Almost commuting self-adjoint operators and measurements

Abstract We study the problem when an n -tuple of self-adjoint operators in an infinite-dimensional separable Hilbert space H with small commutators is close to an n -tuple of commuting self-adjoint operators on $H.$ We give an affirmative answer to the problem when the synthetic-spectrum and the essential synthetic-spectrum are close. Examples are also exhibited that, in general, the answer to the problem when $n\ge 3$ is negative even the associated Fredholm index vanishes. This is an attempt to solve a problem proposed by David Mumford related to quantum theory and measurements.