Could quantum field theories be universal

The findings of Nima Arkani Hamed and his collaborators (see this), in particular Carolina Figueiredo, suggest a universality for the scattering amplitudes predicted quantum field theories. Is it possible to understand this universality mathematically and what could its physical meaning be?

The background for these considerations comes from TGD, where holography = holomorphy principle and M⁸-H duality relating geometric and number theoretic visions fixing the theory to a high degree.

1. Space-time surfaces are holomorphic surfaces in H=M⁴× CP₂ and therefore minimal surfaces satisfying nonlinear analogs of massless field equations and representing generalizations of light-like geodesics. Therefore generalized conformal invariance seems to be central and also the Hamilton-Jacobi structures (see this) realizing this conformal invariance in M⁴ in terms of a pair formed by complex and hypercomplex coordinate, which has light-like coordinate curves.
2. Quantum criticality means that minima as attractors and maxima as repulsors are replaced with saddle points having both stable and unstable directions. A particle at a saddle point tends to fall in unstable directions and end up to a second saddle point, which is attractive with respect to the degrees of freedom considered. Zero energy ontology (ZEO) predicts that the arrow of time is changed in “big” state function reductions (BSFRs). BSFRs make it possible to stay near the saddle point. This is proposed to be a key element of homeostasis. Particles can end up to a second saddle point by this kind of quantum transition.
3. Quantum criticality has conformal invariance as a correlate. This implies long range correlations and vanishing of dimensional parameters for degrees of freedom considered. This is the case in QFTs, which describe massless fields.

Could one think that the S-matrix of a massless QFT actually serves as a model for transition between two quantum critical states located near saddle points in future and past infinity? The particle states at these temporal infinities would correspond to incoming and outgoing states and the S-matrix would be indeed non-trivial. Note that masslessness means that mass squared as the analog of harmonic oscillator coupling vanishes so that one has quantum criticality. What can one say of the massless theories as models for the quantum transitions between two quantum critical states?

1. Are these theories free theories in the sense that both dimensional and dimensionless coupling parameters associated with the critical degrees of freedom vanish at quantum criticality. If the TGD inspired proposal is correct, it might be possible to have a non-trivial and universal S-matrix connecting two saddle points even if the theories are free.
2. A weaker condition would be that dimensionless coupling parameters approach fixed points at quantum criticality. This option looks more realistic but can it be realized in the QFT framework?

QFTs can be solved by an iteration of type DX_n+1= f(X_n) and it is interesting to see what this allows to say about these two options.

1. In the classical gauge theory situation, X_n+1 would correspond to an n+1:th iterate for a massless boson or spinor field whereas D would correspond to the free d’Alembertian for bosons and free Dirac operator for fermions. f(X_n) would define the source term. For bosons it would be proportional to a fermionic or bosonic gauge current multiplied by coupling constant. For a spinor field it would correspond to the coupling of the spinor field to gauge potential or scalar field multiplied by a dimensional coupling constant.
2. Convergence requires that f(X_n) approaches zero. This is not possible if the coupling parameters remain nonvanishing or the currents become non-vanishing in physical states. This could occur for gauge currents and gauge boson couplings of fermions in low enough resolution and would correspond to confinement.
3. In the quantum situation, bosonic and fermionic fields are operators. Radiative corrections bring in local divergences and their elimination leads to renormalization theory. Each step in the iteration requires the renormalization of the coupling parameters and this also requires empirical input. f(X_n) approaches zero if the renormalized coupling parameters approach zero. This could be interpreted in terms of the length scale dependence of the coupling parameters.
4. Many things could go wrong in the iteration. Already, the iteration of polynomials of a complex variable need not converge to a fixed point but can approach a limit cycle and even chaos. In more general situations, the system can approach a strange attractor. In the case of QFT, the situation is much more complex and this kind of catastrophe could take place. One might hope that the renormalization of coupling parameters and possible approach to zero could save the situation.

It is interesting to compare the situation to TGD? First some general observations are in order.

1. Coupling constants are absorbed in the definition of induced gauge potentials and there is no sense in decomposing the classical field equations to free and interaction terms. At the QFT limit the situation of course changes.
2. There are no primary boson fields since bosons are identified as bound states of fermions and antifermions and fermion fields are induced from the free second quantized spinor fields of H to the space-time surfaces. Therefore the iterative procedure is not needed in TGD.
3. CP₂ size defines the only dimensional parameter and has geometric meaning unlike the dimensional couplings of QFTs and string tension of superstring models. Planck length scale and various p-adic length scales would be proportional to CP₂ size. These parameters can be made dimensionless using CP₂ size as a geometric length unit.

The counterpart of the coupling constant evolution emerges at the QFT limit of TGD.

1. Coupling constant evolution is determined by number theory and is discrete. Different fixed points as quantum critical points correspond to extensions of rationals and p-adic length scales associated with ramified primes in the approximation when polynomials with coefficients in an extension of rationals determine space-time surfaces as their roots.
2. The values of the dimensionless coupling parameters appearing in the action determining geometrically the space-time surface (K”ahler coupling strength and cosmological constant) are fixed by the conditions that the exponential of the action, which depends n coupling parameters, equals to its number theoretic counterparts determined by number theoretic considerations alone as a product of discriminants associated with the partonic 2-surfaces (see this and this). These couplings determine the other gauge couplings since all induced gauge fields are expressible in terms of H coordinates and their gradients.
3. Any general coordinate invariant action constructible in terms of the induced geometry satisfies the general holomorphic ansats giving minimal surfaces as solutions. The form of the classical action can affect the partonic surfaces only via boundary conditions, which in turn affects the values of the discriminants. Could the partonic 2-surfaces adapt in such a way that the discriminant does not depend on the form of the classical action? The modified Dirac action containing couplings to the induced gauge potentials and metric would determine the fermioni scattering amplitudes.
4. In TGD the induction of metric, spinor connection and second quantized spinor fields of H solves the problems of QFT approach due to the condition that coupling parameters should approach zero at the limit of an infinite number of iterations. Minimal surfaces geometrizes gauge dynamics. Space-time surfaces satisfying holography = holomorphy condition correspond to quantum critical situations and the iteration leading from one critical point to another one is replaced with quantum transition.

See the article TGD as it is towards end of 2024: part I or a chapter with the same title.

For a summary ofhttps://draft.blogger.com/u/0/blog/posts/10614348 earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.