Quantum Electrodynamics 219
the conservation of energy is meaningless if it cannot be measured. If
there is no way of detecting the photon during the time of its exchange
then we do not have to worry about the conservation of energy since the
total energy before and after the exchange remains the same. We,
therefore, shall assume that the static electric force arises from the
exchange of a “virtual photon” between two charged particles. We call
the photon a “virtual photon” because it cannot be detected directly.
Let us investigate whether the exchange of a virtual photon is
inconsistent with conservation of energy and the uncertainty principle.
Unfortunately, this investigation will require some rudimentary
mathematics, which should not disturb most readers. For those who wish
to skip the mathematics, what we shall essentially show is that the
uncertainty principle does not permit an observation of the exchanged
photon and hence, the violation of energy conservation. We shall also
show that the only form of the force consistent with the hypothesis of
virtual photon exchange, the uncertainty principle and conservation of
energy is the inverse square law.
The uncertainty principle has two forms; one of which states that
the product of the uncertainty in measuring the position and the
momentum must be greater than h, Planck’s constant or ∆p ∆x > h. The
other form of the uncertainty principle states that the product of
the uncertainties in measuring the lifetime and energy of a system is
greater than h or ∆E ∆t > h. It is the latter form that we shall exploit
now. The system of the virtual photon will have a lifetime, t = R/c since
this is the time required for the photon to propagate the distance, R,
between the two charges. We assume the energy of the virtual photon is
related to the potential energy of the two charged particles, which in turn
depends on the force. The force, we learned, is inversely proportional to
the square of the distance between the two charges.
If the two particles are electrons then the force is given by, F = ke^2 /R^2
where k is a constant that we can take equal to one. The energy of the
virtual photon is its potential energy, Ep, that is related to the force, F,
by the expression, Ep = FR and hence Ep = e^2 /R. The lifetime of the
virtual photon, t, is the time it takes the virtual photon to travel the
distance R between the two charges and equals R/c. The product of
the energy of the photon, E = Ep = e^2 /R times its lifetime t = R/c is just
given by Et = e^2 /R × R/c = e^2 /c.
The quantity e^2 /c has the same dimensions as h and just happens to be
equal to h/137. Since the product of the quantities that we wish to