  We study the problem when an almost commuting $n$-tuple self-adjoint
operators in an infinite dimensional separable Hilbert space $H$ 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. In the case that $n=2,$ we show that a pair of almost commuting
self-adjoint operators in an infinite dimensional separable Hilbert space is
close to a commuting pair of self-adjoint operators if and only if a
corresponding Fredholm index vanishes outside of an essential
synthetic-spectrum. This is an attempt to solve a problem proposed by David
Mumford related to quantum theory and measurements.
