Infinite Charged Electron in QFT

[dm4b]

This was many years ago.

I think I can safely say that I understand quantum mechanics, with exceptions of some of the deeper things in rigorous quantum field theory that I hope to understand in the near future.

Are you saying you know QFT better than Feynman did. Has that much really changed since his day? That's a bold statement, even if it has ;-)

 

[A. Neumaier]

In the next couple of sections he proceeds to show these higher order corrections give rise to the anomalous magnetic moment - the difference between what the Dirac equation predicts and what higher order theory gives you.

Looking at this on the surface, it would appear that including the "effects" of the virtual photon explains the origin of the anomalous magnetic moment.

Vague language breeds fuzzy understanding.

Calculating a relativistic effect by a Taylor expansion in 1/c and getting better agreement with the second-order term added, one may conclude that it is the second-order term that ''causes'' or ''explains'' the effect. But this is not a physical cause in the sense that a fire is caused by striking a match. Nothing is caused by a mathematical calculation.

Or, rather explain what physical effects are being left out when they are not included.

Using a more approximate formula produces less accurate results.

The interactions of QED cause the anomalous magnetic moment, and the Feynman expansion is just a way of calculating it.

 

[A. Neumaier]

Alright, so a field is essentially made up of energy and energy comes in discrete bundles. Please explain this: if you take away the discrete bundles of energy, what is left of the field?

There is always the field, which has some expectation value, and there may be particles, which describe localized excitations of this field, modifying it a little. It is like having a drum and beating it. An unbeaten drum is still a drum.

 

[A. Neumaier]

It sounds like you just said forces are sort of transported by particles. Aren't these particles depicted in Feynman diagrams as internal lines? If not, please explain better if you got the time.

Forces are transported by field and currents. Particles are a semiclassical approximation, of the same kind as geometric optics approximates Maxwell equations.

Feynman diagrams without any loops describe precisely the _classical_ field theory, solved by perturbation theory. But nobody ever has taken virtual particles for real in a classical field. Quantum field theory only adds more diagrams - namely those with loops; the k-loop diagrams providing k-th order corrections in hbar.

he depicts Feynman diagrams which contain gravitons (real and virtual) in oder to show how they couple to each other, etc, in opposition to how photons do not in the electromagnetic field. Is he wrong to draw gravitons into Feynman diagrams to show how the gravitational field couples to itself?.

No. Diagrams just represent tensors with as many indices as there are external lines. These occur in all sorts of mathematics. Among others, one can also use them to describe interaction terms in a Lagrangian, and low order expansions of scattering matrix elements. Drawing diagrams abbreviates some complex integral operator in a shorthand way. It is fully legitimate to use it in this way in physics.

But this usage should not be taken as an indication of any causal relationship between the concepts attached to the drawing.

 

[A. Neumaier]

Are you saying you know QFT better than Feynman did. Has that much really changed since his day? That's a bold statement, even if it has ;-)

I said: I understand quantum mechanics.

This is completely independent of Feynman's statement. (I never got to know him.)

 

[dm4b]

There is always the field, which has some expectation value, and there may be particles, which describe localized excitations of this field, modifying it a little. It is like having a drum and beating it. An unbeaten drum is still a drum.

I have always like that analogy. And, it may just be a good jumping off point to illustrate the problem I am having too.

First, I think I basically agree with you, but I'm not sure at this point. I've been sorting of paying devil's advocate to help get me past my own misgivings in the material I am reading.

Here's how I view virtual particles. Basically it boils down to the fact that there are no exactly solvable interacting field theories (in the number of dimensions required to represent reality). So to get around that, we treat the interaction term, Hint, as a perturbation, and treat the problem using perturbative techniques. Feynam diagrams are just really fancy mnemonics used to represent each term in that perturbation series, and saves us from pages of laborius math involving Wick Contractions and the sort. But they represent nothing physical in and of themselves. Hence, neither do the virtual particles.

Do you agree with that? Is it correct?

If so, here is where some of my additional confusion on the subject lies.

(1) I have no problem with saying the field is fundamental, but I just don't see how you can consider the field without considering particles. I mean, that's the whole reason we NEED QFT in the first place - to explain the creation and annihilation of particles. Shrodinger's non-relativistic description is incapable of that. So, as far as reality goes, a field and particles go hand in hand, do they not?

(2) Back to the drum. What does an unbeaten drum do? Nothing! BUT, does an "unbeaten" field do nothing? Formally, the field is composed of an infinite number of harmonic oscillators in momentum space, with each having a quantum "zero-point" energy. This, of course, would eventually lead us to the whole cosmological constant problem and the famous 120 orders of magnitude, but I think it indicates that something is going on - and not nothing like the unbeaten drum. Do you agree?

In relation to that, the vacuum is said to be seething with quantum fluctuations. Aren't these represented as virtual particles? If so, what's really going on? Perhaps wait until you read my #3 next before answering this point.

(3) On page 255 in Peskin and Shroeder, they interpret vacuum polarization as a screening cloud of virtual electron-positron pairs, which gives the electron an "apparent charge" and hiding the "true" charge from us. The virtual electron-positron pairs are "effective" dipoles of ~ 1/m. It's not until you penetrate the screening cloud of virtual electron-positron pairs, that you could "see" the true charge. Once again, after writing off virtual particles as unphysical/unreal, as I did above, here they are again in another interpretation of a certain phenomenon. If virtual particles are truly unphysical, what is really going on here? What is P&S leaving lacking in there interpretive description?

 

[bapowell]

(2) Back to the drum. What does an unbeaten drum do? Nothing! BUT, does an "unbeaten" field do nothing? Formally, the field is composed of an infinite number of harmonic oscillators in momentum space, with each having a quantum "zero-point" energy. This, of course, would eventually lead us to the whole cosmological constant problem and the famous 120 orders of magnitude, but I think it indicates that something is going on - and not nothing like the unbeaten drum. Do you agree?

See: Inflation

 

[dm4b]

I said: I understand quantum mechanics.

This is completely independent of Feynman's statement. (I never got to know him.)

You do realize I was just playing around here? ;-)

 

[dm4b]

See: Inflation

I'm sorry, but your overly verbose post is not helping me ;-)

Seriously though, I have read a little on inflation but not enough to understand or make a connection here. What is your point?

Thanks.

 

[bapowell]

Seriously though, I have read a little on inflation but not enough to understand or make a connection here. What is your point?

Sorry. I suppose my response was a tad terse and now I see premature. I missed your reference to the cosmological constant. I was going to point out that vacuum energy is indeed important in any theory with gravity!

However, in contrast to the CC, inflation is typically implemented via a rolling scalar field with a non-zero vacuum energy. This is one example where the field value, which determines this energy, is particularly relevant without any discussion of particles.

 

[dm4b]

Sorry. I suppose my response was a tad terse and now I see premature. I missed your reference to the cosmological constant. I was going to point out that vacuum energy is indeed important in any theory with gravity!

However, in contrast to the CC, inflation is typically implemented via a rolling scalar field with a non-zero vacuum energy. This is one example where the field value, which determines this energy, is particularly relevant without any discussion of particles.

ahhh, okay. Is this a quantum scalar field? Are there no creation or annihilation operators for the field? Could there be within the framework of the theory?

I guess it sounds different than the QFT discussed above, which was developed for the need to explain particle creation, etc - but maybe it's not?

As you can tell, I'm not familar with the details of inflation theory at all

This also raised about 10 more questions in my head, that I am going to hold off on, since this thread is busy enough already ;-)

 

[bapowell]

Inflation is typically implemented with a quantum scalar field, although the phenomenon of accelerated expansion from the equation of state $p \leq -\rho/3$ is a classical result. The scalar field that drives the expansion is imaginatively called the inflaton. As a quantum field, you do with it what you would any other -- it has a Fourier decomposition, creation/annihilation operators, and perturbative interactions.

QFT is much bigger than the explanation of particle creation -- it's the way we understand particle interactions. This motivated the need for gauge theories and spontaneous symmetry breaking, and along came the first hypothetical scalar in the Standard Model -- the Higgs boson. The inflaton was originally believed to have been one of the Grand Unified Higgs bosons, but these models soon ran into problems. So, scalar fields, whether we use them to break symmetries or drive inflation, are ubiquitous in the Standard Model and theories beyond.

But since we are getting a bit afield of the OP, feel free to start a new thread if you have more questions!

 

[A. Neumaier]

Here's how I view virtual particles. Basically it boils down to the fact that there are no exactly solvable interacting field theories (in the number of dimensions required to represent reality). So to get around that, we treat the interaction term, Hint, as a perturbation, and treat the problem using perturbative techniques. Feynam diagrams are just really fancy mnemonics used to represent each term in that perturbation series, and saves us from pages of laborius math involving Wick Contractions and the sort. But they represent nothing physical in and of themselves. Hence, neither do the virtual particles.

Do you agree with that? Is it correct?

Yes. Classical field theories are usually also not exactly solvable. Solving them by perturbation theory also produces Feynman tree diagrams, with ''virtual particle'' lines, though there are no particles at all.

(1) I have no problem with saying the field is fundamental, but I just don't see how you can consider the field without considering particles. I mean, that's the whole reason we NEED QFT in the first place - to explain the creation and annihilation of particles.

No. QED was invented before particle creation and annihilation was known. Particles can be fully explained as approximations in terms of fields, whereas the other direction meets with unsurmountable difficulties.

Shrodinger's non-relativistic description is incapable of that. So, as far as reality goes, a field and particles go hand in hand, do they not?

The nonrelativistic description is an approximation to QFT in case c^{-1} can be neglected. One can have nonrelativistic particle theories or nonrelativistic field theories, and one can set up equivalences in certain cases.

(2) Back to the drum. What does an unbeaten drum do? Nothing!

It exists. By its existence it exerts gravitational forces upon its environment. It also limits the space where the surrounding air can flow.

BUT, does an "unbeaten" field do nothing? Formally, the field is composed of an infinite number of harmonic oscillators in momentum space, with each having a quantum "zero-point" energy.

No. This is already a computational device that only covers the free case. Formally, a quantum field is just a space-time-dependent operator valued distribution.

In relation to that, the vacuum is said to be seething with quantum fluctuations. Aren't these represented as virtual particles?

The vacuum is completely inert. Nothing happens. The quantum fluctuations are fictions of the imagination. See the entry ''Does the vacuum fluctuate?'' in Chapter A7 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#vacfluc

(3) On page 255 in Peskin and Shroeder, they interpret vacuum polarization as a screening cloud of virtual electron-positron pairs, which gives the electron an "apparent charge" and hiding the "true" charge from us.

The above FAQ entry tells what happens instead on the level of quantum fields:
The vacuum polarization tensor is defined nonperturbatively in terms of it, as
(q^2 eta - q tensor q)Pi(q^2) := Delta_free(q) - Delta_ren(q),
which is equivalent to Dyson's equation (cf. Weinberg,Vol. I, p.451). Its scalar part Pi(p^2) is related to the running fine structure constant as described in
http://en.wikipedia.org/wiki/Vacuum_polarization
This contains all the physics of vacuum polarization, and is completely independent of virtual particles.

What is P&S leaving lacking in there interpretive description?

They are lacking the nonperturbative perspective that would show that the picture they draw is an artifact of perturbation theory.

 

[dm4b]

Y
No. QED was invented before particle creation and annihilation was known.

That's completely besides the point. Lorentz Equations were invented before Special Relativity, but Lorentz Transformations are the tools needed to complete the theory. QFT is the tool we needed to explain particle creation, annihilation and interactions. Fields that are quantized in the theory give rise to particle creation/annihilation operators. As I said, quantum fields and particles go hand-in-hand. I'm not sure why you would disagree with that (and every short history blurb I have seen in physics texts that describe the historical need for QFT)

Y
Particles can be fully explained as approximations in terms of fields, whereas the other direction meets with unsurmountable difficulties.

I never said otherwise.

The vacuum is completely inert. Nothing happens. The quantum fluctuations are fictions of the imagination.

The above FAQ entry tells what happens instead on the level of quantum fields:
The vacuum polarization tensor is defined nonperturbatively in terms of it, as
(q^2 eta - q tensor q)Pi(q^2) := Delta_free(q) - Delta_ren(q),
which is equivalent to Dyson's equation (cf. Weinberg,Vol. I, p.451). Its scalar part Pi(p^2) is related to the running fine structure constant as described in
http://en.wikipedia.org/wiki/Vacuum_polarization
This contains all the physics of vacuum polarization, and is completely independent of virtual particles.

Maybe I'll just quote the Wikipedia article you linked to answer this one:

"According to quantum field theory, the ground state of a system with interacting particles is not simply empty space. Rather, it contains short-lived "virtual" particle-antiparticle pairs which are created out of the vacuum and then annihilate each other.

Some of these particle-antiparticle pairs are charged; e.g., virtual electron-positron pairs. Such charged pairs act as an electric dipole. In the presence of an electric field, e.g., the electromagnetic field around an electron, these particle-antiparticle pairs reposition themselves, thus partially counteracting the field (a partial screening effect, a dielectric effect). The field therefore will be weaker than would be expected if the vacuum were completely empty. This reorientation of the short-lived particle-antiparticle pairs is referred to as vacuum polarization."

Well, I guess I have the same misgivings I had at the start.

Seems like you keep saying some abstract mathematical quantity (like the polarization tensor above) contains all the physics, without being able to give a clear, succinct and intuitive description of what really physically causes vacuum polarization, etc, while at the same time saying mathematical calculations cause nothing.

I'll check out your links though, thanks.

 

[A. Neumaier]

You do realize I was just playing around here? ;-)

In discussions here on PF, I always try to take statements seriously, if possible.
Unless they are explicitly marked by a smiley.

To discuss this seriously beyond just asserting something would require to first agree on a common interpretation of what it means to understand a subject. We can only speculate about the interpretation Feynman gave to the word, but we could try to get agreement between the two of us.

 

[A. Neumaier]

As I said, quantum fields and particles go hand-in-hand. I'm not sure why you would disagree with that

For example, I don't think quarks exist as (localized) particles, only as (delocalized) fields. Particles make sense only in a semiclassical approximation, which never applies for quarks.

Maybe I'll just quote the Wikipedia article you linked to answer this one:
"According to quantum field theory, the ground state of a system with interacting particles is not simply empty space. Rather, it contains short-lived "virtual" particle-antiparticle pairs which are created out of the vacuum and then annihilate each other.

This is a misrepresenting account of what quantum field theory claims. With my link, I didn't endorse the content of the Wikipedia article, but only the relation of vacuum polarization to the fine structure constant.

There hasn't been a single publication about the life-time of virtual particles - there is no such concept.

Seems like you keep saying some abstract mathematical quantity (like the polarization tensor above) contains all the physics, without being able to give a clear, succinct and intuitive description of what really physically causes vacuum polarization, etc, while at the same time saying mathematical calculations cause nothing.

What is intuitive about a formula depends on the training one has.

There is no valid intuition about the quantum regime devoid of mathematics.

 

[dm4b]

For example, I don't think quarks exist as (localized) particles, only as (delocalized) fields. Particles make sense only in a semiclassical approximation, which never applies for quarks.

Of course, that has been a given since the advent of quantum mechanics, and it in no way invalidates what I was saying. In it, when I said "particles" I mean it in the quantum sense of the term. Seems like this is boiling down into a useless argument on semantics?

What is intuitive about a formula depends on the training one has.

There is no valid intuition about the quantum regime devoid of mathematics.

There's an old saying (might have been Feynman): "If you can't explain your theory in layman's terms, there is either something wrong with your theory, or there is something wrong with you".

I don't think it is too much to ask to explain mechanism of some simple things in physical (and not mathematical) terms, such as force, vacuum polarization, etc, in a clear and concise way, that a layman can understand. I have yet to see that done here. Unless the claim is that abstract mathematics actually "drive" the Universe?

Granted the Internet isn't the best tool for this kind of thing somethimes. With that said, I will eventually check out your personal links, which I appreciate you providing.

 

[dm4b]

In discussions here on PF, I always try to take statements seriously, if possible.
Unless they are explicitly marked by a smiley.

To discuss this seriously beyond just asserting something would require to first agree on a common interpretation of what it means to understand a subject. We can only speculate about the interpretation Feynman gave to the word, but we could try to get agreement between the two of us.

This would take us into the realm of philosophy.

A photon cannot "sense the passage of time". What does that truly mean?

Spacetime is warped. What does that truly mean?

An electron passes through both slits in the double-slit experiment. What does that truly mean?

Nobody can give fully satisfactory answers to the above questions, and there are plenty more like them.

You see, just understanding the math, and being able to work the problems isn't enough for me.

I want to understand - in the fullest sense of the term - the reality behind these bizarre behaviors seen in nature.

I believe this is what Feynman meant when he said "he didn't [truly?] understand QM". I know that it's what I mean ;-)

I also think we should always have misgivings about our current theories. It helps us to see beyond them.

 

[lightarrow]

Alright, so a field is essentially made up of energy

I wouldn't say this, because they are two distinct things. Energy is just a "property" of a field (and just one among some).

and energy comes in discrete bundles. Please explain this: if you take away the discrete bundles of energy, what is left of the field?
I'm not trying to say that either the corpuscle OR the field is the only physical concept. I'm trying to say, how can you have one without the other?

You can't, however I see that Neumaier has already explained this (and certainly better than what I could do because I'm just a student).

 

[A. Neumaier]

This would take us into the realm of philosophy.

Indeed. But all talk about the precise nature of the virtuality of virtual particles is philosophy, since there is nothing observable about them.

A photon cannot "sense the passage of time". What does that truly mean?

It means something similar as the phrase ''a blind man cannot sense colors''.

It needs receptors to sense something. Being able to sense the passage of time is very closely related to being able to form the concept of a history.

Spacetime is warped. What does that truly mean?

Whether it is warped is a matter of interpretation. The existence of a symmetric tensor field need not be interpreted as warpedness.

An electron passes through both slits in the double-slit experiment. What does that truly mean?

It means that the electron is a wave, not a particle. There is nothing more behind it.

Nobody can give fully satisfactory answers to the above questions, and there are plenty more like them.

What are your criteria for being fully satisfactory? Can you at least give fully satisfactory criteria? If not, it might be the fault of the latter, not of understanding.

You see, just understanding the math, and being able to work the problems isn't enough for me.

What is missing?

I want to understand - in the fullest sense of the term - the reality behind these bizarre behaviors seen in nature.

Understanding a ghost may mean convincing oneself of its nonexistence, and reinterpreting the signs of ghostiness in a more profitable way.

I also think we should always have misgivings about our current theories. It helps us to see beyond them.

But this is useful only if there is a need to see beyond. Which means - only in those aspects of a theory where its predictions do not match reality.