Virtual particles and propagators

[Jay R. Yablon]

Dear SPR friends:

I am writing to clarify my understanding of what is meant when people
talk about "virtual particles" and their related propagators.

Insofar as I understand the use of this term, it refers to a particle
for which the relation:

p^u p_u = m^2 (1)

does *not* apply, where p^u is the momentum four-vector and m the rest
mass of a massive particle. For a massless particle (e.g., photon), a
virtual particle would have:

p^u p_u = 0. (2)

It is also my understanding that the term "off mass shell" is used to
characterize (1) and (2), and so that the term "virtual particle" is
synonymous with "off-mass-shall" particle. If they are not precise
synonyms, I'd appreciate knowing the points of difference. Also, it
appears that "real, observable" particles are particles which are
"on-mass shell," and that any deviation from equation (2) is thought to
be short-lived, and comes about because of the uncertainty
relationships, i.e., the conjugacy of energy-momentum and spacetime.

For a boson particle, I have seen the term:

p^u - p^u - m (= 0) (3)

referred to as a momentum space operator, while the propagator, which is
a spin sum time the momentum space inverse, contains (3) in its
denominator, i.e., the propagator is proportional to:

1 / (p^u - p^u - m) (4)

Are all of the above fair statements?

Because p^u - p^u - m = 0 for a non-virtual particle, there is, from
what I can see, some amount of ad-hoc heavy lifting needed to write down
a propagator which does not become formally infinite for a real,
on-shell particle. It also appears that it would be desirable if one
could obtain a propagator for an on-shell particle which is: a)
consistent with equations (1) and (2); and, simultaneously, b) remains
formally finite nevertheless, without adding any terms ad-hoc.

I appreciate your thoughts on the above.

Thanks,

Jay.
_____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com
Co-moderator: sci.physics.foundations

[Arnold Neumaier]

Jay R. Yablon schrieb:
> Dear SPR friends:
>
> I am writing to clarify my understanding of what is meant when people
> talk about "virtual particles" and their related propagators.
>
> Insofar as I understand the use of this term, it refers to a particle
> for which the relation:
>
> p^u p_u = m^2 (1)
>
> does *not* apply, where p^u is the momentum four-vector and m the rest
> mass of a massive particle.

Or a tachyion. The exchange photon in the tree approximation of electron
scattering has m^2<0.


For a massless particle (e.g., photon), a
> virtual particle would have:
>
> p^u p_u = 0. (2)
>
> It is also my understanding that the term "off mass shell" is used to
> characterize (1) and (2), and so that the term "virtual particle" is
> synonymous with "off-mass-shall" particle. If they are not precise
> synonyms, I'd appreciate knowing the points of difference. Also, it
> appears that "real, observable" particles are particles which are
> "on-mass shell," and that any deviation from equation (2) is thought to
> be short-lived,

No; deviations from (2) only appear as integration variables in
computing contributions to the S-matrix; they do not have a real life
meaning - measurable is only the S-matrix, not the splitting into
contributions from virtual particles. The latter depends heavily on
the approximation scheme used, hence cannot have a physical meaning.

See the discussion in my theoretical physics FAQ at
http://www.mat.univie.ac.at/~neum/physics-faq.txt
(Section 3; in particular S3c. How real are 'virtual particles'?)


> and comes about because of the uncertainty
> relationships, i.e., the conjugacy of energy-momentum and spacetime.


Arnold Neumaier

[BorisDCLCXVI@gmail.com]

On 31 mai, 18:59, Arnold Neumaier <Arnold.Neuma...@univie.ac.at>
wrote:

> No; deviations from (2) only appear as integration variables in
> computing contributions to the S-matrix; they do not have a real life
> meaning - measurable is only the S-matrix, not the splitting into
> contributions from virtual particles. The latter depends heavily on
> the approximation scheme used, hence cannot have a physical meaning.

I can't understand that paragraph, "depends on the approximation
scheme"?? And what about an exact "scheme"?

Also, why does "integration variable" means "no real life"? Any
quantum state can be written as the sum (integration) of several
states that are
each observable, and projected onto by the measurement procedure.

> See the discussion in my theoretical physics FAQ at
> http://www.mat.univie.ac.at/~neum/physics-faq.txt
> (Section 3; in particular S3c. How real are 'virtual particles'?)

Without that clarification above, I doubt this FAQ will be useful.

[markwh04@yahoo.com]

On May 31, 11:59 am, Arnold Neumaier <Arnold.Neuma...@univie.ac.at>
wrote:
> Or a tachyion. The exchange photon in the tree approximation of electron
> scattering has m^2<0.

This can be put in proper perspective by looking at what the analogue
would be in Newtonian physics under Galilean relativity. The zero-
mass, non-zero momentum representations of the Galilei group (the
"synchrons") represent non-zero transfer of momentum and energy across
distance -- "action at a distance" modes. Thus, if Newtonian gravity
were to be quantized, its quanta would consists of these synchrons.

The synchron is associated with a momentum P, but zero mass. In the
relativistic analogue, one has the generalized mass-shell relation,
P^2 - 2MT + (1/c)^2 T^2 = constant (where M = relativistic mass, T =
kinetic energy). For ordinary particles and luxons, the constant can
be taken to be zero. For tachyons, it must be taken as positive. The
corresponding analogue is then
P^2 - 2MT + (1/c)^2 T^2 = p^2
where p is the "asymptotic momentum"; or the momentum seen in a frame
of reference where the associated velocity is infinity. In that frame,
T = 0; while T approaches infinity as the velocity drops down to light
speed.

So, it's not that you have m^2 < 0; but rather, p^2 > 0. These
"tachyon" modes are the relativistic analogues of the Newtonian action-
at-a-distance modes; the asymptotic momentum giving you the momentum
transfer between the charges. They mediate the 1/r^2 Coulomb part of
the force, just as they would in gravity... and just as they would in
Newtonian Physics.

The mass-energy relations for the sectors in Poincare', Galilei and
Euclidean relativity were worked out in

The Wigner Classification for Galilei/Poincare'/Euclid
http://federation.g3z.com/Physics/index.htm#GeneralizedWigner

For all three sectors, you have the "vacuons" (zero kinetic energy and
momentum), where only the SO(3,1)/SE(3)/SO(4) part of the algebra is
non-trivial.

Then, you have the 3-part classification that comes out the spin-
helicity algebra (part 8); including the "tardions" for all 3 sectors;
the luxons for the Poincare' sector, the synchrons for the Galilei
sector and tachyons for the Poincare' sector. The various attributes
of the synchron, when going over to (Poincare') Relativity, are
divided between the luxon and tachyon.

What's interesting is that not only can you work out the mass-energy-
momentum relations for all these sectors (section 9), but even
*position* space representations for most of the sectors (anywhere
where you have a Heisenberg algebra in the little group).

This includes position space representations for the luxon sector
(i.e. photons), notwithstanding the well-known no-go theorem. In fact,
what you see there (all this is in sections 6 and 7) helps put the
theorem, itself, in perspective.

For each of the sectors, 3 of the 4 coordinates and conjugate momenta
will be operators, while the fourth pair will be classical. For
ordinary particles, the (T,t) pair is classical and does not
participate in the Heisenberg relation. The little group has only H(3)
as a subalgebra, not H(4). For synchrons, (T,t) becomes a conjugate
pair, and t becomes an operator (the "time of action"), while (p_x, x)
becomes classical, where x is the direction of the momentum p. Here,
it's x that plays the role of a classical parameter, in place of t.
For luxons, the situation is a little more complicated. I worked it
out in my notes a while back, but don't think I incorporated it yet
into the article.

What's also interesting is that there's a Dirac equation for all the
sectors. For the Galilean tardions, the Dirac equation reduces to the
Schroedinger equation. For the tachyons, it corresponds to the Dirac
equation with a chiral mass term.

The well-known no-go theorem on the non-normalizability of tachyons is
also put into perspective. Since it's not the (T,t) pair that's
classical for these modes, unlike the case for tardions, but rather
the (p,x) modes, then "normalization" does not take place along
hyperplanes orthogonal to t, but along hyperplanes orthogonal to x!

Tachyons can be normalized as wave functions. But the functions evolve
along the x-direction, and the normalization is an integral of the
form
|psi|^2 = integral (psi* psi dy dz dt).

The so-called tachyonic virtual particles are just the relativistic
version of action-at-a-distance synchrons to mediate the (1/r^2)
Coulomb part of the force.

[markwh04@yahoo.com]

On May 31, 11:59 am, Arnold Neumaier <Arnold.Neuma...@univie.ac.at>
wrote:
> Or a tachyion. The exchange photon in the tree approximation of electron
> scattering has m^2<0.

This can be put in proper perspective by looking at what the analogue
would be in Newtonian physics under Galilean relativity. The zero-
mass, non-zero momentum representations of the Galilei group (the
"synchrons") represent non-zero transfer of momentum and energy across
distance -- "action at a distance" modes. Thus, if Newtonian gravity
were to be quantized, its quanta would consists of these synchrons.

The synchron is associated with a momentum P, but zero mass. In the
relativistic analogue, one has the generalized mass-shell relation,
P^2 - 2MT + (1/c)^2 T^2 = constant (where M = relativistic mass, T =
kinetic energy). For ordinary particles and luxons, the constant can
be taken to be zero. For tachyons, it must be taken as positive. The
corresponding analogue is then
P^2 - 2MT + (1/c)^2 T^2 = p^2
where p is the "asymptotic momentum"; or the momentum seen in a frame
of reference where the associated velocity is infinity. In that frame,
T = 0; while T approaches infinity as the velocity drops down to light
speed.

So, it's not that you have m^2 < 0; but rather, p^2 > 0. These
"tachyon" modes are the relativistic analogues of the Newtonian action-
at-a-distance modes; the asymptotic momentum giving you the momentum
transfer between the charges. They mediate the 1/r^2 Coulomb part of
the force, just as they would in gravity... and just as they would in
Newtonian Physics.

The mass-energy relations for the sectors in Poincare', Galilei and
Euclidean relativity were worked out in

The Wigner Classification for Galilei/Poincare'/Euclid
http://federation.g3z.com/Physics/index.htm#GeneralizedWigner

For all three sectors, you have the "vacuons" (zero kinetic energy and
momentum), where only the SO(3,1)/SE(3)/SO(4) part of the algebra is
non-trivial.

Then, you have the 3-part classification that comes out the spin-
helicity algebra (part 8); including the "tardions" for all 3 sectors;
the luxons for the Poincare' sector, the synchrons for the Galilei
sector and tachyons for the Poincare' sector. The various attributes
of the synchron, when going over to (Poincare') Relativity, are
divided between the luxon and tachyon.

What's interesting is that not only can you work out the mass-energy-
momentum relations for all these sectors (section 9), but even
*position* space representations for most of the sectors (anywhere
where you have a Heisenberg algebra in the little group).

This includes position space representations for the luxon sector
(i.e. photons), notwithstanding the well-known no-go theorem. In fact,
what you see there (all this is in sections 6 and 7) helps put the
theorem, itself, in perspective.

For each of the sectors, 3 of the 4 coordinates and conjugate momenta
will be operators, while the fourth pair will be classical. For
ordinary particles, the (T,t) pair is classical and does not
participate in the Heisenberg relation. The little group has only H(3)
as a subalgebra, not H(4). For synchrons, (T,t) becomes a conjugate
pair, and t becomes an operator (the "time of action"), while (p_x, x)
becomes classical, where x is the direction of the momentum p. Here,
it's x that plays the role of a classical parameter, in place of t.
For luxons, the situation is a little more complicated. I worked it
out in my notes a while back, but don't think I incorporated it yet
into the article.

What's also interesting is that there's a Dirac equation for all the
sectors. For the Galilean tardions, the Dirac equation reduces to the
Schroedinger equation. For the tachyons, it corresponds to the Dirac
equation with a chiral mass term.

The well-known no-go theorem on the non-normalizability of tachyons is
also put into perspective. Since it's not the (T,t) pair that's
classical for these modes, unlike the case for tardions, but rather
the (p,x) modes, then "normalization" does not take place along
hyperplanes orthogonal to t, but along hyperplanes orthogonal to x!

Tachyons can be normalized as wave functions. But the functions evolve
along the x-direction, and the normalization is an integral of the
form
|psi|^2 = integral (psi* psi dy dz dt).

The so-called tachyonic virtual particles are just the relativistic
version of action-at-a-distance synchrons to mediate the (1/r^2)
Coulomb part of the force.

[Oz]

markwh04@yahoo.com writes
>Tachyons can be normalized as wave functions. But the functions evolve along
>the x-direction, and the normalization is an integral of the
>form
> |psi|^2 = integral (psi* psi dy dz dt).
>
>The so-called tachyonic virtual particles are just the relativistic version
>of action-at-a-distance synchrons to mediate the (1/r^2) Coulomb part of the
>force.

My apologies for being stunningly ignorant at this level but can I ask a
basic, and probably poorly-framed question?

Obviously I don't appreciate nearly all of the nuances you here
describe. However it has irritated me for decades that photons propagate
at 45 degrees and I've long wanted to have EM operating entirely in the
spatial directions and inertia in the t direction. That would strike me
as symmetrical in a similar way that I was always irritated with the
asymmetry of Maxwell which is completely cured with the 4D
representation.

It seems to me here that you are (from my naive viewpoint) here
introducing a structure where, crudely and imprecisely, EM propagates in
only spatial directions whilst inertia travels only in the time
direction.

Interactions between EM and inertia are thus local rotations (in a
simplistic sense) which makes the whole setup particularly elegant and
clear.

Is there any chance of a brief discussion at a naive level?

--
Oz
This post is worth absolutely nothing and is probably fallacious.

[Kay zum Felde]

In my opinion, virtual particles appear only in the so-called self energy, which describes the interaction of a given real particle like the electron with its own electromagnetical field. This leads to Feynman diagrams of interactions of virtual photons and virtual electrons in case of QED to yield the corrections of the fine structure constant. The interaction of the real electron with its own field is composed in Feynman diagrams up to infinite order and this sum is called the self-energy and the effect is call. Interactions between two real particles are also yielding Feynman diagrams up to infinite order, but these are then real photons and electrons. Vaccuum polarization arises, if the electron is interacting with the vaccuum leading to the screeing of charge. Hope this is all correct.

Regards Kay

[markwh04@yahoo.com]

On Jun 13, 6:27 am, Oz <O...@farmeroz.port995.com> wrote:
> markw...@yahoo.com writes
> Is there any chance of a brief discussion at a naive level?

Since I generalized the mass shell by bringing in the 11th parameter
and invariant (doing this, in turn, so that a well-defined Galilean
limit of the Poincare' group can be formulated), then there's room to
accommodate these "off-shell" transitions.

The propagator i/(k^2 - m^2) for a scalar virtual particle relates to
the generalized mass shell (P^2 - 2MT + (1/c)^2 T^2), becoming
i/(P^2 - 2MT + (1/c)^2 T^2).
In the Galilean limit, the "synchron" exchanges a momentum |P| = p, (1/
c)^2 = 0 and M = 0. So, one then has i/p^2.

This generalizes in the Relativistic case. In place of "simultaneous
action at a distance" is "faster than light action across a distance".
In place of the "synchron" is the "tachyon". The generalized mass
shell comes out of the mass momentum energy relations
M = v/(v^2 - c^2)^{1/2}, P = Mv, T = U + Mc^2
leading to the propagator
i/(p^2 - (U/c)^2).
This corresponds to an transfer of momentum p and energy U
"simultaneously" (or, the relativistic analogue of "simultaneously":
"faster than light"), across a distance.

The off-shell modes of the propagators are the on-generalized-shell
modes described in the generalized Poincare/Galilei/Euclid group.