  We construct a new mean-field theory for quantum (spin-1/2) Heisenberg
antiferromagnet in one (1D) and two (2D) dimensions using a Hartree-Fock
decoupling of the four-point correlation functions. We show that the solution
to the self-consistency equations based on two-point correlation functions does
not produce any unphysical finite-temperature phase transition in accord with
Mermin-Wagner theorem, unlike the common approach based on the mean-field
equation for the order parameter. The next-neighbor spin-spin correlation
functions, calculated within this approach, reproduce closely the strong
renormalization by quantum fluctuations obtained via Bethe ansatz in 1D and a
small renormalization of the classical antiferromagnetic state in 2D. The heat
capacity approximates with reasonable accuracy the full Bethe ansatz result at
all temperatures in 1D. In 2D, we obtain a reduction of the peak height in the
heat capacity at a finite temperature that is accessible by high-order $1/T$
expansions.
