Do Scientists Believe in Tachyons?

[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.

 

[BiGyElLoWhAt]

Is there a way to get journal access without paying out the arse?

 

[Demystifier]

Is there a way to get journal access without paying out the arse?

Probably not!

 

[JorisL]

You could try interlibrary loan.
Another thing that's sometimes done is directly contacting the authors, asking for a copy.

That are about the only legal ways I can think of.

Edit;
You can request it on http://www.researchgate.net/publication/235453877_Do_tachyons_travel_more_slowly_than_light , don't know the viability of it.
I've read about people getting the paper while others are still waiting :-)