Zariski quantization as second quantization

The Zariski quantization is one of the strong candidates for a quantization of the Nambu-Poisson bracket. In this paper, we reinterpret the Zariski quantization and study physical properties of it. Here, we do not treat the Zariski quantization as a deformation quantization. Instead, we perform a path integral of a theory after the Zariski quantization, which only deforms an action. As a result, we find that second quantized field theories are obtained by performing the Zariski quantization and path-integrals of perturbative superstring and supermembrane theories. Actually, we find flat directions, which indicate that the Zariski quantized theories describe many-body systems. We also find that pair creations and annihilations occur among the many bodies introduced by the Zariski quantization, by studying a simple model. Moreover, the Zariski quantization preserves supersymmetries of the superstring and supermembrane theories.

Figure 1

A classical interaction.

Figure 2

A quantum correction.