The Photon Propagator and Superluminal Speeds
Anamitra Palit
Unfortunately at the moment, I do not have the time to follow up the
below cited paper, but I don't want to delay the posting any further. So I
just put this posting for general discussion into the newsgroup.
The assertions are, however, highly suspicious to be due to the use
of wrong propagators to argue for superluminal signal propagation. Today,
there's no single example of faster-than-light propgation of signals.
Even in the classical electromagnetics, there are apparant faster-than-light
speeds. The most famous is propagation of light in the frequency region
around an absorption line of the material, where anomalous dispersion occurs,
and where the group velocity defined as c \partial_{\omega}/\partial_{\vec{k}}
is greater than c. This has been observed as early as 1907 by Willy Wien and
answered to full satisfaction by Sommerfeld in the same year. Sommerfeld
and Brillouin have wored out this problem further. You can read the theory
in Sommerfeld's brilliant textbook "Lectures on Theoretical Physics, Vol. 4,
Optics". The short answer is that the group velocity has no signal-propagation-speed
meaning in this case at all, and the wave front is always travelling with c and not
superluminal. This is due to the use of the retarded propagator and the relativistic
nature of the dispersion relation.
In QT the linear-response theory also leads to the retarded propagator for observable
signal propagation. So there are also no superluminal speeds in QT either, at least
not under controllable approximations.
HvH.
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The Photon propagator seems to be intimately associated with
superluminal speeds.Let us have a look at it:
DF(x-y)=integral(d^4)q exp[-iq.(x-y)]DF(q^2)
Now the expression exp[-i(x-y)] represents a progressive wave that
travels with the speed given below:
Wave Speed=q(0)/sqrt[q(1)^2+q(2)^2+q(3)^2]= ( omega)/k [wave -
equation comparison]
The above expression may be assigned greater than 'c' values by proper
choice of the q-components. So far as the evaluation of the integral
is concerned the choice of the q-components is quite broad/arbitrary.
The "virtual photon " has been made to incorporate within itself
faster than light components!
Plausible solution:In the above case we have instances where omega/k
>c. But if we make the group velocity (d omega)dk less than 'c' by
some forced condition matters become consistent with Special
Relativity.
A second point of interest:Quantum Mechanics abounds in examples of
Superluminal Speeds
As instances I may state the free particle solutions of the Klein-
Gordon equation and the Dirac equations. If we treat them as
progressive waves they are found to travel faster than light(though
the particle remains in the speed zone permitted by Special
Relativity).
[Reference for the second point:"Superluminal Speeds in Quantum
Mechanics"
European Journal of
Scienitfic Research(Euro Journals)
Vol 37,No3,2009 ]
Rock Brentwood
> The assertions are, however, highly suspicious to be due to the use
> of wrong propagators to argue for superluminal signal propagation. Today,
> there's no single example of faster-than-light propgation of signals.
The problem runs deeper than this. Consider the worldline describing
the center of mass of any system (elementary or composite) that
conforms to the Wigner class 4 (tachyons). If you take seriously the
idea of microcausality, then that means that the different parts of
the worldline are in 0 quantum correlation with one another. So the
usual idea of a propagator for these types of systems is out the
window. And the usual idea that such systems should even be considered
as particles in the first place is out the window.
The longitudinal coordinate remains classical.
So, I take issue with the idea that quantum theory has nothing in it
that corresponds to this Wigner class. As I described in the N-
category Cafe thread on the Wigner classification,
http://golem.ph.utexas.edu/category/2009/03/unitary_representations_of_the.html
the very notion that tachyons are to be interpreted as particles (and
the general idea of irreps being synonymous with the notion of
"particle') is questionable. So, the question as to whether they exist
or not hasn't even been asked correctly, since the "they" that the
question is being asked of may have nothing to do with tachyons at
all!
If every irrep. is to be identified with a type of elementary
particle, for instance, then where are the particles of "vacuum" in
accelerator experiments? (That is: translation-invariant systems,
those conforming to Wigner class 3). The "particle=irrep' creed runs
afoul even when it comes time to describe and interpret the more
general systems of Wigner class 3. Unlike the other Wigner classes
(class 1 = tardion or bradyon, class 2 = luxon, class 4 = tachyon),
they don't even have a name (so I coined the term "vacuon" for them).
So, if the creed fails there, then there is no reason for the
theoretical bias that any of the other irrep. types necessarily have
to have interpretation as "particles".
So, once that question is removed, then the way is paved to better
interpreting what a tachyon is supposed to actually represent; and
then (and only then) can you begin to ask the right questions
regarding what, if any, in the physical world corresponds to tachyons.
The problem, as I mentioned, runs deep. It's the underpinning to 2
major No Go theorems which seem to be specific to relativistic
dynamics, but absent in non-relativistic dynamics.
The best way to understand what these types of systems is to consider
what the non-relativistic version of such propagation and such systems
are. The non-relativistic version of the tachyon (and of the luxon)
does not have a name, but is well-studied -- they are the
representations of the Galilei group; that is, representations of the
10-parameter Galilei group, as opposed to the 11-parameter centrally
extended Galilei group.
The representations have two interesting properties. One, they have a
time operator t = -K.P/P^2, where K is the boost generator and P the
spatial translation generator. This is a canonical coordinate and is
conjugate to the energy.
They have a quadratic invariant P^2 -- the impulse. The impulse is
that which takes place across space at an instant. The time operator t
is an indication of the instant at which the impulse takes place.
The systems, like the vacuons, do not have a name either. So, they
have the same problem as the Millennial Decade has: nobody ever got
around to coming up with a name for them (until I did ... just now).
These are the synchrons.
In non-relativistic dynamics, they underlie the direct simultaneous
action at a distance transfer of impulse across space. This, in turn,
serves as the underpinning to Newton's Third Law.
Their absence is acutely felt in relativistic dynamics. There is no
clear-cut formulation of any relativistic version of Newton's Third
Law. That's because in the very statement of the law "equal and
opposite SIMULTANEOUS actions..." is the tacit appearance of
simultaneity.
So, if you want to replace Newton's Third Law by something
relativistic, you need to generalize it to something that governs
interactions that take place simultaneously. Only here, "simultaneous"
means "simultaneous in somebody's frame of reference".
The absence of the Third Law is felt in two ways. In classical
dynamics, it take the form of the Leutwyler Theorem -- the No
Interaction Theorem. In quantized form, the no go theorem becomes the
Haag Theorem -- the No Interaction Theorem for Fock Spaces and
quantized many-body dynamics.
In relativistic form, the "simultaneous action at a distance" mode
becomes "simultaneous in somebody's frame", and in other frames it
becomes faster than light.
But, as described at the outset, it can't be interpreted as any kind
of quantized particle because its longitudinal coordinate is classical
and must remain classical; it serves as the worldline parameter,
instead of the time t. In the meanwhile, the time t becomes something
more like a canonical coordinate and (as in the case of the synchron)
there is a conjugate relation between it and the energy.
Hence, these modes are closely associated with (a) the Coulomb part of
interactions and (b) the energy-time uncertainty principle.
Is there something in quantum theory that resembles this? Particularly
in quantum field theory?
The best way to get to the answer is the resolve the other major issue
commonly associated with tachyons -- the "negative mass squared"
issue.
The mass shell invariant in non-relativistic dynamics can be written
as: P^2 - 2MH, while a second invariant, M, appears by virtue of the
presence of the central charge. Here, P is the momentum vector, M the
mass and H the KINETIC energy.
In relativistic dynamics, the invariant is written as P^2 - E^2/c^2,
where E is the total energy. If you naively take the non-relativistic
limit, you end up getting the invariant P^2 which is that for the 10-
parameter Galilei group. That's the invariant for synchrons.
So, the correct way to approach the non-relativistic limit is to split
E into the kinetic energy H and the relativistic mass M = E/c^2. Then,
defining a = 1/c^2 for relativity, a = 0 for non-relativistic
dynamics, you have the two invariants:
Modified mass shell invariant: lambda = P^2 - 2MH + a H^2
Modified linear invariant: mu = M - aH.
For tardions and luxons, one has P^2 - E^2/c^2 = (mc)^2, where m is
the rest mass for tardions and m = 0 for luxons. If this is equated to
the linear invariant
m = mu
then the result is that
lambda = 0.
So, this can be referred to as the RESTRICTED mass shell condition. In
relativistic dynamics it's just referred to as the mass shell
condition, so the word "restricted" means we're going to consider
something more general.
The punchline: neither the synchron nor tachyon can conform to this
condition. Instead, they can only satisfy the more general condition
lambda = Pi^2 > 0
with Pi being the impulse.
The inability to satisfy the restricted mass shell condition is what
forces one into the fiction of associating an "imaginary mass" with
them. But here, we see clearly, that it is not imaginary mass lambda
= -(mc)^2 we're talking about here, but real impulse, lambda = +Pi^2.
None of this is possible within the representation theory of the
Poincare' group, since the group has only 10 parameters. What's
described above provides a continuous connection to the centrally
extended Galilei group, which has 11 parameters. The 11th parameter is
the linear invariant mu. It would be the central extension of the
Poincare' group, it the group permitted one. But Poincare' has simple
topology. So the 11-parameter extension is just Poincare' x U(1).
Without mu there is no way to get a consistent interpretation of
tachyons and no way to exploit the fact that the tachyon is the
relativistic version of the synchron.
So, at last, to answer the question: "is there anything in quantum
field theory that resembles the description just given?" The
description just given is of a system that provides an underpinning to
the relativistic form of Newton's Third Law, which supports the
Coulomb mode of the field law, which is not quantizable in the usual
way, which has something representing an energy-time uncertainty
relation, and which is -- strictly speaking -- OFF-SHELL.
We have a name for that, already. It's called a VIRTUAL PARTICLE. The
propagator for the virtual particle is just 1/lambda = 1/Pi^2, up to
constant factors. That's what you see attached to the propagator line
in Feynman diagrams.
So, the fact that lambda is non-zero is not only not a liability, it
may be the most important feature of properly interpreting what a
tachyon is supposed to actually represent -- not a particle, but
simply a medium of a type that supports the Coulomb part of the force
law for a field.
In other words, the notion that field theory somehow got rid of the
action at a distance precept is not only a red herring, but is a
reflection of a significant gap or incompleteness in the theory of
relativity, itself. It is yet another case in point reflecting the
points raised in the "Incompleteness of Relativization" thread
The Incompleteness of Relativization
s.p.r. March 24 - May 22, 2009
http://groups.google.com/group/sci.physics.research/search?hl=en&group=sci.physics.research&q=Incompleteness+of+Relativization&qt_g=Search+this+group