I Confusion about Scattering in Quantum Electrodynamics

[physwiz222]

TL;DR Summary

I am confused about Scattering in QED that no one ever actually solves or describes the actual dynamics of the scattering process using Perturbation Theory of course and why only cross sections and decay rates are computed.

When it comes to scattering in QED it seems only scattering cross sections and decay rates are calculated. Why is that 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.

For example when 2 electrons scatter off each other is it possible to compute the dynamical evolution of the particle states themselves of course using perturbation theory and visualizing it in terms of the expectation value of the charge density.

Also can the scattering amplitudes be used to reconstruct the evolution of the fields themselves if so why isnt it ever mentioned is it because constructing the evolution is a trivial exercise maybe. This bugs me because for a theory that is so praised and can supposedly describe Electromagnetic Fields at the fundamental level it seems awfully limited in what it can do like the fact it doesnt describe the actual dynamics from what I know comes off as hollow and unsatisfying.

For the record I dont know any QFT I simply looked at it a bit out of curiosity of whats the big deal about QFT and found myself kinda puzzled so please forgive me if I made any mistakes I am no expert. I simply want to know if its possible to describe the actual dynamics of scattering processes in QED with Feynman Diagrams for non asymptotic times for momentum eigenstates then use superposition to construct wavepackets like in regular QM or only cross sections can be computed for asymptotic tjmes.

I simply am curious as isnt the actual phenomenon and the dynamics also important as well. I know how well tested QED is as much as the next guy for its experimental confirmation in the 10th digit but shouldnt it also describe the actual dynamics of the phenomena as well.

 

[Nugatory]

I am confused about Scattering in QED that no one ever actually solves or describes the actual dynamics of the scattering process using Perturbation Theory of course and why only cross sections and decay rates are computed.
….
This bugs me because for a theory that is so praised and can supposedly describe Electromagnetic Fields at the fundamental level it seems awfully limited in what it can do like the fact it doesnt describe the actual dynamics from what I know comes off as hollow and unsatisfying.

It’s not just QED that’s that way, it’s all of quantum mechanics. . We are given this astoundingly accurate black box, it gives us such great answers, surely there has to be some mechanism inside that black box that knows the exact dynamics and could tell us exactly what is going on….

Nope. The mathematical formalism of quantum mechanics, which tells us what the result will be but not how and why, is all that is there. If this feels hollow and unsatisfying to you’re in good company, many people have that reaction when they first encounter QM. However, much of this dissatisfaction comes from our lifetime of experience with classical physics - quantum mechanics doesn’t work the way we expect a physical theory to work - and fades as we become more comfortable with mathematical formalism and can appreciate it on its own merits.

It’s worth noting that classical physics has the same problem if we probe at its foundations enough. For example…. No one who uses Newtonian gravity to calculate planetary motion is going to feel hollow and unsatisfied when Copernicus’s laws emerge from the calculation (at least for me “triumphant” would be more accurate). But we’re not seeing any mechanism that explains what is really going on when a planet is pulled towards the sun, we’re just using this $1/r^2$ black box description of the pull because it works. The difference is that we’re more comfortable with the classical model so are less inclined to probe and less concerned by the gaps we find when we do.

 

[physwiz222]

It’s not just QED that’s that way, it’s all of quantum mechanics. . We are given this astoundingly accurate black box, it gives us such great answers, surely there has to be some mechanism inside that black box that knows the exact dynamics and could tell us exactly what is going on….

Nope. The mathematical formalism of quantum mechanics, which tells us what the result will be but not how and why, is all that is there. If this feels hollow and unsatisfying to you’re in good company, many people have that reaction when they first encounter QM. However, much of this dissatisfaction comes from our lifetime of experience with classical physics - quantum mechanics doesn’t work the way we expect a physical theory to work - and fades as we become more comfortable with mathematical formalism and can appreciate it on its own merits.

It’s worth noting that classical physics has the same problem if we probe at its foundations enough. For example…. No one who uses Newtonian gravity to calculate planetary motion is going to feel hollow and unsatisfied when Copernicus’s laws emerge from the calculation (at least for me “triumphant” would be more accurate). But we’re not seeing any mechanism that explains what is really going on when a planet is pulled towards the sun, we’re just using this $1/r^2$ black box description of the pull because it works. The difference is that we’re more comfortable with the classical model so are less inclined to probe and less concerned by the gaps we find when we do.

The problem is NOT the fact its unintuitive in regular QM you do calculate the actual evolution and behavior during scattering like solving the schrodinger equation and using the stationary states to construct more complicated states. The issue is that no one describes the actual dynamical behavior at on asymptotic times in QED like isnt that important. Only cross sections and decay rates are calculated. The issue isnt that im uncomfortable with QM its that I see an issue in not describing the actual dynamical behavior during scattering.

 

[Nugatory]

The issue isnt that im uncomfortable with QM its that I see an issue in not describing the actual dynamical behavior during scattering.

Unless I am misunderstanding your question, the problem is that that dynamical behavior does not exist. There’s no measurement between the preparation of the initial state and the observation of the final scattered state so as far as the mathematical formalism is concerned there’s nothing there.

 

[physwiz222]

Unless I am misunderstanding your question, the problem is that that dynamical behavior does not exist. There’s no measurement between the preparation of the initial state and the observation of the final scattered state so as far as the mathematical formalism is concerned there’s nothing there.

But in regular QM we can describe the dynamical behavior by solving the TISE then constructing time dependent states. Isnt the point of physics to describe how systems in nature evolve. Are you saying QED is simply not capable of describing the dynamical behavior. I am asking because I have heard QED breaks down for non asymptotic times is that true

 

[physwiz222]

It’s not just QED that’s that way, it’s all of quantum mechanics. . We are given this astoundingly accurate black box, it gives us such great answers, surely there has to be some mechanism inside that black box that knows the exact dynamics and could tell us exactly what is going on….

Nope. The mathematical formalism of quantum mechanics, which tells us what the result will be but not how and why, is all that is there. If this feels hollow and unsatisfying to you’re in good company, many people have that reaction when they first encounter QM. However, much of this dissatisfaction comes from our lifetime of experience with classical physics - quantum mechanics doesn’t work the way we expect a physical theory to work - and fades as we become more comfortable with mathematical formalism and can appreciate it on its own merits.

It’s worth noting that classical physics has the same problem if we probe at its foundations enough. For example…. No one who uses Newtonian gravity to calculate planetary motion is going to feel hollow and unsatisfied when Copernicus’s laws emerge from the calculation (at least for me “triumphant” would be more accurate). But we’re not seeing any mechanism that explains what is really going on when a planet is pulled towards the sun, we’re just using this $1/r^2$ black box description of the pull because it works. The difference is that we’re more comfortable with the classical model so are less inclined to probe and less concerned by the gaps we find when we do.

Also I am not looking for deep reasons why things happen that QM doesnt I just want to know if we can describe the Dynamical evolution of systems in QED

 

[PeroK]

TL;DR Summary: I am confused about Scattering in QED that no one ever actually solves or describes the actual dynamics of the scattering process using Perturbation Theory of course and why only cross sections and decay rates are computed.

In the interaction itself the states evolve in a complicated way. This part of the process is essentially mathematically unsolvable. Perturbation theory and the S-matrix allow the final states to be calculated.

Even this becomes mathematically a tough challenge: look at the difficulty in calculating the magnetic moment of the muon.

Laplace is supposed to have said "nature laughs at the difficulties of integration". If so, then she must be splitting her sides at the difficulty of the interaction states in QFT.

For example when 2 electrons scatter off each other is it possible to compute the dynamical evolution of the particle states themselves of course using perturbation theory and visualizing it in terms of the expectation value of the charge density.

The interaction picture, even if you could solve the mathematical challenges, would not simply represent the classical trajectories of two electrons. The clue is in the Feynman diagrams, where the interation is modelled as a sum of various virtual particle interactions. Note that, in particular, in QFT particle number is not fixed, so the interaction states may be a superposition of states with all possible particle numbers.

This bugs me because for a theory that is so praised and can supposedly describe Electromagnetic Fields at the fundamental level it seems awfully limited in what it can do like the fact it doesnt describe the actual dynamics from what I know comes off as hollow and unsatisfying.

Whether it bugs you or not, nature doesn't care. You want a nice, classical, realist picture of an electron-electron interaction and nature serves up a soup of quantum field states.

I simply want to know if its possible to describe the actual dynamics of scattering processes in QED with Feynman Diagrams for non asymptotic times for momentum eigenstates then use superposition to construct wavepackets like in regular QM or only cross sections can be computed for asymptotic tjmes.

There are no "actual dynamics" in the terms you'd like.

I simply am curious as isnt the actual phenomenon and the dynamics also important as well.

When you say "actual phenomenon", you are assuming a realist picture of QM behaviour. You could start by asking precisely what is the orbit of an electron in an atom? The answer is that there is no classical trajectory, and the atom is instead described by an energy eigenstate. If that answer doesn't satisfy you, then you'll find very little in 20th century particle physics that does.

 

[PeroK]

Also I am not looking for deep reasons why things happen that QM doesnt I just want to know if we can describe the Dynamical evolution of systems in QED

In fact, in QCD even the asymptotic states become complicated, chaotic and essentially incalculable. Scattering results in so-called jets of hadrons. These fit the general pattern predicted by the theory, but it's no longer possible to calculate with any precision the final outcome of the scattering.

https://en.wikipedia.org/wiki/Jet_(particle_physics)

 

[physwiz222]

In the interaction itself the states evolve in a complicated way. This part of the process is essentially mathematically unsolvable. Perturbation theory and the S-matrix allow the final states to be calculated.

Even this becomes mathematically a tough challenge: look at the difficulty in calculating the magnetic moment of the muon.

Laplace is supposed to have said "nature laughs at the difficulties of integration". If so, then she must be splitting her sides at the difficulty of the interaction states in QFT.

The interaction picture, even if you could solve the mathematical challenges, would not simply represent the classical trajectories of two electrons. The clue is in the Feynman diagrams, where the interation is modelled as a sum of various virtual particle interactions. Note that, in particular, in QFT particle number is not fixed, so the interaction states may be a superposition of states with all possible particle numbers.

Whether it bugs you or not, nature doesn't care. You want a nice, classical, realist picture of an electron-electron interaction and nature serves up a soup of quantum field states.There are no "actual dynamics" in the terms you'd like.

When you say "actual phenomenon", you are assuming a realist picture of QM behaviour. You could start by asking precisely what is the orbit of an electron in an atom? The answer is that there is no classical trajectory, and the atom is instead described by an energy eigenstate. If that answer doesn't satisfy you, then you'll find very little in 20th century particle physics that does.

I am not saying trajectories by Time Evolution I mean how the Quantum field states evolve throughout scattering. I am not talking about Realism I am talking about the time evolution of the quantum field states. In regular QM you do calculate the time evolution of scattering states as how the wavefunction evolves and can construct wavepacket states from the eigenstates. I just want to know if its possible to calculate how the quantum field states themselves evolve throughout scattering. I am NOT talking about trajectories or realism

 

[physwiz222]

In the interaction itself the states evolve in a complicated way. This part of the process is essentially mathematically unsolvable. Perturbation theory and the S-matrix allow the final states to be calculated.

Even this becomes mathematically a tough challenge: look at the difficulty in calculating the magnetic moment of the muon.

Laplace is supposed to have said "nature laughs at the difficulties of integration". If so, then she must be splitting her sides at the difficulty of the interaction states in QFT.

The interaction picture, even if you could solve the mathematical challenges, would not simply represent the classical trajectories of two electrons. The clue is in the Feynman diagrams, where the interation is modelled as a sum of various virtual particle interactions. Note that, in particular, in QFT particle number is not fixed, so the interaction states may be a superposition of states with all possible particle numbers.

Whether it bugs you or not, nature doesn't care. You want a nice, classical, realist picture of an electron-electron interaction and nature serves up a soup of quantum field states.There are no "actual dynamics" in the terms you'd like.

When you say "actual phenomenon", you are assuming a realist picture of QM behaviour. You could start by asking precisely what is the orbit of an electron in an atom? The answer is that there is no classical trajectory, and the atom is instead described by an energy eigenstate. If that answer doesn't satisfy you, then you'll find very little in 20th century particle physics that does.

It only bugs me if we cant actually calculate this complicated soup of field states. I am ok with there being no classical picture I am not ok with there being no way to describe this NON classical evolution of field states. You didnt understand anything my I said

 

[PeroK]

I am not saying trajectories by Time Evolution I mean how the Quantum field states evolve throughout scattering.

That was answered: it's mathematically too hard. Like most things in physics, outside of a few nice undergraduate problems.

I am not talking about Realism I am talking about the time evolution of the quantum field states. In regular QM you do calculate the time evolution of scattering states as how the wavefunction evolves and can construct wavepacket states from the eigenstates. I just want to know if its possible to calculate how the quantum field states themselves evolve throughout scattering.

David Tong answers this question and makes other relevant points here:

https://www.damtp.cam.ac.uk/user/tong/qft/three.pdf

 

[physwiz222]

That was answered: it's mathematically too hard. Like most things in physics, outside of a few nice undergraduate problems.

David Tong answers this question and makes other relevant points here:

https://www.damtp.cam.ac.uk/user/tong/qft/three.pdf

As for mathematically too hard isnt that what perturbation theory is for. Obviously we cant get exact solutions but can we use Perturbation theory and Feynman diagrams to construct the dynamical evolution of the quantum field states then use superposition to construct wavepacket states. For example the weak scattering of 2 electrons cant we just use perturbation theory to construct how the electron field states evolve throughout scattering. Provided the coupling constant isnt too large.

 

[PeroK]

As for mathematically too hard isnt that what perturbation theory is for. Obviously we cant get exact solutions but can we use Perturbation theory and Feynman diagrams to construct the dynamical evolution of the quantum field states then use superposition to construct wavepacket states. For example the weak scattering of 2 electrons cant we just use perturbation theory to construct how the electron field states evolve throughout scattering. Provided the coupling constant isnt too large.

No. That's not what perturbation theory does. It decomposes a calculation into a series of manageable calculations.

 

[Vanadium 50]

It decomposes a calculation into a series of manageable calculations.

We hope.

 

[physwiz222]

No. That's not what perturbation theory does. It decomposes a calculation into a series of manageable calculations.

So why cant we use these for approximate reconstruction of the time evolution of the electron field states. In regular QM for example the transition amplitudes for Atomic Transitions can be used to reconstruct the dynamical evolution of the atom as it transitions between energy levels. Why cant we do something similar in QFT to Reconstruct the Time Evolution. I mean cant the scattering and transition amplitudes be used for approximate reconstruction of the dynamics.

 

[PeterDonis]

in regular QM we can describe the dynamical behavior by solving the TISE then constructing time dependent states

We calculate time dependent mathematical "states", but that does not mean those time-dependent mathematical states are physical descriptions of the physical states of the systems. There are some QM interpretations that make that claim, but discussion of interpretations belongs in the interpretations subforum. As far as the actual math of QM is concerned, the math that is used to make actual predictions, what @Nugatory said about the status of quantum systems between measurements in post #4 applies just as much to "regular" (i.e., non-relativistic) QM as to quantum field theory (i.e., relativistic QM). We would expect that to be the case since non-relativistic QM is just an approximation to QFT in the non-relativistic limit.

 

[PeterDonis]

am ok with there being no classical picture

Then why do you keep talking about "reconstructing time evolution"? That only makes sense if you are not ok with there being no classical picture, since "time evolution" is a classical picture.

 

[physwiz222]

Then why do you keep talking about "reconstructing time evolution"? That only makes sense if you are not ok with there being no classical picture, since "time evolution" is a classical picture.

How is Time evolution a classical picture there is a Time Evolution Operator in QM all I was asking is if there is a way to describe the dynamics of the Quantum Fields throughout scattering at non asymptotic times. Also I was more so asking because in regular QM you can reconstruct the time evolution of the wavefunction with the transition amplitudes cant the same be done in QFT where you describe the dynamics of the quantum field states throughout scattering.

 

[PeterDonis]

there is a Time Evolution Operator in QM

There is a time evolution operator in non-relativistic QM, but it doesn't mean the same thing as time evolution in classical mechanics.

all I was asking is if there is a way to describe the dynamics of the Quantum Fields throughout scattering at non asymptotic times.

You have been told repeatedly in this thread that the answer to this question is no.

I was more so asking because in regular QM you can reconstruct the time evolution of the wavefunction with the transition amplitudes

Again, "time evolution" here does not mean the same thing as time evolution in classical mechanics. It is not telling you "this is what really happens to the atom during the transition". That has already been stated in this thread.

cant the same be done in QFT where you describe the dynamics of the quantum field states throughout scattering.

You have been told repeatedly the answer to this question as well: it is no.

 

[physwiz222]

There is a time evolution operator in non-relativistic QM, but it doesn't mean the same thing as time evolution in classical mechanics.You have been told repeatedly in this thread that the answer to this question is no.Again, "time evolution" here does not mean the same thing as time evolution in classical mechanics. It is not telling you "this is what really happens to the atom during the transition". That has already been stated in this thread.You have been told repeatedly the answer to this question as well: it is no.

so why cant you the evolution of the quantum field states throughout scattering be Computed with Perturbation Theory. Why cant we describe QED scattering with perturbation theory for Non asymptotic times like does the theory break down or something like that. This isnt about wether the wavefunction really describes reality or our ignorance but why cant the evolution of the quantum field states be computed with perturbation theory the same way as the evolution of the wavefunction in regular QM can

 

[PeterDonis]

Thread closed for moderation.

 

[WernerQH]

Hi @physwiz222,

I only became aware of your question this morning, when the thread had already been closed. I think I understand (and even sympathize) with your question, but it seems to rest on a false premise. A field, spread over all of space and being in some "state" that evolves continuously in time, is an entirely classical picture. The real world, at the microscopic level, is discontinuous and random. QED is a statistical theory that allows us to calculate how frequently specific patterns of events (like the scattering of two electrons) are likely to occur. Sure, many QFT textbooks emphasize the equations of motion satisfied by the field operators, and they seem to suggest that "dynamical evolution" must always be continuous and deterministic. But in my view this can't be the whole story, this "continuity" and "determinism" is an illusion because it refers only to averages, expectation values, and not the specifics of two "individual" eletrons. The kind of detailed "dynamical evolution" that you envision could be calculated in principle, but it would be an unnecessarily complicated description of an average collision of two electrons, on a time scale that is not possible to resolve experimentally. (Or at most indirectly, through QED, by its effect on the observed statistical distributions.)

I think the idea that the world, at the fundamental level, must evolve continuously and deterministically, is one of the biggest misconceptions about QFT.

Hope this helps.

Best wishes,
Werner

 

[Demystifier]

I simply want to know if its possible to describe the actual dynamics of scattering processes in QED with Feynman Diagrams for non asymptotic times for momentum eigenstates then use superposition to construct wavepackets like in regular QM

Yes, in principle it's possible. However, it's much more complicated, and the differences in the measurable predictions are negligible.

 

[Nugatory]

The original post and responses bring up some of the conceptual difficulties that people face when they first encounter QM. The thread has been reopened to allow more responses from people who have useful things to say about these difficulties.

 

[hutchphd]

The OP talks about scattering yet seems to not understand how one does even the ordinary quantum calculation.
Suppose two particle interact (without explicit time dependence). In principle, one can write a complete set of energy eigenstates for the system and these will provide a complete solution for the stationary states for this two particle universe for all time. Scattering, bound states, etc. This is true but practically irrelevant for a host of reasons.
None of them will provide the information that the OP desires. As soon as you ask a question about a spatially or temporally localized event, the actual system in question will be described by a very complicated sum of amplitudes. Some considerable simplification can be obtained only looking at states with certain asymptotic character and making a series of other approximations that lead to our various formulations of perturbation theory (and the S-Matrix which also pushes the nasty stuff behind the curtain). These techniques are designed to do that and are typically not universally valid.
Does the OP really think that with God's microscope (maybe from Ebay?) we could "see" the Feynman diagrams? The motivation for the questions is not foolish but the answers are in fact quite subtle. Nobody (I surely don't) understands

 

[Vanadium 50]

The OP is clearer about his unhappiness than what would make him happy. One way to hopefully simplify and clarify is to consider things in terms of measurements. Anything that can be measured can be calculated, at least in a probabilistic manner.

If that is not satisfactory, there are only two remaining lines of attack:
(1) Not liking the fact that it's probabilistic. Welcome to the club, but that's QM.
(2) Wanting to know what is happening when we are not measuring (if you like, in between measurements). That's a question impossible to answer scientifically.

 

[LittleSchwinger]

I am understanding the OP's question to refer to the fact that QFT texts rarely discuss what form finite time evolution might take in QFT.

In other words why we generally seem to avoid formulating a state $\rho(T)$ at some time $T$ in even a semi-explicit manner and solving for its evolution over some finite time interval: $U(t-T)\rho(T)$. This is quite unlike non-relativistic QM where this is often sketched, even though in many cases it could not be explicitly computed. Non-relativistic QM texts will cover cases outside of the overlap of asymptotic states.

I would give four initial reasons for this:

1. In QFT it is very difficult to characterise states that are not asymptotic. In non-relativistic QM of N-particles, pure states at any given time have the form $\psi(x_{1},\ldots,x_{N})$, i.e. some function in $L^{2}(\mathbb{R}^{3N})$.
   In QFT however we don't even know the Hilbert space for states in general and thus cannot even guess their rough functional form. At asymptotic times we know they are like free field states and thus can be represented as states in a Fock space of a finite number of particle species. At intermediate times it is an open problem of mathematical physics what form the states take, since we know that in general they cannot be understood in terms of particle states.
2. For asymptotic times we have the LSZ formalism that tells us we may compute the overlap between asymptotic states, i.e. the S-matrix, by computing Amputated n-point correlation functions. Since these n-point functions are expressions involving only finite products of local fields, we have further theorems showing they may be computed via Feynman graph techniques perturbatively.
   We have no such calculational recipe in general for finite time evolution.
3. Finite time evolution requires further renormalisations beyond those of asymptotic S-matrix calculations. A finite time propogator $U(t_{2},t_{1})$ requires formulating the theory on constant time slices, which due to the highly singular nature of the fields introduces new singularities. The finite time Green's functions that replace the usual propogator don't have the nice properties that make all-order BPHZ style renormalisation possible. Thus we don't have a general renormalisation prescription.
4. For QED in particular we have to deal with superselection sectors caused by an infrared cloud of photons surrounding a given electron. This means that nonperturbatively the relevant states are not Lorentz irreps and thus we lose all the simplifications such a simple representation theory brings.
   In fact non-perturbatively the evolution is most likely some non-unitary contractive Markovian evolution* about which very little is currently know.
   
   *One may understand this evolution in simpler terms as one involving a "perfect/exact" form of decoherence.

This is just a brief introduction to the issues involved. It's hard to explain in one go, let me know if you have questions.

 

[hutchphd]

So

1. there are no "forces"
2. there is no "finite time"
3. The particles are in drag

I need to learn QFT.......

 

[WernerQH]

(More at the level of the OP, hopefully.)

Wanting to know what is happening when we are not measuring (if you like, in between measurements).

It's because of the unshakable belief in the wavefunction. If you can calculate it for all moments of time, it ought to mean something, no? Many people seem to forget that the processes described by quantum theory always take a finite amount of time, e.g. the time it takes for a particle to travel from the double-slit to the screen. It's not the $| \text{ket} \rangle$ at some time that is relevant, but the probability $| \langle \text{final} | \text{initial} \rangle |^2$. The theory does not tell us through which slit the particle went, but the calculation must of course take into account that the particle could go through both slits.

 

[Haborix]

I am understanding the OP's question to refer to the fact that QFT texts rarely discuss what form finite time evolution might take in QFT.

In other words why we generally seem to avoid formulating a state $\rho(T)$ at some time $T$ in even a semi-explicit manner and solving for its evolution over some finite time interval: $U(t-T)\rho(T)$. This is quite unlike non-relativistic QM where this is often sketched, even though in many cases it could not be explicitly computed. Non-relativistic QM texts will cover cases outside of the overlap of asymptotic states.

I would give four initial reasons for this:
[...]

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.

 

[Vanadium 50]

The particles are in drag

Well, we say they are "dressed". We don't say how. Non-judgemental and all that.

 

[hutchphd]

No judgement intended. Just wanted to decorate the prose.

 

[meopemuk]

There is an approach called "dressed particle" QFT or "clothed particle" QFT, which succeeds in formulating a well-defined finite Hamiltonian in the Fock space. With this Hamiltonian one can
1. Reproduce S-matrix of the traditional renormalized QFT in all perturbation orders.
2. Form a well-defined unitary evolution operator, so that time evolution of Fock states with variable numbers of particles can be followed, similar to how it is done in non-relativistic QM.

The best way to access literature on this subject is to use Google Scholar and find papers quoting

O. W. Greenberg, S. S. Schweber, "Clothed particle operators in simple models of quantum field theory", Nuovo Cim. 8 (1958), 378.

Eugene.

 

[vanhees71]

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.

 

[Demystifier]

So

1. there are no "forces"
2. there is no "finite time"
3. The particles are in drag

I need to learn QFT.......

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.
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.
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).

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

 

[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.