Coherent Soft-Photon States and Infrared Divergences. I. Classical Currents

As a first step toward a treatment of soft‐photon processes which is free of infrared divergences and avoids the necessity of introducing a fictitious photon mass, the specification of asymptotic photon states belonging to non‐Fock representations is discussed. As in the work of Chung, a basis consisting of generalized coherent states is used, but in contrast to his work, these states are rigorously defined in terms of von Neumann's infinite tensor product. It is shown that the states must be given an additional label which serves to distinguish various ``weakly equivalent'' vectors, and which corresponds formally to an infinite phase factor. A nonseparable Hilbert space ${\mathcal{H}}_{\mathrm{IR}}$ is defined (as a subspace of the infinite tensor‐product space) which may be regarded as the space of all possible asymptotic photon states. The interaction of the electromagnetic field with a prescribed classical current distribution is discussed, and it is shown that a unitary S operator, all of whose matrix elements are finite, may be defined on ${\mathcal{H}}_{\mathrm{IR}}$ .