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We consider noncommutative rational functions as well as matrices inpolynomials in noncommuting variables in two settings: in an algebraic contextthe variables are formal variables, and their rational functions generate the"free field"; in an analytic context the variables are given by operators froma finite von Neumann algebra and the question of rational functions is treatedwithin the affiliated unbounded operators. Our main result shows that for a"good" class of operators - namely those for which the free entropy dimensionis maximal - the analytic and the algebraic theory are isomorphic. This meansin particular that any non-trivial rational function can be evaluated as anunbounded operator for any such good tuple and that those operators don't havezero divisors. On the matrix side, this means that matrices of polynomialswhich are invertible in the free field are also invertible as matrices overunbounded operators when we plug in our good operator tuples. We also addressthe question how this is related to the strong Atiyah property. The aboveyields a quite complete picture for the question of zero divisors (or atoms inthe corresponding distributions) for operator tuples with maximal free entropydimension. We give also some partial results for the question of existence andregularity of a density of the distribution.

The free field: zero divisors, Atiyah property and realizations via unbounded operators