A General Non-Commutative Limit Theorem
2001, v.7, Issue 2, 215-224
One of the authors recently proved two non-commutative versions of the central limit theorem. Both of these theorems concerned a pair of creation and annihilation operators. The first theorem is a statement about the limit of the expectation value of a function of the scaled difference of the corresponding number operators with respect to an n-particle state as n tends to infinity. Here we extend this theorem to the case of an arbitrary (but fixed) number of creation and annihilation operators. We also formulate a conjecture extending the second theorem about the limit of the trace of a certain exponential of operators over the n-particle subspace as n tends to infinity.
Keywords: central limit theorem,creation and annihilation operators,Gaussian measures,Feynman - Kac formula