Quantum Physics as Non-Commutative Geometry

For a person in mathematical physics, notions of non-commutative geometry (NCG) seem very natural. Related ideas to those in NCG occur in quantum theory — especially supersymmetric quantum theory — and also in statistical mechanics. One can interpret NCG as a quantization of geometry, in the sense that quantum theory is a quantization of classical physics. Many basic notions of non-commutative geometry can be understood by thinking of NCG as a way to define and to integrate differentials a 0 da 1 ··· da n in a framework more general than that of differential forms on manifolds. The quantum functions a0,..., a n are operators; their integrals can be thought of as quantum mechanical expectation values. What results is a theory in which classical notions of geometry carry over. In particular, there is a natural interpretation of NCG in terms of a cohomology theory. This cohomology reduces to de Rham theory in the usual commutative case.