Monday, July 28, 2014

Adiabatic Constraint of the Scalar Field

In this post I give an detailed description of the process of the Revolution of Matter.

The Hartle-Hawking no-boundary proposal holds that the identity of time and space become indistinguishable as the universal wave function approaches $\mathbb{U}_i$, this requires transforming from $\mathbb{R}^{1,3}$ to $\mathbb{R}^{4}$ as the Klein-Gordon equation reduces in the ground state of the potential well to the non-relativistic Schrödinger's equation, and I have shown this leads to a geometric phase shift,

$$\mid \psi(T) \rangle_\mathbb{U_i}  = e^{i[\gamma(t) - \theta(t)]} \quad \mid  \psi(T(0)) \rangle $$

where the $\gamma$(t) depends on how the particle evolves through space,

$$\gamma_n(t) =  i \int \langle \psi_n \mid \nabla_R \mid  \psi_n  \rangle \cdot dR$$

The vacuum energy does not change from undergoing a geometric phase shift, as from the vacuum expectation value,

$$\langle O \rangle = \langle \psi \mid e^{-i \gamma_n (t)} e^{ \theta(t)}  e^{i \gamma_n (t)}  e^{ -\theta(t)} \mid \psi \rangle = \langle \psi^* \psi \rangle$$

In the case of the geometric phase shift the antiparticle $\psi_1$ upon reflection into $\psi_2$ does not return to its original point of origin $\mathcal{I}$ for the space has expanded, so $\psi_2$ now is attempting to move into the state of another particle  $\psi_3$ , because the state of the particle is defined by not only its mass, charge, spin, energy but also its place in space and time.

This is an example of the dynamical Casimir Effect within the context of a potential of time varying width.

This is clearly a nonholonomic process, driven by two different phases,

1: The dynamic phase $e^{-\theta(t)}$ of the particles evolving cyclically along $X_i$

2: The geometric phase  $e^{\gamma(t)}$  of the potential evolving adiabatically along  $X_i$

The two phases are equivalent to two different clocks, the internal clock $T_i$ or the frequency of the virtual particle as it evolves through  $\mathbb{R}^{1,3}$, and the external clock  $T_e$ for the potential well derived from the adiabatic cycle caused by the intrinsic uncertainty of width R, and allows an exact solution of the universal wave function that includes admixtures of other states,

$$\Psi(x,t) = \psi_n(x,t) e^{i (\gamma (t)-\theta (t))} + \epsilon \sum_{m\neq n} c_m \psi_n(x,t)$$

$$\epsilon =\frac{T_{\text{internal}}}{T_{\text{external}}}$$

"Where $\epsilon$ characterizes the departure from adiabaticity (it goes to zero in the adiabatic limit)" - Griffiths Introduction to Quantum Mechanics

In contradistinction to symmetric and antisymmetric states these extra terms -which I label as asymmetric wave functions- are intrinsically Off Mass-Shell. This is fairly easy to show by taking standing waves in a potential well and adjusting the boundaries non-adiabatically, L $\to$ L + $\delta$L, the original standing waves are now the asymmetric non-solutions in the adjusted well, these asymmetric terms violate conservation of energy and are therefore non-physical. This is a important realization, for it divides the universal wave function into two parts, the first part constitutes the real On Mass-Shell universe and the second the virtual Off Mass-Shell universe of the quantum vacuum. It can been seen real particles lie on the axis of the wave function matrix and virtual particles off the axis, hence I will also refer to this division as on-axis and off-axis particles.

To show the exact formula of the universal wave equation includes the off axis or virtual terms for the quantum vacuum,

 $$\Psi(x,t) = \psi_n(x,t) e^{(i \gamma (t)-\theta (t))} + \epsilon \sum_{m\neq n} c_m \psi_n(x,t)$$

start from a static universal wave function,

 $$\Psi(x,t) =  \Psi_0 e^{ i ( \gamma (t)-\theta (t))} =   \Psi_0 e^{i 0} \to \Psi =\Psi_0 : \forall (x,t)=0$$

$$\Psi = \sum_{m\neq n}  c_{m n} \psi_{m n}=0$$

give this the adiabatic parameter $\epsilon$,

$$\Psi =  \epsilon  \sum_{m\neq n}  c_{m n} \psi_{m n}=0$$

This infinite sea of oscillators is identical to a quantum field, and matches the idea the universe is like a bubble of Hamiltonian action that fluctuates back and forth in time and space, where $\epsilon$ is the variable that shapes and changes the global structure. I can show this by assuming the internal energy of the universe equals the external energy then internal time and external time are in the same proportion as internal action $S_i$ and external action $S_e$,

$$\epsilon =\frac{T_i}{T_e}=\frac{T_i E_i }{T_e E_e }=\frac{S_i}{S_e}$$

Since the internal action is comprised of a set of oscillators $\psi$, I can take n as the number of  $\psi$  and write $\sigma$ for the action of an internal particle,

$$\epsilon =\frac{n \sigma }{S_e}$$

drop n and consider the action for an arbitrary volume with arbitrary external action S,

$$\epsilon \geq \frac{\sigma }{S}$$

introduce the uncertainties,

$$\epsilon \geq \frac{\sigma }{\text{$\delta $E} \text{$\delta $t}}\geq \frac{\sigma }{E t}\geq \frac{\sigma }{S}$$

$\epsilon$ is a pure number so absorb $\epsilon$  into $\sigma$ and write it as $\hbar$, the 2$\pi$ comes from the cyclic evolution of the volume,

$$1\geq \frac{\hbar }{\text{$\delta $E} \text{$\delta $t}}\geq \frac{\hbar }{E t}\geq \frac{\hbar }{S}$$

since the volume is arbitrary I can contract this to the size of an elementary particle and derive the energy-time Heisenberg Uncertainty principle,

$$\text{$\delta $t}\geq \frac{\hbar }{\text{$\delta $E}}$$

similarly the exact same technique leads to the momentum-space Heisenberg Uncertainty principle,

$$\text{$\delta $x}\geq \frac{\hbar }{\text{$\delta $p}}$$

So in this model the Heisenberg Uncertainty principle and the adiabatic parameter are directly related, and the evolution of internal particles is directly related to the evolution of the universe as a whole, which on the surface appears similar to Dirac's Large Number Hypothesis (LNH), however, the LNH requires a varying gravitational constant which violates the Perfect Cosmological principle, so for the moment I'm ignoring this possibility. The important part is that the individual dynamics of the internal particles which is governed by the dynamic and geometric phases are derived from the global adiabatic parameter.

Returning to the universal wave equation under the adiabatic parameter and expanding $\Psi$,

$$\Psi =\epsilon \begin{pmatrix}
 c_{11} \psi _{11} & c_{12} \psi _{12} & \cdot & c_{1 m} \psi _{1 m}\\
  c_{21} \psi _{21} & c_{22} \psi _{22} & \cdot & c_{2 m} \psi _{2 m}\\
  \cdot & \cdot & \cdot & \cdot \\
  c_{n1} \psi _{n1} & c_{n2} \psi _{n2} & \cdot & c_{n m} \psi _{n m}\\ \end{pmatrix}$$

Separate the wave functions of the matrix into on-axis (real matter) and off-axis (virtual particles),

$$\Psi =\epsilon \begin{pmatrix}
 c_{11} \psi _{11} & 0 & \cdot & 0\\
  0 & c_{22} \psi _{22} & \cdot & 0\\
  \cdot & \cdot & \cdot & \cdot \\
  0 & 0 & \cdot & c_{n m} \psi _{n m}\\ \end{pmatrix} + 
\epsilon \begin{pmatrix}
 0 & c_{12} \psi _{12} & \cdot & c_{1 m} \psi _{1 m}\\
  c_{21} \psi _{21} & 0 & \cdot & c_{2 m} \psi _{2 m}\\
   \cdot & \cdot & 0 & \cdot \\
  c_{n1} \psi _{n1} & c_{n2} \psi _{n2} & \cdot  & 0\\ \end{pmatrix}

or more conveniently,

 $$\Psi =  \epsilon  \sum_{m = n}  c_n \psi_n= \epsilon  \sum_{m\neq n}  c_m \psi_m $$

Make the substitution $\epsilon = e^{i(\gamma-\theta)}$ where the phases arise from fluctuations in the universal potential as equivalent to fluctuations in $\epsilon$, and I can justify this from by the derivation of the Heisenberg Uncertainty principle from the adiabatic parameter, I will also demonstrate exactly how this is possible in another post.

Group the real wave-functions together and sum over the on-axis terms,

  $$\Psi =  e^{i(\gamma-\theta)}   \psi_n +  \epsilon  \sum_{m\neq n}  c_m \psi_m $$

Calculate the expectation for $\Psi$,

$$\langle \Psi \mid \Psi \rangle =  (  e^{i(\gamma-\theta)}   \psi_n +  \epsilon  \sum_{m\neq n}  c_m \psi_m )^* (  e^{i(\gamma-\theta)}   \psi_n +  \epsilon  \sum_{m\neq n}  c_m \psi_m )$$

$$ =  (e^{-i(\gamma-\theta)}   \psi_n +  \epsilon  \sum_{m\neq n}  c_m \psi^*_ {m})(e^{i(\gamma-\theta)}   \psi_n +  \epsilon  \sum_{m\neq n}  c_m \psi_m)$$


By definition the m terms are orthogonal to the n terms, so the inner product of $\psi_n$ and $\psi_m$ is zero, reducing $\langle \Psi \mid \Psi \rangle$,

 $$ \langle \Psi \mid \Psi \rangle =  ( \psi_n e^{-i(\gamma-\theta)} \psi_n  e^{i(\gamma-\theta)}     )  + (    \epsilon  \sum_{m\neq n}  c_m \psi^*_ {m}   \epsilon  \sum_{m\neq n}  c_m \psi_m)$$

  $$ \langle \Psi \mid \Psi \rangle  = \mid \psi_n \mid ^2  +  \quad \epsilon^2 \mid  \sum_{m\neq n}  c_m \psi_{m}  \mid ^2$$

In the adiabatic limit the second term vanishes leaving only On-Mass and On-Axis matter, it is not that the quantum vacuum does not exist but rather its asymmetric expectation is trivial, the on-axis terms are by definition on mass-shell. Remembering "Where $\epsilon$ characterizes the departure from adiabaticity (it goes to zero in the adiabatic limit)" - Griffiths Introduction to Quantum Mechanics, and this is the critical idea, the adiabatic parameter regularizes  the scalar field and places special constraints on how the Scalar Field can behave, and this now leads to the mechanism by which virtual particles are transformed into matter - which I will address in another post.