Quantization (physics) attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in. One of the earliest attempts at a natural quantization was Weyl quantization, proposed by Hermann Weyl in 1927. Here, an attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space.
Heisenberg group connection with the theta functions. If , are integers (in the ring Z) then one has the discrete Heisenberg group H(Z). It is a non-abelian nilpotent group. It has two generators, and relations where is the generator of the center of H. (Note that the inverses of "x", "y", and "z" replace the 1 above the diagonal with −1.) By Bass's theorem, it has a polynomial growth rate of order 4. One can generate any element through If one takes "a, b, c" in Z/"p" Z for an odd prime "p", then one has the Heisenberg group modulo p. It