolet infinities and as a result you have a finite well-defined Hamiltonian that can be used in routine quantum mechanical calculations without worries about asymptotic states, renormalizations, etc.

Eugene.

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#### A. Neumaier

Luckily, it is possible to bring any QFT theory (I leave aside theories with massless particles for which some technical problems remain unresolved) to the "dressed particle" form by making a unitary "dressing" transformation. This transformation absorbs all ultraviolet infinities and as a result you have a finite well-defined Hamiltonian that can be used in routine quantum mechanical calculations without worries about asymptotic states, renormalizations, etc.
The UV worries about renormalizations are instead in the ''unitary'' dressing transformation, which isn't unitary at all, but has infinite coefficients that must be rendered finite by a similar renormalization as in other approaches to QFT.

Once one has the Hamiltonian, there are no UV worries left, but the IR worries still exist. In addition, there are homebred worries about causality because of the lack of manifest covariance.

The Wightman approach to QFT also proceeds in two stages: In the first stage, one constructs the Wightman functions; this is the formal part where infinities must be cancelled. Once one has the Wightman functions, there are neither UV worries nor IR worries left. In the second stage, one constructs the Hilbert space and the Hamiltonian from the Wightman functions, along the lines discussed in the early part of the thread https://www.physicsforums.com/showthread.php?t=476412 (up to post #28).

The advantage of the Wightman approach is that the final Hamiltonian is much nicer to work with, and that everything is manifestly covariant. This is why the dressing approach (invented in 1963 by Faddeev) fell out of favor with the mainstream particle physicists.

#### meopemuk

Once one has the Hamiltonian, there are no UV worries left, but the IR worries still exist.
It is true that IR sector is not well-developed in the "dressed particle" approach. However, I don't think this is an unsurmoutable obstacle. In my opinion, this is just a technical problem, which has not been resolved yet for the lack of sufficient mathematical efforts.

In addition, there are homebred worries about causality because of the lack of manifest covariance.
There is a proof that the "dressed" version of QFT remains relativistically invariant (=different inertial observers are equivalent). Moreover, there are no problems with causality:
The cause always precedes the effect in all frames of reference.

It is true, the "dressed" version lacks "manifest covariance" (=simple linear universal action of boost transformations). But this is not a "bug", this is a "feature". In my opinion, the "manifest covariance" is not a required physical principle, but just a wishful thinking.

Eugene.

#### paweld

I'm not sure if it's posssible to describe all observed physical phenomena using dressed approach. Could one comupute a liftime of particles (e.g. muon) if there is an assumption that particles are stable? Probably it's more subtle and I don't understand sth.

#### meopemuk

I'm not sure if it's posssible to describe all observed physical phenomena using dressed approach. Could one comupute a liftime of particles (e.g. muon) if there is an assumption that particles are stable? Probably it's more subtle and I don't understand sth.
Sure, this can be done. See chapter 14 in the book.

Eugene.