# Darboux theorem and the canonical decomposition of a two-fermion wave function

It is a classical theorem in quantum mechanics or quantum chemistry or quantum information that a two-fermion wave function has a beautiful canonical expansion:

$$f(x_1, x_2) = \sum_{j=1}^N \sqrt{p_j} \left( \phi_{2j-1}(x_1)\phi_{2j}(x_2) -\phi_{2j}(x_1)\phi_{2j-1}(x_2) \right).$$

Here $f$ is the wave function which is antisymmetric, i.e., $f(x_1, x_2 ) = - f(x_2, x_1)$. The orbitals $\phi_i$ are the natural orbitals, i.e., eigenstates of the single-particle density matrix. They are orthonormal. The weights $p_j$ are related to the population in the natural orbitals.

This theorem was first obtained in quantum chemistry by lowdin in 1956 as far as I know. See http://journals.aps.org/archive/abstract/10.1103/PhysRev.101.1730

I discovered it myself when i was studying the best Slater approximation of a fermionic wave function. Its simplicity and beauty surprised me. It is so disappointing that I never found a quantum mechanics textbook mentioning it, although almost every quantum mechanics textbook would mention the two-electron wave function.

Now reading differential geometry, I find the Darboux theorem:

http://en.wikipedia.org/wiki/Darboux%27s_theorem

It seems that Darboux theorem is basically the same stuff as the canonical decomposition above. Is that right? I am not familiar with differential geometry.

-

Comments to the question (v2):

1. The canonical expansion of the two-fermion wave function seems more related to the canonical form of antisymmetric real matrices in the framework of vector spaces and linear algebra.

2. The Darboux' theorem in the framework of manifolds and differential geometry (which is a surprisingly potent result) is overkill for the mentioned formula.

The former (1) relies mainly on the properties of skewsymmetry, while the latter theorem (2) also depends deeply on the Jacobi identity$^1$ (JI) for the Poisson bracket.

Phrased in another way: The two-fermion system in the formulation of Ref. 1 possesses no relevant analogue of a non-trivial JI, and no arguments analogous to the proof of Darboux' theorem were used to establish the mentioned formula.

References:

1. P-O. Löwdin and H. Shull, Phys. Rev. 101 (1956) 1730.

--

$^1$ The JI is equivalent to the closedness of the symplectic 2-form (if the structure is non-degenerate). See also this related Math.SE post.

-
No. If you look at the formula in the Darboux theorem: $\omega = dx_1 \wedge dy_1 + dx_2 \wedge dy_2 + \ldots dx_n \wedge dy_n$, you see it is very similar to the canonical decomposition of the wave function. – Jiang-min Zhang Jul 10 '14 at 9:55
Darboux' theorem is overkill. I updated the answer. – Qmechanic Jul 10 '14 at 12:25
I agree that Darboux theorem can be an overkill. But, it is instructive to see the link. – Jiang-min Zhang Jul 11 '14 at 12:45