collective field methods

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Example: System of N Bose particles

This follows Jevicki and Sakita (1980). The multiparticle wavefunction \psi(x_1,...x_N)\, can be written as a functional \Psi[\rho]\, of the density function

\rho(x) = \sum_{i=1}^N \delta(x-x_i)\,.

E.g., if \psi(x_1, x_2) = \psi_1(x_1)\psi_2(x_2) + \psi_1(x_2)\psi_2(x_1)\,, then \psi(x_1, x_2) = \left(\int\!dx\, \rho(x)\psi_1(x)\right) \left(\int\!dx\, \rho(x)\psi_2(x)\right) - \left(\int\!dx\, \rho(x)\psi_1(x) \psi_2(x)\right) = \Psi[\rho]\,.


References

[1] [2]

  1. ↑ A. Jevicki and B. Sakita (1980). "The quantum collective field method and its applications to the planar limit". Nuclear Physics B 165: 527. DOI:10.1016/0550-3213(80)90046-2. 
  2. ↑ A. Jevicki and B. Sakita (1981). "Collective field approach to the large-N limit: Euclidian field theories". Nuclear Physics B 185: 100. DOI:10.1016/0550-3213(81)90365-5. 
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