A particle physicist's view of virtual particles (with which I agree)
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Hendrik van Hees
http://theorie.physik.uni-giessen.de/~hees/
wrote in an article from March 24, 2010 in sci.physics.research:
I think the trouble with "virtual particles" is again just a
mystification of rather simple facts. The problem is to speak in plain
(English) language about things one cannot express in plain language but
has to use mathematics. Another complication is, that this mathematics
is quite dirty, but we have nothing better in physics than quantum
(gauge) field theory, and in fact it's the best kind of theories we have
so far.
The Stueckelberg-Feynman trick is in fact no trick but a necessary
mathematical construction, given the fact that (by definition!) physics
deals with causal descriptions of nature, i.e., for any observer an
observable fact "now" can depend only on events in his "past". That's
also inherent in the very definition of time. So it doesn't make sense
to have "particles" running backwards in time. That's why I wrote "modes
of negative frequency" and not "plane-wave functions of negative energy"
as is sometimes done even in textbooks, adding to the confusion rather
than helping with the understanding. This forces us to write these modes
with a creation operator instead of an annihilation operator in the
expansion of the field operator. Nothing could be simpler than this!
However, it makes relativistic "vacuum QFT" manifestly different (and
more complicated) than non-relativistic "vacuum QFT". Solving these
difficulties in turn solves a nearly one century old obstacle of
classical field theory (i.e., Maxwell-Lorentz electromagnetics), namely
the question of the back reaction of a particle's own electromagnetic
field on its (accelerated) motion (at least in a perturbation-
theoretical sense, and I only talk about interacting field theories in
the perturbative sense in this posting!). The particle simply gets a
self energy with a diverging mass and a diverging renormalization factor
of the corresponding single-particle state.
That these infinities appear in perturbation theory is no big surprise
anymore nowadays, but was a bit obstacle in the early 30ies when QFT has
been invented by Pauli, Born, Jordan, and Dirac. The solution came in
1948 with the discovery of renormalization theory (which was developed
in full glory only in the mid 60ies or so with the works of
Bogoliubov+Parasiuk, Hepp, and Zimmermann with their famous BPHZ
formalism). Although it's not a very easy task in the concrete
calculations, the physical meaning is simple. In perturbation theory,
everything is expanded in terms of free-particle states. In QED that
means that at 0th order perturbation theory there are non-interacting
electrons and non-interacting photons. This already shows that this is a
men-made construct and not a description of what we observe in nature!
Particularly the electron in this order has no electromagnetic field
around it and in the same way the photon (i.e., the elementary
electromagnetic excitations) have no electron-positron cloud around it.
Thus, one speaks (in the usual funny slang of QFT physicists) about
"bare particles" and/or "bare fields".
Now, taking into account the electromagnetic interaction, we calculate
things with a "point of view" (technically speaking we formulate the
theory in a free-particle Fock basis) which handles fictitious objects,
namely the bare particles and bare fields. In reality, these objects
don't exist. Thus, order by order perturbation theory, we have to
construct the Fock basis of the real-particle/fields states, and this is
done by expressing everything in terms of physical parameters (like
wave-function normalization factors, masses and coupling constants),
approximated in the sense of perturbation theory (which is formally an
expansion in the coupling constant and/or \hbar). These parameters are
perfectly finite and can be adjusted by observable facts. Fortunately
there's a class of QFTs, called Dyson-renormalizable, where there's a
finite number coupling and mass parameters which is sufficient to absorb
all infinities of perturbation theory, i.e., the model has only a finite
number of parameters. I don't speak about the other case, namely
"effective theories" that are not renormalizable in the Dyson sense but
can be used in a certain (mostly low-) energy range and are expansions
with respect to powers in energy/momentum rather than in terms of
coupling constants/hbar (although both expansions are closely related,
which makes these theories very useful, like in chiral perturbation
theory to describe the low-energy behavior of strongly interacting
particles in terms of the hadronic "relevant" degrees of freedom rather
than with quarks and gluons, which makes the issue of hadron
interactions a very complicated matter).
Another fortunate property of the perturbative expansion of interacting
QFTs is that everything can be expressed in a *formal* diagrammatical
way, invented first by Feynman. He presented this diagram technique to
his astonished theory colleagues at the famous Shelter-Island
conference. Nowadays this technique is nearly an own special subject in
theoretical physics and far developed, particularly it's also applied in
a much wider range of applications, particularly in many-body QT (be it
non-relativistic of relativistic) in and out off equilibrium. Here, I
stick to the special case of the "vacuum theory", dealing with a small
number of particles (usually two in the initial and a few in the final
state of a collision process).
The original Feynman diagrams stand for terms in the perturbative
expansion of scattering-matrix elements. This said, it is clear that one
should not put too much physical interpretation into them although they
are very intuitive, if one keeps their abstract meaning in mind, and S-
matrix elements are pretty abstract objects. They are the probability
amplitudes for the transition of an asymptotically free initial state
(usually two particles which are prepared far away from each other with
well-adjusted momenta (and sometimes also spin/polarization)) to an
asymptotically free final state (whatever one decides to detect after
the collision about the then approximately freely streaming reaction
products). These kind of Feynman diagrams lead indeed to observable
quantitities, namely these transition-probability rates which are
usually expressed in terms of a cross section.
The Feynman diagrams themselves are built out of elementary Feynman
diagrams which stand for the bare-particle propagators (lines connecting
two points, where the particle or antiparticle is created and reabsorbed
again) and the interaction points where three or more lines meet at a
space-time point where "a perturbation" occurs in the sense of
perturbation theory. This also shows that the number of vertices in a
diagram counts the number of coupling constants in the mathematical
expression of the corresponding contribution to the S-matrix element.
Unfortunately the internal propagator lines are also called "virtual
particles" in the funny jargon of QFT physicists, and this gives all the
trouble we are discussing about, but the words "virtual particles" in
fact have no other meaning than that given by the mathematical
expressions. It stands for a propagator of bare particles no more no
less.
A bare-particle propagator is of course not observable in principle
since we cannot do experiments with bare particles/fields, because these
are fictitous inventions of the human mind to organize the perturbation
series! What is observable in principle are the properties of single
real particles, meaning in a more accurate language, single-particle
states for real particles. The properties of the single-particle states
can be reconstructed from the full two-point Green's function which can
also be calculated with the Feynman-diagram technique. Again, each
Feynman diagram stands for a certain correction term to the bare
propagator no more no less, and one has to ask what is observable. A
single-particle state is characterized by some set of compatible
observables, e.g., momentum and spin states. If it comes to
asymptotically free (stable or quasi-stable) particles, there's also a
relation between energy and momentum, and this is observable in
principle (in case of a stable particle, it's a strict relation
omega=omeg(\vec{p}), in the case of an unstable but long-lived single-
particle excitation it's the spectral function). The spectral function
is given by the imaginary part of the full propagator. Approximately
(for not too broad spectral functions) the spectral function is
characterized by the possition of the mass peak and its width, which in
turn are given by the poles of the propagators in the complex energy
plane, and this is what is observable in principle (e.g., looking at
elastic pion-nucleon scattering one observes a strong resonance, called
Delta(1232) or lepton scattering giving the W- and Z-boson peaks from
weak interaction etc.).
The notion of virtual particles doesn't play a big role in the
interpretation of QFTs, it's only part of a convenient jargon of
experts. Unfortunately in many attempts to popularize this fascinating
theory by crude simplifications about the meaning of Feynman diagrams
"virtual particles" are made to something intuitively "real", and that's
imho a big didactical mistake. One shouldn't speak about the theory in
these terms. To popularize particle physics it's much wiser to
concentrate on phenomenology, i.e., the observable facts about particles
rather than to chat about highly specialized theories which are not
suitable for popularization since the real meaning of the abstract
objects (including Feynman diagrams, as I hope to have made clear above)
is incomprehensible without the appropriate training in theoretical
physics (that's why relativistic QFT is taught only in higher semesters
for physics majors, and renormalization theory and all that perhaps even
only at graduate-student level). What one can popularize, however, is
the general world view which follows from the application of quantum
theory to real phenomena, but for that one shouldn't (and doesn't need
to) talk about "virtual particles"!