### Continuity and Discontinuity of Definite Properties in the Modal
Interpretation.

#### Helvetica Physica Acta, Volume 68, pages 679-704 (1995)

#### Guido Bacciagaluppi, Matthew J. Donald, Pieter E. Vermaas

** Abstract ** Technical results about the time dependence of eigenvectors
of reduced density operators are considered, and the relevance of these
results is discussed for modal interpretations of quantum mechanics which
take the corresponding eigenprojections to represent definite properties.
Continuous eigenvectors can be found if degeneracies are avoided. We show
that, in finite dimensions, the space of degenerate operators has co-dimension
3 in the space of all reduced operators, suggesting that continuous
eigenvectors almost surely exist. In any dimension, even when degeneracies are
hit, we find conditions under which a theorem due to Rellich can provide
continuous eigenvectors. We use this result to formulate an extended version
of the modal interpretation. We also discuss eigenvector instability which we
argue poses a serious problem for the modal interpretation, even in our
extended version. Many examples are given to illustrate the mathematics.

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