Quantum mechanics without state vectors

Because the state vectors of isolated systems can be changed in entangled states by processes in other isolated systems, keeping only the density matrix fixed, it is proposed to give up the description of physical states in terms of ensembles of state vectors with various probabilities, relying only on density matrices. The density matrix is defined here by the formula giving the mean values of physical quantities, which implies the same properties as the usual definition in terms of state vectors and their probabilities. This change in the description of physical states opens up a large variety of new ways that the density matrix may transform under various symmetries, different from the unitary transformations of ordinary quantum mechanics. Such new transformation properties have been explored before, but so far only for the symmetry of time translations into the future, treated as a semigroup. Here, new transformation properties are studied for general symmetry transformations forming groups, not just semigroups. Arguments that such symmetries should act on the density matrix as in ordinary quantum mechanics are presented, but all of these arguments are found to be inconclusive.