Does Schrödinger evolution fix Planck’s constant?

Obtaining an evolution law and fixing its physical scale are different achievements

An oriented real dynamics can select a compatible complex structure and be written as a Schrödinger-type evolution. That is a substantive structural result: it explains a phase-bearing carrier, its time orientation and a self-adjoint frequency generator. To interpret the generator as measured energy, however, one needs an action scale connecting phase accumulation to an independently calibrated energy measurement.

For DTT, this raises a constructive question. Can the same source dynamics determine both the phase rate and the work recorded by a physical energy transducer, with one universal conversion between them? The Schrödinger and action guide develops the distinction. A successful source relation would be stronger than multiplying a frequency operator by a named constant.

A finite example leaves the action scale free

Let K=diag(0,ω), where ω is measured in radians per second, and let U(t)=exp(−itK). For any positive action scale a, define H=aK. The same evolution then satisfies ia dψ/dt=Hψ. Changing a and H by the same positive factor changes no state trajectory or completion probability calculated from U.

Prepare |+⟩ and measure X after time t. The positive outcome probability is [1+cos(ωt)]/2. This experiment determines ω relative to the chosen clock. It does not, by itself, determine whether the corresponding energy separation is aω, 2aω or another value connected by the rescaling. An independent energy measurement is needed to distinguish those physical identifications.

Adding bI to H gives only an additional common phase in this isolated model. Energy differences, work comparisons and gravitational couplings have different operational roles, so their calibration must be discussed explicitly. A source phase equation cannot settle all of them by a choice of zero.

The same example explains a common circularity in a speed-of-evolution argument. The state first becomes orthogonal to |+⟩ after t=π/ω, and its energy standard deviation is aω/2. Their product is πa/2. This exact relation connects an orthogonalization time to an energy spread, but every positive a satisfies it. Unless the energy spread is calibrated independently, the relation cannot also determine the action scale that was used to define that energy.

What positive-frequency selection establishes

For a finite real skew generator A, define Ω=(−A²)¹ᐟ² on its nonzero-frequency sector and J=AΩ⁻¹. Then J²=−I and A=JΩ. With the ordinary imaginary scalar chosen as i=−J, the forward equation becomes iaψ̇=aΩψ with H=aΩ nonnegative. The construction fixes a compatible phase orientation and positive frequency operator under its stated hypotheses.

The factor a remains arbitrary in those identities. The published real-commutant phase-selection study addresses compatible complex structures, dynamic-sector phase and composition conditions. Its phase result should not be read as a numerical derivation of the physical quantum of action.

Likewise, six nested inner levels organize dependencies rather than supply six independent energy units. To turn their count into an action, a model must identify a measurable conjugate pair or a work integral. The dimensional units and the physical readout are part of that construction; they cannot be recovered from a dimensionless stage count alone.

What a full phase cycle does and does not imply

If phase is related to a classical action S by φ=S/a, a full phase change of 2π corresponds to an action change 2πa. Calling that change h is consistent with a=ħ. However, the relation between S and phase has already introduced a. Periodicity fixes the dimensionless factor 2π; it does not determine the size of a in physical work and time units.

A classical oscillator illustrates the remaining freedom. Its action integral is J=∮p dq=2πE/ω. Classical trajectories allow a continuous range of E, so they allow a continuous range of J. Quantized action spacings require an additional quantum or source selection rule. Declaring only one allowed phase winding does not establish which oscillator energy is physically retained unless the action–phase coupling is also specified.

This observation gives a useful test for proposed derivations from winding, compactification or tick counting. Rewrite the derivation using an unspecified a at every phase–energy conversion. If the final equation becomes an identity for arbitrary a, it has organized the units or the spectrum but has not fixed the action scale. A successful additional principle must remove that continuous freedom using independent source quantities.

Universality is a meaningful target even before a numerical value

Suppose two independently calibrated systems have phase generators K₁ and K₂, while their energy transducers report H₁=a₁K₁ and H₂=a₂K₂. A shared physical time determines the product evolution through K₁⊗I+I⊗K₂. To identify the additive total energy with one universal action times that generator, the model must make a₁ and a₂ agree. This is a substantive relation between independently measured quantities.

One could investigate whether a source conservation law, a common conjugate structure or a restricted interaction enforces the same conversion across distinct carriers. Such a result would explain universality within the stated class even if the overall dimensional scale remains a calibration. It must include interactions; agreement between two copies whose parameters were set equal at preparation does not demonstrate a general law.

In the present SI, the numerical value of h is fixed exactly as part of the unit definitions, as the BIPM defining-constants table explains. That metrological fact does not remove the theoretical question. It changes how an explanation should be expressed: specify a physical relation among independently defined standards or predict a dimensionless ratio, rather than treating a number in chosen units as an invariant consequence.

What a non-circular action calibration would require

Start with a source state space, an autonomous update and a clock whose readings are defined without the target action relation. Add an energy transducer described through work on another admitted degree of freedom. Derive its response and compare it with the source's interferometric phase accumulation. Every coupling introduced to operate that transducer should appear explicitly in the result.

Then vary an internal frequency, a preparation and a carrier type while holding the independently specified transducer mechanism fixed. The target is a theorem that the energy-to-phase-rate ratio remains common because of the source law, together with an account of any surviving overall scale. A useful failure result would exhibit a family of equally admissible transducers that changes the ratio while leaving the phase dynamics intact.

This is an inviting mathematical problem because its first stage can be finite and exact. Matrices, work registers and conservation laws are enough to identify the calibration freedom. The later physical identification can then be attached to a result whose actual content is clear, rather than to a rearrangement of the Schrödinger equation.

Continue reading

• From an oriented generator to Schrödinger evolution →
• Worked derivation: complex structure from real dynamics →
• Selecting phase by polar decomposition →
• The constants programme: structure, response, and common scales →

Read this discussion as part of a path

• From structural counts to physical constants →