  We represent in this note the solutions of the electronic Schr\"odinger
equation as traces of higher-dimensional functions. This allows to decouple the
electron-electron interaction potential but comes at the price of a degenerate
elliptic operator replacing the Laplace operator on the higher-dimensional
space. The surprising observation is that this operator can without much loss
again be substituted by the Laplace operator, the more successful the larger
the system under consideration is. This is due to a concentration of measure
effect that has much to do with the random projection theorem known from
probability theory. The text is in parts based on the publications [Numer.
Math. 146, 219--238 (2020)] and [SIAM J. Matrix Anal. Appl., 43, 464--478
(2022)] of the author and adapts the findings there to the needs of quantum
mechanics. Our observations could for example find use in iterative methods
that map sums of products of orbitals and geminals onto functions of the same
type.
