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.
Quantum-mechanics theories {matrix quantum mechanics} {S-matrix theory} can use linear-equation systems, with indexed terms, to model electronic-transition energies.
transition matrix
Square matrices can represent linear-equation systems. Infinite square matrices can represent Hilbert spaces with infinitely many dimensions. Matrix rows and columns represent the same energy levels. Matrices are infinite, because particles can go to any energy level, and energy levels can go higher infinitely. Matrix cells represent possible particle-energy-level transitions and their probabilities. Matrix elements are time-dependent complex numbers in infinite Hilbert space. Squared-amplitude absolute values give probabilities of energy-level transitions.
Matrix cells include all direct and cross-channel electronic transitions. Cells (linear-equation terms) with both indices the same are for directly emitted or absorbed photons. Cells (linear-equation terms) with different indices are for cross channels.
Because transition-matrix amplitudes are renormalized, sum of all state probabilities is one. Transition matrices are mathematically equivalent to Schrödinger wave equations, because time-dependent complex numbers represent anharmonic oscillators.
quanta
Matrix cells represent discrete energy changes and so quanta. Matrices are not continuous.
deterministic
Particles move from energy state to energy state deterministically, with probabilities.
space
Transition matrices are not about space. There is no position or trajectory information.
space: no fields
Energy and momentum transfers are quanta. There are no fields.
space: uncertainty
Matrices use non-commutative symbol algebra, not wave-equation Hamiltonian-equation variables. The uncertainty principle depends on wave behavior. Non-commuting operators are certain, so matrix theory does not account for uncertainty.
time
Transition matrices can change over time.
tensor
Quantum-mechanical matrices are similar to general-relativity symmetric tensors. Hermitean-matrix principal-axis transformation is a unitary-Hilbert-space tensor. If transformation is independent of time, tensor is a diagonal matrix. However, quadratic distance form is invariant, so transformations are unitary, not orthogonal as in general relativity.
S-matrix theory additions {Regge calculus} can group hadron mesons and baryons. Hadron masses and angular momenta have groups {Regge hypothesis}. Hadron groups lie on a line {Regge trajectory} plotting angular momenta versus mass squared. Because mesons and baryons have same relation between mass and angular momentum, and both depend on quarks, their internal dynamics must be the same.
simplexes
Flat simplexes joined edge to edge, face to face, and vertex to vertex can approximate continuous space. For two-dimensional spaces, all curvature is at vertexes. For four-dimensional spaces, all curvature is at triangles. Curvature is where masses and particles are.
For hadrons, exchange-transition scattering-amplitude sum equals direct-channel-transition scattering-amplitude sum {dual resonance theory}. Hadrons are zero-rest-mass-string quantum states. String ends move at light speed. Strings can break, rejoin, rotate, and oscillate. String tension is potential energy. Quarks are at string ends, so strings are one-dimensional gauge fields. Dual-resonance theory requires hadrons {pomeron} with no quarks. Dual-resonance theory predicts infinite hadrons, with heavier masses {Regge recurrences}. Dual-resonance theory predicts that maximum temperature is 10^12 K.
Perhaps, rather than calculus of continuous variables, discrete algebra {algebraic physics} can describe physical laws using groups or matrices.
Perhaps, rather than calculus of continuous variables, spins or other quanta can be space, time, energy, and/or mass units, making discrete-number physics {combinatorial physics}.
5-Physics-Quantum Mechanics-Theory
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Date Modified: 2022.0225