Introduction The Dirac hole theory was developed in response to a growing crisis over the Dirac theory of the electron. It predicted the existence of antiparticles in relativistic quantum theory; the antiparticle came into existence as a 'hole' in a sea of negative-energy particles. In 1972 Heisenberg said, 'I think that this discovery of antimatter was perhaps the biggest jump of all the big jumps in physics in our century'. He was speaking of the phenomenology of pair creation and annihilation processes, the basic mechanisms of relativistic dynamics, but the conceptual basis, the concept of antimatter, has a corresponding importance…
Nowadays no one would regard the use of this Fourier expansion, or of the normal-ordering, as logically dependent on the Dirac hole theory; they are supposed to stand in their own right. At the same time, the entire theory can be regarded as a quantum field theory, and the link with the 1-particle theory becomes hopelessly tenuous. One looks upon the Dirac equation as a classical field equation, itself derived from a classical Lagrangian. The quantization of this theory is to yield the standard formalism (correctly interpreted), as above. There is, however, a connection between the antimatter fields and the negative-energy solutions. The latter contribute negative energy to the total field energy. (The use of anticommutators on quantization allows us to change the sign of this contribution, depending on whether we consider it a creation field or an annihilation field.) This part of the field is the antiparticle (creation) field. But the negative-energy solutions disappear from the Foc
id: dcefba131c9216f20b57797367ccc710 - page: 20
What were before calculations of transition amplitudes at the level of the states become analysis at the level of the fields; particularly, it is analysis on the c-number bispinors that occur in the plane-wave expansion.
id: 2e0251e80b55bad801307a328547a3ec - page: 20
It does not appear possible to understand antimatter at the level of the states. As a result, certain questions, such as the definition of states when one does not have a scattering situation, or the meaning of the Wigner negativeenergy representations of the Lorentz group (which now appear to be excluded by fiat), cannot even be formulated. It is the canonical theory that imparts precision to these questions.
id: 4274867f17e6f37b6554505602c65113 - page: 20
Let me pursue the question of the independence of the standard formalism from the Dirac hole theory. The problem is to justify the planewave expansion of the fields. It turns out that the necessary assumptions have a natural interpretation in field theory: the field must be a linear combination of creation and annihilation operators. This is implicit in some earlier discussions, but (so far as I know) there is no very clear statement prior to Weinberg (1964). One reason for this neglect is that the precise definition of the creation and annihilation operators namely an explicit action on a concrete Fock space was not and could not have been available prior to the late 1950s, because of the difficulty in relating the Dirac bispinors to the Wigner spinor representations. This problem (cf. fn. 17) requires the distinction between representations on Hilbert space and those on Hilbert space bundles;21 the explicit bispinor c-numbers that occur in the plane-wave expansion should be und
id: 85dcbd77003c9b6187976fb59dadd64e - page: 20