In the early 1970's, attention was drawn to the remarkable similarity between the excitation spectra exhibited by S(Q, ω) in solid 4He and superfluid 4He at low temperatures (Werthamer, 1972; Horner, 1972a; Glyde, 1974), as shown very dramatically in the theoretical results of Figs. 11.1 and 11.2. While various suggestions have been made as to the origin of this similarity, it remains an unresolved and intriguing problem. In this brief chapter, we compare the theoretical description of excitations in a quantum solid with those of a Bose-condensed liquid. While we review the key ideas, we assume that the reader has some familiarity with an introductory account of quantum crystals. (The modern theory of excitations in quantum crystals was essentially completed in the early 1970's. For background and a more detailed discussion of solid 4He than we give in this chapter, we recommend the review by Glyde, 1976.)

In both condensed phases, it is important to distinguish clearly between the elementary excitations and the density fluctuations. We argue that the phonons in solid 4He are the natural analogue of the single-particle excitations in liquid 4He. In Section 11.1, defining the phonons as the poles of the displacement correlation function, we briefly review theories which start with the self-consistent harmonic (SCH) approximation or something similar. In Section 11.2, we discuss the relation between the displacement–displacement and the density–density correlation functions in solid 4He.