  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.
