Abstract We implement the so-called Weyl–Heisenberg covariant integral quantization in the case of a classical system constrained by a bounded or semi-bounded geometry. The procedure, which is free of the ordering problem of operators, is illustrated with the basic example of the one-dimensional motion of a free particle in an interval, and yields a fuzzy boundary, a position-dependent mass, and an extra potential on the quantum level. The consistency of our quantization is discussed by analyzing the semi-classical phase space portrait of the derived quantum dynamics, which is obtained as a regularization of its original classical counterpart.
We present a comprehensive comparison of spin and energy dynamics in quantum and classical spin models on different geometries, ranging from one-dimensional chains, over quasi-one-dimensional ladders, to two-dimensional square lattices. Focusing on dynamics at formally infinite temperature, we particularly consider the autocorrelation functions of local densities, where the time evolution is governed either by the linear Schr\odinger equation in the quantum case, or the nonlinear Hamiltonian equations of motion in the case of classical mechanics. While, in full generality, a quantitative agreement between quantum and classical dynamics can therefore not be expected, our large-scale numerical results for spin-
1/2 systems with up to
N=36 lattice sites in fact defy this expectation. Specifically, we observe a remarkably good agreement for all geometries, which is best for the nonintegrable quantum models in quasi-one or two dimensions, but still satisfactory in the case of integrable chains, at least if transport properties are not dominated by the extensive number of conservation laws. Our findings indicate that classical or semi-classical simulations provide a meaningful strategy to analyze the dynamics of quantum many-body models, even in cases where the spin quantum number
S=1/2 is small and far away from the classical limit
S→∞.