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The collective field theory (see nLab for a list of main historical references) which came up as a generalization of the Bohm-Pines method in treating plasma oscillations often used in the study of large N asymptotics (e.g. of matrix models). I have seen few cases where the equations for collective fields are derived; the equations are obtained using approximations and look more systematically understood by other methods than those available at the dawn of collective field theory (i.e. since the works of Sakita and Jevicki).

Is there now a systematic method to derive collective field theory equations to any order in 1/N ? Also if the Yaffe's methods concerning coherent state methods in large N asymptotics are ever used in collective field theory? The subject looks interesting for doing some work (I have some background in coherent state techniques), but I do not know what the current state of advance is.

EDIT: added a bounty to renew interest.

golden_hawk21
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Yes, there is now a systematic method, let me know if you need a DOI. In this approach, one reformulates pilot‐wave dynamics so that the entire ensemble is encoded in an effective mass field—a collective variable representing the cumulative influence of all microscopic degrees of freedom. Starting from a unified variational principle in flat 4D Euclidean space, one then projects along an observer’s four-velocity to recover conventional 3+1 spacetime (including familiar gravitational corrections) while simultaneously yielding an effective action for a scalar collective field. This action can be expanded systematically in powers of 1/N, capturing corrections to all orders.

Moreover, because the derivation employs a polar (or Madelung) decomposition of the wavefunction, it naturally leads to a coherent state description. This directly connects with Yaffe’s coherent state methods in large N asymptotics, providing further control over the 1/N expansion. While traditional techniques (e.g., saddle-point or bosonization methods) remain viable for some models, this projection-based framework offers a unified, first-principles alternative that systematically derives collective field theory equations to any desired order.

gstieger
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