Zariski Quantization as Second Quantization

TL;DR

This paper explores Zariski quantization as a method for second quantization of superstring and supermembrane theories, revealing its physical meaning, gauge invariance, and ability to describe many-body systems while preserving supersymmetry.

Contribution

It demonstrates that Zariski quantization can serve as a second quantization method for superstring and supermembrane theories, introducing a gauge-invariant metric and revealing many-body dynamics.

Findings

Zariski quantization acts as a second quantization.

It preserves supersymmetry in the quantized theories.

The quantized theories exhibit many-body features like pair creation.

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.