  We investigate the quantum theory of closed systems based on the linear
positivity decoherence condition of Goldstein and Page. A quantum theory of
closed systems requires two elements; 1) a condition specifying which sets of
histories may be assigned probabilities that are consistent with the rules of
probability theory, and 2) a rule for those probabilities. The linear
positivity condition of Goldstein and Page is the weakest of the general
conditions proposed so far. Its general properties relating to exact
probability sum rules, time-neutrality, and conservation laws are explored. Its
inconsistency with the usual notion of independent subsystems in quantum
mechanics is reviewed. Its relation to the stronger condition of medium
decoherence necessary for classicality is discussed. The linear positivity of
histories in a number of simple model systems is investigated with the aim of
exhibiting linearly positive sets of histories that are not decoherent. The
utility of extending the notion of probability to include values outside the
range of 0 to 1 is described. Alternatives with such virtual probabilities
cannot be measured or recorded, but can be used in the intermediate steps of
calculating real probabilities. Virtual probabilities give a simple and general
way of formulating quantum theory.
