Conformal Invariance with Boundaries

Quantum field theories become relevant to statistical physics for systems at or near their critical points. Under such circumstances physical observables obey simple scaling relations and the parameters or critical exponents describing the power law relations are essentially independent of the microscopic details of the interactions and depend only on general features of the system such as the overall symmetry invariance. This crucial property of universality allows critical exponents to be calculated by using continuum quantum field theories with the same symmetry content. The couplings of the quantum field theory must be tuned to a critical point, or surface, when the theory becomes scale invariant and correlation functions of various operators have a power law dependence on their spatial separations determined by the dimensions of the operators which are directly related to the statistical mechanical critical exponents. Thus the Ising model, describing simple magnetic systems on a lattice, is related to the field theory with a single scalar field ∅. For the renormalisable ∅ 4 theory there is one coupling g and at the critical point, determined by the vanishing of the β function, β(g *) = 0, the theory becomes scale invariant.