Zariski Quantization as Second Quantization

Abstract: The Zariski quantization is one of the strong candidates for a quantization of the Nambu-Poisson bracket. In this paper, we apply the Zariski quantization for first quantized field theories, such as superstring and supermembrane theories, and clarify physical meaning of the Zariski quantization. The first quantized field theories need not to possess the Nambu-Poisson structure. First, we construct a natural metric for the spaces on which Zariski product acts in order to apply the Zariski quantization for field theories. This metric is invariant under a gauge transformation generated by the Zariski quantized Nambu-Poisson bracket. Second, we perform the Zariski quantization of superstring and supermembrane theories as examples. 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. These facts imply that the Zariski quantization is a second quantization. Moreover, the Zariski quantization preserves supersymmetries of the first quantized field theories. Thus, we can obtain second quantized theories of superstring and supermembranes by performing the Zariski quantization of the superstring and supermembrane theories.