Chaos theory, a underlying model of every Quantum particle

Oh YoungJong 

1.      Introduction

I strongly argue that scientist must consider the chaos theory as a basis to understand the probabilistic nature of subatomic particle if they really want to unify quantum mechanics and general relativity into a theory of quantum gravity.

Why is that? The reason is that only chaos theory can provide a working example of the deterministic dynamical system having unpredictability but statistically stability. As we will see soon, the dynamical system of chaos system and subatomic particle show many common behaviors such as determinism, unpredictability and probabilistic nature. For better explanation on these topics, let me take two examples as follows.

2.     Similarities between Chaos system and Quantum particle system

First example is the equation for Lorenz attractor, which is a famous simple chaos system introduced by ([1] Lorenz Edward, 1962), shown in Eq.1. Even though the x, y, z in the original equation for Lorenz attractor is not relevant with the space coordinates x, y, z in 3 dimensional space, I believe that the time evolution of a certain subatomic particle in 4 dimensional space and time coordinate should have a set of simple Ordination Differential Equations (ODE) similar to Lorenz equation shown below and I will take it as a postulate for further discussion. For now, instead of sticking to search the exact solution for the “wave function”, I will focus to share a common understanding about the characteristics what ODEs can show.

Equation 1: Lorenz equations

Second example is the wave function Ψ(x,t) which is the solution of Schrodinger equation shown in Eq. (2). Equation (2)  with two variables x,t can be, without loss of generality, easily extended to include other 3 dimension space variable y, z and I assume that (2) is defined in the 4 dimensional state space – Cartesian coordinate and time. In Quantum Mechanics, it is well known that all properties of such a particle determined by the wave function Ψ(x,t) of the system.

Equation 2: Schrödinger equation and wave function

2.1.      Determinism

Both (1) and (2) have the form of time dependent differential equations. If the complete equation with all initial values is known completely, then the time evolution of all states of the system is unambiguously determined by calculation in principle. Under the condition that every parameters and initial values are available, prediction using the equations mentioned above will match exactly with the observation of the experiment.

2.2.      Unpredictability

They are both unpredictable. As can be seen in the double slit experiment with individual particles, we know the particles do not arrive at the screen in a predictable order even though it is known that the Schrodinger equation (2) fully determine the all-time future motion of the particle and the exact solution for the wave function available. The uncertainty principle prevents us to know all initial conditions of the particle system so that predicting the state of the particle in time is impossible. 

The chaos system, often characterized as deterministic non-periodic dynamical system, is predictable in principle but in practical point of view unpredictable because we cannot know the exact initial values. It is understandable if we think a case when we try to quantize an initial value of irrational number. Predicting means we need to complete some calculations requiring quantized numbers for the initial values and quantizing any irrational number with non-zero error is impossible. So for both cases, we can say that we are allowed to get only approximated initial values and it prevent us predicting the future states of the particle system.

2.3.      Probability - Statistical distribution

Many quantum mechanics textbooks explain that the wave function does not have any physical interpretation but it gives a way to calculate the probability density which is the probability to find a particle at certain time instance t within an unit volume area centered at position (x,y,z). This idea has been tested and current understanding is there is no exception violating the probability calculation. This weird feature, deterministic and probabilistic at the same time, can be found in Chaos system too.

For the chaos system, in a video clip (refer to [2] (Jos LeysÉtienne)), a computer simulation result using Lorenz equations shows the probability to find an event in a certain area in this state space is getting apparent as time flows. In Figure 1 captured from the video clip, the butterfly shaped trajectory of Lorenz equation is displayed and all point along the trajectory is considered as the set of all possible observable events. It is guaranteed that at any instance time t, the solution point (x,y,z) will be a point along the trajectory. 

To find the probability to find a observable event within a specific area, in Figure 1, they selected three different regions which are ball shaped areas colored yellow, green, pink and measured the frequency of event to be included in those area with three different initial points. As time flows in horizontal direction and samples are accumulated, three graphs for frequency measure are getting stabilized to 5.103, 14.033 and 7.54 per a unit time. The computer simulation result shows clearly that there exists a time independent probability distribution in the long run. I argue that comparing this simulation results with the time invariant probability to find an electron within an electron orbit cloud supports my view that the subatomic particle system should be the system of a sort of strange attractor which has deterministic and probabilistic features together.

 Figure 1: Statistical distribution of states in Lorenz attractor

2.4.      Electron-positron pair creation by photon

Another interesting similarity is the matter-antimatter pair production. The discovery of the positron based on Dirac’s equation shed some light on the existence of anti-matter. In particle physics, antimatter is almost same to ordinary matter. Only one difference between matter and anti-matter is that they have opposite charge. It is known that the electron-positron pair creation can be observed to occur in nature when a photon, of greater than 1.02 MeV crosses near the electric field of a heavy atom as can be seen in Figure 2.

 (From [3] Electron-positron Pair Creation)

Figure 2: Electron-positron pair production.

From the pair production, my understanding is that a photon, a quanta of light, having quantum energy  higher than the rest energy of an electron plus a positron 1.02 MeV=2*0.51MeV consists of electron and positron pair. The inner structure of a photon formed by an electron-positron pair gave me a hint for the analogy between particle system in quantum mechanics and Chaos system. From Figure 3, the time evolution of Lorenz equations are displayed. The interesting part is that there are three critical points. One is located at the origin and other two critical points are located at the opposite side from the origin. 

This observation gives me an idea that the trajectory is the trail of a virtual particle and two non-zero critical points act like matter-antimatter pair. As we will see later, two non-zero critical points shares many characteristics such as distance from origin and frequency related stuffs, which will be discussed in other material soon. Only difference between them is the opposite sign which is similar to the matter-antimatter pair. This similar pattern found in both fields also supports my current view.

Figure 3: Particle movement governed by Lorenz equation with two critical points

3.      Problem of Time can be solved with the help of Chaos theory

In previous section, I listed some similarities between two dynamical systems and argued that unifying quantum mechanics and general relativity will be impossible unless physicists consider the strange attractor style dynamics as a fundamental working framework of any basic building block. 

In addition, I argue that this mechanical framework of strange attractor style may be the key to solve the problem of time which has been the most fundamental unsolved mystery in nature. To understand time nature such as what time is and why time flows is always one directional, I think the idea that chaos theory is the underlying model of every quantum particle or basic building black is required to understand it. I will discuss this topic in another writing soon.

To be continued ...