equation that wasfirst order in the time derivative, he hoped to have an equation that behaved
like the Schr ̈odinger equation, an equation for a single particle. TheDirac equation also has
“negative energy” solutions. While the probability is positive, theflux that we have derived
is in the opposite direction of the momentum vectorfor the “negative energy” solutions.
We cannot discount the “negative energy” solutions since thepositive energy solutions alone
do not form a complete set. An electron which is localized in space, will have components of its
wave function which are “negative energy”. (The infinite plane wavesolutions we have found can
be all positive energy.) The more localized the state, the greater the “negative energy”
content.
One problem of the “negative energy” states is that an electron in apositive energy (bound or free)
state should be able toemit a photon and make a transition to a “negative energy” state.
The process could continue giving off an infinite amount of energy. Dirac postulated a solution to
this problem. Suppose thatall of the “negative energy” states are all filledand the Pauli
exclusion principle keeps positive energy electrons from making transitions to them.
The positive energies must all be bigger thanmc^2 and the negative energies must all be less than
−mc^2. There is an energy gap of 2mc^2. It would be possible for a“negative energy” electron to
absorb a photon and make a transition to a positive energy state. The minimum photon
energy that could cause this would be 2mc^2. (Actually to conserve momentum and energy, this must
be done near a nucleus (for example)). Ahole would be left behindin the usual vacuum which
has a positive charge relative to the vacuum in which all the “negativeenergy” states are filled. This
hole has all the properties of a positron. It has positive energy relative to the vacuum. It has
momentum and spin in the opposite direction of the empty “negative energy” state.
The process of moving an electron to a positive energy state islike pair creation; it produces both
an electron and a hole which we interpret as a positron. The discovery of the positron gave a great
deal of support to the hole theory.
The idea of aninfinite sea of “negative energy” electronsis a strange one. What about all
that charge and negative energy? Why is there an asymmetry in thevacuum between negative and
positive energy when Dirac’s equation is symmetric? (We could also have said that positrons have
positive energy and there is an infinite sea of electrons in negative energy states.) This is probably
not the right answer but it has many elements of truth in it. It also gives the right result for some
simple calculations. When the Dirac field is quantized, we will no longer need the infinite “negative
energy” sea, but electrons and positrons will behave as if it were there.
Another way to look at the “negative energy” solution is as apositive energy solution moving
backward in time. This makes the same change of the sign in the exponential. The particle would
move in the opposite direction of its momentum. It would also behave as if it had the opposite
charge. We might just relabel~p→−~psince these solutions go in the opposite direction anyway and
change the sign ofEso that it is positive. The exponential would be then change to
ei(~p·~x−Et)/ ̄h→e−i(~p·~x−Et)/ ̄hwith e positive and~pin the direction of probability flux.
36.8 Equivalence of a Two Component Theory
The two component theory withψA(andψBdepending on it) isequivalent to the Dirac theory.
It has a second order equation and separate negative and positiveenergy solutions. As we saw in