Natural Science Forum / Physics / Particle Physics / October 2004

A brief primer on "virtual" particles, for those who are uncomfortable
with the topic.

It's actually an arbitrary line between real and virtual particles.
Real particles are ones that have been around infinitely long and
therefore have no uncertainty in their invariant mass, but that's an
asymptotic condition, something like "perfect". In reality, there's
only various degrees of near-perfect observed but never perfect.

In a Feynman diagram, you are tracing a possible "route" between
initial and final state particles, which may include internal lines
representing particles neither in the initial or final state. These
particles are called virtual particles in this context, though note
that in a real experiment, even the final state particles have to
interact with the detector, which makes them virtual... But let's set
that last point aside and just say we want to know the rate at which a
particular final state emerges from a particular initial state.

A given Feynman diagram may be one of several of the same complexity,
and in general one of infinitely many of different complexities, that
are all possible "routes" between the initial and final states. What
this means is that the initial and final states might be (and in fact
are) created by a vast number of different possible processes. To get
the right answer for how often the final state gets created from the
initial state, we add up the amplitude (related to the likelihood) of
each one and -- here's where it gets a little weird -- we only get the
right answer if we add them up as though they were all happening at
the same time.

Another interesting facet of a single Feynman diagram is that it
really represents a sum (integral) over all possible momentum and
energy values of the internal lines, or rather, those that obey the
conservation of energy and momentum. What this means is, you can't
tell by looking at a diagram what the momentum and energy of all the
internal lines are going to be, and so you have to sum over all the
possible momenta they can have -- again as though they were all
happening at the same time.

Every "vertex" in a Feynman diagram represents part of an interaction
-- a photon is emitted from an electron, a W+ is emitted from a u
quark, etc. What's important to recognize is, virtual or not, every
vertex strictly conserves energy and momentum. Taking a step back,
this means that energy and momentum conservation apply rigorously and
*exactly* throughout the Feynman diagram. Of course, this means that
the initial and final states will have the same total energy and the
same total momentum.

But here's the rub. Even though energy and momentum are individual
conserved, the invariant mass of the virtual particle do NOT have to
correspond to what you would expect the invariant mass of that
particle to be if it were real. For example, an internal line in a
Feynman diagram can be an electron, identified by its quantum numbers
(electric charge, lepton number) and the vertex can conserve momentum
and energy exactly, but the invariant mass of the virtual electron
need not be 511 keV. Likewise, an internally radiated photon will
conserve momentum and energy at every vertex in which it participates,
but its invariant mass will not necessarily be zero. Of course, the
invariant mass of the final state has to be the same as the invariant
mass of the initial state, but that's not the same constraint.

There are three arguments why we should allow this behavior.
* First, the Heisenburg Uncertainty Principle says that it's not
unreasonable and in fact it's expected.
* Second, if you don't allow it, you don't get the right answers for
the rates from the initial to final states from the calculated sum;
and you do if you do allow it.
* Third, we have a big library of particles that only live a short
lifetime, some long enough for us to spot directly and track directly
in a detector, others only indirectly. In all such cases, we can still
reconstruct the momentum and energy of the transient beast, either
directly or by measuring the momentum and energy of the final state it
produces, and then we can calculate the beast's invariant mass. And
what we find is that the invariant mass from a collection of such
measurements does not produce a single number but a distribution of
numbers -- a peak in a distribution vs mass, with a definite width.
And once again, the width of that peak depends on how short-lived the
particle is, again consistent with the uncertainty principle. Note
that I do not have to ASSUME the uncertainty principle to confirm it
in this measurement. I just have to assume energy and momentum
conservation and calculate the invariant mass. There is no circular
logic. In any event, the noted behavior is, the shorter the lifetime
of the particle, the wider the distribution of mass -- the more
"off-shell" the particle is likely to be found. In this sense,
particles that happen to have the central mass value in this
distribution are no less "virtual" than the ones that are far
off-shell. They just happen to be in that part of the distribution.
This width has nothing to do with experimental uncertainty. Even after
the resolution of the measurement has been deconvoluted, a width of
physical origin remains.

Nearly-real particles, ones with long lifetimes, have very narrow mass
distributions, so that the peak mass is very nearly always the mass
you will measure in all measurements. For example, for photons that
have traveled long distances, the invariant mass -- as calculated from
momentum and energy -- will always be indistiguishable from zero, and
this has other implications about helicity states and so on.

What this means is that the value of mass is not an intrinsic property
of a particle. The value of electric charge, the value of color
charge, the value of baryon number, the value of lepton number all
might be intrinsic properties of a particle, but not the value of its
mass.

In short, there is nothing magic about virtual particles. If anything,
their prevalence tells us that our notions that we attach to "real"
particles are things we should let go of as being overly constraining,
overly confining.

PD