Fock space: A bridge between Fredholm index and the quantum Hall effect

Abstract

We compute the quantized Hall conductance at various Landau levels by using the classic trace. The computations reduce to the single elementary one for the lowest Landau level. By using the theories of Helton-Howe-Carey-Pincus, and Toeplitz operators on the classic Fock space and higher Fock spaces, the Hall conductance is naturally identified with a Fredholm index. This brings new mathematical insights to the extraordinary precision of quantization observed in quantum Hall measurements.