A number of properties of the exchange operator are discussed in detail. (i) The problem of a harmonic crystal with a one- and two-phonon exchange operator is solved exactly. The matrix elements of the exchange operator are related to experimental quantities. The one- and two-phonon exchange operator does not provide an explanation of the experiment of Kirk and Adams [Phys. Rev. Lett. 24, 392 (1971)]. (ii) The surface integral formulas for the one- and two-phonon exchange operators are shown to give the wrong magnitude and sign. It is argued that the exchange operator must undergo the same kind of renormalization as the spring constants of the phonons. (iii) A four-spin contribution to the generalized exchange operator is identified that has the qualitative features required to explain the anomalous low-temperature properties of solid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$.

• Received 3 January 1974

DOI:https://doi.org/10.1103/PhysRevA.10.1785