Atomic Theory of Collective Excitations in Bose-Einstein Condensation and Spontaneously Broken Gauge Symmetry

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A theory of collective excitations in Bose-Einstein condensation in a trap is developed based on the quantum Hamilton-Jacobi equation of Bohm and the phase coherence along with the idea of off-diagonal long range order of Penrose and Onsager. First, we show that a free surface behaves like a normal fluid - a breakdown of superfluidity. Second, inside the free surface it is shown that the spectrum of phonons is of the form ω=ck scaled with the external potential, where the speed of (first) sound, c=[4πaρℏ²]¹/²/M and k is the wave number. Third, in the limit a→ 0, the hard spheres in the Bose-Einstein condensation collapse to a close-packed classical lattice with the zero-point vibrational motion about fixed points.