Re: what determines virtual particles energy?

James,

All particles obey the rules of Quantum Mechanics, and one of the rules
called the Heisenberg Uncertainty Principle states that it is impossible
to know precisely the energy of a state of a system and simultaneously
the lifetime of the state.  Mathematically, the Principle looks like

               dE * dt >= hbar/2

where dE is the uncertainty in the energy, dt is the uncertainty in the
time, and hbar is Planck's constant divided by (2 pi).

In perturbation theory, a "virtual" particle is emitted by a real
particle and absorbed by a real particle.  The virtual particle can
violate conservation of energy by an amount dE, but only for an
amount of time dt given by the Uncertainty Principle, before it
vanishes.  The energy of the virtual particle can be anything from
a minimum energy E_min that depends on the particle type to infinity.

For example, the energy of the vacuum (zero) can fluctuate away from
zero because the energy can not be determined with infinite precision.
If the uncertainty in the energy dE is large enough, virtual
electron-positron pairs can form.  (Remember that energy can be
converted into mass, and vice versa, through Einstein's famous formula
E=mc^2.)  In this case, the minimum energy E_min is (2 m_e c^2) or about
1.022 MeV, where m_e is the mass of an electron (same as the mass of the
positron) and MeV is an energy unit, the mega-electron-volt.

          1 MeV = 1.6 x 10^(-13) joules

Using the Uncertainty Principle, the virtual electron-positron pair
can be seen to last at most about 10^(-21) seconds.  Of course, if the
virtual electron-positron pair is created with more energy than E_min
then the virtual state will not last this long.

As another example, an electron's energy can fluctuate by an amout dE
allowing the electron to emit a virtual photon.  Because the photon is
massless, the minimum energy E_min in this case is zero.  The virtual
photon can have any energy from near zero to infinite energy.  If the
energy of the virtual photon is very small, the photon can last for a
long time (even millions of years!) before being absorbed by a real
particle.

When one computes the effects of virtual particles on a process, such as
the emission of light when a real electron changes states in a hydrogen
atom, one integrates over all possible virtual particle energies from
E_min to infinity.  The Lamb shift calculation takes into account all
possible virtual particle energies; the amount of the shift therefore
does not depend on the virtual particles' energies because these have
been "integrated out".

--Randall J. Scalise    http://www.phys.psu.edu/~scalise/