SR, LET, FTL & Causality Violation

[Dale]

qft.

AFAIK all modern QFT are fully relativistic. So it seems impossible from first principles that a prediction from a relativistic theory would be non relativistic.

 

[TrickyDicky]

AFAIK all modern QFT are fully relativistic. So it seems impossible from first principles that a prediction from a relativistic theory would be non relativistic.

Do you consider GR a relativistic theory?
Do you know that certain predictions of QFT are incompatible with GR and viceversa and therefore some folks are desperate to find a theory to unify GR and quantum theory because both theories can't be completely right at the same time?

"Those are my principles, and if you don't like them... well, I have others. "
Groucho Marx

 

[Dale]

Do you know that certain predictions of QFT are incompatible with GR and viceversa and therefore some folks are desperate to find a theory to unify GR and quantum theory because both theories can't be completely right at the same time?

I am aware of that, but I don't see how GR is relevant to a prediction of QFT. If you are discussing a prediction of theory X then only theory X is important. If theory X is relativistic then the predictions of theory X must necessarily be relativistic.

 

[TrickyDicky]

I am aware of that, but I don't see how GR is relevant to a prediction of QFT. If you are discussing a prediction of theory X then only theory X is important. If theory X is relativistic then the predictions of theory X must necessarily be relativistic.

Non-linear QED (where I think the prediction about vacuum breakground comes from) is not "fully" relativistic in the restricted sense I think you are using then. (Neither is GR BTW)

 

[Dale]

Non-linear QED (where I think the prediction about vacuum breakground comes from) is not "fully" relativistic

That is why I asked the above question. Normal QED is relativistic, but I don't know about non-linear QED.

Can anyone confirm exactly which theory is being discussed?

 

[TrickyDicky]

That is why I asked the above question. Normal QED is relativistic, but I don't know about non-linear QED.

Can anyone confirm exactly which theory is being discussed?

I thought I confirmed it in the last post. How else do you think the high energies necessary to produce the breakground are dealt with?
I'm afraid you have a somewhat naive conception about theories in general and QFT in particular, non-linear QED is as "normal" as "normal" (linear) QED. It would be good if you defined what you mean by "relativistic" in this context, but the whole QFT includes procedures like Renormalization to adapt the "relativistic linearity" that produces nonsensical infinite results to the "non-linearity" of the real physical situations.

 

[Dale]

It would be good if you defined what you mean by "relativistic" in this context

If the basic equations that govern the experimental predictions of a theory are unchanged under a Lorentz transform then the theory is Lorentz covariant. If the basic equations that govern the experimental predictions of a theory are unchanged under arbitrary coordinate transforms then the theory is generally covariant. That is what I mean by "relativistic".

My knowledge of QM in general is not strong, but I am certainly not alone in the belief that QFT (including QED) is relativistic: https://qed.princeton.edu/main/Principedia/PHY_509:_Relativistic_Quantum_Theory

 

[Q-reeus]

Let's take this argument of yours to its logical conclusion. I set up an E field source and turn it on. In the source's rest frame, the field is a pure E field in a single direction (say along the x-axis). Since I am saying that E must equal E_crit in the source's rest frame (well, actually it's the frame in which the pairs are created at rest, but I think we've established that for a single source this is the source rest frame--we'll leave aside the lasers and electron beams for a bit, but I'll come back to them below), I do not expect to see breakdown until I measure E = E_crit (we're assuming that I'm not moving relative to the source.)
But you, standing next to me,...

That could be risky!

...can, it seems to me, argue as follows: as soon as I turn on the E field source, as soon as it is producing *any* nonzero E field at all, there will be *some* frame in which E > E_crit. (The field won't be a static, pure E field in that frame, since there will be a large B component, but we've agreed that B doesn't directly affect breakdown, and the E *component* will be greater than E_crit. More on this below.) And since the vacuum can "detect" E > E_crit in *any* frame, breakdown should occur immediately when I turn on the field source.

No no no, and honestly how can you not be well aware by now how many times I have emphasized *duration* as a necessary ingredient. Time to put just a bit of meat on that - been getting hand-wave fatigue. Consider then an evacuated parallel plate cap, 1cm x 1cm area, x 0.1cm plate separation. Charged to 10² v -> E = 10³ v/cm, that's a nice figure to start with. We know to get significant pair creation E = E_crit = 1.3*10¹⁶ v/cm. So wrt our lab frame, the cap has to be propelled to a relative velocity gamma factor of 1.3*10¹⁶/10³ ~ 1.3* 10¹³. By the LT's the plates have thus contracted in direction of motion in lab frame to l = 1cm/gamma. So any patch of vacuum in lab frame experiences Ecrit from cap for a duration of dt = l/c = 1cm/(gamma*c) = 1/(1.3*10¹³*3*10¹⁰cm/s) = 2.6*10^-24 seconds. Not long. A virtual electron can gain in that blip of time at most a normalized velocity of v/c = a*dt/c = (e*Ecrit/me)*dt/c = (1.6*10^-19C*1.3*10¹⁸v/m/9.1*10^-31Kg)*2.6*10^-24s/(3*10⁸m/s) = 0.02. That calc assumed relativistic mass would not become significant - justified in the circumstances. Square that and one has roughly the fractional shortfall before pair creation is likely.

So do I accept your devastating critique? Not likely. Recall that from the outset in #255 I emphasized this is a gedanken experiment and practical realization of any test along such lines is utterly unfeasable. What matters is the logic involved. If Ecrit as intensive factor makes any sense then it inevitably leads to the notion of an underlying reference frame(s) in order to avoid absurdities. If I follow your reasoning continued in the #321 post we have a situation where Ecrit *cannot* simply be an intensive that vacuum responds to in an unmysterious way. Maybe nature is that weird, but I prefer otherwise. You really, really don't want to contemplate the rotating hoops scenario in #318, I can tell that. But give it more thought maybe.

 

[TrickyDicky]

If the basic equations that govern the experimental predictions of a theory are unchanged under a Lorentz transform then the theory is Lorentz covariant. If the basic equations that govern the experimental predictions of a theory are unchanged under arbitrary coordinate transforms then the theory is generally covariant. That is what I mean by "relativistic".

My knowledge of QM in general is not strong, but I am certainly not alone in the belief that QFT (including QED) is relativistic: https://qed.princeton.edu/main/Principedia/PHY_509:_Relativistic_Quantum_Theory

Ok, I see where you are coming from wrt QM. There's an interesting paper by fellow PFer Demystifier that might help you clarify some things: http://arxiv.org/abs/quant-ph/0609163 points 7 and 8

 

[Haelfix]

Quantum field theory is fully relativistic. It has to be, as it is trivially built out objects that transform properly under lorentz transformations.

For instance, in the path integral formulation of the theory, the lagrangian is required to be a lorentz scalar. You can also show that all the other objects (creation/annihilation operators, commutation relationships, scattering amplitudes etc) are necessarily LI.

You can of course drop some of these assumptions, and the resulting field theories can break lorentz invariance, but they will always suffer from pathologies (for instance CPT is broken and a great deal of finetuning is required to prevent certain non LI operators from producing effects which are clearly not observable)

 

[TrickyDicky]

Of course, but frankly this is not what this thread is about.
You might want to take a look at Nikolic's paper too.

 

[Dale]

Ok, I see where you are coming from wrt QM. There's an interesting paper by fellow PFer Demystifier that might help you clarify some things: http://arxiv.org/abs/quant-ph/0609163 points 7 and 8

OK, I read those points but I don't get your implication. Are you saying that the math for deriving the experimental predictions of QED is indeed covariant, but that there are "interpretational" problems?

 

[PeterDonis]

No no no, and honestly how can you not be well aware by now how many times I have emphasized *duration* as a necessary ingredient.

But without ever explaining why. However...

Time to put just a bit of meat on that

Now you have, and what your explanation amounts to is: the vacuum has to "detect" E_crit for a minimum amount of time. What is this minimum amount of time? It's the average lifetime of the virtual particles. But time is frame-dependent; so in order to apply this criterion, we *have* to know *in what frame* to apply it!

Now go back and read what I've said several times about the second paper you linked to, and how it shows that the duration is what *determines* E_crit. What did I say? I said: the strength of E determines how fast the field can pump energy into the virtual particles; the average lifetime of the virtual particles determines how fast the field *has* to pump energy into the virtual particles; put these two things together and you get a critical value for the field, E_crit.

Now, once again: *in what frame* is the "average lifetime" defined? In the frame in which the virtual particles are created at rest, which for a single source is the same as the frame in which the source is at rest, which is what I've been saying all along. And this is basically what you are now saying! You have a single source: a parallel plate capacitor. You argue that in a frame in which the source is moving, the E field of the capacitor is higher, but the vacuum experiences it for a shorter time. So if the field is E_crit in a frame in which the source is moving, the vacuum won't detect it for long enough to induce breakdown. The field has to be E_crit in a frame in which the source is at rest for the duration requirement to be satisfied.

In other words, you have basically been in agreement with me all along.

If Ecrit as intensive factor makes any sense then it inevitably leads to the notion of an underlying reference frame(s) in order to avoid absurdities.

Yes, exactly: the "underlying reference frame" is the rest frame of the source.

If I follow your reasoning continued in the #321 post we have a situation where Ecrit *cannot* simply be an intensive that vacuum responds to in an unmysterious way.

You have obviously not been following my reasoning. As I showed above, my reasoning is the same as yours: the vacuum has to sense E >= E_crit for a certain minimum duration. The formula for E_crit is *defined* in such a way that it equates to the E field that needs to be sensed for the minimum duration *as seen in the frame in which the source is at rest*. If you LT into a frame in which the source is moving, both E and the duration change in concert to keep the covariant expression of the breakdown criterion the same.

 

[PeterDonis]

Nope, I was under the impression that I was clear that the source should be at rest in that frame (maybe you are mixing my position with Q-reeus' ). Actually IMO what you say suggests a way to check experimentally the putative existence of an absolute frame, which would be the only one where "vacuum breakdown" in earnest could be achieved.

I'm certainly not trying to say that would be the case; I'm saying that any source which can achieve E >= E_crit in the frame in which it (the source) is at rest should be able to induce breakdown. If we did in fact discover that only sources in a particular state of motion, for example a state of motion in which the CMBR was isotropic, could induce breakdown by achieving E >= E_crit, while sources in other states of motion (such as at rest on the Earth) either couldn't induce breakdown at all, or had to achieve E >> E_crit in the source rest frame to induce breakdown, that would indeed be an experimental refutation of SR *and* current quantum field theory, both of which say that the vacuum is Lorentz invariant, implying E > E_crit in the rest frame of the source is all that is required.

 

[TrickyDicky]

OK, I read those points but I don't get your implication. Are you saying that the math for deriving the experimental predictions of QED is indeed covariant, but that there are "interpretational" problems?

Oh, no. I just perceived you might be saying quantum relativistic mechanics is exactly the same thing as QFT.
IMO discerning whether nonlinear QED qualifies as relativistic or not is maybe a theoretically debatable point but outside the scope of this thread's discussion or even this relativity subforum.

 

[TrickyDicky]

I'm certainly not trying to say that would be the case; I'm saying that any source which can achieve E >= E_crit in the frame in which it (the source) is at rest should be able to induce breakdown. If we did in fact discover that only sources in a particular state of motion, for example a state of motion in which the CMBR was isotropic, could induce breakdown by achieving E >= E_crit, while sources in other states of motion (such as at rest on the Earth) either couldn't induce breakdown at all, or had to achieve E >> E_crit in the source rest frame to induce breakdown, that would indeed be an experimental refutation of SR *and* current quantum field theory, both of which say that the vacuum is Lorentz invariant, implying E > E_crit in the rest frame of the source is all that is required.

Ok, I see, we agree about the empirical dependence of this whole issue. Absent that evidence, I understand that surely you are not saying that would be the case, I myself am merely exploring logical alternatives. Just in case...

 

[TrickyDicky]

Also note that the hypothetical existence of an absolute frame would be compatible with a Lorentz invariant vacuum, the violation of LI would only pertain to 1/2 spin particles.

 

[PeterDonis]

Also note that the hypothetical existence of an absolute frame would be compatible with a Lorentz invariant vacuum, the violation of LI would only pertain to 1/2 spin particles.

But the vacuum contains virtual spin 1/2 particles, so any violation of LI with respect to them would imply a violation of LI for the vacuum as well.

 

[TrickyDicky]

But the vacuum contains virtual spin 1/2 particles, so any violation of LI with respect to them would imply a violation of LI for the vacuum as well.

They only come in virtual pairs. I don't think you can consider virtual particles as spin 1/2 but a reference would help. All virtual bosons obviously don't have spin 1/2 and they are interchangeable with any virtual pair.
Real pairs of course do have spin 1/2.

 

[Q-reeus]

Now you have...

And thankfully for me no-one did close checks. My calcs were out by a factor of 10⁴ the wrong way (exponentiation errors - damm calculator), so have edited initial voltage to give the final correct result. By bumping up the voltage or lengthening the capacitor the required gamma factor goes down, but one is still widely outside anything experimentally attainable.

, and what your explanation amounts to is: the vacuum has to "detect" E_crit for a minimum amount of time. What is this minimum amount of time? It's the average lifetime of the virtual particles. But time is frame-dependent; so in order to apply this criterion, we *have* to know *in what frame* to apply it!

Average lifetime of vp's is *not* frame dependent from vacuum pov - things look the same in every frame - same spectrum of vp's regardless. Otherwise, why are we even arguing about detecting the 'ether frame' - there would be an easily detectable 'flow' not needing breakdown to discover.

Now go back and read what I've said several times about the second paper you linked to, and how it shows that the duration is what *determines* E_crit. What did I say? I said: the strength of E determines how fast the field can pump energy into the virtual particles; the average lifetime of the virtual particles determines how fast the field *has* to pump energy into the virtual particles; put these two things together and you get a critical value for the field, E_crit.

That passage is ok - on it's own.

Now, once again: *in what frame* is the "average lifetime" defined? In the frame in which the virtual particles are created at rest, which for a single source is the same as the frame in which the source is at rest, which is what I've been saying all along.

And imo it doesn't hold water for the reason noted earlier that the vacuum, thanks to it's frequency spectrum, looks the same in any frame (until breakdown that is!).

And this is basically what you are now saying! You have a single source: a parallel plate capacitor. You argue that in a frame in which the source is moving, the E field of the capacitor is higher, but the vacuum experiences it for a shorter time. So if the field is E_crit in a frame in which the source is moving, the vacuum won't detect it for long enough to induce breakdown. The field has to be E_crit in a frame in which the source is at rest for the duration requirement to be satisfied.
In other words, you have basically been in agreement with me all along.

No. The (corrected) example I gave showed a shortfall, but adjusting as earlier described and we would be in breakdown regime. From SR perspective this leads to a bizarre picture. Positing an local LET preferred frame in which breakdown is maximal allows a sensible restraint on what is possible.

You have obviously not been following my reasoning. As I showed above, my reasoning is the same as yours: the vacuum has to sense E >= E_crit for a certain minimum duration. The formula for E_crit is *defined* in such a way that it equates to the E field that needs to be sensed for the minimum duration *as seen in the frame in which the source is at rest*.

That last bit is simply wrong. Again - the normal vacuum has a vp spectrum that means it appears the same in every frame. And for that reason I maintain it cares not about whether an applied E is from a source at rest or moving in any given frame.

If you LT into a frame in which the source is moving, both E and the duration change in concert to keep the covariant expression of the breakdown criterion the same.

Can you make sense of that position for the example of rotating hoops capacitor I gave in #318 (last paragraph)? Do that and maybe we see eye to eye. Yes, a Xmas present wil be yours!

 

[PeterDonis]

They only come in virtual pairs. I don't think you can consider virtual particles as spin 1/2 but a reference would help. All virtual bosons obviously don't have spin 1/2 and they are interchangeable with any virtual pair.
Real pairs of course do have spin 1/2.

Virtual electrons and positrons are spin 1/2, just like real ones. The fact that they are virtual particles doesn't affect their spin. Also, virtual bosons aren't "interchangeable" with virtual fermions; they are separate kinds of virtual particles, just as they're separate kinds of real particles. I'll try to find a reference, but AFAIK this is basic QFT; virtual particle states are described by the same kinds of fields as real ones, just off the mass shell (which means their energy and momentum don't satisfy the equations of motion); being on or off the mass shell doesn't affect spin, since spin has to do with a different set of symmetries (pure spatial rotations vs. pure boosts).

 

[PeterDonis]

Average lifetime of vp's is *not* frame dependent from vacuum pov - things look the same in every frame - same spectrum of vp's regardless.

<rest of post mainly along similar lines>

And taken to its logical conclusion, this would again imply that breakdown should occur *immediately* upon turning on any field source, regardless of its state of motion, since there will always be *some* vp that will sense E > E_crit. That's obviously false, so again something must be wrong with your argument. It's true that the virtual particle spectrum "looks the same" in every frame, but which particular modes of that spectrum get energy transferred into them from the source is *not* the same in every frame; it depends on the state of motion of the source.

What does "the vp spectrum looks the same in every frame" actually mean? A Lorentz transformation acts on the virtual pairs just like it does on everything else: if I pick out a particular virtual pair that is at rest in frame F, and I apply a LT, that particular pair will *not* be at rest in frame F'; it will have a much higher energy, and hence a much shorter lifetime. There will be some *other* pair that gets shifted by the LT into being at rest in frame F', so the totality of all the virtual pair field modes (since there are an infinite number) will look the same, including the *average* lifetime integrated over the infinite spectrum of vp modes.

So the lifetime of any *particular* virtual pair depends on which frame it is seen from; the higher the pair's energy in a given frame, the shorter its lifetime in that frame. So the pair with the *longest* lifetime in any given frame will be the pair which has minimum energy in that frame, which is the pair at rest in that frame. That will be the pair that is easiest for the field to turn into a real pair. But since we are dealing with a quantum phenomenon, there is some (smaller) probability of the field being able to turn a vp with higher energy in that frame into a real pair; when you average over all the modes, you get that the average lifetime of vp's is what determines E_crit, as the second paper you linked to says.

But again, that reasoning depends on the fact that we are applying E >= E_crit in the source's rest frame. Put another way, it depends on the fact that the field produced by the source does not affect the minimum energy required to create a vp. That is true in the source's rest frame (assuming that the field is a pure E field in that frame). But how do things look from a frame in which the source is moving? As I said in a previous post, in that frame the field is no longer a pure E field, so there will be a nonzero current in that frame due to the nonzero B field. That means the minimum energy of a pair in that frame, *given the field produced by the source*, is no longer just the pair rest energy; it is the pair rest energy plus the kinetic energy added by the current. Which in turn means that the average lifetime of pairs in a frame in which the source is moving is *smaller* than it is in the source's rest frame. So to calculate what the critical E field would be in *that* frame, we have to *increase* the required field strength in that frame to compensate for the reduced average lifetime.

Can you make sense of that position for the example of rotating hoops capacitor I gave in #318 (last paragraph)?

I haven't even considered it yet. One scenario at a time.

 

[Dale]

IMO discerning whether nonlinear QED qualifies as relativistic or not is maybe a theoretically debatable point but outside the scope of this thread's discussion or even this relativity subforum.

Linear QED certainly is, but I can't contribute to the theoretical debate beyond that. In any case, I think that the "handwaving" nature of the vacuum breakdown argument is rather absurd. This is something that simply needs to be calculated out rigorously.

 

[TrickyDicky]

Virtual electrons and positrons are spin 1/2, just like real ones. The fact that they are virtual particles doesn't affect their spin. Also, virtual bosons aren't "interchangeable" with virtual fermions; they are separate kinds of virtual particles, just as they're separate kinds of real particles. I'll try to find a reference, but AFAIK this is basic QFT; virtual particle states are described by the same kinds of fields as real ones, just off the mass shell (which means their energy and momentum don't satisfy the equations of motion); being on or off the mass shell doesn't affect spin, since spin has to do with a different set of symmetries (pure spatial rotations vs. pure boosts).

Read carefully what I wrote, I talked about exchange with virtual pairs, not individual virtual particles, hopefully you have previously seen a diagram of a virtual photon with a loop representing the virtual pair. I have never heard of a virtual electron or positron on their own in the context of QED and vacuum polarization. In this physical context virtual particles are considered to have 0 spin. You may choose to think of the virtual electrons and positrons in a virtual pair as having each spin 1/2 with different sign but the result is the same, the virtual pair has spin 0. Real, measurable electrons and positrons have spin 1/2 but fotunately for us they are not all paired, there are many more electrons than positrons otherwise the universe would be in real trouble.

 

[TrickyDicky]

I'll correct my initial statement so the subsequent argument is not necessary.

Also note that the hypothetical existence of an absolute frame would be compatible with a Lorentz invariant vacuum, the violation of LI would only pertain to REAL 1/2 spin particles.

 

[Q-reeus]

And taken to its logical conclusion, this would again imply that breakdown should occur *immediately* upon turning on any field source, regardless of its state of motion, since there will always be *some* vp that will sense E > E_crit. That's obviously false, so again something must be wrong with your argument.

Peter - you seem to have memory problems. Above is practically verbatim what you claimed in #321 - what did I demonstrate in #338 re that false reasoning? A specific setup chosen at whim, but it showed your position is simply not true. It should have been evident there that merely boosting to any higher gamma factor will not change the result. Seems you were not convinced. If you really need it, I can easily prove there is an invariant |E|*|l| product (E and source characteristic length l being orthogonal) that must be exceeded before breakdown is possible in *any* other inertial frame. No rocket science is required. Unless you want a fairly lengthy and messy but otherwise straighforward derivation presented, please concede and do not repeat the false reductio ad absurdum arguments of #321 and #352.

It's true that the virtual particle spectrum "looks the same" in every frame, but which particular modes of that spectrum get energy transferred into them from the source is *not* the same in every frame; it depends on the state of motion of the source.

Which means no more than: singling out any particular vp (particular mode in that frame), it transforms according to LT's. Sure - but askew of the real issue.

What does "the vp spectrum looks the same in every frame" actually mean? A Lorentz transformation acts on the virtual pairs just like it does on everything else: if I pick out a particular virtual pair that is at rest in frame F, and I apply a LT, that particular pair will *not* be at rest in frame F'; it will have a much higher energy, and hence a much shorter lifetime. There will be some *other* pair that gets shifted by the LT into being at rest in frame F', so the totality of all the virtual pair field modes (since there are an infinite number) will look the same, including the *average* lifetime integrated over the infinite spectrum of vp modes.

And that's my point; vacuum looks the same, acts the same, in any inertial frame (again - true only till breakdown condition). It follows that vacuum cannot care whether source of E is in motion or not in any frame - E is acting on the same vp spectrum regardless. The rest of your #352 gets back to minimum energy argument I dealt with many times earlier and showed were false (e.g. in #304,313,318).

There's a possible way out for both of us here, but only partially I suspect. While a frame dependent B field makes no contribution to the energy requirements of pair creation, it might just effectively act in another way as a frustration mechanism. In plasma physics it is well known that motion of plasma particles under crossed E and B static fields can lead to cycloidal drift in a direction normal to both. Without working out the details, maybe in a certain regime vacuum vp's are thus prevented from separating in sufficient time. It would at most be partial cycloidal motion, but maybe enough to make a substantial difference. That needs balancing though against the reduction in E_crit associated with relative motion that provides the B in the first place. This idea likely fails badly though when a system is just short of E_crit in source rest frame and needs just a mild velocity boost to exceed E_crit. Needs a closer look.
[On further thought, it would seem to afford effective suppression of breakdown current magnitude for large l, and large gamma where E/B ~ c, but not prevent breakdown as such]

 

[PeterDonis]

Peter - you seem to have memory problems.

I remember perfectly well what I wrote, but it's obvious that we are having communication problems. I suspect we are partly talking past each other; we use similar words but mean different things by them, and we describe things that are actually similar using very different words. Also, we are using English instead of math to express our ideas, which is much less precise. I'll make an attempt at fixing that below.

As an example of the communication problem, take this:

It should have been evident there that merely boosting to any higher gamma factor will not change the result.

Which is exactly what I've been saying all along: "boosting to a higher gamma factor", in the sense of making the source of the EM field move faster and faster, will not change the result: you will still need E > E_crit *in the source's rest frame*, regardless of the source's state of motion. And yet you are evidently disagreeing with me.

Seems you were not convinced. If you really need it, I can easily prove there is an invariant |E|*|l| product (E and source characteristic length l being orthogonal) that must be exceeded before breakdown is possible in *any* other inertial frame...Unless you want a fairly lengthy and messy but otherwise straighforward derivation presented...

I'll present my own below; it will not be "lengthy and messy", but will only take a few lines.

And that's my point; vacuum looks the same, acts the same, in any inertial frame (again - true only till breakdown condition).

And this is the key thing you missed from my post #352: once you turn on a source of an EM field, *the vacuum no longer looks the same in any inertial frame*. What the papers you linked to call "vacuum polarization" happens *as soon as you turn the field on*. What you are calling "breakdown" is not the *start* of the polarization: it's the point at which the polarization gets strong enough to make virtual pairs into real ones.

But that's all English again; here's the math. This is the breakdown condition, in covariant terms, for a single source such as a capacitor:

$$F_{ab} u^{a} e^{b} >= E_{crit}$$

where $F_{ab}$ is the EM field tensor, $u^{a}$ is the source's 4-velocity, $e^{b}$ is the spacelike 4-vector that defines the orientation of the source (for example, if the source is a capacitor it points in the direction normal to the plates, from one plate to the other), and $E_{crit}$ is defined purely in terms of physical constants, as in the first paper you linked to. (Btw, in so far as any "properties of the vacuum" affect the result, they do so purely through their contribution to $E_{crit}$. This includes any effect of "minimum duration", as I said before.)

The LHS of the above equation is a scalar invariant; i.e., you can compute it using the components of $F_{ab}$, $u^{a}$, and $e^{b}$ in any frame you like, and it will give the same result, a number. Call that number $E_{0}$. Then we can re-write the above, for additional clarity, as

$$F_{ab} u^{a} e^{b} = E_{0} >= E_{crit}$$

Since we can calculate the invariant in any frame, I'll pick the easiest one: the source's rest frame. In that frame, the components of all three geometric objects are simple. $F_{01} = - F_{10} = E_{0}$, with all other components of the EM field tensor zero. $u^{0} = 1$, with all other components of the source 4-velocity zero. And, if we define the "x" axis of the source rest frame as the one along which the source is oriented (which we can do without loss of generality), then $e^{1} = 1$, with all other components of the orientation vector zero. Compute the contraction of these three objects and you will obtain $E_{0}$.

Now in a frame where the source is moving, the computation won't be as simple. But it will still give the same final answer. Apply the same Lorentz transform to $F_{ab}$, $u^{a}$, and $e^{b}$, and then contract the transformed objects; the result will still be $E_{0}$. So the covariant expression above is the precise way of saying what I've been saying all along, that E > E_crit *in the source's rest frame* is the correct breakdown criterion.

 

[harrylin]

Oh, no. I just perceived you might be saying quantum relativistic mechanics is exactly the same thing as QFT.
IMO discerning whether nonlinear QED qualifies as relativistic or not is maybe a theoretically debatable point but outside the scope of this thread's discussion or even this relativity subforum.

I also think that there's no need to go into non-linear theories, linear theories suffice: the "LET" of this thread closely corresponds to a major (or "mainstream") interpretation of relativistic QM (see my post #272).

 

[PeterDonis]

Read carefully what I wrote, I talked about exchange with virtual pairs, not individual virtual particles, hopefully you have previously seen a diagram of a virtual photon with a loop representing the virtual pair. I have never heard of a virtual electron or positron on their own in the context of QED and vacuum polarization. In this physical context virtual particles are considered to have 0 spin.

Yes, the *pair*, considered as a single quantum system, has zero spin; it has to, by conservation of angular momentum. Although I suppose, since we're dealing with virtual particles that can violate conservation laws within the limits of the uncertainty principle, that there could also be virtual particle states where the spins of the electron and positron were parallel instead of antiparallel, so the pair as a whole would have spin 1 instead of spin 0. I haven't really seen a good discussion of this in a reference.

You may choose to think of the virtual electrons and positrons in a virtual pair as having each spin 1/2 with different sign but the result is the same, the virtual pair has spin 0.

The *pair* does. But the individual electron and positron each have spin 1/2; the pair as a whole has spin 0 because the two spins are antiparallel (at least, in what I think would be the usual case--see above). So the pair as a whole being spin 0 *depends* on each of the individual particles being spin 1/2 and on the spins cancelling--which means that if there were a violation of Lorentz invariance for spin 1/2, then the pair would not be spin 0 in all frames, so there would be a violation of LI for spin 0 as well.

 

[Q-reeus]

Which is exactly what I've been saying all along: "boosting to a higher gamma factor", in the sense of making the source of the EM field move faster and faster, will not change the result: you will still need E > E_crit *in the source's rest frame*, regardless of the source's state of motion. And yet you are evidently disagreeing with me.

That's because you are not saying the same thing as me at all. I accept you think we are just talking past each other, but not so. Your criteria is simply E >= E_crit in source rest frame - period. The scalar invariant you express later in #357 follows from that imposed condition. Whether source of E there is a micron or a million miles long is irrelevant in such a view. Your statement further down "This includes any effect of "minimum duration", as I said before." is true only in the limited sense that, with the source rest frame E >= E_crit imposed, LT's naturally determine a frame dependent vp 'duration' that will be [STRIKE]less[/STRIKE] greater seen in another frame, as will E there - offset by a higher vp energy (thus inertia) seen there. That invariance recipe seems right because there is no 'moving observer causes physics' paradox. But it fails to consider the pov from a vp pair that cares only that E >= E _crit for a minimum HUP time span in it's own rest frame, and can't care less what the source rest frame sees. [And btw, it's still ok to accept my request to apply your position to the rotating hoops (or annulus pair) capacitor scenario I gave in #318 (oh yeah, that Xmas present can still be yours).]

I am claiming, based on example in #338 and codified in #356 as |E|*|l| >= 'volts'_min (even though it is not really volts per se), something quite different. That perspective that matters is that of vp's in their rest frame. Which in turn means that a source must have a minimal product of apllied E *and* length normal to E, measured in any given frame, before, seen in any other inertial frame, a 'rest' vp pair will receive a minimal impulse capable of boosting to real status. And then only if E >= E_crit in that vp 'rest' frame, or alternately where gamma factor of any vp pair passing through in source rest frame yields the equivalent. This criteria is very different from demanding E >=E_crit in source rest frame. On that basis it remains the case your reductio ad absurdum argument that immediately upon switching on any source of E breakdown should occur is wrong. Putting it more concretely, from #338 example, a capacitor must roughly have M = |E|*|l|>= 10⁴ (v/cm)cm before a vp pair passing through at any relative gamma factor whatsoever can be elevated to real pair status.

Our criteria are thus fundamentally different - if nothing else on an elementary dimensional analysis level. For you, the source E 'does all the energy pumping'. For me, it is often the inherent KE of relative lateral motion (vacuum vp spectrum alone can provide that) that mostly 'does the pumping' - applied E has more the role of catalyst. Yes the E source discharges when breakdown current flows, but the source power drain per vp-pair->rp-pair creation is vastly different depending on relative motion of source. From my outlook there is a transverse energy budget that cannot be ignored. In past entries I have argued that transverse energy is a result of KE energy pumped into gross motion of the source making it move relative to the underlying local LET rest frame. But some more thought and it becomes obvious there are relevant transverse motions omnipresent and inherent in vacuum as vp 'sea'.

But this leads up to my new outlook. A logical conclusion from M = |E|*|l|>= ~ 10⁴ (v/cm)cm is that there must be some breakdown occurring just having a high voltage structure sitting around with such a perfectly achievable parameter mix. Why? Courtesy of the vacuum. We all agree vp spectrum is frame invariant. This immediately requires that in any frame there must be a finite ultra-relativistic random flux component of vp's, and some fraction will be exceeding breakdown criteria merely by passing between the plates of a suitably dimensioned and charged capacitor in the right directions. Some will strike the plates, some will simply pass through as newly created ultra-relativistic real pairs - subsequently smashing into say air molecules, or recombining to produce a presumably faint but finite ultra-high energy gamma-ray flux. The latter in particular seems disturbing because there is no evident energy drain from the E source involved at all. The vacuum itself seems to provide all the 'oomph' needed there. My own 'magnetic suppression' counter-argument in #356 cammot be germaine I think because seen in source rest frame, a high gamma factor vp passing through and elevated to real status simply executes an ever so slightly parabolic path; that's all.

Bizarre as that seems, imo above is the logical conclusion if the vacuum vp picture we have been working from is true. Somehow I doubt gamma rays emanating from certain electrostatically charged structures would have gone undetected till now, so not rushing to patent a 'free-energy-from-vacuum' device. After all the vacuum is generally believed to have either precisely or next to zero real energy density - despite the notorious ~ 10¹²⁰ order of magnitude problem.

Upshot is my new perspective renders prior arguing for LET over SR on vacuum breakdown criteria sort of moot. If there is a top notch QFT expert here that can knock the above on it's head - please step up now. Otherwise, maybe someone can pick the above matter up, wrap it a bit differently, and run it under their own banner on another thread. So be it. I'd rather end this windy-twisty saga on that note.