  We introduce and analyze symmetric infinite-body optimal transport (OT)
problems with cost function of pair potential form. We show that for a natural
class of such costs, the optimizer is given by the independent product measure
all of whose factors are given by the one-body marginal. This is in striking
contrast to standard finite-body OT problems, in which the optimizers are
typically highly correlated, as well as to infinite-body OT problems with
Gangbo-Swiech cost. Moreover, by adapting a construction from the study of
exchangeable processes in probability theory, we prove that the corresponding
$N$-body OT problem is well approximated by the infinite-body problem.
  To our class belongs the Coulomb cost which arises in many-electron quantum
mechanics. The optimal cost of the Coulombic N-body OT problem as a function of
the one-body marginal density is known in the physics and quantum chemistry
literature under the name SCE functional, and arises naturally as the
semiclassical limit of the celebrated Hohenberg-Kohn functional. Our results
imply that in the inhomogeneous high-density limit (i.e. $N\to\infty$ with
arbitrary fixed inhomogeneity profile $\rho/N$), the SCE functional converges
to the mean field functional.
  We also present reformulations of the infinite-body and N-body OT problems as
two-body OT problems with representability constraints and give a dual
characterization of representable two-body measures which parallels an
analogous result by Kummer on quantum representability of two-body density
matrices.
