Levy flight and TGD

I encountered an interesting Science Daily article (see this) about Levy flight (see this), which is the analog of Brownian motion characterized by the distribution for the time durations for the periods during which the motion occurs with a constant velocity. In Brownian motion the distribution is Gaussian but in Levy motion it can be more complex, and have even several peaks.

Could random motion reflects the structure of space-time at the fundamental level?

The key point is that in TGD point-like particles are replaced by 3-surfaces and their orbits define space-time surfaces in H=M⁴× CP₂ as analogs of slightly non-deterministic Bohr orbits satisfying holography forced by the realization of general coordinate invariance without path integral (see this and this). Holography= holomorphy principle implies that the solution of field equations is a minimal surface irrespective of the action principle as long as it is general coordinate invariant and constructed in terms of the geometry induced from H. One can wonder whether Brownian motion and Levy flight could at the fundamental level relate to the slight failure of the classical determinism in the TGD framework.

Space-time surfaces as solutions of field equations (see this) are determined as a root f=(f₁,f₂)=(0,0) of two analytic functions of one hypercomplex coordinate and 3 complex coordinates of H defining what I call Hamilton-Jacobi coordinates (see this). Space-time as a minimal surface is expected to be slightly non-deterministic. The non-determinism would be located at 3-D surfaces analogous to the 1-D frames spanning 2-D soap film where the same occurs. At the frames the space-time surface can branch in several ways.

This could give rise to an analog of Brownian motion and Levy flight with point-like particles replaced with a 3-surface and the orbits with the preferred extremals as a minimal surface defining the analog of Bohr orbits.

How to describe classical non-determinism elegantly?

There are strong constraints to be satisfied at the loci of non-determinism since the field equations code for conservation laws of the isometry charges of H and these must be satisfied. This kind of conditions must be satisfied also for 2-D minimal surfaces at frames and they pose very strong conditions for what can happen at the frames. A natural guess is that the space-time surface decomposes to regions charracterized by different function pairs f=(f₁,f₂) having the loci of non-determinism as interfaces.

In TGD number theoretic vision is complementary to the geometric vision of physics. It involves p-adic physics for various values of p-adic prime characterizing the p-adic numbers field in question. Also the extensions of p-adic numbers, induced by extensions E of rationals, are allowed. In the holography= holomorphy vision, the extension of rationals would characterize the Taylor coefficients of analytic functions f₁,f₂ of the Hamilton-Jacobi coordinates of H. This is not enough to give p-adic primes without additional assumptions.

For ordinary polynomials of a complex argument the product for the difference of the roots defines what is known as discriminant. For rational coefficients it decomposes to a product of powers of so called ramified primes which define special p-adic primes and the proposal is that these primes define the p-adic primes characterizing the system. These p-adic primes near powers of 2 were found to characterize elementary particles in p-adic mass calculations that I performed around 1995 (see this).

How could the roots of polynomials of complex coordinates emerge?

One should somehow introduce naturally the roots of 2-D complex polynomials in order to get p-adic primes as ramified primes.

1. The basic observation is that the analytic maps g: C²→ C² allow to generate new solutions from the existing ones by the map f=(f₁,f₂)→ g(f)= (g₁(f₁,f₂),g₂(f₁,f₂)).
2. As a special case, one has g= (P,Id), P a polynomial, giving rise to map f=(f₁,f₂)→ (P(f₁),f₂). There are physical motivations for considering this restriction: the proposal is that f₂=0 represents a very long length scale constraint on the physics (see this), defining a slowly varying cosmological constant. The "world of classical worlds" (WCW) as the space of these 4-surfaces inside causal diamond (CD) would decompose to sub-WCWs inside with f₂ is fixed. One can calculate the roots for the polynomial P and assign to it ramified primes. The roots (g₁(f₁),f₂)=(0,0) give 4-surfaces representing roots of r of P as 4-surfaces f₁=r.
3. The maps g play a key role in the TGD based view about the physical analog of metamathematics and Gödel's incompleteness theorem (see this). Space-time surfaces as almost deterministic systems would be analogous to theorems with the assumptions of the theorem represented by holographic data at the so called passive boundary of CD remaining unaffected in the sequence of small state function reductions (SSFRs) defining the analog of Zeno effect.

   The axiom system would be represented by the variational principle, which also gives minimal surfaces as its solutions by holography= holomorphy correspondence therefore one can say that classical physics defines the analog of logic, which cannot depend on axiomatics. The map g would mean a transition to a meta level and the surfaces obtained in this way would represent theorems about theorems. One would obtain hierarchies of composite maps representing statements about statements about....

   If this picture is correct, the p-adic primes as ramified primes would be associated with a metalevel accompanying already elementary particles and represented by g. Hierarchies of these metalevels are predicted the iterations of g gives rise to the analogs of Mandelbrot fractals and Julia sets and for the approach to chaos as increase of complexity.

Does the p-adic non-determinism of differential equations correspond to the classical non-determinism of Bohr orbits?

p-Adic differential equations differ from their real counterparts in that the integration constants are pseudo constants having vanishing p-adic derivatives. They depend on a finite number of pinary digits. This generalizes also to partial differential equations. Suppose that the space-time surfaces can be characterized by a p-adic prime p, or its analog for algebraic extension of rationals, identified as a ramified prime of f=P.

1. Could the classical non-determinism of field equations correspond to the p-adic nondeterminism for some ramified prime p in the sense that the coefficients of (f₁,f₂) are p-adic pseudo constants for p? Could the real coefficients and their p-adic counterparts be related by a canonical identification x_p=∑ x_npⁿ→ x_R= ∑ x_np^-n or be identical? Canonical identification would guarantee a continuous correspondence: otherwise one would obtain huge fluctuations.
2. Could the Brownian motion and Levy flight reflect the underlying p-adic non-determinism at the fundamental level at which particles are replaced by 3-surfaces which in turn are replaced by the analogs of Bohr orbits which are slightly non-deterministic.
3. Could the parameters characterizing the TGD analog of Brownian motion and Levy flight be p-adic pseudo constants depending on finite number of pinary digits for the Hamilton-Jacobi coordinates of H. Could they change their values at points with Hamilton-Jacobi coordinates with a finite number of pinary digits: this would conform with the idea of discretization as a representation for a finite measurement resolution.

One can consider several ways of modelling ordinary random walks from this perspective. One can consider two basic options.

1. The above picture suggests that one considers random walks as a smooth dynamical evolution at the p-adic level and that the replacement of the initial values or other parameters characterizing the orbits with p-adic pseudo constants as functions of time gives rise to discontinuities of say velocity in the random walk. The p-adic scale should make itself visible in statistical sense as a natural scale associated with the motion.

   This seems to require that first the real configuration space is mapped to its p-adic counterpart by the inverse of canonical identification. After the smooth p-adic orbit, say free motion or motion in gravitational field, is mapped to its real counterpart by canonical identification. One can require the orbit x₊vt is continuous (x₀ is constant) but v is pseudo constant. The orbit would be a zigzag curve having also the characteristic p-adic fractality.

2. Probability distributions suggests an alternative, perhaps more elegant approach. Also the p-adic calculations based on p-adic thermodynamics define this kind of approach. The key idea would be that everything is smooth at the p-adic level and canonical identification brings in fractallity.

   The probability distributions of durations T_n= t_n-t_n-1 and velocities v is a way to statistically characterize the random walk. Brownian walk and Levy flight serve as basic examples.

   Restrict first the consideration to the durations T_R of motion with a constant velocity. The inverse I^-1 of the canonical identification would map the real durations T_R to p-adic durations T_p=I^-1T_R. The p-Adic distribution function P_p(T_p) would be a smooth function in accordance with the idea that the fractality at the real side derives from p-adic smoothness via canonical identification. On the real side, canonical identification would give the real distribution as P_R(T_R)= (P_p(T_p))_R by I. Normalization would be required to get probability interpretation.

   One would have T_R→ T_p → P_p(T_p)→ (P_p(T_p))_R= P_R(T_R) One would obtain a hierarchy of distributions labelled by the parameters of P_p and by p. A similar map v_R→ v_pP_p(v_p)→ (P_p(v_p))_R= P_R(v_R) would give a fractal velocity distribution. For instance, Brownian walk with a Gaussian distribution would be replaced with its fractal counterpart.

For a summary of 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.