  Non-uniform strain applied to graphene's honeycomb lattice can induce
pseudo-Landau levels in the single-particle spectrum. Various generalizations
have been put forward, including a particular family of hopping models in $d$
space dimensions. Here we show that the key ingredient for sharp pseudo-Landau
levels in higher dimensions is dimensional reduction. We consider particles
moving on a $d$-dimensional hyper-diamond lattice which displays a semimetallic
bandstructure, with a $(d-2)$-dimensional nodal manifold. By applying a
suitable strain pattern, the single-particle spectrum evolves into a sequence
of relativistic Landau levels. We develop and solve the corresponding field
theory: Each nodal point effectively generates a Landau-level problem which is
strictly two-dimensional to leading order in the applied strain. While the
effective pseudo-vector potential varies across the nodal manifold, the
Landau-level spacing does not. Our theory paves the way to strain engineering
of single-particle states via dimensional reduction and beyond global minimal
coupling.
