# Is Bose-Einstein condensate a good example of a classical massive boson field?

Physically, we know that a BEC has formed if a macroscopic number of bosons occupy a single quantum state. The wave-function $\Psi(x)$ of the latter, normalized to the total number of condensed atoms $N \gg 1$, gives the macroscopic description of the condensate. $\Psi(x)$ will satisfy the ususal Schrodinger euqation in the trapping potential (which is the special case of the Gross-Pitaevskii equation with zero interactions).

On the other had, a macroscopic limit of a boson field should be described by the (classical) Klein-Gordon equation. In the relevant non-realtivistic limit, dispersion of the Klein-Gordon field is quadratic, as in the Schrodinger equation, but I struggle to derive the former from the latter.

Despite this technical difficulty of mine, is it a valid claim that BEC is a physical realization of the non-relativistic, classical-wave-limit of a boson field? I see it in the same vein as electric-field description of the laser radiation being a classic limit of the quantum coherent state.

(The difference between number states and coherents states in the limit $N \gg 1$ is, hopefully, a mere technicality here).

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The way to do this is to expand the dispersion to second order in the scalar Klein Gordon field, and (in the free theory) to identify $\psi(k,t)$ with $e^{imt}a(k)$ (throwing away the rest mass), where $a(k)$ is the nonrelativistically normalized creation operator for mode k. Then $\psi(x,t)$ is the Fourier transform of $\psi(k,t)$, and unlike the relativistic theory, has an immediate interpretation as an annihilation operator removing a particle at x, and from the expansion of the dispersion, it obeys the Schrodinger equation, and from complex conjugation, it's a complex field whose conjugate involves modes near -m energy in the relativistic theory.