Weyl-Heisenberg Quantization of the Continuous Torus and its Applications

We introduce covariant integral quantization on the continuous 2-Torus, $\T = S^1 \times S^1$, as a configuration space. Using Weyl-Heisenberg analysis, we construct the phase space, dubbed the double-discrete cylinder as $\Gamma = \Z^2 \times \T$, quantizing functions on $\Gamma$ into quantum operators that act on $\T$. This allows us to deconstruct a myriad of novel examples that can be mapped to the continuous 2-Torus, from those in quantum mechanics such as magnetic coupling of spin-spin interactions, to the analysis of two-dimensional periodic signals. Some interesting examples of two-dimensional periodic signals are significant cultural patterns such as those found in Iranian temples and Persian rugs.