Nonlinearity of Hartree-Fock and Landau-Ginzburg equations contrasted

However, when one writes them in second-quantization, we regain linearity in the sharp number orthogonal basis Fock space.

In first quantization

http://vergil.chemistry.gatech.edu/notes/hf-intro/img63.png

note the 3rd term on the LHS that couples the different base eigenfunctions in the integral.

http://vergil.chemistry.gatech.edu/notes/hf-intro/node7.html

compare to the emergent SSB Landau-Ginzburg L-G equation e.g.

http://upload.wikimedia.org/math/9/4/5/945a3f42246130d5daba8e3c6417aec4.png

http://en.wikipedia.org/wiki/Ginzburg–Landau_theory

But the L-G equation is fundamentally zero-quantization i.e. a c-number emergent nonlinear local equation that trivially solves the emergence of classicality problem of quantum reality explaining why no big Schrodinger Tigers.

For example, crystal lattices come from spontaneous broken T3 symmetry in the ground states of systems of atoms in thermal equilibrium at low enough temperature - the lattice positions are a set of emergent "phonon" Goldstone boson condensate macro-quantum coherent order parameters. See Hagen Kleinert's books for the details.

http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=booklist&details=9