A few years ago, Bost and Connes [3] discovered a surprising relationship between the class field theory of Q and quantum statistical mechanics.Mathematically, a quantum statistical mechanical system consists of a pair (A, σ t ), where A is a C * -algebra and σ t is a one-parameter group of automorphisms of A;physically, A is the algebra of observables and σ t is the time evolution of the physical system. The physical states of the system are given by certain linear functionals on A.The analogy between classical and quantum statistical mechanics can be described by Table 1.1.The statistical content means that one singles out the equilibrium states at a given temperature T = 1/β on A, and these are characterized by the so-called KMS β condition. The set of these equilibrium states may have symmetries. Changing the temperature of a system can produce a phase transition phenomenon with spontaneous symmetry breaking, meaning that the symmetry changes radically with an arbitrary small change of temperature. For example, the formation at zero temperature of a snowflake from water is a phase transition, for which we can observe a symmetry breaking phenomenon: a snowflake has much more symmetry (it has crystal structure) than a drop of water (which consists of a random collection of molecules).

We prove an operator-theoretic reconstruction of the Beilinson-Bloch regulator for compact Riemann surfaces, using loop operators and the Connes-Karoubi character for Fredholm modules. The proof includes a new computation of the Connes-Karoubi character for Steinberg symbols of the circle, which relies on the Helton-Howe determinant theory, but not on the Carey-Pincus theory of joint torsion.