I What is the basic scheme of quantum field theories?

[jonjacson]

Can you show me where is the edit button? I feel so stupid at this moment.

If I click reply I will add a new answer, Is that correct?

 

[A. Neumaier]

Can you show me where is the edit button?

Below any of your posted entries (except in closed threads),
there are 4 buttons: edit - delete -report -bookmark
(Below other's entries, there are only the last two of these.)

If I click reply I will add a new answer, Is that correct?

Yes, to edit your old posts, click on edit.

If you want to reply to several parts of an answer, delete first the unneeded stuff, then copy and duplicate the whole quote environment, then edit out the parts not commented.

 

[jonjacson]

Well, I can't see it.

 

[stevendaryl]

QFT does not allow you to calculate the time evolution, because the Hamiltonian (=the generator of time evolution) of QFT is badly damaged by the presence of divergent counterterms. QFT can only calculate the S-matrix, which is a mapping of states from the infinite past to the infinite future. From the calculated S-matrix you can extract scattering cross-sections and energies/lifetimes of bound states. That's all you can do with the textbook QFT.

Lattice quantum field theory can in principle calculate time evolution. There might be problems in taking the continuum limit, though, so this calculation might not tell us anything about real quantum fields.

 

[A. Neumaier]

Lattice quantum field theory can in principle calculate time evolution. There might be problems in taking the continuum limit, though, so this calculation might not tell us anything about real quantum fields.

Textbook QED has no sensible relation to lattice QED. At large lattice spacing it is way too inaccurate, and at small lattice spacing, at a cell size still amenable to simulation, it shows already all numerical signs of triviality, i.e., it converges to a free theory rather than to QED. Thus there is no domain where lattice QED approximates QED in any meaningful way.

Eugene's book is about a particle approach to QED, so your comment is irrelevant for his purposes.

 

[A. Neumaier]

Well, I can't see it.

oh, this is different from what I see. Then currently you cannot edit old posts. Maybe editing is allowed only for science advisors? @Greg Bernhardt

 

[Greg Bernhardt]

oh, this is different from what I see. Then currently you cannot edit old posts. Maybe editing is allowed only for science advisors? @Greg Bernhardt

SA's have no time limit whereas regular membership has 60min.

 

[king vitamin]

Textbook QED has no sensible relation to lattice QED. At large lattice spacing it is way too inaccurate, and at small lattice spacing, at a cell size still amenable to simulation, it shows already all numerical signs of triviality, i.e., it converges to a free theory rather than to QED. Thus there is no domain where lattice QED approximates QED in any meaningful way.

Do you have a reference for this?

 

[bolbteppa]

Apologies for the nested quotes, for context:

... QFT can only calculate the S-matrix, which is a mapping of states from the infinite past to the infinite future... "The more one thinks about this situation, the more one is led to the conclusion that one should not insist on a detailed description of the system in time. ... Essentially only scattering experiments are possible,

You may think this is the case.

But on the level of rigor customary in theoretical physics, quantum field dynamics at finite time is actually well-defined in terms of the so-called closed time path (CTP) approach

Trying to describe the time evolution of a relativistic quantum system contradicts the most fundamental principles of quantum mechanics + relativity:

"The momentum can figure in a consistent theory only for free particles; for these it is conserved, and can therefore be measured with any desired accuracy. This indicates that the theory will not consider the time dependence of particle interaction processes. It will show that in these processes there are no characteristics precisely definable (even within the usual limitations of quantum mechanics); the description of such a process as occurring in the course of time is therefore just as unreal as the classical paths are in non-relativistic quantum mechanics. The only observable quantities are the properties (momenta, polarisations) of free particles: the initial particles which come into interaction, and the final particles which result from the process"

But not the principles of relativistic quantum field theory. Given a realization of the Wightman axioms, time evolution is given by shifting all field arguments in their time argument by the same amount.

I was referring to the principles of relativistic quantum field theory - it is one thing for the fields to shift in their field arguments in time, which of course has to happen, it is another to actually describe (i.e. measure) what is going on at every time in the course of that time evolution in interacting theories, "quantum field dynamics at finite time" for anything not equivalent to scatting between free particle states, which amounts to free particles scattering "from the infinite past to the infinite future", is "just as unreal as the classical paths are in non-relativistic quantum mechanics", contradicting this is not permissible in the absence of a suitable level of rigor.

 

[bolbteppa]

It seems to me that these considerations only make sense if one either (1) defines "particles" in a strict sense which is not useful in an interacting QFT (which generically does not have particles in the sense often meant), or (2) demands that the theory is "relativistic" in the sense that one does not have a UV cutoff, in which case I do not know of any realistic interacting relativistic QFT in (3+1) dimensions.

If I define a quantum field theory with the appropriate regulators, I can describe time-dependent phenomena (equilibrium or non-equilibrium) just fine. You may say that I have broken Lorentz invariance, which is true (after all Lorentz invariant theories have not been proven to exist), but for suitably large cutoffs all low-energy phenomena exhibit emergent Lorentz invariance anyways. And these calculations agree with experiment (see the many thousands of papers using time-dependent methods in QFT published every year).

(I also don't trust a QFT textbook/lecture which does not cover the renormalization group. This field has evolved in the past 50 years!)

If we can only measure things like free particle momenta, with anything else "just as unreal as the classical paths are in non-relativistic quantum mechanics", why should we expect the notion of particles in interacting qft to be useful if all that matters, according to the fundamental principles of QM and relativity, is the outcome (i.e. particles scattering off into free states) of such processes?

Without Lorentz/Poincare invariance, we have nothing - apart from, at most, non-relativistic QFT, i.e. second quantized non-relativistic quantum mechanics - the idea that "Lorentz invariant theories have not been proven to exist" is a realistic justification for ignoring one of the most important/successful principles in science is as realistic as saying 'no theory in science has been proven to exist'.

Concepts like cutoffs and regulators are related to the necessity of renormalization which is also a huge problem in classical ('relativistic') electromagnetism where, again, the issue is the point-particle model of physics, something relativity simply makes mandatory - again, without relativity we have nothing, especially dependent concepts like cutoffs (which also arise in 'relativistic' classical electromagnetism), and with relativity we unavoidably have a point particle model (and so renormalization issues, which is why it's not surprising string theories with their non-point-particle model are the only known way to deal with QM-GR renormalization issues).

That book does mention the renormalization group (of Gell-Mann and Low) when deriving the Landau pole issue in the second last chapter, advances in renormalization are pretty much not going to do anything but better bypass the more fundamental fact that infinities come from the point-particle nature of our models of (most) fundamental physics, which are inherently linked to relativity - a point which is undeniable in modern mainstream physics, and something modern research (strings etc) tries to deal with.

 

[DarMM]

Lorentz invariant theories have not been proven to exist

I think @king vitamin is referring to the fact that no 4D interacting quantum field theory has been proven to be well defined mathematically.

quantum field dynamics at finite time...just as unreal as the classical paths are in non-relativistic quantum mechanics

I don't see this. In any rigorously constructed QFT one has finite time evolution as a well-defined concept, how is it unreal?

 

[bolbteppa]

I don't see this. In any rigorously constructed QFT one has finite time evolution as a well-defined concept, how is it unreal?

Time evolution of the fields is of course obviously necessary, measuring a system at a point in time when the system is still interacting and expecting "characteristics precisely definable" is the issue, as described in the reference cited here, which is why standard concepts like LSZ are framed in terms of free states.

 

[king vitamin]

If we can only measure things like free particle momenta, with anything else "just as unreal as the classical paths are in non-relativistic quantum mechanics", why should we expect the notion of particles in interacting qft to be useful if all that matters, according to the fundamental principles of QM and relativity, is the outcome (i.e. particles scattering off into free states) of such processes?

I don't understand the question. I did not say that all we can measure is particle scattering, nor did I say that all we can measure free particle momenta. In fact I would argue that "particles" are no longer a useful notion for a general interacting QFT. Part of me thinks that this is really the point that the authors of the textbook were making, in which case I agree. I don't agree that there aren't interesting and experimentally relevant processes in QFT involving time evolution.

You've discussed divergent integrals, free-particle states at infinity, and the LSZ theorem, so I want to ask you about a family of relativistic field theories where none of this applies. What about (1+1)-dimensional conformal field theories? There are an infinite number of these theories which are exactly solvable, they are relativistic (Lorentz is a subgroup of conformal), they generically have no notion of "particles" even at asymptotic spacetime*, it does not even make sense to define asymptotic states due to dilation symmetry, and they may be defined in a mathematically rigorous manner without a cutoff.

We can write down the time-dependence of these interacting relativistic quantum field theories exactly and explicitly. How do these not constitute a simple counter-example to your claim in post #44? If somehow working in lower spatial dimensionality avoids the issues mentioned in your source, can you detail how?

*Completely free theories are also conformal, but most conformal theories are not free.

edit: i cleared up why asymptotic states don't exist in CFTs

 

[bhobba]

That book does mention the renormalization group (of Gell-Mann and Low) when deriving the Landau pole issue in the second last chapter, advances in renormalization are pretty much not going to do anything but better bypass the more fundamental fact that infinities come from the point-particle nature of our models of (most) fundamental physics, which are inherently linked to relativity - a point which is undeniable in modern mainstream physics, and something modern research (strings etc) tries to deal with.

I think our modern understanding of renormalisation also sheds light on interesting phenomena I would not have expected like triviality. It has been suggested, and I do not as yet know the details, the standard model itself may be trivial - but I do not think anyone knows for sure. I think there is likely riches to be mined here especially considering string theory has morphed:
https://www.quantamagazine.org/string-theorys-strange-second-life-20160915/

Time will tell. At 63 my only regret is I am unlikely to be around to see it's resolution.

Thanks
Bill

 

[jonjacson]

I think our modern understanding of renormalisation also sheds light on interesting phenomena I would not have expected like triviality. It has been suggested, and I do not as yet know the details, the standard model itself may be trivial - but I do not think anyone knows for sure. I think there is likely riches to be mined here especially considering string theory has morphed:
https://www.quantamagazine.org/string-theorys-strange-second-life-20160915/

Time will tell. At 63 my only regret is I am unlikely to be around to see it's resolution.

Thanks
Bill

That last sentence reminded me Horace Lamb.

https://en.wikipedia.org/wiki/Horace_Lamb

 

[bhobba]

That last sentence reminded me Horace Lamb.

Well I am not in his class, but yes you just have to look at what has been discovered since his death - I have no doubt when I leave this mortal realm even greater advances at a startling pace will be made. It really is sobering when you realize that.

Thanks
Bill

 

[Peter Morgan]

That last sentence reminded me Horace Lamb.

https://en.wikipedia.org/wiki/Horace_Lamb

Thanks for the Wikipedia link for Horace Lamb (which is enjoyable enough for me to recommend it). I'm rather taken with the final quote, "I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic", but to me that seems not quite as pessimistic as @bhobba's "At 63 my only regret is I am unlikely to be around to see it's resolution".

 

[MathematicalPhysicist]

Even if there would some problems that will be presumably resolved, they will just raise more problems which would be unresolved by then.

 

[A. Neumaier]

I was referring to the principles of relativistic quantum field theory - it is one thing for the fields to shift in their field arguments in time, which of course has to happen, it is another to actually describe (i.e. measure) what is going on at every time in the course of that time evolution in interacting theories, "quantum field dynamics at finite time" for anything not equivalent to scatting between free particle states, which amounts to free particles scattering "from the infinite past to the infinite future", is "just as unreal as the classical paths are in non-relativistic quantum mechanics", contradicting this is not permissible in the absence of a suitable level of rigor.

Lots of stuff can be computed in QFT at finite times - field expectation values and field correlations but not particle properties.

 

[bolbteppa]

I don't understand the question. I did not say that all we can measure is particle scattering, nor did I say that all we can measure free particle momenta. In fact I would argue that "particles" are no longer a useful notion for a general interacting QFT. Part of me thinks that this is really the point that the authors of the textbook were making, in which case I agree. I don't agree that there aren't interesting and experimentally relevant processes in QFT involving time evolution.

The very specific claim in section 1 of the reference of 44 is not that particles are just not useful in interacting QFT, it's that nothing is precisely definable in interacting QFT (not the 'non-precisely-definable-in-the-QM' sense, rather the 'nothing-makes-any-sense' sense): "the theory will not consider the time dependence of particle interaction processes. It will show that in these processes there are no characteristics precisely definable (even within the usual limitations of quantum mechanics); the description of such a process as occurring in the course of time is therefore just as unreal as the classical paths are in non-relativistic quantum mechanics. The only observable quantities are the properties (momenta, polarisations) of free particles: the initial particles which come into interaction, and the final particles which result from the process", which is a really interesting/bold claim (right or wrong).

Claiming that there are experimentally relevant processes involving time evolution really does directly contradict the claim of this textbook, without addressing any of the fundamental arguments given in section 1 of that textbook.

You've discussed divergent integrals, free-particle states at infinity, and the LSZ theorem, so I want to ask you about a family of relativistic field theories where none of this applies. What about (1+1)-dimensional conformal field theories? There are an infinite number of these theories which are exactly solvable, they are relativistic (Lorentz is a subgroup of conformal), they generically have no notion of "particles" even at asymptotic spacetime*, it does not even make sense to define asymptotic states due to dilation symmetry, and they may be defined in a mathematically rigorous manner without a cutoff.

It seems like you're implying we can't interpret modes in CFT's as particles, which would imply we can't interpret the 4-D conformal field theory known as free electromagnetism in terms of particles... I am not sure why any of the arguments in section 1 of the ref of 44 don't apply in 2-D or 10/11/26-D as well as in 4-D, it would not surprise me if 2-D behaved differently (e.g. Coleman-Mandula, anyons), but I don't see a contradiction - it would be interesting to see specific claims about measurements in CFT and how they could potentially invalidate such fundamental claims of QFT...

The arguments in section 1 of the ref of 44 are not framed in terms of asymptotic states as a means for measurements, they are framed in terms of measuring things like free particle momenta, because "for these it is conserved, and can therefore be measured with any desired accuracy" - asymptotic states is just a way to still end up with states with measurable properties when dealing with interactions, but the claim couched in terms of symmetries/conservation-laws which is very strong.

We can write down the time-dependence of these interacting relativistic quantum field theories exactly and explicitly. How do these not constitute a simple counter-example to your claim in post #44? If somehow working in lower spatial dimensionality avoids the issues mentioned in your source, can you detail how?

I don't know why exact solvability matters, but I presume you mean we can exactly write down solutions of these theories i.e. write down (in principle or in practice) the exact wave function solutions of these exactly solvable theories like we can with the non-relativistic harmonic oscillator or hydrogen atom. We don't even need to involve interactions to show why exact solvability is not good enough - in free electromagnetism we can write down the exact wave functions of the theory (e.g. pages 1 - 2 of this or this), but this kind of exact solvability literally destroyed the meaning of non-relativistic wave functions in QFT (and nearly ended QFT as a subject before it got started) as explained in those references and (partially?) led to this whole idea of only measuring free particle properties in the first place (a serious issue that makes alternatives like dBB shocking ideas when they just ignore these things, or worse try to make them emergent).

 

[king vitamin]

Claiming that there are experimentally relevant processes involving time evolution really does directly contradict the claim of this textbook, without addressing any of the fundamental arguments given in section 1 of that textbook.

I agree, which makes me doubt the arguments of the textbook.

It seems like you're implying we can't interpret modes in CFT's as particles, which would imply we can't interpret the 4-D conformal field theory known as free electromagnetism in terms of particles...

My claim isn't that we can never interpret the modes of CFTs as particles - clearly free theories contradict that. But in most cases we cannot, at least not locally (in space).

I am not sure why any of the arguments in section 1 of the ref of 44 don't apply in 2-D or 10/11/26-D as well as in 4-D

I agree, which makes me doubt the arguments of the textbook.

I don't know why exact solvability matters

This is probably me misreading your arguments - you've mentioned various aspects of perturbative field theory, and I wanted to give an example which avoids all that baggage.

We don't even need to involve interactions to show why exact solvability is not good enough - in free electromagnetism we can write down the exact wave functions of the theory (e.g. pages 1 - 2 of this or this), but this kind of exact solvability literally destroyed the meaning of non-relativistic wave functions in QFT (and nearly ended QFT as a subject before it got started) as explained in those references and (partially?) led to this whole idea of only measuring free particle properties in the first place (a serious issue that makes alternatives like dBB shocking ideas when they just ignore these things, or worse try to make them emergent).

Can you give a gist of the references and how they render my counterexamples moot? I simply don't believe you when you say that one cannot compute time-dependent properties in relativistic QFTs, as I feel like the given counterexamples show that this claim is false. If you understand the arguments in these references, can you explain to me what goes wrong when I claim that I can compute the exact time dependence of the correlation functions of a relativistic QFT?

I have a feeling there is some major misunderstanding happening between us here. The reason I have mentioned things like particle states and perturbation theory is because I suspected the arguments of the textbook only applied to some limiting cases or in perturbation theory, but you seem adamant that they are completely general. In which case I would like to understand what they say about relativistic QFTs where I can write down time dependence.

 

[bolbteppa]

Can you give a gist of the references and how they render my counterexamples moot? I simply don't believe you when you say that one cannot compute time-dependent properties in relativistic QFTs, as I feel like the given counterexamples show that this claim is false. If you understand the arguments in these references, can you explain to me what goes wrong when I claim that I can compute the exact time dependence of the correlation functions of a relativistic QFT?

I have a feeling there is some major misunderstanding happening between us here. The reason I have mentioned things like particle states and perturbation theory is because I suspected the arguments of the textbook only applied to some limiting cases or in perturbation theory, but you seem adamant that they are completely general. In which case I would like to understand what they say about relativistic QFTs where I can write down time dependence.

Maybe there is a misunderstanding - are you saying (something equivalent to) that being able to take some initial state $|i,t_0>$ in some relativistic quantum theory (CFT) and then being able to abstractly or even concretely write down, with $U$ as the time evolution operator, the state $U(t,t_0)|i,t_0>$ (with t in some frame being a time in the middle of an interaction) contradicts what I'm saying? I think you're bringing in CFT to give an example of somewhere where we can do the equivalent of writing down $U(t,t_0)|i,t_0>$ explicitly without the baggage of perturbation theory, so let's for the sake of argument assume we could explicitly compute $U(t,t_0)|i,t_0>$ for the QED Hamiltonian, especially because we need to relate it to actual physical measurements. Do you think I am saying that we can't write down $U(t,t_0)|i,t_0>$ and that no we can (do something equivalent to) write(ing) this down explicitly?

 

[king vitamin]

Do you think I am saying that we can't write down U(t,t0)|i,t0>U(t,t0)|i,t0>U(t,t_0)|i,t_0> and that no we can (do something equivalent to) write(ing) this down explicitly?

Of course I think you are saying that. After all, quoting your post #44:

Trying to describe the time evolution of a relativistic quantum system contradicts the most fundamental principles of quantum mechanics + relativity

I assume that "time evolution" refers precisely to writing down $U(t,t_0)|i,t_0 \rangle$, i.e. applying a time evolution operator to some initial state. I don't see what else your statement could possibly mean.

 

[A. Neumaier]

the theory will not consider the time dependence of particle interaction processes. It will show that in these processes there are no characteristics precisely definable (even within the usual limitations of quantum mechanics); the description of such a process as occurring in the course of time is therefore just as unreal as the classical paths are in non-relativistic quantum mechanics.

This is correct. QFT says nothing at all about particles at finite times. QFT is quantum field theory, and talks at finite time only about fields. There it has to say a lot. Particles in QFT are only semiclassical (and hence only approximately defined) objects.

 

[meopemuk]

This is correct. QFT says nothing at all about particles at finite times. QFT is quantum field theory, and talks at finite time only about fields. There it has to say a lot. Particles in QFT are only semiclassical (and hence only approximately defined) objects.

When we do scattering experiments (both experimentally and theoretically), we specify particle properties (momenta, spin projections) before and after the collision and measure or calculate the probabilities of transitions between these asymptotic particle states. Quantum fields are useful mathematical tools for calculating these probabilities.

You may say that the notion of particles makes sense only in the asymptotic regime, when particles are free. I can agree that experimentally it is very difficult to penetrate into the small and short collision region and see how the time evolution of the colliding particles looks like there. But in very low energy collisions of charged particles (when the collision time is measured in seconds and the size of the collision region is measured in centimeters) there should be no problem in measuring their time-evolving wave functions or even curved classical trajectories. In our low-energy macroscopic life we are surrounded by particles of various kinds (atoms, molecules, dust, billiard balls). They move around, interact with each other and are describable by quantum mechanics at incredible level of accuracy. If you insist that particle properties (momenta, positions, spins, etc.) do not make rigorous sense in QFT, then how QFT is going to describe the particle dynamics that we see so clearly in the world around us?

Eugene.

 

[DarMM]

If you insist that particle properties (momenta, positions, spins, etc.) do not make rigorous sense in QFT, then how QFT is going to describe the particle dynamics that we see so clearly in the world around us?

This is not @A. Neumaier 's personal theory or insistence. The quantum field theories which we use in particle physics and have been experimentally confirmed to incredible accuracy simply do not admit a well-defined notion of particle in general.

Haag and Ruelle, with their rigorous description of particle detection devices (see either Haag's Local Quantum Physics or Wald's QFT is Curved Spacetime text for details) already explains how particle dynamics is a feature of QFT in the experiments where one expects it to be relevant.

 

[PeterDonis]

The quantum field theories which we use in particle physics and have been experimentally confirmed to incredible accuracy simply do not admit a well-defined notion of particle in general.

And even when they do, those notions can be observer-dependent, as illustrated by, for example, the Unruh effect.

 

[bolbteppa]

Of course I think you are saying that. After all, quoting your post #44:

I assume that "time evolution" refers precisely to writing down $U(t,t_0)|i,t_0 \rangle$, i.e. applying a time evolution operator to some initial state. I don't see what else your statement could possibly mean.

The (not my, the ) claim is that the outcome of actually evaluating $U(t,t_0)|i,t_0 \rangle$ is just as irrelevant to physics (i.e. it can never, in principle, lead to measurements in an experiment) as the outcome of finding/computing the position-space wave function for the free electromagnetic field as given in the two links of post #80 is.

In the two links of post #80 the explicit solution of the first quantized position-space Schrodinger equation is explicitly presented and it is pointed out that the wave function is completely useless, and one does not need to use any arguments related to second quantization or interactions or anything to see this, it's non-locality was enough to destroy our understanding of position-space wave functions - in other words, just because we can do some math, e.g. solve some equations or evaluate some expression, that doesn't necessarily make it physics unless things are consistent.

The fundamental reason that wave function is useless is given in section 1 of the reference of post #44, i.e. it could have been (was) predicted in advance to be pathological, and similarly the fundamental reason why actually evaluating $U(t,t_0)|i,t_0 \rangle$ at a time when the system is not a bunch of free particles will give at best something irrelevant to physics is also presented there - there is a very simple argument around equation 1.2 for why one could never, in principle, measure something like a momentum related to a state like this that's not a system of free particles.

Why is it not at all shocking that blindly solving the position-space Schrodinger equation for the free EM field can in principle say nothing about the real world yet completely shocking when it is claimed the outcome of evaluating a state $U(t,t_0)|i,t_0 \rangle$ at a time during an interaction and then looking for something measurable like a momentum can in principle say nothing about the real world, when both are justified by basically the same arguments (section 1 of the reference of post #44), so shocking as to doubt such direct consequences of relativity and the uncertainty principle (section 1)?

These results are apparently so strong that if it were not for the fact that QM inherently assumes the existence of classical physics it would literally destroy QFT as a subject, as realized over 80 years ago (section 3).