### Finiteness for the number of non-vanishing Wick contractions, quantum criticality, and coupling constant evolution

Finiteness for the number of non-vanishing Wick contractions, quantum criticality, and coupling constant evolution

The consistency with number theoretic vision requires that the number of terms in the super-Taylor expansion of action is finite - otherwise one is led out from the extension: this applies both to the action determining space-time surfaces and to the corresponding modified Dirac action. There are several options that one can consider.

1. Normal ordering of the fermionic oscillator operators would be a straightforward manner to handle the situation. One would obtain finite number of terms since the number of quark oscillator operators is d=4+4=8. The maximal degree mmax of multiple partial derivative of action with respect to gradient of H-coordinate h would be mmax= d=8 and correspond to monomial with 4+4 quark oscillator operators. Note that the normal ordering of this term gives rise to c-number.

It however seems that the natural solution of the problem must involve cancellation of the Wick contractions when the degree m of the multiple partial derivative satisfies m>mmax. Some cancellation mechanism for m≥ mmax should guarantee that Wick-contractions give in this case a vanishing contribution to each of the d= 8 monomials in the super-action.

2. The strongest condition would be that all Wick contraction terms coming from the normal ordering vanish. The contraction terms are expressible as divergences of currents and the interpretation would be in terms of Noether current associated with some symmetry. Super-symplectic symmetry is the best candidate in this respect. Note that besides these currents also the Noether currents coming from the super-symplectic variations should have a vanishing divergence.
3. One can consider also a weaker condition. Wick contractions vanish for m>mmax such that mmax>8 is possible. This would give rise to the analog of radiative corrections, and if mmax can vary, one obtains the analog coupling constant evolution and discrete coupling constant evolution corresponds to the variation of mmax.
How the value of mmax could be determined?
1. M8-H duality requires that M8- and H-pictures are structurally similar. Octonionic polynomials are characterized by their order n and also the super-extremals should be characterized by n and even the individual terms of super-polynomial should have counterparts at H-level.

One can define super-octonionic polynomials at M8-level and also for these normal ordering terms appear. Ordinary derivatives of P(o) with respect to o replace those of the action with respect to the gradients of H coordinates, and one obtains only finite number of Wick contractions. There is no need to require their vanishing now, and the hierarchy of degrees n=heff/h0 for P defines a discrete coupling constant evolution with each level corresponding to its own values of coupling constants differing by the number of Wick contractions. This gives a connection with the ordinary coupling constant evolution with Wick contractions taking the role of loops.

This picture should have direct image at H-side. In particular, one should have mmax=n.

2. The cancellation of Wick contractions for the action containing both Kähler term and cosmological term probably happens only for critical values of cosmological constant determined dynamically from the mechanism of dimensional reduction reducing 6-D surface in the product of twistor spaces T(M4)= M4× S2 and T(CP2)= SU(3)/U(1)× U(1) to S2 bundle over space-time surface representing induced twistor structure. The cancellation condition for the higher terms could fix the value of cosmological constant emerging from the mechanism.
3. The picture could be interpreted in terms of quantum criticality. The polynomials P(o) characterize quantum critical phases. Also Taylor series can be considered but they would not be critical and infinite amount of information would be required to specify them whereas the specification of critical dynamics requires by its universality only a finite number of parameters coded by the rational coefficients of polynomial.

Criticality corresponds to the vanishing of not only function but also some of its derivatives at critical point. The criticality would be now infinite in the sense that all derivatives of P(o) higher than n would vanish. This is indeed the view about quantum criticality that I ended up to long time ago. This implies that the parameter space for the functions describing criticality is finite-dimensional.

In Thom's catastrophe theory which essentially describes a hierarchy of criticalities concretely, the finite-dimension of the space of control parameters is essential. For cusp catastrophe this space is 2-dimensional and catastrophe graph is defined by a fourth order polynomial so that all higher order derivatives vanish identically also now.

4. At the level of H criticality would mean that m-fold partial derivatives of action only up to m=mmax=n-fold partial derivatives contribute to the radiative corrections. The action would be polynomial of finite order in the multi-spinor components of super-coordinates and discrete coupling constant evolution would be realized. The ordinary variations of the action would be of course non-vanishing to arbitrary high order.

Coupling constant evolution would reduce to the hierarchy of extensions of rationals since the degree n of P determines the dimension of extension. Evolution in terms of the hierarchy of extensions of rationals would dictate also coupling constant evolution. This evolution would also dictate the preferred p-adic length scales if preferred p-adic primes are identifiable as ramified primes. Ramified primes at the lowest level of hierarchy are ramified primes at higher levels if P(0)=0 condition is true for them. Evolutionary hierarchies correspond to functional composition hierarchies for polynomials with degrees ni such that ni+1 is divisible with ni that is ni+1/ni=ki.

Remark: Functional composition occurs also in the construction of fractals like Mandelbrot fractal and as a special case one iterates single polynomial to get a hierarchy in powers of integers n1. This interpretation would conform with the interpretation of the symmetries guaranteeing the cancellation of Wick terms as super-symplectic symmetries.

5. A connection with the inclusion hierarchies for super-symplectic algebra is highly suggestive. The fractal hierarchy of super-symplectic sub-algebras (fractality and conformal symmetry - now in generalized sense - are essential for quantum criticality) with levels labelled by n would naturally give rise to counterparts of the functional composition hierarchies.

Inclusion hierarchies would correspond to sub-hierarchies of super-symplectic algebras formed by sequences of sub-algebras with weights divisible by integer ni such that ni divides ni+1. ni would correspond to a degree of polynomial in the hierarchy formed by their compositions in accordance with functional composition of polynomials.

6. The inclusion hierarchies of super-symplectic algebras would have interpretation in terms of inclusions of hyper-finite factors of type II1. The ratios ni+1/ni= ki appearing in the composition hierarchies would correspond to the integers labelling the inclusions of HFFs and defining quantum phases U=exp(iπ/ki) characterizing quantum algebras and quantum spaces as analogs of state spaces modulo finite measurement resolution.

The interpretation of finite measurement resolution as an ability to detect only space-time sheets characterized by polynomials of order n below some fixed integer is natural. n would characterize the measurement resolution.

To sum up, this picture rather neatly fuses together several speculative visions about quantum TGD. The reduction of dynamics to polynomial dynamics at the level of M8 has interpretation in terms of quantum criticality with finite-D space of control parameters implying universal dynamics involving very few coupling parameters, which are fixed points of coupling constant evolution for given value of n. M8-H duality maps M8 dynamics to the level of H, where it is realized in terms of a hierarchy of sub-algebras of super-symplectic algebra and sub-hierarchies correspond to sequences of integers ni dividing ni+1. A connection with the inclusions of HFFs and finite measurement resolution emerges. The notion of discrete coupling constant evolution finds a precise formulation, and the notion of radiation correction is realized in terms of Wick contractions.