Fokker-Planck differential operators {density matrix} represent quantum-measurement processes. Discrete phase-space states (eigenstate) are independent and orthogonal and have real-number probabilities. States are phase-space vectors (state vector). State vectors have complex-number amplitudes, and probabilities are positive real-number absolute values of amplitude squares. State probability is tensor product of normalized state vector with complex conjugate, which eliminates phase. Tensor products are planes through complex Hilbert space. Renormalization can make sum of state probabilities equal one, and density-matrix-trace sum is one.
measurement
Measuring instruments are density-matrix projectors with one state vector, with real-number probability 1 = 100%. Product of physical-system density matrix and measuring-instrument density matrix makes density matrix with one trace value, the measurement.
transformations
Coordinate transformations do not change density matrices, because they are linear.
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Date Modified: 2022.0224