The temporal operators (implicit in temporal curl) written into these wave equation sets represent a species of operator-time in a field-theoretical context. Were these equation sets to be transposed from Maxwellian to Schrödinger form, the first inclination of the physicist would be to conjugate the temporal operators to the Hamiltonian, H, the total energy operator. (In classical particle mechanics, H is a function of generalized coordinates and momenta minus a Lagrangian function, L. If time is not explicit in L, then H represents the total energy of the system. In quantum mechanics, H is an operator which provides the equation of motion for the wave function.) Indeed, conjugating operator-time to the Hamiltonian was considered and rejected as early as the late 1920’s by Pauli. More recently, Misra, Prigogine, and Courbage have reconsidered this proposition and reached the conclusion that operator-time is incompatible with the standard interpretation of quantum mechanics. (“Lyapounov Variable, Entropy, and Measurement in Quantum Mechanics”. Preprint provided during personal communication. No date given; sometime in the early 1980’s.) They say, “… the generator of the time evolution group is also the operator representing the energy observable and hence is required to be bounded from below.” Which is to conclude that operator-time would generate negative values of the energy, and this is regarded as meaningless. But conjugating operator-time to the Hamiltonian is to treat it simply as a measurable, not as an active topological operator. Authentic operator-time would have to enter the Schrödinger format in a much more complex manner involving a related reinterpretation of the wave function.
When Dirac revised Schrödinger’s original time-dependent wave equation (which was unbalanced in regards to time reference), the resulting quadratic invoked a spin coordinate used to explain the observed split in spectral lines. No one quite knows, even yet, what this spin coordinate is whose value, according to Dirac’s formulation, must be either plus or minus 1/2. Spin 1/2 is agreed not to be simple physical spine upon an axis, because the spin value cannot vary over a range as h (Planck’s constant) tends to zero. Such continuous variation would be required to yield a classical rotational analogy. It is the property of remaining either plus or minus 1/2 that identifies Dirac’s spin coordinate as a shadow of active operator-time.
During the cascade, angular momentum is transferred from one limited spacetime domain to another, such that the horizontal component of velocity is twisted into an imaginary dimension. This 90-degree twist (mediated by an operator with the
, an imaginary number, as a coefficient) in the spatial axis of rotation can be regarded either as the emergence of complex angular momentum, or, alternatively, as the temporal curl of horizontal velocity. As the cascade ensues from one scale level to the next, the spatial axis of spin undergoes repeated 90-degree twists into and out of imaginary dimensions. Temporal curl (which performs a function similar to Maxwell’s “demon”) is time actively imparting imaginary spin moments to the system. That time can accomplish this feat of imparting spin moments to a system was first publicly suggested by Russian physicist Nicolai Kozyrev in 1958. As in Dirac’s spin 1/2, the imaginary spin moments imparted during tornado genesis occur only by 90-degree jumps; there is no continuous variation over a range. Transposition from Maxwellian to Schrödinger format of the temporal curl involved in energy-momentum cascade leading to tornado genesis would involve the appearance of operator-time as a generalization of Dirac’s spin coordinate. A complete exposition, likely, would involve not only active 3-fold imaginary time, but also Roger Penrose’s twistor as quantization of temporal curl.