Scale Invariance and Emergence

Fractal, or fractional dimensional, relations in physics describe a lot of fundamental physical processes. For the moment, entertain the idea of the Euler beta function, an expression that not only describes a self-similar pattern, but one that has inherent symmetry in the abscissa. What is important to understand is that the functional form of a fractal has a kernel of logic that expresses a nascent logical causative property of what are best expressed as new field axioms. Here, to make a complete Gödel-like logical semantic representation, the new field axioms must involve expressions isomorphic to natural phenomena inherent in universal systems. This is why quantum mechanics is critical to the discussion of free will and analytic mechanics and the least action path have engendered determinism. The expressions reflecting nature are found in the CMB, inflation, tunneling, the fundamental force relations, general relativity, and spooky action. Here, the panoply of physical states that saturate the description of nature are leading to thermodynamics, just statistical physics. The mathematics use analytic continuation and asymptotic freedom to find finite values in infinite series. Truncating terms means uncovering that kernel of logic nested in intuitive everyday observation. This is what is so difficult in the development of physics and mathematics. What we have is the loud cacophony of cognitive dissonance emerging as turbulence in the statistical description. Essentially, the progenitors are things like Euler beta function regularity in string states, quantum states, and microstates of thermodynamics which saturate all physicalism. There is no mental world meaning all logical fractal and self similar expressions of logic are the building blocks of larger scale invariant systems. Nature behaves according to laws that are small and large scale, with phenomena emerging from laws of regularity and patterning of things like prime numbers. The randomness of nature is best exemplified by the Riemann zeta function which can be expressed as a fractal Euler beta function. General Physics scientists are working on novel integral expressions of the Riemann zeta function and the Euler beta function that involve quantum field theory and expressions of scale invariant fractals to generate new quantum mechanics with more deterministic scattering amplitudes from supersymmetry. What needs to happen is a statistical determinism approach to the scattering phase shift of inelastic and elastic particle collisions; collisions are deterministic even in the Poincaré analysis of multi-body physics. The future holds many interesting directions for self-similar and emergent studies from fundamental Euler beta function relations to turbulence and scattering amplitude functions from primordial physical systems. General Physics will keep updating scale invariant analysis and emergent property studies in the upcoming phase of research and development. New waveform antennas, quantum sensors, and pattern recognition software are in the pipeline.