-----------------
Virtual particles
-----------------
Virtual particles are part of the imagery of quantum field theory.
They are figurative language for abstract mathematics, used by experts
and laymen as imagery for giving abstract recipes for calculating
scattering amplitudes an appearance of intuitive meaning.
However, any attempt to take this language literally gives a very
misleading and unscientific view of the microscopic world.
The virtual particle imagery stems from the 1940s and 1950s when people
tried to understand how quantum electrodynamics and its generalizations
can make sense. For the experts of today, the term is fully exchangable
with ''internal lines in a Feynman diagram'', without any intended
meaning beyond that.
The collection of Feynman diagrams without loops describes _exactly_
the scattering of classical fields in a perturbation theoretic
treatment; the diagrams with k loops describe quantum corrections of
order O(hbar^k).
If virtual particles had a meaning, then they would already exist
in classical field theory, since tree diagrams have internal lines.
But nobody ever claimed that predictions of classical field theories
are caused by virtual particles.
Popularizations often take the imagery for real since on the surface it
seems far more understandable that the formal stuff. But these
popularization have to pay for it (and get paid for it by the public,
in terms of sold copies of their books) by having to ascribe to the
virtual particles very strange properties far from both ordinary
experience and measurable facts - solely to be able to maintain a
pseudo-realistic view of virtual particles.
Nowhere in science is such a way of proceeding deemed respectable.
Indeed, quantum mechanics is much more rational and intelligible if one
avoids such spurious imagery. So it is best to unlearn it as soon as
possible.
Physicists talk about virtual particles as illustrative language for
internal lines in so-called Feynman diagrams. A Feynman diagram is a
mnemonic graphical representation of a multiple integral contributing
to a scattering amplitude in a collision process between real (i.e.,
measurable) particles. The collision process is characterized by
ingoing and outgoing real particles, represented in the Feynman diagram
as external lines, which suggests an interpretation of the remaining,
internal, lines as sort of short living intermediated products of the
collision, made up of virtual (i.e., nonmeasurable) particles.
While this gives the Feynman diagrams an intuitive interpretation,
it is impossible to give this intuition a deeper foundation in terms
of processes happening in space and time. The attempt to do so leads
to a phantastic view of the microscopic world.
In this phantastic world view, all sorts of unobservable, nonphysical
behavior (e.g., imaginary mass, violation of the conservation of energy,
violation of causality, traveling faster than light or backwards in
time, popping in and out of the vacuum via ''vacuum fluctuations'',
etc.) must be postulated in order to reconcile the mathematical
properties of virtual particles with their alleged existence in space
and time.
None of these speculative aspects can be verified by experiment, which
places them outside the realm of science and into the realm of fiction.
Since virtual particles are unobservable, one can attribute to them
whatever properties one likes, without any real consequence or
testability. This explains the phantastic aura surrounding virtual
particles, and it also explains their name - they are called virtual
since they are not real in any strong sense of the word.
If one clearly distinguishes between reality and virtual reality, one
finds that the physics of the former is much more rational than that
of the latter, where everything goes, and where (as the Wikipedia
article on virtual particles shows) inconsistent statements stand
undisputed side by side.
On the other hand, those accustomed to the view that virtual particles
are ''really there'' have later a difficult time unlearning it when
they want to get real understanding and want to work with the concepts.
Below the surface talk, nothing but internal lines in diagrams is
associated with the concept. No states, no positions, no motion
(forward or backward in time), no times, no creation or annihilation
- nothing.
People are sometimes invoking Heisenberg's uncertainty relation that
allegedly allows the violation of conservation of energy for a very
short time, thus apparently making room for seemingly nonphysical
processes. However, the uncertainty relation is based on the existence
of operators satisfying the canonical commutation rule, and while
there are such operators for spatial position and spatial momentum,
there are no such operators for time and energy, or for 4-position
and 4-momentum. Indeed, there is no time operator in either quantum
mechanics or quantum field theory, and since the energy operator (the
Hamiltonian) of a physical system is always bounded below, it cannot
be part of a pair of operators satisfying the canonical commutation
rule. Therefore the time-energy uncertainty relation is without a
formal basis.
What can be verified with high accuracy are physical effects derivable
from the scattering theory of the particles, i.e., from the fully
summed and renormalized perturbative calculations involving an
evaluation of the multiple integrals represented by the Feynman
diagrams. Plenty of experiments establish without doubt the correctness
of the scattering theory and the phenomena predicted by it, such as
Coulomb scattering and the Casimir effect.
But (in spite of frequent claims in the popular physics literature
and sources from the internet) none of these experiments verify
anything of the unobservable phantastic scenarios frequently associated
with virtual particles. The claims simply rest on taking the successes
of perturbation theory with its Feynman diagrams as proof of the
validity of the virtual particle picture. But these successes are
based on the multiple integral interpretation of the Feynman diagrams,
not on their virtual particle interpretation. No evidence at all
exists that the latter has any roots in space and time.
There is plenty of evidence that sums of Feynman diagrams, interpreted
as renormalized multidimensional integrals, correctly predict many
phenomena. But to interpret this as evidence for the existence of
virtual particles manifesting themselves in space and time is
stretching the interpretation too far -- something perhaps acceptable
at the layman's level to provide some sort of intuition for otherwise
too abstract things (which is what one can find in popularizing
accounts by some well-known physicists), but unacceptable on a more
scientific level.
Indeed, there are strong arguments that loudly speak against assigning
reality to virtual particles. If virtual particles were real, they
would leave their trace in all methods of predicting certain phenomena,
and they would assign the same properties to the virtual particles no
matter which approximation method is used.
However, the literature readily shows that the details of Feynman
diagrams strongly depend on the perturbation scheme used:
In light front calculations, one gets a completely different set of
diagrams than in the more traditional covariant form. And in
nonperturbative approaches such as lattice gauge theory or conformal
field theory, the predictions do not involve virtual particles at all.
How can anything be real if its existence depends on a particular way
of viewing the world? How can an experiment (verifying the Casimir
effect, say) can be said to prove the existence of virtual particles
if the same experiment can be explained by a method of calculation not
involving virtual particles at all?
The nonexistence of virtual particles in nonperturbative calculations
(whether conformal field theory or lattice gauge theory) is proof that
the virtual particle concept is an artifact of perturbation theory.
Something whose existence depends on the method of calculation cannot
exist in a strong sense of the word.
Thus from the scientific point of view, the concept of virtual
particles is quite shallow. The latter is not the level of science but
the level on which science can be presented to laymen.
For them, physicists put their intuition (which often is quite
imprecise) into ordinary language since it is intended only to
give a rough orientation of what happens. But when they do real
science, they ignore the superficial imagery and work on the level of
formulas - not virtual particles.
The huge differences in the answers given by experts to questions such
as ''Are virtual particles really there?''only shows how vague these
questions are: Each physicist understands them in a different way,
because they have a different conception of the meaing of 'real'.
See the thread on the Physics Forum at
http://www.physicsforums.com/showthread.php?t=75307
where a number of well-known physicists are cited with widely differing
answers related to this question.
Therefore virtual particles ''exist'' as lines on paper, as intuition
in people's minds, as superficial but catchy allusions to images that
make abstract things concrete, but not as tangible, verifiable entities.
On the level of physics, virtual particles are quite similar to what
ghosts are on the level of ordinary experience. One cannot ascribe
to them most properties that real things have. One can only ascribe
to them the properties of internal lines in diagrams and
multidimensional integrals in perturbative computations. Once one
attempts to ascribe to them more, one gets nonsense.
A real particle is generally taken as an elementary system (described
by an irreducible representation of the Poincare group) separated well
enough from the environment to be tractable with creation and
annihilation operators (e.g., as an in or out state in scattering).
This separation makes it distinguishable enough from the environment
to merit the designation ''particle''. Note that it is only an
approximate concept, but a very useful one. When the separation gets
poorer (as during scattering or in a many-body context), the notion of
a real particle becomes less and less useful. In particular, in the
solid state, one has no longer identifiable particles but only
so-called quasi-particles. Again their characteristics is that they
are described by (effective) creation and annihilation operators.
On the other hand, there are no creation and annihilation operators
for virtual particles, not even in theory. This makes them unreal -
they cannot be created or annihilated, not even in theory.
They can only be used to write down Feynman diagrams!
That calculations of perturbative effects involve integrals
corresponding to internal lines of Feynman diagrams (which may be
interpreted loosely as virtual particles) doesn't make these virtual
particle real. (Nowhere in physics is reality ascribed to diagrams
related to mathematical techniques that help one evaluate the terms of
a series.)
One cannot write down a state vector containing a virtual particle -
a physical Hilbert space does not contain such states. But one can
easily write down state vectors for the usual, real objects, such as
quarks, nuclei, electrons, or photons.
What is left of the virtual particle concept when stripped of the
phantastic superstition surrounding it?
On the scientific level, the formal definition of a virtual particle
is as an object associated with an internal line of a Feynman diagram.
As such, a virtual particle corresponds to a particular representation
of the Poincare group, hence has a definite value of rest mass, spin
or helicity, and maybe other quantum number such as charge or color.
Associated with the internal line is a 4-dimensional momentum variable
p over which an integration is performed. The fact that we have an
internal line means that the virtual particle is ''exchanged'' between
other (virtual or real) particles; because of the effective force
resulting in the scattering, one says that the virtual particles
''mediate'' the force between the real particles in volved in the
scattering process.
The mommentum is not specified but takes all possible values consistent
with the boundary conditions imposed by the scattering process.
Energy-momentum conservation is part of the formal framework of
Feynman diagrams; it allows one in the simplest (H-like) exchange
diagram between two real particles to relate the possible momenta
of the virtual particle to the measurable ingoing and outgoing momenta.
If the ingoing momenta are p and p' then the outgoing momenta are
p+q and p'-q, where q is the momentum exchanged, i..e, the momentum
transported by the virtual particle. In particular, one can determine
q from measurements.
That's all; this makes up virtual particles and their alleged influence
on real (observable) particles. Everything else is superstition.
And this much is needed to relate virtual particles to QFT: without the
interpretation as internal lines of Feynman diagrams, QFT would be
completely silent about virtual particles, and none of the successes
of QFT could be interpreted as evidence for virtual particles.
On the surface, this virtual particle picture is attractive and has
an appearance of explanatory power. This is the reason why it is
frequently used as an illustrative tool for laymen and when introducing
the abstract formalism of Feynman diagrams.
But the limitations of the virtual particle picture become immediate
when one realizes that the H-like Feynman diagram only gives the
lowest-order contribution to the scattering amplitude. Higher-order
contributions involve more complex diagrams with the same external
lines but arbitrary internal structure involving loops, constrained
only by the general principles of conservation of charges.
Thus, depending on the Feynman diagram considered, the two particles
exchange one or more virtual particles, which again may exchange one
or more virtual particles, etc.. Unlike in the simple case of the
H-like diagram, the exchange momenta in the more complex diagrams are
no longer determined by the external momenta; so there are infinitely
many different situations that are possible. Which of these possible
exchanges actually happened? It is impossible to answer to this, since
the exchange is unobservable, ficticious.
A common intuitive response is to argue that in quantum theory,
things happen simultaneously in a superposition of all the
possibilities. But unlike in the simple nonrelativistic quantum systems
used to illustrate the superposition principle, it is impossible in
relativistic theories such as QED to calculate probabilities for the
possible exchange momenta defining the virtual particles. The reason
is that the Feynman integrals containing loops all diverge! Thus
there is not even a formal support for assigning probabilities of
virtual particles exisitng in a specified number and with specified
momenta. Of course, there are renormalization prescriptions for
rendering the scattering amplitudes finite, but these prescriptions
don't render the contributions of individual diagrams finite but only
the total sum with a given number of loops.
Another common response is to fall back on Feynman's path integral,
according to which the scattering amplitude of a single particle in
an external potential is represented as an integral over all posssible
paths of the particle. This suggests that the scattering amplitude of
a collision process is represented as an integral over all posssible
histories of the input, allowing for pair creation and destruction.
The histories appearing in this way look like Feynman diagrams,
except that they live in space and time. But in the path integral,
all histories appear, whether or not they satisfy energy-momentum
conservation, while they can be interpreted as Feynman diagrams only
in case of energy-momentum conservation. Thus the correspondence is
only apparent.
Moreover, the path integrals actually used in quantum field theory
are always integrals over histories of a quantum field, not of a
collection of particles.
The virtual particle view is dynamically not coherent. There is no
theory how the state of a virtual particle changes with time, not even
in the simplest situations. Virtual particles make sense only at a
very superficial level comparable to a billiard ball view of quantum
particles. Both are very inadequate to describe reality.
Trying to interpret virtual particles in space and time by giving
the quantum field formalism a well-defined Hamiltonian formulation
also failed. The Feynman diagram techniques only tell about the
behavior of a process starting in the infinite past and proceeding
to the infinite future, and says nothing at all about the time in
between. Nobody so far succeeded to give a valid definition of
QED at finite times. Since virtual particles are defined only in
terms of the Feynman diagrams, they describe asymptotic properties
of the scattering, not an actual motion (which would be described by
some process at finite times). Thus virtual particles don't ''move''.
They are ''exchanged'', but it makes no scientific sense sense to talk
about their motion, their speed, or about the direction they travel,
This is meaningless talk, and asking about such properties is like
asking for the speed of a ghost.
Thus it seems impossible to place the superficial virtual particle
picture on a sound scientific footing. It is a picture valid only
if restricted to the superficial level where no detailed inquiries
are made. It is like ordinary people using the word ghost to describe
a fleeting but fear-provoking experience. It makes sense only as long
as you don't ask about their precise properties. But once you start
asking how fast a ghost is traveling, things no longer make sense,
since the concept of a ghost is not intended to be applied literally.