Friday, July 4, 2014

A Derivation of Berry's Geometric Phase from the Geometric Potential

Excerpts from The Revolution of Matter

Following on from the previous derivation of the Higgs Mechanism from the Geometric Potential I'm going to show Berry's Geometric Phase is also derivable from the Geometric Potential.

Assume a global phase $\gamma$ for the Geometric Potential

$$\phi \to e^{i \gamma} \phi$$

Take the new Lagrangian for Euclidean space,

$$\mathcal{L}_O = \mathbb{T} (\phi) +  \mathbb{U} (\phi) $$

this results in a new action $S_O$ for $\mathbb{R}^4$,

$$S_O = \int  \mathcal{L}_O dt = \int   \mathbb{T} (\phi) +  \mathbb{U} (\phi) dt$$

for the ground state of $\mathbb{U} (\phi)$ the kinetic term $ \mathbb{T} (\phi) $ tends to zero, allowing the geometric phase to be determined in terms of  $\mathbb{U} (\phi)$,

$$e^{i \gamma (t)} = e^{i \int \mathbb{U} (\phi)dt } $$

Taking time t as an independent variable, this phase successively becomes,

$$ \int  \mathbb{U} dt = - \int -   \mathbb{U} dt = - \int (- \frac{\partial \mathbb{U}}{\partial x}) dt dx$$

By Ehrenfest's Theorem and taking $\mathbb{P}$ as the geometric equivalent of momentum in the same way as $\mathbb{T}$ is the geometric equivalent of kinetic energy, (and this is the clever bit),

$$\frac{d \langle \mathbb{P}\rangle }{{dt}} =  \langle - \frac{\partial \mathbb{U}}{\partial x} \rangle $$

$$ - \int (- \frac{\partial \mathbb{U}}{\partial x}) dt dx = - \int \frac{d \langle \mathbb{P}\rangle }{{dt}} = - \int \langle \mathbb{P}\rangle \cdot dx$$

Substitute the momentum operator for the n'th level of the infinite square potential in the vicinity of $\mathbb{U}_i$, even though $\hbar$ is dimensionless in $\mathbb{R}^4$ it is included for completeness,

$$ \int  \mathbb{U} dt =   - \int \langle \mathbb{P}\rangle \cdot dx = - \frac{\hbar }{i} \int \int \psi_n^* \frac{\partial }{\partial x} \psi_n dx \cdot dx$$

Simplify and use the Dirac notation,

 $$ \int  \mathbb{U} dt = - i \hbar  \int  \langle  \psi_n  \mid \nabla_x \mid \psi_n   \rangle \cdot dx$$

to determine the phase $\gamma$ of the integral $\int \mathbb{U}$ dt divide by $\hbar$ and the  $\hbar$ drops out, then integrate over all space,

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


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

so the phase is real,

$$i \gamma \in \mathcal{R}$$

giving the wavefunction in terms of a geometric phase,

$$\Psi = \int dx \Psi_0 e^{i \gamma}$$

the extra term applies globally to the potential $\mathbb{U}$ as $\Psi$ evolves, this is equivalent to a global geometric phase change $\gamma$,

$$\phi \to e^{i \gamma} \phi$$

Since this is a global phase change I expect it to apply in both $\mathbb{R}^4$ and $\mathbb{R}^{ (1,3) }$, returning to the new Lagrangian $\mathcal{L}_M$ to include the dynamic phase $e^{\int\mathcal{L}dt}$,

$$e^{\int\mathcal{L_M}dt} =  e^{\int\mathbb{U} (\phi)dt} \quad  e^{\int\mathcal{L}dt} $$

and finally the universal wave function can be written,

$$\Psi = \int  \Psi_0 e^{[\gamma (t) - \theta (t)]}$$

It can be seen that integrating the new potential over time is identical to Berry's Geometric Phase factor from his work on the Adiabatic Theorem, where he showed from the geometrical properties of the parameter space the Hamiltonian of a cyclic quantal adiabatic process will acquire an additional phase $\gamma (C)$. This can be generalized by writing for a Hamiltonian $\hat{\mathcal{H}}$ (X(T)) on a parameter space R = (X,Y,Z...), where C is the circuit over R(T) = R(0), and quantal adiabatic limit T $\to$ $\infty$. Since the natural basis of discrete eigenstates under the Schrödinger equation with energies $E_n$(X) is,

$$\hat{\mathcal{H}} (R(t)) \mid n(R) \rangle =  E_n(X) \mid n(R) \rangle $$

with dynamic phase,

$$\theta (T) = - \frac{i}{\hbar } \int _0^T dt  E_n(R(t)) $$

and geometric phase over a closed cycle C,

 $$ \gamma_n(C) = i \oint   \langle  \psi_n  \mid \nabla_R \mid \psi_n   \rangle \cdot dR$$

where Minkowski Spacetime is assumed to be a continuously transformable from Euclidean space and noting the geometric phase is a pure number, it is now possible without loss of generality to use the geometric phase as an additional factor of the wave function in $\mathbb{R}^{1,3}$ as it affects all points in $\mathbb{R}^{1,3}$ equally, allowing,

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

The idea that the dynamic phase disappears in the Euclidean domain is consistent with the idea the physical universe having a beginning in Time, where transforming from Euclidean space to Minkowski Spacetime under the Wick rotation at $\mathbb{U}_i$ is equivalent to the switching from a geometric system to a dynamic system, which is essentially the idea behind the Hartle-Hawking no boundary proposal, so remarkably the ideas of Hartle-Hawking and Berry can be combined into a single model.

Importantly this transformation is only possible for a cyclic space in its lowest energy level, and this will be of crucial importance in the construction of a Big Bang model to be addressed later in this paper, - very importantly this additional phase factor in $\mathbb{R}^{1,3}$ is homogeneous and isotropic and affects all particles equally and this crucial idea will be returned to in the section on Newton's First Law.

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