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
Explain how a positive-time generator can select a compatible complex structure on the dynamic sector. Make clear why zero-frequency modes need separate treatment.
- Real-commutant phase selection: a paper guide
Introduce the paper’s phase-selection question in real Hilbert representations. Prepare an accessible route from the commutant to compatible complex structures and their physical interpretation.
- A quantum source for generation and completion
Introduce the positive phase–dilation model as an explicit realization within quantum mechanics. Explain how it distinguishes inner work from completed observation without deriving quantum kinematics itself.
- Completion, commutants, and positive frequency
A reading guide to the published paper’s relationship between completion principles, real representations, and positive-frequency geometry. Focus on the theorem assumptions before discussing physical interpretations.
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.