Do Scientists Believe in Tachyons?

[ScientificMind]

For this question, I should probably clarify my question. I want to know, more or less, what the general consensus is on the existence of tachyons. What do most scientists in fields that are relevant to this question think about tachyons, and - as a side question - what might their reasoning be for their positions?

 

[rootone]

Tachyons are hypothetical particles which can only travel faster than light.
They are not part of the standard model of particle physics and there is no experimental evidence of them.
If they do exist then the whole ediface of SR and GR would be wrong, yet there are mountains of persuasive evidence confirming the predictions of relativity.
As far as I know they mostly are considered as a mathematical artifact.

 

[mfb]

In particle physics, they are considered to be even more exotic than those searches.
I don't think anyone really expects them to exist.

 

[Demystifier]

Tachyonic fields are not really so exotic as usually thought. For example, the Higgs field in the standard model before symmetry breaking is a tachyonic field, in the sense that the mass term in the Lagrangian has a negative sign. But such a tachyonic field configuration is unstable, so it quickly roles down to the minimum of the potential, thus breaking the symmetry and becoming a stable a massive Higgs.

 

[Avodyne]

Tachyonic fields are not really so exotic as usually thought. For example, the Higgs field in the standard model before symmetry breaking is a tachyonic field, in the sense that the mass term in the Lagrangian has a negative sign.

Yes, but not in the sense that it goes faster than light, which is what people usually mean by a tachyon.

 

[Demystifier]

Yes, but not in the sense that it goes faster than light, which is what people usually mean by a tachyon.

If you solve the classical field equation in the vicinity of the maximum of the Higgs potential, you will see that the field does propagate faster than light.

More precisely, the oscillatory modes propagate faster than light. There are also modes exponentialy growing with time, which do not propagate faster than light. Since the exponential modes describe instability which results in settling down to the minimum of the potential, it is believed that this exponential not-faster-than-light solution is the physical solution. Yet, it is not clear if there is any physical principle which could generally prevent a realization of the oscillatory faster-than-light solutions.

 

[Avodyne]

Do you have a reference for this claim?

I would expect that the classical initial-value problem does not allow signals to propagate faster than light, even if the initial data includes modes with wavenumber $k>|m|$ (which is what I assume you mean by "oscillatory modes").

EDIT: here is paper that states that there is no superluminal propagation for the classical initial-value problme even with negative mass-squared (see p.8):
"No superluminal propagation for classical relativistic and relativistic quantum fields"
John Earman
http://www.sciencedirect.com/science/article/pii/S1355219814000811
pdf: http://philsci-archive.pitt.edu/10945/1/NSP_SHPMP_Final_Version.pdf

 

[Demystifier]

Do you have a reference for this claim?

Most of what I claim can be easily derived from the tachyon dispersion relation and assumption that group velocity is the velocity of propagation.

I would expect that the classical initial-value problem does not allow signals to propagate faster than light, even if the initial data includes modes with wavenumber k>|m| (which is what I assume you mean by "oscillatory modes").

Yes, this is what I mean by oscillatory modes. For such modes the group velocity is larger than c. So why do you expect that classical initial-value problem does not allow signals to propagate faster than light?

 

[ShayanJ]

Yes, this is what I mean by oscillatory modes. For such modes the group velocity is larger than c. So why do you expect that classical initial-value problem does not allow signals to propagate faster than light?

What about this?
It says:

localized tachyon disturbances are subluminal and superluminal disturbances are nonlocal.

 

[Demystifier]

What about this?

I agree with this. Indeed, it is easy to understand it intuitively. For tachyons, space and time exchange their roles (this is the most obvious in 1+1 dimensions). Normal massive matter is local in space but nonlocal in time (matter lives "forever" due to conservation in time), and has subluminal velocity dx/dt<1. Likewise, tacyon matter is local in time but nonlocal in space, and has subluminal inverted velocity dt/dx<1. The Cauchy surface must be spacelike for normal matter ("initial" condition), but timelike for tachyons ("boundary" condition). Normal conscious beings experience "the flow of time", while tachyon conscious beings would probably experience "the flow of space".

The real problem appears when you have an interaction between normal and tachyon matter. Should the Cauchy surface be spacelike or timelike? Locally it doesn't matter (Cauchy-Kovalevska theorem provides the local existence of a solution near the hypersurface), but globally the Cauchy problem is not well defined. This means that the initial/boundary condition cannot be arbitrary, because otherwise you can obtain inconsistencies (e.g. the grandfather paradox).

The above were features of classical fields, but quantization of tachyons leads to additional challenges. Canonical quantization gives a special status to the time coordinate, but for tachyons time and space should exchange their roles. Even if this is not a problem in 1+1 dimensions (at least if there are no interactions with normal matter), this is very problematic in 3+1 dimensions because, even in a fixed Lorentz frame, now we have 3 natural "canonical momenta" and 3 natural "Hamiltonians". How to perform canonical quantization with this? An option is to do covariant path-integral quantization, but it only works for calculation of n-point functions. If you want to calculate physical probabilities via something like LSZ formula, the problems of canonical quantization are turned back. For instance, should probabilities be conserved (in the sense of unitarity) in time or in space? Both options are problematic for tachyons.

Finally, let me emphasize that all this is not purely academic. Suppose that I want to describe the Higgs field before symmetry breaking, i.e. at sufficiently large energies. In that regime the Higgs field looks tachyonic, it is a quantum field, and it interacts with non-tachyonic matter. If one wants to describe the actual experiment with an accelerator such as LHC, the quantization and all the theory can still be done in the laboratory frame, which may be sufficient for a comparison with the experiment. But the theory (describing quantum interacting tachyons in the laboratory frame) will not be Lorentz invariant. Perhaps this is not inconsistent in an instrumental interpretation of quantum theory (according to which the role of quantum theory is not to describe nature as such, but to describe what can we say about nature in a given experimental setup), but it is at least disturbing.

 

[Demystifier]

I've just discussed with a colleague the idea of producing such tachyons in LHC. We finally agreed that such a tachyon could be described as a complicated bound state of a large number of massive Higgs particles, analogous to a QCD glueball made of a large number of gluons. To stress the analogy, he invented a suggestive name for such a tachyon - the higgsball.

Another related thought. Even if such tachyons cannot in practice be created in LHC, perhaps something similar could be created in condensed matter. What I have in mind is an acoustic tachyon, an excitation moving faster than sound described by a tachyon dispersion relation associated with the velocity of sound. The dispersion relation is Lorentz invariant with respect to Lorentz transformations with an invariant velocity of sound. The violation of Lorentz invariance due to quantization would have a clear physical interpretation; there is a preferred frame, the one with respect to which the condensed-matter material is at rest.

 

[Avodyne]

Finally, let me emphasize that all this is not purely academic. Suppose that I want to describe the Higgs field before symmetry breaking, i.e. at sufficiently large energies. In that regime the Higgs field looks tachyonic, it is a quantum field, and it interacts with non-tachyonic matter. If one wants to describe the actual experiment with an accelerator such as LHC, the quantization and all the theory can still be done in the laboratory frame, which may be sufficient for a comparison with the experiment. But the theory (describing quantum interacting tachyons in the laboratory frame) will not be Lorentz invariant.

This disagrees with textbook QFT. A theory with a Lorentz-invariant lagrangian density that results in spontaneous symmetry breaking (such as the Standard Model) remains Lorentz invariant (at all energies and for all processes), and there are no tachyons (that is, localized excitations of any sort that move faster than light).

 

[mfb]

Another related thought. Even if such tachyons cannot in practice be created in LHC, perhaps something similar could be created in condensed matter. What I have in mind is an acoustic tachyon, an excitation moving faster than sound described by a tachyon dispersion relation associated with the velocity of sound. The dispersion relation is Lorentz invariant with respect to Lorentz transformations with an invariant velocity of sound. The violation of Lorentz invariance due to quantization would have a clear physical interpretation; there is a preferred frame, the one with respect to which the condensed-matter material is at rest.

A preferred frame in solid matter is not really surprising.

A technical remark: a group velocity larger than c is possible in special relativity if there is nonlinear absorption or emission. This has been observed with light, see the references there.

 

[Demystifier]

This disagrees with textbook QFT. A theory with a Lorentz-invariant lagrangian density that results in spontaneous symmetry breaking (such as the Standard Model) remains Lorentz invariant (at all energies and for all processes), and there are no tachyons (that is, localized excitations of any sort that move faster than light).

That is a valid objection, but see post #11. A tachyon excitation should be a non-perturbative effect that cannot easily be described by the standard textbook techniques, very much like a glueball in QCD. And I am not saying that it is easy to create such a tachyon in LHC collisions. I am only suggesting that it should be possible in principle, even if the probability of creation is too small to consider this possibility more seriously. Certainly the possibility of such a tachyon is, at the moment, too exotic to be discussed in textbooks. But it can be an interesting idea for a more detailed research paper. And it sholud be appropriate to discuss such exotic stuff at the Beyond the Standard Model forum.

 

[Demystifier]

A preferred frame in solid matter is not really surprising.

Of course it's not. But what if QFT in condensed matter is how QFT should generally be interpreted? What if all SM particles are only quasi-particles like phonons? What if Lorentz-invariant QFT of the standard model is only an effective theory, while the fundamental theory has a preferred frame? That could solve many conceptual problems like those concerning tachyons, non-local hidden variables for EPR correlations, the problem of time in quantum gravity, UV completeness of quantum gravity (Horava theory), and so on ...

A technical remark: a group velocity larger than c is possible in special relativity if there is nonlinear absorption or emission. This has been observed with light, see the references there.

Interesting!

 

[Sexton Blake]

We have enough trouble trying to detect conventional particles like neutrinos. If tachyons existed, we and everything around us might have them flying through us, like neutrinos do, and never know it.

Everything conventional has a speed of light speed limit. Tachyons don't so may not be particle, wave or anything else we know about. I don't think anything we currently know about could detect them, unless maybe one day a neutrino detector detected a collision which went way off the normal energy scale.

 

[phinds]

Everything conventional has a speed of light speed limit. Tachyons don't ...

Actually, tachyons DO have a "speed of light speed limit", it's just that it's a minumum speed limit rather than a maximum which is what normal matter has.

 

[Sexton Blake]

I know that an idea is that the more energy they have, the slower they go, till they reach light speed but that just does not make sense. If tachyons have any conventional energy, they will be limited to conventional speeds. I would find it more believable if they, like photons, had a set speed.

 

[phinds]

I know that an idea is that the more energy they have, the slower they go, till they reach light speed but that just does not make sense. If tachyons have any conventional energy, they will be limited to conventional speeds. I would find it more believable if they, like photons, had a set speed.

And what do you propose that "set" speed is? It can't be c because they have to travel faster than c. Should it be 1.1c? How about 9.62c? How would you justify such as set speed?

 

[mfb]

Such a fixed speed is impossible: relative to what? Spacetime transformations allow the existence of exactly one fundamental speed that appears the same to all observers. We found this speed, it is typically called "speed of light" as light in vacuum moves at this speed.

There are many observed things that don't appear to make sense when you first hear about them. Everyday experience is a bad judge.

 

[ScientificMind]

I know that an idea is that the more energy they have, the slower they go, till they reach light speed but that just does not make sense. If tachyons have any conventional energy, they will be limited to conventional speeds. I would find it more believable if they, like photons, had a set speed.

If you're going for what would make it make sense, I would assume that the "imaginary energy," however that would manifest itself in reality, would be more or less equivalent to negative "conventional" energy, meaning that by adding conventional energy, it would lose imaginary energy and thus get slower, and by adding conventional energy, it would lose imaginary energy. Unfortunately though, considering how much normal science can contradict common sense, if tachyons do exist, they probably make no sense without a lot of time and effort having gone into understanding them, so I expect that this isn't actually the case.

 

[ScientificMind]

Of course it's not. But what if QFT in condensed matter is how QFT should generally be interpreted? What if all SM particles are only quasi-particles like phonons? What if Lorentz-invariant QFT of the standard model is only an effective theory, while the fundamental theory has a preferred frame? That could solve many conceptual problems like those concerning tachyons, non-local hidden variables for EPR correlations, the problem of time in quantum gravity, UV completeness of quantum gravity (Horava theory), and so on ...

Your comments have been very interesting to read, and I sincerely appreciate your help, but I did find much of this very difficult to understand, and I feel as though I only barely understood most of what you said, so would it be possible for you to explain this a slight bit more simply? While I probably have a better understanding that many in my position due to my strong interest and enjoyment of science as a hobby and future carrier path, I still only have as much formal education as a High School Senior (and only a week of school as a senior so far). Still, I understand if this is not a topic that can simplify very much or very easily.

 

[Demystifier]

Your comments have been very interesting to read, and I sincerely appreciate your help, but I did find much of this very difficult to understand, and I feel as though I only barely understood most of what you said, so would it be possible for you to explain this a slight bit more simply? While I probably have a better understanding that many in my position due to my strong interest and enjoyment of science as a hobby and future carrier path, I still only have as much formal education as a High School Senior (and only a week of school as a senior so far). Still, I understand if this is not a topic that can simplify very much or very easily.

In the text you quoted I mentioned a dozen of otherwise not directly related concepts. To explain those to someone with high school education, each would require a few teextbook pages of thorough explanation. I hope you understand that I cannot provide such explanations here. But I can answer a more specific question, if you have one.

For a start, you can read more about the Lorentz interpretation of relativity, e.g. here
https://www.physicsforums.com/threads/lorentzian-relativity.298185/
http://www.worldsci.org/pdf/abstracts/abstracts_6205.pdf

 

[Urs Schreiber]

I agree with what Demystifier says, this here is just to amplify that the issue is one of which vacuum one is perturbing about. Perturbation theory, by construction, is (up to issues of resummation) the Taylor expansion of what really ought to be a globally defined function (the S-matrix) around some point of its parameter space. Usually one takes that point to be the minimum of the relevant potential. Then all particle quanta, which are the small oscillations around that minimum, have positive mass squared, determined by the second derivative of the potential at that point: at a minimum, the first derivative vanishes, the second is positive and gives positive mass squares to particles.

Now, even though it is often not useful, it is nevertheless mathematically possible to instead look at the peturbation series around not a minimum but a maximum of a potential. Then its second derivative is negative there, and accordingly the mass squared of the small fluctuations around that points come out negative. These are the tachyons. Their presence signifies that the perturbation is not done around a potential minimum, but around a maximum. This is where the example of the Higgs particle comes in. Its Mexican hat potential does have a maximum, and even though one usually does not do perturbation theory around this maximum, because by design that's not the point of interest, it is in principle possible and doing so will show that there is a tachyon in the spectrum at that point, signifying the tendency of the theory to roll down away from that point.

Notice that the theory doesn't change as we change the point where we base its perturbation series, just the approximation to it changes.

One of the best studied cases of this is the open string tachyon. Consider it as an academic example if you wish, just to help understand the situation. In open string theory there is a tachyonic excitation, and it was early on voiced by Ashoke Sen that this hence means that the open string perturbation series is the perturbation about a potential maximum (namely that of the space-filling D25 brane, for what it's worth), hence that the stable/true vacuum of open bosonic string theory should be elsewhere. Eventually that picture was verified by detailed computations in open string field theory, one of the big computational successes in the field. References are here.

 

[Avodyne]

Now, even though it is often not useful, it is nevertheless mathematically possible to instead look at the peturbation series around not a minimum but a maximum of a potential. Then its second derivative is negative there, and accordingly the mass squared of the small fluctuations around that points come out negative. These are the tachyons.

You can call these excitations "tachyons" if you like, but they do not move faster than light. The same is true of the open-string "tachyon".

 

[Urs Schreiber]

Sure, the concept of "moving faster than light" does not even make sense in relativity, beyond moving lightlike there is only being spacelike, i.e. having negative rest mass squared.

 

[Demystifier]

You can call these excitations "tachyons" if you like, but they do not move faster than light.

Do they have group velocity larger than c?

 

[Demystifier]

Sure, the concept of "moving faster than light" does not even make sense in relativity, beyond moving lightlike there is only being spacelike, i.e. having negative rest mass squared.

I don't understand this claim. You admit that velocity can be spacelike, but you deny that such velocity corresponds to a motion faster than light!? Then what is a spacelike velocity if not a motion faster than light?

 

[Urs Schreiber]

A spacelike vector is not a velocity, but a direction in space.

 

[Avodyne]

You can call these excitations "tachyons" if you like, but they do not move faster than light. The same is true of the open-string "tachyon".

Do they have group velocity larger than c?

This is the wrong question, because a group velocity greater than $c$ does not mean that particles actually move faster than light.

Consider a quantum state $|\psi\rangle$ of the Standard Model with the property that the expectation value of the Higgs field has its usual nonzero value outside a particular spherical region of radius $R$, but inside goes smoothly to zero at a radius $R/2$, then remains zero inside $R/2$. In this interior region, we have a quantum state that supports the existence of "tachyons", modes of the Higgs field with negative mass-squared.

I claim that if the state of the world is $|\psi\rangle$ at time zero, and you sit a distance $L$ away from the center of the sphere, you will see nothing happen until a time $(L-R)/c$. That is, the "tachyons" inside the sphere cannot move away from it faster than light.

 

[Demystifier]

... a group velocity greater than $c$ does not mean that particles actually move faster than light.

There is a rather general result in classical theory of waves that group velocity is the velocity of the wave packet. Do you deny it?

Consider a quantum state $|\psi\rangle$ of the Standard Model with the property that the expectation value of the Higgs field has its usual nonzero value outside a particular spherical region of radius $R$, but inside goes smoothly to zero at a radius $R/2$, then remains zero inside $R/2$. In this interior region, we have a quantum state that supports the existence of "tachyons", modes of the Higgs field with negative mass-squared.

I claim that if the state of the world is $|\psi\rangle$ at time zero, and you sit a distance $L$ away from the center of the sphere, you will see nothing happen until a time $(L-R)/c$. That is, the "tachyons" inside the sphere cannot move away from it faster than light.

Maybe so, but consider a different initial configuration. Higgs field is zero outside $R$, while inside $R$ it is nonzero but small. (By small, I mean much smaller than the "usual" nonzero value corresponding to the minimum of the Higgs potential.) I claim that for this state there will be motion faster than light.

 

[Demystifier]

A spacelike vector is not a velocity, but a direction in space.

That's wrong. In general, a spacelike vector is a direction in spacetime, not merely in space. It is a direction in space only in the special case, when the time-component of the 4-vector is zero. And even in this special case it describes a 3-velocity - an infinite one.

 

[mfb]

There is a rather general result in classical theory of waves that group velocity is the velocity of the wave packet. Do you deny it?

As discussed before, a superluminal group velocity does not have to break Lorentz invariance. The propagation speed of changes (the propagation speed of the front of a wave packet) has the fundamental limit.

 

[Avodyne]

Maybe so, but consider a different initial configuration. Higgs field is zero outside $R$, while inside $R$ it is nonzero but small. (By small, I mean much smaller than the "usual" nonzero value corresponding to the minimum of the Higgs potential.) I claim that for this state there will be motion faster than light.

No.

First consider the problem classically. Assume that the initial time derivative of the Higgs field is zero (and all other fields have zero initial value and zero initial time derivative). Then it is a theorem of classical wave analysis that the fields at a distance $L$ from the center of the sphere remain at zero until a time $(L-R)/c$, when they are hit by the front of a shock wave from the sphere. As noted by mfb, the leading edge of a wave cannot travel faster than $c$, even in the "tachyonic" region outside the sphere.

Quantum mechanically, the state outside the sphere is subject to quantum fluctuations, and the Higgs field immediately starts spontaneously "rolling down the hill" everywhere at once (but in different directions/rates at different points, depending on the precise details of the initial quantum state). So everything is going nuts everywhere, and I don't know what you would use to diagnose "motion faster than light".

 

[Demystifier]

Then it is a theorem of classical wave analysis that the fields at a distance L from the center of the sphere remain at zero until a time (L-R)/c, when they are hit by the front of a shock wave from the sphere. As noted by mfb, the leading edge of a wave cannot travel faster than c, even in the "tachyonic" region outside the sphere.

Can you give a reference for this claim? (Make sure that the reference includes a discussion of the tachyonic case.)

 

[Avodyne]

I already gave a reference in post #7:

here is paper that states that there is no superluminal propagation for the classical initial-value problem even with negative mass-squared (see p.8):
"No superluminal propagation for classical relativistic and relativistic quantum fields"
John Earman
http://www.sciencedirect.com/science/article/pii/S1355219814000811
pdf: http://philsci-archive.pitt.edu/10945/1/NSP_SHPMP_Final_Version.pdf

See also this less formal discussion (cited by Shyan in post #9), "Do tachyons exist?":
http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/tachyons.html

See also this discussion (with references):
http://scienceworld.wolfram.com/physics/Superluminal.html
Key quote: "For example, the phase velocity and group velocity of a wave may exceed the speed of light, but in such cases, no energy or information actually travels faster than c."

 

[Demystifier]

Thanks for the references, I need some time to read it and think about it.

 

[Demystifier]

See also this less formal discussion (cited by Shyan in post #9), "Do tachyons exist?":
http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/tachyons.html

It says:
"(2) If we don't permit the second sort of solution, we can't solve the Klein-Gordon equation for all reasonable initial data, but only for initial data whose Fourier transforms vanish in the interval [−| m |, | m |]. By the Paley-Wiener theorem this has an odd consequence: it becomes impossible to solve the equation for initial data that vanish outside some interval [−L, L]! In other words, we can no longer "localize" our tachyon in any bounded region in the first place, so it becomes impossible to decide whether or not there is "unit propagation velocity" in the precise sense of part (1). Of course, the crests of the waves exp(−iEt + ipx) move faster than the speed of light, but these waves were never localized in the first place!"

Let me propose a possible loophole in this argument. I agree that in this case we cannot localize initial data. I also agree that consequently it is impossible to decide whether or not there is "unit propagation velocity" in the precise sense of part (1).

However, I do not think that the precise sense of part (1) is the only possible precise sense to define propagation velocity. Even though the initial data cannot vanish outside a finite interval, the initial data can be small outside the finite interval. So one can choose to encode the initial information into large crests (where "large" can be defined precisely). A crest is a field configuration which is large within the finite interval and small outside of it. As said in the quotation above, the crests do move faster than light. (Indeed, the velocity of the crest is given by the group velocity, which is larger than the velocity of light). The receiver of information may choose to ignore all waves except those that come in the form of large crests. In this way, it seems that information encoded in the crests can travel faster than light.

If one wants a precise way to incode information, here is one possibility. One can arrange that a crest is either high (an "I" shape) or very high (an "I" shape). Then a typical chain of signals looks like
..._________IIIIIIIIIIIIIIIIIIIII________ ...
By interpreting I's and I's as binary 0's and 1's, one can encode any information one wants. The lines ___ are tails corresponding to a field which is non-vanishing but small. The dots ... denote that the tails do not stop there, but stretch to infinity. The chain of crests, however, is finite and moves faster than light.

 

[Avodyne]

There is no question that wave crests can move faster than light. But I do not see that there is anything very interesting about this. For example, we could get a "stadium wave", created by people holding placards, to move faster than light! We just tell everyone in advance exactly when to hold up their placards. Thus, if you get to set initial conditions everywhere, you can get all sorts of results.

What you cannot do, even with access to a field with negative mass squared, is set up a field configuration in a bounded region of space (with $\phi=0$ outside), and then use it to send a signal to someone outside that arrives before a light ray would arrive.

 

[Demystifier]

Let me propose a possible loophole in this argument ...

I think I understand now what was my mistake.

There is a general mathematical theorem about domain of influence for hyperbolic partial differential equations. It is irrelevant whether the field is zero or not outside the interval. The relevant thing is a relative difference between two solutions with two different initial conditions. If two initial conditions differ only inside a finite interval, how fast that difference will propagate? The general theorem says that the difference never propagates faster than c, irrespective of the sign of the mass term.

But then why do crests move faster than c? Suppose that at t=0 we have a crest at x=0, and suppose that after a short but finite time dt we have a crest at dx>cdt. The point is that, despite the appearance, the crest at dt is not caused by the crest at t=0. In fact, by a suitable initial condition at t=0, the crest at dt may appear even if there was no crest at all at t=0.

So for tachyon fields information cannot propagate faster than c, provided that information is defined in terms of initial conditions (e.g. Cauchy data at t=0 for all x).

Alternatively, if information was defined in terms of boundary conditions (e.g. Cauchy data at x=0 for all t), then information would "propagate" only faster than c, irrespective of the sign of the mass term. But with boundary conditions instead of initial conditions it is perhaps more natural to redefine velocity as dt/dx (rather than dx/dt). The redefined velocity is always slower than 1/c.

 

[Avodyne]

I agree with your summary! And thanks for providing the term "domain of influence". I knew there was a name for it but couldn't remember or find it ...

 

[Demystifier]

I agree with your summary! And thanks for providing the term "domain of influence". I knew there was a name for it but couldn't remember or find it ...

Thank you for helping me to correct my misconceptions!

Perhaps the most surprising result in my summary is that even a positive mass squared may lead to a kind of superluminal "propagation", provided that Cauchy data is defined in terms of boundary rather than initial conditions. So whether the solutions are subluminal-or-superluminal does not depend on whether the mass squared is positive-or-negative. It depends on whether the Cauchy data are given on a spacelike-or-timelike hypersurface. In physics we usually consider spacelike Cauchy hypersurfaces (leading to subluminal phenomena), but mathematically nothing forbids to consider timelike Cauchy hypersurfaces.

I knew from the beginning that there must be a kind of mathematical symmetry between subluminal and superluminal solutions, but I was looking for that symmetry at the wrong place.

 

[Demystifier]

I have found a simple intuitive heuristic argument why, when the initial condition is specified, the propagation of the initial condition cannot be faster than light. To emphasize the idea and not the technicalities, my argument will be sketchy.

Let us have a Lorentz-covariant partial differential equation for the function $\phi(x)$, where $x$ is the spacetime coordinate $x=(x^0,{\bf x})$. Let us solve the Cauchy problem by the Green function method. In the spirit of DeWitt it is useful to think of $\phi(x)=\phi_x$ as components of a vector in an infinite dimensional space, so in the abstract index notation the solution can be written in a schematic form as
$$\phi_x=G_{xy}\phi^{in}_{y} \;\;\;\;\; (Eq. 1)$$
where $G$ is the Green function, $\phi^{in}$ is the initial condition, and sum over repeated indices is understood. Since the differential equation is Lorentz-covariant, the Green function is Lorentz invariant.

Now consider (Eq. 1) at the initial time $x^0=y^0$. In this case the solution $\phi$ must reduce to the initial condition $\phi^{in}$, so the Green function must be proportional to $\delta^3({\bf x}-{\bf y})$. In other words, for the Cauchy problem to be well posed, the Green function must vanish for equal time coordinates and non-equal space coordinates. But the Green function is Lorentz invariant, so this statement is valid in all Lorentz frames. This implies that the Green function vanishes everywhere outside the lightcone. (Eq. 1) then implies that the solution vanishes everywhere outside of the lightcone generated by the localized initial condition. In other words, the initial condition cannot propagate faster than light. Q.E.D.

Note that I have not specified whether the differential equation is first order (Dirac equation), second order (Klein-Gordon equation with postitive, zero or negative mass squared), or even higher order. Without specifying it, one cannot specify the precise form of the schematic (Eq. 1). As a specific case, let me only note that in the Klein-Gordon case the precise manifestly-covariant form of (Eq. 1) is
$$\phi(x)=\int_{\Sigma}d\Sigma'^{\mu}G(x,x') \!\stackrel{\leftrightarrow\;}{\partial'_{\mu}}\! \phi(x')$$
where ${\Sigma}$ is the initial spacelike hypersurface.

 

[Demystifier]

As a specific case, let me only note that in the Klein-Gordon case the precise manifestly-covariant form of (Eq. 1) is
$$\phi(x)=\int_{\Sigma}d\Sigma'^{\mu}G(x,x') \!\stackrel{\leftrightarrow\;}{\partial'_{\mu}}\! \phi(x')$$
where ${\Sigma}$ is the initial spacelike hypersurface.

For the case someone wants to know more technical details how did I arrive at this very elegant formula (valid even in curved spacetime), and how to determine the Green function G itself, here are some hints.

First, note that this formula is very similar to Eq. (1.42) in
J.D. Jackson, Classical Electrodynamics, 3rd edition.
It was Eq. (1.42) in Jackson that gave me the idea.

Second, take a look at my own paper
[1] http://lanl.arxiv.org/abs/hep-th/0205022
Add the two equations in (7), use (4), and define $G=i(W^+ -W^-)$. That gives you the quoted equation above.

For further insight, see also my
[2] http://lanl.arxiv.org/abs/hep-th/0202204
and the books
[3] N.D. Birrell, P.C.W. Davies, Quantum Fields in Curved Space
[4] S.A. Fulling, Aspects of QFT in Curved Space-time

Eq. (2.68) in [3] corresponds to my Eq. (6) in [1]. To see that, take a look at (26)-(27) in [2]. Thus we have $G^+=W^+$, $G^-=W^-$. Eq. (2.67) in [3] is (up to a different sign convention) my definition of $G$ above.

Read also the first two paragraphs of page 79 in [4]. From those one can learn that $G^+$ and $G^-$ depend on the split into "positive and negative frequencies", which is observer dependent, but $G$ does not depend on it and is observer independent.

 

[Demystifier]

If there were non-local truly hidden variables, as annoying as that might be, might they not define a non vanishing difference between all spacetime points?

What hidden variables? We are discussing mathematical methods for solutions of partial differential equations. If physical hidden variables exist, they are descibed by some other equations, so they are irrelevant in this context.

 

[BiGyElLoWhAt]

Tachyons are hypothetical particles which can only travel faster than light.
They are not part of the standard model of particle physics and there is no experimental evidence of them.
If they do exist then the whole ediface of SR and GR would be wrong, yet there are mountains of persuasive evidence confirming the predictions of relativity.
As far as I know they mostly are considered as a mathematical artifact.

I know I've linked this before, but I'll leave this here, anyways.
http://arxiv.org/abs/1408.2804

 

[Demystifier]

I know I've linked this before, but I'll leave this here, anyways.
http://arxiv.org/abs/1408.2804

The paper is also published in a peer-reviewed journal:
http://inspirehep.net/record/1310694?ln=en

I don't understand why the author uploaded this only to the [physics.gen-ph] section of arXiv, without crosslists to more relevant sections such as hep-ph.

 

[mfb]

"Consistent with"... all observations are consistent with pink unicorns in the Kuiper belt, but there is no reason to speculate about their existence without experimental hints.
I know the paper and the content does include things that can be considered as experimental hints, but I really don't like the title.

 

[BiGyElLoWhAt]

Pink unicorns in the Kuiper belt is very arbitrary, and observations here are "consistent with" them by virtue that they are unrelated. These observations are, however, directly related to the contention.

 

[Demystifier]

I have found a great paper (for those who have access to Phys. Rev. D) on classical propagation of tachyon fields:
http://journals.aps.org/prd/abstract/10.1103/PhysRevD.18.3610

Title:
Do tachyons travel more slowly than light?

Abstract:
The propagation of solutions of the Klein-Gordon equation with an arbitrary complex mass is investigated. Owing to the strict hyperbolicity of the Klein-Gordon operator, the global Cauchy problem with initial data on a spacelike hyperplane is well posed. In the process of constructing the solution of this Cauchy problem, a Lorentz-invariant retarded Green's function is calculated. Thus the Klein-Gordon causal order relation between pairs of events is invariant under any orthochronous Lorentz transformation and the field propagates no faster than light. In particular, this limitation on the propagation speed is valid for the imaginary-mass Klein-Gordon field. The relation between the causal order and Green's functions is examined. It is shown that only those Green's functions associated with the global Cauchy problem are relevant to the causal order. Finally, it is shown that the group velocity is not physically significant when the dispersion is anomalous.