Phonons in liquid helium

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By writing a dispersion relation for the density propagation function we obtain its most general form consistent with known sum rules, for any macroscopic system of bosons or fermions. The Fourier transform of the pair correlation S k appears explicitly in the dispersion relation. An inequality for S k is derived for any macroscopic system. We specialize the results to liquid He 4 by making the single assumption that for k → 0, S k “saturates” the inequality. This assumption is consistent with experiments and is later proved theoretically. The nature of the low-lying excited states of He 4 can then be deduced. The result is that for given small momentum k, there is a group of states having an energy distribution peaked about ck, with a Lorentzian shape. The constant c is the velocity of sound at absolute zero, defined in terms of the macroscopic compressibility. The wave functions of these states are in some average sense Feynman's phonon wave function. We prove the assumption mentioned above by assuming Bose-Einstein condensation and by making essential use of the gauge invariance associated with the conservation particles. The mathematical technique is simple and does not require perturbation expansions, or summation of diagrams.