  We consider an isotropic spin-degenerate interacting uniform $D$-dimensional
electron gas (DDEG) with $D > 1$ within the Luttinger-Ward (LW) formalism. We
derive the asymptotically exact semiclassical/infrared limit of the LW
functional at large distances, $r \gg \lambda_F$, and large times, $\tau \gg
1/E_F$, where $\lambda_F$ and $E_F$ are the Fermi wavelength and the Fermi
energy, respectively. The LW functional is represented by skeleton diagrams,
each skeleton diagram consists of appropriately connected dressed fermion
loops. First, we prove that every $D$-dimensional skeleton diagram consisting
of a single fermion loop is reduced to a one-dimensional (1D) fermion loop with
the same diagrammatic structure, which justifies the name dimensional
reduction. This statement, combined with the fermion loop cancellation theorem
(FLCT), agrees with results of multidimensional bosonization. Here we show that
the backscattering and the spectral curvature, both explicitly violate the FLCT
and both are irrelevant for a 1DEG, become relevant at $D > 1$ and $D > 2$,
respectively. The reason for this is a strong infrared divergence of the
skeleton diagrams containing multiple fermion loops at $D > 1$. These diagrams,
which are omitted within the multidimensional bosonization approaches, account
for the non-collinear scattering processes. Thus, the dimensional reduction
provides the framework to go beyond predictions of the multidimensional
bosonization. A simple diagrammatic structure of the reduced LW functional is
another advantage of our approach. The dimensional reduction technique is also
applicable to the thermodynamic potential and various approximations, from
perturbation theory to self-consistent approaches.
