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11.8: Positron Theory

We have already mentioned that the Dirac equation admits twice as many solutions as it ought to, half of them belonging to states with negative values for the kinetic energy \(c\,p^0 + e\,{\mit\Phi}^{\,0}\). This difficulty was introduced when we passed from Equation (1113) to Equation (1114), and is inherent in any relativistic theory.

Let us examine the negative energy solutions of the equation

\[\left[\left(p^0+\dfrac{e}{c}\,{\mit\Phi}^{\,0}\right)-\alpha_1\left(p^1+\dfrac{e}{c}\,{\mit\Phi}^{\,1}\right)-\alpha_2\left(p^2+\dfrac{e}{c}\,{\mit\Phi}^{\,2}\right)-\alpha_3\left(p^3+\dfrac{e}{c}\,{\mit\Phi}^{\,3}\right)-\beta\,m_e\,c\right]\psi=0. \tag{1290}\]

a little more closely. For this purpose, it is convenient to use a representation of the \(\alpha\)'s and \(\beta\) in which all the elements of the matrices \(\alpha_1\), \(\alpha_2\), and \(\alpha_3\) are real, and all of those of the matrix representing \(\beta\) are imaginary or zero. Such a representation can be obtained from the standard representation by interchanging the expressions for \(\alpha_2\) and \(\beta\). If Equation (1290) is expressed as a matrix equation in this representation, and we then substitute \(-i\) for \(i\) , we get [remembering the factor \(i\) in Equation (1111)]

\[\left[\left(p^0-\dfrac{e}{c}\,{\mit\Phi}^{\,0}\right)-\alpha_1\left(p^1-\dfrac{e}{c}\,{\mit\Phi}^{\,1}\right)-\alpha_2\left(p^2-\dfrac{e}{c}\,{\mit\Phi}^{\,2}\right)-\alpha_3\left(p^3-\dfrac{e}{c}\,{\mit\Phi}^{\,3}\right)-\beta\,m_e\,c\right]\psi^\ast=0. \tag{1291}\]

Thus, each solution, \(\psi\), of the wave equation (1290) has for its complex conjugate, \(\psi^\ast\), a solution of the wave equation (1291). Furthermore, if the solution, \(\psi\), of (1290) belongs to a negative value for \(c\,p^0 + e\,{\mit\Phi}^{\,0}\) then the corresponding solution, \(\psi^\ast\), of (1291) will belong to a positive value for \(c\,p^0-e\,{\mit\Phi}^{\,0}\). But, the operator in (1291) is just what we would get if we substituted \(-e\) for \(e\) in the operator in (1290). It follows that each negative energy solution of (1290) is the complex conjugate of a positive energy solution of the wave equation obtained from (1290) by the substitution of \(-e\) for \(e\). The latter solution represents an electron of charge \(+e\) (instead of \(-e\), as we have had up to now) moving through the given electromagnetic field.

We conclude that the negative energy solutions of (1290) refer to the motion of a new type of particle having the mass of an electron, but the opposite charge. Such particles have been observed experimentally, and are called positrons. Note that we cannot simply assert that the negative energy solutions represent positrons, since this would make the dynamical relations all wrong. For instance, it is certainly not true that a positron has a negative kinetic energy. Instead, we assume that nearly all of the negative energy states are occupied, with one electron in each state, in accordance with the Pauli exclusion principle. An unoccupied negative energy state will now appear as a particle with a positive energy, since to make it disappear we would have to add an electron with a negative energy to the system. We assume that these unoccupied negative energy states correspond to positrons.

The previous assumptions require there to be a distribution of electrons of infinite density everywhere in space. A perfect vacuum is a region of space in which all states of positive energy are unoccupied, and all of those of negative energy are occupied. In such a vacuum, the Maxwell equation

\[ \nabla\cdot {\bf E} = 0 \tag{1292}\]

must be valid. This implies that the infinite distribution of negative energy electrons does not contribute to the electric field. Thus, only departures from the vacuum distribution contribute to the electric charge density \(\rho\) in the Maxwell equation

\[\nabla\cdot{\bf E} = \dfrac{\rho}{\epsilon_0}. \tag{1293}\]

In other words, there is a contribution \(-e\) for each occupied state of positive energy, and a contribution \(+e\) for each unoccupied state of negative energy.

The exclusion principle ordinarily prevents a positive energy electron from making transitions to states of negative energy. However, it is still possible for such an electron to drop into an unoccupied state of negative energy. In this case, we would observe an electron and a positron simultaneously disappearing, their energy being emitted in the form of radiation. The converse process would consist in the creation of an electron positron pair from electromagnetic radiation.

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