1. The outline of the argumentIn a very concise form the argument goes as follows.
- Generalization of the basic facts about the dependence of Dirac operator on hbar leads to the realization that the freedom to choose the value of hbar corresponds to the freedom to choose the overall scaling λ= hbar/hbar0 of M4 metric associated with various copies of M4 obtained identified as various algebraic extensions of rational M4 glued together along common set of rationals consistent with the isometric identification. This generalization of number concept is one of the basic ideas of number theoretic vision and allows to fuse real and p-adic physics to single coherent whole by requiring that all of these physics are obtained by an algebraic continuation from rational physics.
- The modified Dirac operator for space-time surfaces depends on λ via the induced metric determining the fermionic oscillator operator basis. Dirac determinant determines Kähler function. λ cannot be chosen freely. The dependence of λ on the algebraic extension for a given p-adic prime p is fixed by the quantum criticality condition stating that the critical Kähler coupling strength gK2 is same for various algebraic extensions associated with given p-adic prime p.
- Number theoretic ideas allow to make good guesses concerning the dependence of λ on the algebraic extension.
2. Hierarchy of spinors structures in the world of classical worlds defined by algebraic extensions of rationals
What the rational physics and its algebraic extensions mean in practice is following.
- The modified Dirac operator D associated with the induced spinor structure at space-time surfaces which depends on λ via the scale of M4 metric. The eigen values associated with the eigen spinors of D label the fermionic oscillator operators in turn determining the configuration space spinor Clifford algebra and spinor structure. Rational physics means restriction to a subset of rational eigenvalues and various extensions restriction to eigenvalues in that extension so that one obtains a family of configuration space spinor structures. There is no need for discretization as one might naively think: just a restriction to eigenvalues in a given extension.
- The inclusion of rationals to a given extension induces an inclusion of type II1 factors analogous to Jones inclusion characterized by the index M:N telling the dimension of the extended configuration space ("world of classical worlds") Clifford algebra as a rational Clifford algebra module. This generalized index M:N is what contains information about hbar for that extension.
- Kähler function for a given space-time sheet in given extension can be defined as a Dirac determinant involving only the eigenvalues in that extension. It is quite possible that this restriction automatically guarantees the finiteness of the determinant. The Dirac determinant is exponent of Kähler function with certain value of Kähler coupling gK2. The condition determining the dependence of λ on algebraic extension is that the value of λ belonging to the extension is such that gK2 is same as for rational case.
- The beauty of this picture is that since M4 metric is inversely proportional to λ the induced metric contains λ Kähler action and function and thus configuration space geometry depends highly nonlinearly on λ (hbar). This means that classical theory codes all non-perturbative effects although it is quite possible that loop corrections to configuration space functional integral vanish. Classical theory allows also to obtain information about quantum corrections: for instance, quantization of the magnetic flux classically implied by the basic variational principle uses λ as a unit and thus it might be possible to deduce λ in this manner for a given algebraic extension.
3. λ as homomorphism from space of algebraic extensions of rationals to reals
- The generating units of extension can be classified to phases, the number of which is at most one, and set E of generating units for which complex norm differs from unity. A natural assumption is that λ is product of term λa depending on the phase via Beraha number defined by the phase and of term λb depending on E only through the number of units in E. λa can be identified either as the inverse of the fractal spinor dimension d or the corresponding space dimension given by 2-based logarithm of d. The latter option is more plausible physically since it is consistent with the assumption that M:N is integer power of M:N for ordinary Jones inclusion meaning that a tensor power of M:N= Bn fractal-dimensional Clifford algebra as rational Clifford algebra module is obtained. This option is also consistent the earlier physics inspired guess.
- The general form for the dependence of λb on the extension can be deduced by requiring that it is a multiplicative homomorphism from the family of algebraic extensions to real numbers. What this means is that if one combines basis of linearly independent extensions E1 and E2 with d1 and d2 linearly independent generating "imaginary" units with complex norm different from unity, the value of λb for the combined extension is product.
This gives a large factor which comes as a power of N= (1/v0), 1/v0≈ 211. The Clifford algebra associated with the algebraic extension is Nk:th tensor power of minimal M:N= Bn dimensional module or tensor product of these. The powers of 1/v0 define a hierarchy of values of hbar and thus a hierarchy of macroscopically quantum coherent systems with increasing size. Dark matter hierarchy corresponds to the hierarchy of algebraic extensions of rationals and ordinary matter to rational physics. This should translate more or less easily to the p-adic context so that each p-adic number field defines this kind of hierarchy of extensions.
For more details see the chapter Was von Neumann Right After All? of "TGD".