Field Operators for Bosons with Impenetrable Cores. I. Equations Which Replace the Commutation Rules

Field operators representing particles with impenetrable cores cannot satisfy the usual commutation rules. While the customary derivation of the second quantization formalism cannot be applied to the case of particles with nonintegrable interaction potentials, the field operators ψ(x) and ψ^(x) can be defined by a matrix representation which exhibits them explicitly as transformations of functions of N position vectors into functions of N-1 and $N+1$ position vectors, respectively. The assumption of impenetrable cores is introduced in this definition by taking the matrix elements which lead to prohibited configurations of position vectors equal to zero. Equations which replace the usual set of commutation rules are derived from this definition for the case of hard sphere interaction. Conversely it is shown that results which follow from the commutation rules in the standard formalism, follow from the new set of equations with the changes obviously required by the assumption of impenetrable spherical cores. For example, the operator for the number of particles in a finite domain has as eigenvalues the non-negative integers not exceeding the largest number of hard spheres which can be placed into the domain.