Monday, June 30, 2014

An Introduction to the Dynamical Casimir Effect and the Big Bang

I recently published on-line a physics paper I've worked on for the last three years.

The central idea is that the Scalar Field used for the Higgs field, the Hartle-Hawking State and Berry's Geometric Phase are all aspects of the same underlying field. 

In this paper a mechanism is proposed which has its foundation in the Wick Rotation, the Dynamical Casimir effect and the Geometrical Phase, it does this by first considering the relation of the Hartle-Hawking no boundary proposal to Berry's Geometric Phase in the context of the scalar field - which Vilenkin and others have labelled the Inflaton. To do this requires pulling apart the Inflaton as a dynamic infinite potential well and applying Heisenberg Uncertainty relations to the zero state of the quantum vacuum (Intrinsic Quantum Uncertainty) or the minisuperspace of the Hartle-Hawking model. This immediately leads to a model for the Big Bang and a mechanism for the production of universally identical fermions. 

 By modeling the dynamic infinite potential well as a Dynamical Casimir Effect where the walls of the potential are varying adiabatically a mechanism is give showing how virtual matter-antimatter pairs are revolved into real on mass-shell matter, in so doing giving a possible solution for The matter-antimatter asymmetry problem

I've found in certain circumstances, e.g., during the Hartle-Hawking state at the Big Bang and in deep space far from energetic matter, the off-axis matrix of the Geometric phase becomes dominant in quantum mechanical systems, and in contrast near the energetic matter the virtual particle component which is contained within the off-axis matrix falls to zero leaving only on-mass real matter to dominant matter interactions.  

Normally in quantum mechanics a particle is described by a wave function equation involving the amplitude and phase, indeed Feynman once said its all in the phase, with the following equation

$$ \Psi _0  e^{-i \theta } $$

Where the  $\Psi _0$ is the original amplitude of the wavefunction, and the $e^{i \theta }$ determines the phase.

note: if your browser is having trouble rendering the Latex try viewing this blog in Chrome.

 Fock and Born's examination of the Adiabatic theorem allowed them to write the wave equation in full by including the off axis terms to produce

  $$\Psi =\sum _{m=n} e^{ - i \theta } c_n \psi _n +  \epsilon \sum _{m\neq n} c_m \psi _m $$  

Where $\epsilon$ is the Fock and Born's Adiabatic parameter which is only significant in low energy or infinitesimally slow evolutions of the system.

Sir Michael Berry extended this by adding what he termed the Geometric Phase

$$ \Psi_0 e^{i (\gamma -\theta )} $$ 

where the Geometric Phase $\gamma$ is given by

$$ \gamma(C) =  i\oint  \langle   \psi _n   |   \nabla   _R  | \psi _n  \rangle   \cdot dR $$   

where the C represent a geometric circuit over the integral.

This allows the Adiabatic wavefunction to be written as

  $$\Psi =\sum _{m=n} e^{i(  \gamma  -\theta )} c_n \psi _n +  \epsilon \sum _{m\neq n} c_m \psi _m $$

Now the key idea I've come up with is that using this more general form of the wave function by including the adiabatic parameter and the geometric phase and applying that to the principle of the Big Bang and low energy gravity fields - allows me to construct the set of physical models as I listed above, from Wheeler's Single Electron Model to the a solution to the Vacuum Catastrophe.

In succeeding posts to this blog I'm intending to discuss each of these models in turn, drawing mainly on my paper Dynamical Casimir Effect and the Big Bang, and within this framework I've found it possible to construct models for

   A solution to the problem of renormalization,
   Origin of Gravitational field,
   Newton's First Law,
   A MOND for Dark Matter,

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