Polar decomposition separates rotation from rate. Converting that rate to energy requires an action scale; selecting complex phase does not determine its numerical value or a physical Hamiltonian.
Derive the form before assigning the physics
The SMM/DTT programme seeks a connection between generative recurrence and quantum dynamics. A useful first result is to identify when real evolution can take Schrödinger form. This is a question about generators, phase conventions, and dimensions. It is distinct from deriving a particular particle Hamiltonian, a probability rule, or the value of Planck's constant.
Start on a finite real inner-product space with ẋ=Ax and Aᵀ=−A. The norm is conserved because d(xᵀx)/dt=xᵀ(Aᵀ+A)x=0. Such a flow supplies coherent reversible motion. To identify it as physical time evolution, its parameter t must eventually be related to an operational clock.
Separate the phase orientation from the rate
On the nonzero-frequency sector, define Ω=(−A²)1/2 and J=AΩ−1. The finite polar calculation gives A=JΩ, J²=−I, and Jᵀ=−J. The positive symmetric operator Ω carries the magnitudes of the angular rates.
The operator J is selected by the orientation of A. Reversing the real flow reverses J and leaves Ω unchanged. Zero modes are excluded from this inverse formula and require their own phase extension if one is wanted. Nothing in this step assigns energy units to Ω: with t measured in seconds, its eigenvalues have units of inverse seconds.
Keep the Schrödinger sign convention explicit
The conventional Schrödinger equation uses evolution exp(−iHt/ħ). To match that convention with positive H in the polar example, represent the scalar i by the real operator Ic=−J. Choose a positive action scale σ and define H=σΩ. Multiplying the real equation by σIc gives:
In the reconstructed complex notation this is iσ∂tψ=Hψ. The sign is a convention that must be carried consistently: if i is instead represented by J while the same oriented flow is kept, the corresponding Hamiltonian is −σΩ. One cannot simultaneously keep that choice, reverse no sign, and claim the positive Schrödinger Hamiltonian.
Why the action scale is still an input
An energy multiplied by a time has units of action. The equation E=σω converts an angular frequency into an energy only after σ is specified. The real generator determines ω, while a physical calibration or an additional source law determines the energy assigned to that frequency.
Choosing σ=ħ reproduces the familiar quantum conversion. It does not derive ħ from the existence of a recurrence. One can multiply σ and H by the same positive factor without changing the flow generated by H/σ. A proposed derivation of the numerical action scale must remove that freedom using independently defined physical resources and units, not by inserting a measured energy into a frequency relation and then calling the conversion predicted.
A two-level example with a measurable relative phase
Supply a two-level Hamiltonian H=(σΩ₀/2)(I−X), where Ω₀>0 and X interchanges |0⟩ and |1⟩. Its eigenvalues are zero and σΩ₀. The initial state |0⟩ is an equal superposition of the two energy eigenvectors, which acquire a relative phase Ω₀t.
The probability follows from the supplied quantum measurement rule. A common energy shift adds only a global phase to this isolated two-level experiment, while the energy difference sets its oscillation frequency. The zero eigenvalue also illustrates the kernel issue: this model supplies a complex structure on the whole carrier, including the stationary sector; that extension was not selected by a vanishing generator.
Where the usual spatial equation adds assumptions
To obtain iσ∂tψ=[−σ²∇²/(2m)+V]ψ, one must additionally supply spatial coordinates, translation generators p=−iσ∇, a mass parameter m, and the kinetic-energy law p²/(2m). Substitution then gives the Laplacian form. The polar theorem alone contains none of those physical choices.
For a plane wave exp[i(k·x−ωt)] in the free model, p=σk and E=σω=σ²|k|²/(2m). This relation combines the action conversion with a chosen nonrelativistic dispersion law. A source-based derivation would explain why the completed geometry, matter propagation, and clock calibration jointly lead to that law, and describe the regime where relativistic or lattice corrections become relevant.
What a stronger SMM/DTT result would accomplish
The present calculation establishes a clean route from oriented real evolution to complex dynamical notation. Its next stage should construct a persistent matter excitation and compute its generator from the nested inner operations. The same model should supply the completion mechanism and the physical clock used to measure its frequencies.
A further independent result is needed for the action scale. It could involve a quantized source resource, an operational energy definition, and a proof fixing their conversion across several observables. Until then, using ħ in a quantum realization is legitimate and useful, provided it is recognized as an input. The constants programme tracks that normalization problem alongside the structural questions.
Continue with the articles
- Selecting phase by polar decomposition
Polar decomposition of an oriented real dynamical generator selects an ordinary complex structure on its nonzero-frequency sector. Zero modes, symmetry compatibility, and discrete logarithm branches remain separate inputs.
- Real-commutant phase selection: a paper guide
An oriented real orthogonal flow can select an ordinary complex structure on its dynamic sector. Planar examples explain residual zero-mode freedom, time-reversal antilinearity and the extra balancing rule needed for composite systems.
- A quantum source for generation and completion
A positive phase–dilation Hamiltonian ties one phase encoder to a powered coordinate and controlled work. It gives an explicit quantum realization of generation and completion while retaining its constitutive interfaces as inputs.
- Completion, commutants, and positive frequency
This guide follows completion, compatible complex structures and positive-frequency selection through their separate mathematical roles. Finite examples connect the framework to SMM/DTT while keeping causal readout and physical calibration as additional constructions.
Sources and reading
- Completion principles, real commutants, and positive-frequency geometry for orthogonal Hilbert representations
- Real-commutant phase selection from real Hilbert representations
Read the original source for its exact formulation and edition. The examples on this page are explanatory reconstructions.