Determination of an Operator Algebra for the Two-Dimensional Ising Model

A previous publication showed how the critical indices for the two-dimensional Ising model could be derived from an assumed form of an operator algebra which describes how the product of two fluctuating variables may be reduced to a linear combination of the basic fluctuating variables. In this paper, the previously used algebra is derived from the Onsager solution of the two-dimensional Ising model. The calculation makes use of a "disorder" variable which is mathematically the result of applying the Kramers-Wannier transformation to the Ising-model spin variable. The average of products of spin and disorder variables are evaluated at the critical point for the special case in which all the variables lie on a single straight line. The ordering of these variables on the line determines a "quantum number" $\Gamma$ such that the average is nonzero only for $\Gamma =0\mathrm{}$. Composition rules for this quantum number are derived and used to develop an algebra for the multiplication of complex variables at the critical point. Arguments are given to suggest the identifications of elements of the algebra as the spin, the energy density, the Kaufman spinors, and a stress density. The result of this calculation is the operator algebra which formed the starting point of the previous paper.