Product ansatz

A product ansatz has the form

$$\phi (z) = \prod_l \phi_l (z_l) ,$$ (403)

where $z_l$ = ${\bf I}_l^{(z)} z$ represents projections of the vector $z$ consisting of all $x$ and $y$ components. The ansatz can be made more flexible by using sum of products

$$\phi (z) = \sum_k \prod_l \phi_{k,l} (z_l).$$ (404)

The restriction of the trial space to product functions corresponds to the Hartree approximation in physics. (In a Hartree-Fock approximation the product functions are antisymmetrized under coordinate exchange.)

For additive ${{\bf K}}$ = $\sum_l {{\bf K}}_l$ with ${{\bf K}}_l$ acting only on $\phi_l$, i.e., ${{\bf K}}_l$ = ${{\bf K}}_l \otimes \left( \bigotimes_{l^\prime \ne l} {\bf I}_{l^\prime} \right)$, with ${\bf I}_l$ the projector into the space of functions $\phi_l$ = ${\bf I}_l \phi$, the quadratic regularization term becomes, assuming ${\bf I}_l$ ${\bf I}_{l^\prime}$ = $\delta_{l,l^\prime}$,

$$(\,\phi,\, {{\bf K}}\,\phi\,) = \sum_l (\,\phi_l,\, {{\bf K}... ...d_{l^\prime \ne l} (\, \phi_{l^\prime},\, \phi_{l^\prime}\,) .$$ (405)

For ${{\bf K}}$ = $\bigotimes_{l} {{\bf K}}_{l}$ with a product structure with respect to $\phi_l$

$$(\,\phi,\, {{\bf K}}\,\phi\,) = \prod_{l} (\,\phi_l ,\, {{\bf K}}_l \, \phi_l\,).$$ (406)

In both cases the prior term factorizes into lower dimensional contributions.