Quantum Chaos Theory: Nature's Emergence of Fractal Complexity Using Quantum Mechanics Across Scales

One of the puzzling aspects of the different theoretical models in physics is the question of why should the laws of physics behave differently

at different scales if there should be no absolute frames of reference in the universe as a whole. 

It sort of the opposite to the argument given in favour of the turtle-stacking model of suspension of the Earth where a person who believes that

the earth is suspended on the back of a giant turtle and is asked "but what suspends the turtle?" to which he or she answers: "simple. It's turtles all the way down!"

this example, though humorous, does reflect our need for sensible models to describe non-local behaviour, be it the position of the earth in space or electrons

in an atom.

If we are to accept, with equal but scientifically verifiable passion, how the fundamental particles that make up atoms, molecules and matter are themselves suspended 

to the rules of quantum theory could we not also make a similar argument to turtle stacking, but in reverse?  That is to ask: 

is it really quantum all the way from down to up? 

The correspondence principle states that classical mechanics is merely the classical limit of quantum mechanics, specifically in the limit as the ratio of 

Planck's constant as the action of the system tends to zero. 

So in certain interpretations of Quantum theory, such as the path integral interpretation, we see the cancelling out the quantum effects 

by the separate particle histories causing decoherence with one another. 

So that in effect we are expected to assume that we loose some of the quantumness "in the wash" so to speak

from the microscopic world to the mesoscopic and macroscopic. 

The question that still arises however is that if quantum theory suspends the behaviour of atoms, matter and everything else from the ground up then 

shouldn't there be some quantum effects that translate across the scales? Even relatively small quantum effects?

In a sense this is the sameness problem in physics, where we can ask why do we not 

see many of the features that govern quantum effects reappear at larger scales in some aspects of structure?

Unlike the turtle stacking analogy, this is not an unreasonable question to ask, after all as quantum effects occur at small scales so their must be a lot of 

these effects that  add up even in an ordinary piece of matter. Macroscopic Magnetic effects for example is the amplification of the 

individual magnetic moments of atoms in a a material which are related directly to the quantum spin of individual atoms and electrons.

If there is not a sameness across all physical theories then we may be forced to admit that many of the separate subsets of physics are not branches as much as they 

are disjointed appendages assumed to be glued together by some insofar unwitnessed "unification". 

It should be recognized that just because we have a set of theories that are able to grind up 

experimental data and churn out predictions does not mean we truly understand the principles behind the theory and we may be just working with 

the mathematical physicists equivalent of a black box that only lets us witness the inputs and outputs in a blinding "shut-up and calculate" fashion.

  

 

In Mathematics, the application of Fractals in geometry provides a clear insight to this concept of sameness being apparent across scales, 

being made popular by the mathematics of Gaston Julia, Benbot Mandelbrot and others. 

In Fractal Geometry we see that complex geometical patterns, some of which even begin to imitate the kind of patterns we see in the natural world, have 

this principle of sameness of general shape operating at different scales. 

What is most surprising is how these patterns can emerge from simple rules of continuous 

iteration, without the need for specific coded instructions to create the precise shapes of the patterns. 

In bifurcation diagrams for example we see patterns emerge from a combination of following a computation with a randomly varying term or

set of terms added to it. The potentially infinite, often repeating patterns could not have been explicitly coded, as infinite amounts of instructions

would be required to be translated in an arcane fashion into such code. They arise from the balance between

exploration and exploitation in the system: the core feature of a meta-heuristic procedure.

We can also begin to see that unlike the relatively abstract and idealized geometry we are forced to learn in school about perfect cubes, spheres, cones and so on, 

this fractal geometry seems to create the kind of shapes and patterns that are seen in the physical phenomena of the real world- those of the 

shapes of mountains, coastlines, river systems, blood vessels, clouds, continents and even the vast networks of galaxy clusters as seen in the large scale universe. 

Shapes that are visually invariant across scales. 

It becomes apparent when studying the mathematics of scalar fields and how they couple that we can see kinds of physical power law systems emerging 

from networks of scalar fields that are coupled with one another and how similar this appears to the mathematics of discontinous pas-coupling, 

a metaheuristic technique that involves a defined signalling term with a randomly oscillating delay term to achieve an emergent equilibrium.

In effect we see a duality between such scalar fields and self-synchronizing quantum networks.

The emergent synchronization and equilibrium of these systems is favored as being the energetic ground sate of the system, that the state evolves toward over time.

Even in systems that are not explicitly programmed to achieve this kind of emergent adaptive network, such as naturally occurring quantum systems

(for example nanoribbons, bose-einstein condensates, networks of quantum spins in magnetic materials to name a few) it can arise naturally by 

simply having the individual nodes exploit a power law for coupling with a randomly varying term.

As we discussed in a previous video, many of the fundamanetals of quantum theory behave this way, exploiting a very precise and simple series of 

momentum exchange rules with a randomly varying probabilistic term to create the kind of emergent behaviour that we see in so-called pure quantum systems.

 

If this is true, then there must be quantum mechanisms underlying classical chaos in such systems.

In our previously discussed model of the 2D Quantum Newton’s cradle, the balls are replaced by our signalling atoms or electrons, confined in rows.

Adding additional momentum, such as a photon from a laser, can kick the atoms into motion, causing them to oscillate back and forth just as in 

the classical Newton's Cradle. 

However, unlike the toy, the atoms in a quantum newtons cradle can both collide and pass through one another because of the oddities of quantum physics, such as 

quantum tunneling.

This leads to a sum over histotires of the different paths a particle can travel. 

Just as in classical mechanics, With our quantum Newton's Cradle as the strength of the interaction increased, the motion of each cradle’s atoms in the arrangement

can transitioned from periodic to chaotic.

This is equivalent of the the momentum space distribution of the atoms approaching a thermal distribution over a frequency of time

indicating the system is reaching some equilibirium. In effect we have a synchronized system of atom to responds collectively.

In certain quantum transition effects, with thermalisation of groups of atoms, we also see the effects of quantum chaos which create

complex structure from simple momentum transition rules. A small section of electrons in the thermalised system, when perturbed, can cause

interactions which have effects that iterate out into the entire system even without direct contact between individual electrons. non-local behaviour in effect.

the spectral properties of non-interacting two-dimensional electrons in a magnetic field in a lattice can also create self-similar fractals

first discovered in the 1976 Ph.D. work of Douglas Hofstadter. Hofstadter described the structure in 1976 in his modelling of the energy levels 

of Bloch electrons in magnetic fields.[1] It gives a graphical representation of the spectrum of Harper's equation at different frequencies. 

The intricate mathematical structure of this spectrum was independently discovered by Soviet physicist Mark Azbel in 1964 (the Azbel-Hofstadter model),

[4] but Azbel did not plot the structure as a geometrical object.

The fact is that the non-linear effects of imperfections and random behavior in additional to the quantum exchanges during photon-electron interactions 

and electron-electron interactions will inevitably lead to the same kind of emergent chaos that is seen in classical systems such that a small change 

in the position of an atom arrangement in a crystal lattice or the random excitation of a quasiparticles momentum state will inevitably lead to 

an emergent pattern which will be drastically different to the original pattern

by creating a grid of 2D atoms we can simulate this effect and make a relationship between the system behaviour across the map of the fractal pattern.

The grid is the energy surface the atoms are binded to. Mathematically this is a matrix, the hamiltonian.

It is a general principle of Quantum Mechanics that there is an operator for every physical observable, for energy and momentum operators 

for example, which can be measured.

In a system that is defined by a wavefunction, which is an eigenfunction, acts on an operator

then the system is said to be in an eigenstate. The values for energy or momentum operators are therefore eigenvalues.

In the 2d square grid we can represent the evolution of the eigenstate as an emergent fractal pattern with the pattern being highly ordered

and dependent on the slight tweaking of the initiral conditions of the atoms topology in the lattice.

for example lets take the idea of 2d electrons in a square lattice and compare with a hexagonal lattice

In The arrangement of atoms The momentum exchange rules are the same in each case however due to the position of the atoms creating changes in 

the small and randomised position of the particles, that emergent pattern will be completely different.

Even though our change in arrangement was simple, the emergent fractal nature of the spectrum shows completely different results.

this is not decoherence of any kind, the system is still behaving as an isolated thermal bath. However a local variation causes perturbations that 

reverberate throughout the network in a complex and adaptive system that reinforces itself. 

The emergent nature of the different energy level structure is also apparent, with the electron transition regime in the hexagonal lattice

now appearing much more relativistic as compared to the 2D lattice.

the onset of chaotic behavior in the system can be be used to describe how interacting quantum particles drive certain materials,such as graphene or 

superconducting crystals to a thermal equilibrium. 

This insight is important to note as many quantum technological devices are being considered that rely on nonequilibrium quantum effects. Of particular interest are 

devices that use the cuprate high temperature superconductors, most notably BSCCO which has a crystal structure that behaves as

a natural form of josephson junction. These are considered as one of the most promising elements in quantum sensors and as potential processors in 

the much touted field of neuromorphic quantum computing, using scalar coupling that can occur between separate josephson junctions in quantum circuits.

However many of the design considerations of both quantum-based sensors and as hardware for quantum computing seem to ignore the effects of quantum chaos, 

or else think that the description of quantum chaos is somehow not important to the development of their "machines" quote un quote.

Fundamentally it is still an unsettling definition as to why quantum mechanical systems is framed to be in the domain of what we arbitrarily call "small" 

especially when we see effects in the laboratory, such as quantum entanglement, superconductivity, superfluidity and bose-einstein condensation, 

that have nothing to do with small length or time scales.

So the foundations of quantum theory may not really be dependent on what we refer to as the size scale the physical laws operate upon but as a more underlying factor

that operates independently across different scales, the factor of an emergent quantum chaos that goes on to define a quantum systems behavior.

A quantum sameness that is conceptually just as impressive , and may even be complimentary to, the concept of fractal sameness across scales.