The invariance of angular momentum under the scaling of hbar

As explained in the previous posting, number theoretic vision in principle allows to predict the spectrum of quantized values of Planck constant in TGD framework. Basically the scaling of hbar reduces to the overall scaling of the Minkowski metric of an algebraic extension of rational M⁴ factor of M⁴×CP₂.

The assumption that four-momentum is invariant in the scaling of M⁴ metric by λ combined with Poincare invariance implies that also angular momentum is invariant under the scaling by λ. Hence the analog of wave vector defined as L_z/hbar would scale down by a factor 1/λ giving L_z/hbar=m/λ. The one-valuedness of the wave function however requires L_z/hbar=m.

A possible resolution of the paradox is based on following observations.

1. Number theoretic vision suggests strongly that λ is integer. Effective 2-dimensionality means that 2-dimensional partonic surfaces are basic structures to be considered. By conformal invariance scalings and rotations are generated by L₀ and iL₀. Hence it would not be surprising if the radial scaling by λ^k would be accompanied by a similar angular scaling. If M⁴ projection of the partonic 2-surface has dimension D> 0, this scaling would at space-time level mean that partonic 2-surface becomes analogous to λ^k-sheeted Riemann surface defining λ^k-fold covering of E² identified as Euclidian plane of M⁴. Using M⁴ coordinates for the space-time sheet as local coordinates, one finds that induced spinor fields have fractional angular momentum eigenvalues L_z=m/λ^k.

2. In this picture the approach to quantum criticality would correspond to the emergence of classical chaos at space-time level by a step-wise process in which the step hbar--> λ×hbar can be regarded as generalization of the period doubling bifurcation. As hbar increases by a factor λ^k, the space-time sheet representing an orbit of particle closing after one turn transforms to an orbit closing only after λ^k turns. Note that the volume of space-time sheet remains finite only if the orbit closes after finite number of turns. The step k--> k+1 would correspond to a local fractal operation making each sheet of the λ^k sheeted surface λ-sheeted so that λ^k+1 sheeted surface would result. Instead of period doubling one would have period λ-folding with the value of λ depending on p-adic prime p≈ 2^k.

3. The model of Nottale for planetary orbits requires besides the harmonics of λ₀≈ 2¹¹ also some of its sub-harmonics, at least third and fifth one. This conflicts with the assumption that λ is integer unless λ₀ is divisible by 3 and 5. This would be the case for λ₀= 2175=3× 5²× 29 consistent with the mean value λ= 2174 deduced with one per cent accuracy from the model of Nottale for planetary orbits. Only integer factors of λ₀(p≈ 2^k) would be allowed as sub-harmonics for given p for this option and integer valuedness of λ₀ would also pose strong conditions on the values of p if the proposed formula for λ₀ is accepted.

4. If one allows the quantization of angular momentum using integer multiple of hbar₀ as a unit, λ need not be an integer. In this case one could have λ=(r/s)×λ₀, where λ₀≈ 2¹¹ is integer. The orbit represented by the space-time sheet would close after (rλ₀)^k turns and angular momentum would be quantized as L_z=ms^k×hbar₀. Even in this case the proposed formula for λ(p) would give strong constraints on the values of p.

For more details see the chapter Was von Neumann Right After All? of "TGD".