Quantum Man: Richard Feynman's Life in Science (12 page)

In order to calculate the quantum mechanical amplitude for such a process, we would have to consider all possible space-time paths corresponding to the exchange of a virtual photon between the two particles. Since we don’t observe the photon, the following process, in which the photon is emitted at
B
and absorbed at
A
, when
B
is earlier than
A
, also contributes to the final sum:

Now, there is another way of thinking about the two separate diagrams. Remember that we are dealing with quantum mechanics, and therefore in the time between measurements, anything that is consistent with the Heisenberg uncertainty principle is allowed. Thus, for example, the virtual photon is not restricted to travel at exactly the speed of light for the entire time it travels between the two particles. But special relativity says that if it is traveling faster than light, then in some frame of reference it would appear to be going backward in time. If it is going backward in time, then it can be emitted at
A
and absorbed at
B
. In other words, the second diagram corresponds to a process identical to the first, except that in the latter case the virtual photon is traveling faster than light.

In fact, while Feynman never explicitly described it at that time, as far as I know, this same effect explains why a relativistic theory of electrons—that is, Dirac’s theory—
requires
antiparticles. A photon, which doesn’t have any electric charge, traveling backward in time from
A
to
B
, just looks like a photon traveling forward in time from
B
to
A
. But a charged particle traveling backward in time looks like a particle of
opposite
charge traveling forward in time.

Thus, the simple process where an electron (e–) travels between two points, pictured in this space-time diagram:

must also be accompanied by this process:

But the latter process could also be described with an intermediate positron (e+), as follows:

In other words, it appears that a single electron begins its journey, and at another point an electron-positron pair is created from empty space, and a virtual positron travels forward in time, ultimately annihilating with the first electron, leaving only the single final electron at the end of the journey.

Feynman later beautifully described this situation in his 1949 paper “The Theory of Positrons.” His famous analogy was a bombardier looking down at a road from his scope on an airplane (the recent world war had undoubtedly influenced Feynman’s choice of analogies here): “It is as though a bombardier watching a single road through the bomb-sight of a low flying plane suddenly sees three roads and it is only when two of them come together and disappear again that he realizes that he has simply passed over a long switchback in a single road.”

Thus particle number in a relativistic quantum theory has to be indeterminate. Just when we think we have one particle, a particle-antiparticle pair can pop out of the vacuum, making it three. After the antiparticle annihilates one of the particles (either its partner or the original particle), once again there is only one, just as what the bombardier would see through his bomb-sight if he were counting roads. The key point again is not merely that this is
possible
but that it is
required
by relativity, so that with hindsight we see that Dirac had no other choice but to introduce antiparticles in his relativistic theory of electrons and light.

That Feynman was the one to point out the possibility of treating positrons as time-reversed electrons in his diagrams is fascinating because it immediately implies that his earlier aversion to quantum field theory was misplaced. His diagrammatic space-time expansion for calculating physical effects in QED implicitly contained within it the physical content of a theory where particles could be created and destroyed and particle number during intermediate steps of a physical process was therefore indeterminate. Feynman had been forced by physics to reproduce the physical content of quantum field theory. (In fact, in an obscure 1941 paper that predated Feynman’s by eight years, the German physicist Ernst Stückelberg had independently been driven to consider space-time diagrams, and positrons as time-reversed electrons, although he was not sufficiently driven to carry through the program that Feynman ultimately carried out with these tools.)

Now that we are familiar with the diagrams that would eventually become known as
Feynman diagrams
, we can depict the events that correspond to the otherwise infinite processes associated with the electron self-energy and vacuum polarization:

Self-energy (an electron interacting with its own electromagnetic field)

Vacuum polarization (splitting of a virtual photon into electron-positron pair)

For Feynman there was a fundamental difference between these two diagrams, however. The first diagram he could imagine occurring naturally as an electron emitted a photon and later reabsorbed it. But the second diagram seemed unnatural because it would not result from the trajectory of a single electron moving and interacting backward and forward in space and time, and he felt that such trajectories were the only appropriate ones to incorporate in his calculations. As a result, he was wary of the need to include these new processes, and did not originally do so. This decision caused Feynman a number of problems as he tried to derive a framework in which all of the infinities of Dirac’s theory could be obviated, and in which predictions for physical processes could be unambiguously derived.

The first great success of Feynman’s methods involved calculating the self-energy of the electron. Most important, he found a way to alter the interaction of electrons and photons at very small scales and very high energies in a manner that was consistent with the requirements of relativity. Pictorially this results from considering the case where the loop in the self-energy diagram becomes very small, and then altering the interactions for all loops that are small and smaller. In this way a provisional result could be derived, which is finite. Moreover, this result could be shown to be independent of the form of the alteration of the interactions for small loops in the limit that the loops become smaller and smaller. Most important, as I stressed earlier, because the loops take into account an arbitrary time of emission and absorption and at the same time include objects going forward and backward in time, the form of his alteration did not spoil the relativistic behavior of the theory, which should not depend on any one observer’s definition of time.

As Kramers and others had predicted, the key was making these altered-loop contributions finite, or
regularizing
them, as it was later called, in a way that was consistent with relativity. Then if one expressed the corrections to physical quantities, such as the energy of an electron in the field of a hydrogen atom, in terms of the physical mass and physical charge of the electron, the remainder, after canceling out the term that would otherwise become infinite without the alteration for small loops, was both finite and independent of the explicit form of the alteration one made. More important, it remained finite even if the size scale at which one alters the loop diagrams is decreased to zero, where the loop diagram would otherwise become infinite. Renormalization worked. The finite correction agreed reasonably well with the measured Lamb shift, and electrodynamics as a quantum theory was vindicated.

Unfortunately, however, the same type of procedure that Feynman used to change the theory on small distance scales when considering the self-energy of the electron did
not
work when considering the infinite impact of vacuum polarization diagrams. Feynman could find no alteration for small electron-positron loops that maintained the nice mathematical properties of the theory without such an alteration. This is probably another reason, in addition to his sense that these diagrams might not be physically appropriate to the problem at hand, why he ignored them in his original Lamb shift calculation.

Feynman wrestled with this problem on and off during 1948 and 1949. He was able to derive quantitatively accurate results by intuitively supposing that various extra terms in his equations, induced by altering the form of the electron-positron loop, were likely to be unphysical, because they did not respect the mathematical niceties of QED, and therefore could be safely ignored. This unsatisfactory situation was resolved when late in 1949 Hans Bethe informed Feynman of a trick Wolfgang Pauli discovered that allowed a mathematically consistent alteration to be introduced in vacuum polarization diagrams.