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1. What the question is asking (plain‑language restatement)

The asker wants to know how to obtain the equations of a collective‑field theory (CFT) in a controlled expansion in powers of (1/N), where (N) is the number of microscopic degrees of freedom (e.g. the size of a matrix or the number of particles).

  • In the original Sakita–Jevicki papers the collective Hamiltonian was written down and then approximations (large‑(N) limit, quadratic fluctuations) were made “by hand”.
  • The question is whether a systematic, algorithmic procedure exists that produces all terms in the (1/N) expansion, order by order.
  • The asker also wonders whether Yaffe’s coherent‑state large‑(N) formalism can be employed to achieve the same systematic expansion.

In short: Give a step‑by‑step method that starts from a microscopic Hamiltonian, rewrites it in terms of collective fields, and generates the full (1/N) series (or equivalently the loop expansion) in a transparent way. Explain how Yaffe’s coherent‑state approach fits into this picture.

Below is a complete worked solution that shows every stage of the construction, the bookkeeping of powers of (N), and the connection to coherent states.

2. Systematic derivation of collective‑field equations to any order in (1/N)

We will present the method for a one‑dimensional bosonic many‑body system; the extension to higher dimensions, to fermions, or to matrix models follows the same steps with only notational changes.

2.1 Microscopic model

Consider (N) identical particles on a line with canonical variables ((x_i,p_i)) and Hamiltonian

[ H = \sum_{i=1}^{N}\frac{p_i^{2}}{2m} +\sum_{i=1}^{N} V(x_i) +\frac{1}{2N}\sum_{i\neq j}U!\bigl(x_i-x_j\bigr) . \tag{2.1} ]

The factor (1/N) in the two‑body term makes the total interaction energy (\mathcal O(N)) (the standard ’t Hooft scaling).

Our goal is to rewrite this theory in terms of the density field

[ \rho(x) \equiv \frac{1}{N}\sum_{i=1}^{N}\delta\bigl(x-x_i\bigr), \qquad \int!dx\,\rho(x)=1, \tag{2.2} ]

and its canonically conjugate momentum (\pi(x)).

2.2 Change of variables in the phase‑space path integral

The quantum dynamics is encoded in the phase‑space (Hamiltonian) path integral

[ Z = \int!\prod_{i=1}^{N}{\cal D}x_i {\cal D}p_i\; \exp!\Bigl{ i!\int!dt \Bigl[\sum_i p_i\dot x_i - H(x,p)\Bigr]\Bigr}. \tag{2.3} ]

Insert a functional representation of the identity that enforces the definition (2.2):

[ 1 = \int!{\cal D}\rho\; \delta!\Bigl[\rho(x)-\tfrac1N!\sum_i!\delta(x-x_i)\Bigr] \det!\Bigl[\frac{\delta}{\delta\rho}\Bigl(\rho-\tfrac1N\sum_i\delta\Bigr)\Bigr], \tag{2.4} ]

where the determinant is the Jacobian of the change of variables.

Perform the Fourier representation of the delta functional,

[ \delta[\ldots]=\int!{\cal D}\pi\; \exp!\Bigl{iN!\int!dt\,dx\,\pi(x,t) \bigl[\rho(x,t)-\tfrac1N!\sum_i!\delta(x-x_i(t))\bigr]\Bigr}, \tag{2.5} ]

so that (\pi(x,t)) appears as the conjugate momentum to (\rho).

Carrying out the Gaussian integral over the particle momenta (p_i) and over the coordinates (x_i) subject to the constraint produces the collective‑field path integral

[ Z = \int!{\cal D}\rho\,{\cal D}\pi\; J[\rho]\, \exp!\Bigl{ iN^{2}!\int!dt\,L_{\rm coll}[\rho,\pi]\Bigr}, \tag{2.6} ]

where

  • (L_{\rm coll}= \int!dx\;\bigl(\pi\dot\rho - {\cal H}_{\rm coll}\bigr)),
  • ({\cal H}_{\rm coll}) is the collective Hamiltonian density, and
  • (J[\rho]=\exp{N^{2}S_{!J}[\rho]}) is the Jacobian factor (sometimes called the “Vandermonde term”).

Because each particle contributes a factor (N) in the definition (2.2), the overall prefactor of the exponent is (N^{2}); this is the source of the (1/N) loop expansion (the saddle point is of order (N^{2}), each quantum fluctuation costs a factor (1/N)).

2.3 Exact collective Hamiltonian

Carrying out the integrations in (2.6) (see Sakita–Jevicki, Ann. Phys. 140 (1982) 406) yields

[ \boxed{ \begin{aligned} {\cal H}{\rm coll}[\rho,\pi] &= \frac{1}{2m}\int!dx\;\frac{\bigl[\partial_x!\bigl(\rho\pi\bigr)\bigr]^{2}}{\rho} \;+\;\int!dx\,V(x)\,\rho(x)
&\qquad +\frac12!\int!dx\,dy\;U(x-y)\,\rho(x)\rho(y) \;+\;{\cal H}{\rm J}[\rho] . \end{aligned} } \tag{2.7} ]

The first term is the kinetic energy written entirely in terms of (\rho) and its conjugate (\pi).

The Jacobian contribution (the only term that carries explicit (\hbar)–type quantum corrections) is

[ {\cal H}_{\rm J}[\rho] = \frac{\hbar^{2}}{8m}\int!dx\; \frac{\bigl[\partial_x\rho(x)\bigr]^{2}}{\rho(x)^{3}} . \tag{2.8} ]

For bosons the (\hbar) can be set to 1; for fermions the sign flips. The crucial point is that ({\cal H}_{\rm J}) is already of order (N^{0}), whereas the other pieces scale as (N^{2}) (because (\rho\sim\mathcal O(1)) and the overall factor in the exponent of (2.6) is (N^{2})). This separation of scales makes the systematic (1/N) expansion transparent.

2.4 Large‑(N) saddle point (classical collective field)

Define the classical background (\rho_{0}(x)) by the stationary‑phase condition

[ \frac{\delta}{\delta\rho(x)}\Bigl[ N^{2}{\cal S}{\rm cl}[\rho]\Bigr]{\,\rho=\rho_{0}}=0, \qquad {\cal S}{\rm cl} = \int!dt!\int!dx\;\Bigl[\pi\dot\rho-{\cal H}{\rm coll}\Bigr] . \tag{2.9} ]

Since (\pi) appears linearly in the action, the variation w.r.t. (\pi) gives the continuity equation

[ \dot\rho_{0}(x,t)+\partial_{x}!\bigl[\,\rho_{0}(x,t) v_{0}(x,t)\bigr]=0, \qquad v_{0}\equiv\frac{1}{m}\partial_{x}\pi_{0}, \tag{2.10} ]

and variation w.r.t. (\rho) yields the Euler (force) equation

[ m\dot v_{0}+v_{0}\partial_{x}v_{0} = -\partial_{x}V(x)-\int!dy\,U’(x-y)\,\rho_{0}(y) +\frac{\hbar^{2}}{2m

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