Unitarity of Dynamical Propagators of Perturbed Klein-Gordon Equations

Abstract

After discussing the basic notions of quantizations as representations of the Weyl relations, a criterion is established for a symplectic transformation on a classical linear system to be unitarily implementable in the free (zero‐interaction) representation. The result is applied to the temporal propagators of ${\mathrm{\square \phi =m}}^2\mathrm{\phi +K\phi }$ to obtain a condition which is sufficient to ensure that they are unitarily implementable in the free representation of the quantized Klein‐Gordon field of mass m. Necessary conditions are also obtained when K commutes with m²I − Δ. Several examples are discussed, the most interesting of which is that of a mass jump (i.e., K = m′²I), where the results given are fairly complete.