N-Body Bose System with a Finite Number of States. I. Irreducible Representations

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Abstract
For the purpose of investigating the N‐Body Bose System with a finite number of states, we analyze Xij = Ai†Aj, i, j ≤ K. (A and A† are the usual annihilation and creation operators of the second quantization formalism). The Hamiltonian for a fixed number of particles may be expressed in terms of the (finite‐dimensional) irreducible representations of the Xij. A set of fundamental equations is defined for the irreducible representations Xij(N, K), i, j ≤ K and is solved for arbitrary N and K. The analysis of the structure of the Xij(N, K) yields a simple, systematic method for listing all possible ways in which n1+ … ni+ … + nK = N may be satisfied for ni positive integral or zero. This leads to a particularly simple method for constructing the Xij(N, K), i, j ≤ K.

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