Relativistic Lee Model

A relativistic version of the well-known exactly soluble Lee model of quantum field theory is constructed, which—unlike the original nonrelativistic Lee model—fulfills also the condition of microcausality (field operators commuting or anticommuting for space-like distances). Microcausality is, however, found compatible with solubility only if an indefinite metric is introduced on a much larger scale than was necessary in the nonrelativistic case. Notwithstanding the fact that a formalism of this type would seem to bear still less resemblance to physical reality than does the original Lee model, we arrive at a theory almost identical with the usual version of relativistic charged scalar (or pseudoscalar) meson field theory. From a certain point of view these two formalisms can even be considered identical. In fact, the difference between the realistic theory and the present Lee model hinges merely on whether one uses, for the representation of the Dirac standard ket $|0\mathrm{}〉$ a physical vacuum or a suitably defined "completely empty space." The form of the presentation follows closely that of the well-known Pauli-Källén description of the nonrelativistic Lee model, up to the point where it is shown that some of the main results can also be obtained with help of a corresponding Chew-Low equation.