The Continuous-Spin, Ising Model of Field Theory and the Renormalization Group

Abstract

This paper sketches briefly the ideas of the ferromagnetic critical point, or bifurcation point, as exemplified in the Ising model. Historically, the only reliable, general methods available to attack this problem have been series expansion methods. The development of scaling ideas and the realization that Euclidean, Boson, quantum field-theory is the same problem as the scaling limit of critical phenomena has led to the development of the renormalization group approach to critical phenomena. For clarity, attention is focused on that continuous- spin Ising model which is equivalent to a g₀:ɸ⁴:_d field theory. A review of the ideas of the renormalization group approach in statistical mechanical language shows that the key assumption is that the limits as the bare coupling-constant goes to infinity and the ultra-violet cut-off is removed are independent of order. Calculations show that this assumption is satisfied in one and two dimensions, but fails in three and four dimensions where the renormalized coupling-constant is not a monotonic function of the bare coupling constant. The renormalization group approach to critical phenomena is seen as a theory of the maximum of the renormalized coupling-constant which may, or may not, be a theory of the Ising-model critical-point.