Confusion about Scattering in Quantum Electrodynamics

[gentzen]

3. I have no idea what "particles are in drag" is supposed to mean, but it sounds to me as a manifestation of a force (see 1. above).

I read it as "dressed particles"/"non-naked particles".
By the way, I guess hutchphd's comment is tongue in cheek, at least partly. I guess he is more experienced with QFT than me, even so of course he is far below your level, or that of vanhees71 or A. Neumaier.

 

[vanhees71]

1. There are forces, but in quantum physics they do not manifest in the same way as they do in classical physics. See e.g. the Ehrenfest theorem.

It depends on, how you define "forces". I'd say there are no forces in relativistic physics, because there are only local interactions through fields. For me a "force" is a Newtonian action-at-a-distance concept, but that's a matter of how you define words. Physicists are often pretty sloppy with that ;-)).

2. There is finite time, all experiments are done during finite times. But these times are long compared to typical times during which the scattering interaction is significant, so calculations are simpler when the long time is approximated with infinite time.

That's true. It's, however, a matter of time scales. To have a particle interpretation you need to be sufficiently close to a situation, where the fields become asymptotically free. "Interpolating fields" (in the Heisenberg picture) generally don't admit a particle interpretation.

3. I have no idea what "particles are in drag" is supposed to mean, but it sounds to me as a manifestation of a force (see 1. above).

I guess it's meant "interpolating field".

First one needs to properly learn QM. The crucial chapters are
a) axioms of QM, the role of measurement

Foundational issues don't play much of a role as far as the hard scientific facts are concerned. A measurement is done in the lab with detectors, not on the desk of the theoretician.

b) time evolution in Schrodinger, Heisenberg and Dirac (interaction) picture
c) quantum scattering theory
After that, QFT should be easy at the conceptual level, while the new difficulties are mostly technical.

I think the trick to really understand relativistic QFT is to really understand the representation of the Poincare group in terms of local quantum fields. For the intuitive concepts the best book I know is

S. Coleman, Lectures of Sidney Coleman on Quantum Field
Theory, World Scientific Publishing Co. Pte. Ltd., Hackensack
(2018), https://doi.org/10.1142/9371

Another great source is

R. U. Sexl and H. K. Urbantke, Relativity, Groups, Particles,
Springer, Wien (2001).

For all the mathematical details about the representation theory of the Poincare group in all generality of fields with arbitrary spin, see

S. Weinberg, The Quantum Theory of Fields, vol. 1,
Cambridge University Press (1995).

 

[vanhees71]

I read it as "dressed particles"/"non-naked particles".
By the way, I guess hutchphd's comment is tongue in cheek, at least partly. I guess he is more experienced with QFT than me, even so of course he is far below your level, or that of vanhees71 or A. Neumaier.

I would say "dressed particles" are rather the "true asymptotic free states" also known as "infra particles", i.e., The naked free particle + the "cloud of soft photons" around them (for QED).

 

[Demystifier]

It depends on, how you define "forces". I'd say there are no forces in relativistic physics, because there are only local interactions through fields. For me a "force" is a Newtonian action-at-a-distance concept, but that's a matter of how you define words. Physicists are often pretty sloppy with that ;-)).

Sure. What I mean by force in relativistic field theory is best explained on the example of Klein-Gordon equation $\ddot{\phi}-\nabla^2\phi+m^2\phi=0$. It can be written as
$$\ddot{\phi}=F[\phi]$$
where
$$F[\phi]\equiv \nabla^2\phi - m^2\phi$$
is naturally interpreted as the "force".

 

[vanhees71]

This is not a force on a particle. I've never seen anybody calling this a force.

 

[Demystifier]

This is not a force on a particle. I've never seen anybody calling this a force.

It's a force on the field. If you think of field as a continuum limit originating from a lattice of atoms (emergent field from a condensed matter point of view), then the force on the field originates from forces on particles. Or even without such a condensed-matter point of view, thinking of it as force is helpful to understand how macroscopic forces on macroscopic bodies emerge from field theory. In particular, I have used such a point of view in my analysis of Casimir force: https://arxiv.org/abs/1605.04143

 

[hutchphd]

After that, QFT should be easy at the conceptual level, while the new difficulties are mostly technical.

Well I think I know (or at least once knew!) the three prereqs you indicate but I never had a formal course in QFT and my (admittedly half-hearted) attempts to learn it have not caught fire. Could be I'm juuust tooo daamned old.

 

[Demystifier]

Well I think I know (or at least once knew!) the three prereqs you indicate but I never had a formal course in QFT and my (admittedly half-hearted) attempts to learn it have not caught fire. Could be I'm juuust tooo daamned old.

May I ask, how old?
BTW I'm 53, look like 43, and feel like 33, for the case someone wonders.

 

[hutchphd]

May I ask, how old?

I am 71.
I look just about 71
Lately I feel... about ...71

But last year I felt about 61

 

[LittleSchwinger]

Very nice response, thank you for sharing! Do you have any suggestions for references that you think someone coming more from the physics side of QFT, as opposed to the math side, could read as an introduction to these finite time problems? But I'd be happy to know of more mathematical introductions also.

A very nice paper is:

Buchholz, Detlev. (1986). Gauss' law and the infraparticle problem. Physics Letters B. 174. 331–334. 10.1016/0370-2693(86)91110-X.

I recommend this because it is short and close to typical QFT language as opposed to mathematical physics language. Buchholz gives an accessible proof of how the electron (or any charged particle) has no precise mass value.

Toward then end he gives a brief account of QED's complicated superselection structure, leading to a spontaneous breakdown of Lorentz symmetry . Ultimately this leads the non-unitary time evolution QED has at the nonperturbative level, although this is usually discussed in far more technical papers.

 

[LittleSchwinger]

There is an approach called "dressed particle" QF....

Eugene.

Wow, we get to meet celebs on this forum.

I just wanted to say I loved the account of Quantum Logic in your textbook. Several points that many people miss about the relations of logics in general. The book in general is wonderful.

 

[Demystifier]

A little googling led me to this:
https://academic.oup.com/ptp/article/86/1/269/1867326

Title: Quantum Field Theory with Finite Time Interval: Application to QED

Abstract:
Diagrammatical expansion of the quantum field theory with the finite time interval is discussed by evaluating the time evolution kernel. The coherent state representation is adopted which is the most convenient formalism for this purpose. It is applied to quantum electrodynamics (QED) to derive the Feynman rule for the kernel. The rule for the matrix element of any operator is also given. Our method can also be a systematic approach to the time dependent perturbation theory. As an example, a full order formula of the transition amplitude between the number states is given in terms of the connected Green's functions.

 

[Demystifier]

Also something new https://inspirehep.net/literature/1788843

The point is, it's not that finite time effects in QFT are not studied at all. It's not a mainstream, but some people do it. Mathematically, it should not be much different from studying finite temperature effects.

 

[LittleSchwinger]

Title: Quantum Field Theory with Finite Time Interval: Application to QED

Note that this approach does not produce an actual Hamiltonian evolution. The "unitary" evolution produced does not obey:
$U(t_{3},t_{1}) = U(t_{3},t_{2})U(t_{2},t_{1})$

Hence it's an approximation of the true time evolution breaking some of the true evolution's properties. This goes for many other such finite time papers. Very commonly momenta are treated in the non-relativistic approximation.

So just to note for anybody trying to read papers on this topic if the paper deals with:
1. Unitary time operators
2. Particle states

It is a severe approximation to the true finite time evolution which has been proven to not involve either. It's contractive Markovian on non-particle states. And by "non-particle" these states are not even an infinite sum/superposition of particle states. They're in what is called a separate folium of states.

 

[WernerQH]

Note that this approach does not produce an actual Hamiltonian evolution. The "unitary" evolution produced does not obey:
$U(t_{3},t_{1}) = U(t_{3},t_{2})U(t_{2},t_{1})$

Hence it's an approximation of the true time evolution breaking some of the true evolution's properties.

Does "true" evolution mean unitary evolution? But you can't describe the real world with unitary evolution alone. When you apply quantum theory, at some point you have to introduce "measurements" and the Born rule.

 

[meopemuk]

@LittleSchwinger: Thank you for the kind words about my work.

Title: Quantum Field Theory with Finite Time Interval: Application to QED

The authors of this work used the "standard" Lagrangian of QED in equation right after eq. (2.56). I have two major issues with their approach:

1. This is the Lagrangian before renormalization. At page 295 the authors expressed a belief that renormalization can be added to their finite-time description. But I doubt that it can be done, because the whole point of renormalization is to fix time evolution in the infinite time interval (a.k.a. the S-matrix) by adding infinite counterterms to the Hamiltonian, thus destroying the possibility to obtain a realistic time evolution at finite time intervals.

2. This Lagrangian/Hamiltonian is formulated in the "bare" particle representation. But time evolution of bare particles (even if they are alone) is meaningless, because they are not eigenstates of the total mass operator. If you start with a single bare electron at time t=0, then at the next time instant the electron will "dress" itself with a coat of virtual photons. Real life electrons don't do that.

The "dressed particle" theory fixes both these problems. It rewrites the Hamiltonian of the renormalized QED in terms of physical or dressed particles (e.g., the bare electron plus its virtual coat). The new Hamiltonian appears to be finite and can be used as the generator of time evolution for physical multiparticle states. The S-matrix obtained with this Hamiltonian coincides with the S-matrix of renormalized QED in all perturbation orders.

Eugene

 

[Demystifier]

the whole point of renormalization is to fix time evolution in the infinite time interval

I've seen various views of renormalization, but I have never seen a claim before that this is the whole point of renormalization. In particular, lattice QCD also uses renormalization, but time (described on a finite 4D lattice) is finite.

 

[meopemuk]

I've seen various views of renormalization, but I have never seen a claim before that this is the whole point of renormalization. In particular, lattice QCD also uses renormalization, but time (described on a finite 4D lattice) is finite.

I was talking about renormalization in the context of removing UV divergences from the S-matrix of QED as explained by Tomonaga, Schwinger, Feynman and Dyson.

Eugene.

 

[Demystifier]

I was talking about renormalization in the context of removing UV divergences from the S-matrix of QED as explained by Tomonaga, Schwinger, Feynman and Dyson.

Eugene.

But infinite time gives rise to IR divergences, not UV divergences.

 

[meopemuk]

But infinite time gives rise to IR divergences, not UV divergences.

S-matrix is a mapping from (free) states in the remote past to (free) states in the distant future. So, basically S-matrix can be regarded as a result of integrating the time evolution in an infinite time interval.

The Hamiltonian of renormalized QED contains divergent counterterms. So, dynamics in finite time intervals cannot be obtained, because the counterterms produce infinite frequencies. But all infinities cancel out happily when the dynamics is averaged over infinite time interval in the S-matrix.

Eugene.

 

[AndreasC]

A bit more to the points is, I think, the idea of "infraparticles" in QED, i.e., to use the "true asymptotic free states" rather than naive "plane waves". The point is that in QED the photon is massless, and the asymptotic free states are in fact not plane waves due to the long-rangedness of the em. interaction (aka the masslessness of the photon). That solves the IR problems in a physical way. A very pedagogic paper about this is

P. Kulish and L. Faddeev, Asymptotic conditions and infrared
divergences in quantum electrodynamics, Theor. Math. Phys.
4, 745 (1970), https://doi.org/10.1007/BF01066485

or the series of papers by Kibble

T. W. B. Kibble, Coherent Soft-Photon States and Infrared
Divergences. I. Classical Currents, Jour. Math. Phys. 9, 315
(1968), https://doi.org/10.1063/1.1664582

T. W. B. Kibble, Coherent Soft-Photon States and Infrared
Divergences. II. Mass-Shell Singularities of Green’s Functions,
Phys. Rev. 173, 1527 (1968),
https://doi.org/10.1103/PhysRev.173.1527.

Kibble:1969ep[Kib68b]T. W. B. Kibble, Coherent Soft-Photon States and Infrared
Divergences. III. Asymptotic States and Reduction Formulas,
Phys. Rev. 174, 1882 (1968),
https://doi.org/10.1103/PhysRev.174.1882.

Kibble:1969kd [Kib68c] T. W. B. Kibble, Coherent Soft-Photon States and Infrared
Divergences. IV. The Scattering Operator, Phys. Rev. 175,
1624 (1968), https://doi.org/10.1103/PhysRev.175.1624.

I learned about it from this nice relatively recent paper by Holmfrodur Hannesdottir and Matthew Schwartz: https://arxiv.org/abs/1911.06821

 

[A. Neumaier]

I am confused about Scattering in QED [...] why only cross sections and decay rates are computed. [...] does anyone calculate the actual evolution of the field states or operators themselves like how the particles and fields evolve throughout a scattering process not just asymptotic times.

This is a very reasonable question that I had puzzled over for a long time before understanding what actually happens.

In textbook QED one starts with a Fock space and finds out that all quantities of interest are ill-defined without renormalization. By Haag's theorem, this is necessarily so for relativistic QFTs based on the interaction picture (which is needed for perturbation theory) - the latter simply does not exist. To circumvent Haag's theorem while retaining Fock space one has to introduce renormalization, which works smoothly only at the level of the S-matrix.
[added after discussion:] However, Haag's theorem only implies that the Fock space concept is inadequate to capture interacting QFT. Indeed, there are many rigorous (non-Fock) constructions of interacting QFTs in 2 and 3 spacetime dimensions, in spite of the validity of Haag's theorem there.

There is a widespread view that in relativistic QFTs, only the asymptotic descriptions (i.e., the S-matrix) makes quantitative sense, and can be computed using renormalization, together with IR regularization via cumulative cross sections or coherent states.
The reason is the difficulty to give QED (and other physically relevant relativistic QFTs) in 4 spacetime dimensions a rigorous mathematical basis: In a logically impeccable sense (in principle being able to produce model predictions to infinite accuracy), not a a single interacting relativistic QFT in 4 spacetime dimensions is known. One of the Clay Millenium Problems asks for a construction for 4D Yang-Mills, which is though to be the simplest relativistic QFT in 4 spacetime dimensions potentially in reach by current methods.

On the other hand, suppose a mathematical construction of QED (or any relativistic QFT in 4 spacetime dimensions) exists. This would imply a realization of the Wightman axioms. Any such realization produces a Hilbert space (by Haag's theorem not a Fock space) with a unitary representation of the Poincare group, and the generator $H=P_0$ of the time translation acts as a Hamiltonian producing the temporal evolution of the states.

Currently only free fields (and their quasi-free cousons) - such as QED with zero electron charge - are known to realize the Wightman axioms in 4 spacetime dimensions. Their states indeed have an explicit temporal evolution, given by the traditional free Hamiltonian on Fock space.

Thus the reason why textbooks are silent about the temporal evolution is because theory is not developed enough to be able to say more than trivialities about it.

 

[Demystifier]

Thus the reason why textbooks are silent about the temporal evolution is because theory is not developed enough to be able to say more than trivialities about it.

Yes, but theorists would work harder to better develop it, if experimentalists were able to measure it.

 

[vanhees71]

Thus the reason why textbooks are silent about the temporal evolution is because theory is not developed enough to be able to say more than trivialities about it.

It's true that there are many mathematical problems unsolved in QFT. However, one must also say that in the vacuum QFT the calculation of the time-evolution (the initial-value problem) is pretty useless since there are no observables it depicts.

Things change, when it comes to the question of many-body physics. In relativistic heavy-ion collisions in fact we have to calculate the time evolution to a certain extent. For realistic simulations of the hot and dense fireball the most "fundamental" thing that's feasible computationally is the solution of quantum Boltzmann-Uehling Uhlenbeck equations on both the partonic and the hadronic level. There has a lot of progress over the last few decades. Among other things multi-particle collision terms (beyond the 2->2 level) can now be simulated accurately. Newest developments are mostly centered around "spin transport" and the derivation of hydrodynamics from relativistic kinetic theory. Also there there was tremendous progress over the last few years. Among other things now there's also a stable causal formulation of first-order ("Navier-Stokes") relativistic hydrodynamics as well as systematic approximations of higher orders. Of course also on the hydro level there's a lot of interest in spin and magneto-hydrodynamics.

On a more fundamental level one can solve toy models, using the real-time formalism of many-body off-equilibrium QFT, leading to Kadanoff-Baym equations (from which the above mentioned transport equations can be derived through gradient expansion of the Wigner representation of the contour Green's functions).

Two recent books about these topics are

G. S. Denicol and D. H. Rischke, Microscopic Foundationsof
Relativistic Fluid Dynamics, Springer, Cham (2021),
https://doi.org/10.1007/978-3-030-82077-0

W. Cassing, Transport Theories for Strongly-Interacting
Systems: Applications to Heavy-Ion Collisions, Springer,
Cham (2021),
https://doi.org/10.1007/978-3-030-80295-0

 

[LittleSchwinger]

Does "true" evolution mean unitary evolution?

No, in fact that's the point. Time evolution in QED is not unitary, but a contractive Markovian process. This is at the non-perturbative level. Perturbatively time evolution is unitary.

 

[A. Neumaier]

No, in fact that's the point. Time evolution in QED is not unitary, but a contractive Markovian process. This is at the non-perturbative level. Perturbatively time evolution is unitary.

Please give a reference for this strange claim.

 

[LittleSchwinger]

Please give a reference for this strange claim.

A nice worked model is here, done in the non-relativistic limit:
https://arxiv.org/abs/2212.02599

The general theory is covered in Frohlich's proceeding papers, which include references to Buchholz's previous works on the topic. I prefer this because the other papers are "advanced" mathematical physics. I think most would find this easier to follow.

Roughly speaking due to a contraction of the algebra of observables we have a time evolution that is most likely unitary on the observables, but non-unitary/Markovian on the states.

 

[A. Neumaier]

A nice worked model is here, done in the non-relativistic limit:
https://arxiv.org/abs/2212.02599

The general theory is covered in Frohlich's proceeding papers, which include references to Buchholz's previous works on the topic. I prefer this because the other papers are "advanced" mathematical physics. I think most would find this easier to follow.

Roughly speaking due to a contraction of the algebra of observables we have a time evolution that is most likely unitary on the observables, but non-unitary/Markovian on the states.

However, this is not QED but a nonrelativistic approximations. Approximations often violate unitarity, due to neglect of degrees of freedom that can absorb energy or entropy.

 

[physwiz222]

This is a very reasonable question that I had puzzled over for a long time before understanding what actually happens.

In textbook QED one starts with a Fock space and finds out that all quantities of interest are ill-defined without renormalization. By Haag's theorem, this is necessarily so for relativistic QFTs based on the interaction picture (which is needed for perturbation theory) - the latter simply does not exist. Thus one introduces renormalization, which works smoothly only at the level of the S-matrix. There is a widespread view that in relativistic QFTs, only the asymptotic descriptions (i.e., the S-matrix) makes quantitative sense, and can be computed using renormalization, together with IR regularization via cumulative cross sections or coherent states.

The reason is the difficulty to give QED (and other physically relevant relativistic QFTs) a rigorous mathematical basis: In a logically impeccable sense (in principle being able to produce model predictions to infinite accuracy) of a single interacting relativistic QFT in 4 spacetime dimensions is known. One of the Clay Millenium Problems asks for a construction for 4D Yang-Mills, which is though to be the simplest relativistic QFT in 4 spacetime dimensions potentially in reach by current methods.

On the other hand, suppose a mathematical construction of QED (or any relativistic QFT in 4 spacetime dimensions) exists. This would imply a realization of the Wightman axioms. Any such realization produces a Hilbert space with a unitary representation of the Poincare group, and the generator $H=P_0$ of the time translation acts as a Hamiltonian producing the temporal evolution of the states.

Currently only free fields (and their quasi-free cousons) - such as QED with zero electron charge - are known to realize the Wightman axioms in 4 spacetime dimensions. Their states indeed have an explicit temporal evolution, given by the traditional free Hamiltonian on Fock space.

Thus the reason why textbooks are silent about the temporal evolution is because theory is not developed enough to be able to say more than trivialities about it.

I did some research on this issue and it seems if my understanding is right the fundamental issue to describing the finite time dynamics of Interacting Fields is Haag’s Theorem because it states that the Interacting States cant be described by a combination of Fock States.

Thus there is no way to describe how the occupation numbers evolve and how the states morph as the Interacting states are not describable in terms of the Fock States at finite time so at least for finite time prohibits the idea of Perturbation theory and series expansions in general.

Its akin to how a combination of plane waves with spatial variation only in the x direction no matter how many cant describe a shape like a sphere as its not describable in terms of solely plane waves with variations in the x directions as it also varies along y and z. Interacting states are similar to this.

 

[A. Neumaier]

I did some research on this issue and it seems if my understanding is right the fundamental issue to describing the finite time dynamics of Interacting Fields is Haag’s Theorem because it states that the Interacting States cant be described by a combination of Fock States.

Thus there is no way to describe how the occupation numbers evolve and how the states morph as the Interacting states are not describable in terms of the Fock States

Your argument is not conclusive as there is no necessity for the Hilbert space of a quantum field theory to be a Fock space. It only follows that the Fock space concept is inadequate to capture interacting QFT.

Note that there are many rigorous constructions of interacting QFTs in 2 and 3 spacetime dimensions, in spite of the validity of Haag's theorem there. But in no case is the finite-time dynamics given in a Fock space.

 

[physwiz222]

Your argument is not conclusive as there is no necessity for the Hilbert space of a quantum field theory to be a Fock space. It only follows that the Fock space concept is inadequate to capture interacting QFT.

Note that there are many rigorous constructions of interacting QFTs in 2 and 3 spacetime dimensions, in spite of the validity of Haag's theorem there. But in no case is the finite-time dynamics given in a Fock space.

Well what I said was that the reason no one computes a time dependent state for interacting QFT by using the fock basis to construct a state pf the form c0(t)|0>+c1(t)|k> is because the Interacting states arent describable by Fock states.
My argument was that Haag’s Theorem effectively prohibits constructing a time dependent approximate state using the Fock basis.

 

[Morbert]

However, this is not QED but a nonrelativistic approximations. Approximations often violate unitarity, due to neglect of degrees of freedom that can absorb energy or entropy.

He presents a relativistic account here which involves the same theorem by Buchholz. (see equation 7)

 

[A. Neumaier]

He presents a relativistic account here which involves the same theorem by Buchholz. (see equation 7)

But there Fröhlich doesn't make the strange claim that

Time evolution in QED is not unitary, but a contractive Markovian process.

Indeed, in a very recent paper, Buchholz and Fredenhagen explicitly discuss the loss of information entailed in constructing an arrow of time for QED. The contractivity is explained by an approximation that restricts to what an observer can know.

 

[Morbert]

Indeed, in a very recent paper, Buchholz and Fredenhagen explicitly discuss the loss of information entailed in constructing an arrow of time for QED. The contractivity is explained by an approximation that restricts to what an observer can know.

I can see how using loss of access to information to construct a family of event algebras that satisfies a contractivity is in a sense an approximation. On the other hand, Fröhlich makes an implicit distinction between quantum theory and a quantum theory of events. Perhaps a quantum theory of of events (whether the theory is quantum-mechanical or quantum-electrodynamic) cannot otherwise be constructed.

 

[A. Neumaier]

using loss of access to information to construct a family of event algebras that satisfies a contractivity is in a sense an approximation.

This is the general mechanism; nothing is special here. See any derivation of decoherence or Lindblad master equations:

Restricting attention to less than complete information (absolutely necessary for an observer since complete information is never available) always turns the exact unitary evolution into a contractive evolution.