Thursday, July 24, 2014

Monopole Suppression

It is possible to show Monopole Suppression takes place at the Big Bang as follows, in discussing the Aharonov-Bohm effect Berry showed for $\mathbb{R}^3$ that $\mathbb{B}$ can be written as the magnetic flux,

$$\Phi _B = \oint_\Sigma  \mathbb{B}\cdot d\Sigma =   \oint_\Sigma (\nabla \times \mathbb{A})   \cdot d\Sigma$$

where $\mathbb{A}$ is written in terms of the Berry connection,

$$\mathbb{A} = i \langle \psi_n \mid \nabla _R \mid  \psi_n \rangle$$

and noting the relation to Berry's geometric phase,

$$\gamma (t) = i \int  \langle \psi_n \mid \nabla_R \mid  \psi_n \rangle  \cdot  d\mathbb{R}$$

Given $ \mathbb{B} =  \nabla \times  \mathbb{A}$ 

$$\mathbb{B} =  i  \nabla_R \times   \langle \psi_n \mid \nabla_R \mid  \psi_n \rangle$$

and Gauss's Theorem,

 $$\Phi _B = \oint_\Sigma  \mathbb{B}\cdot d\Sigma =  \int \int \int_V  (\nabla  \cdot  \mathbb{B} ) dV$$

By the gHH the volume in $\mathbb{R}^3$  is arbitrarily small, so the divergence theorem applies, and passing the imaginary i over the Del,

$$\nabla_R \cdot  \mathbb{B} = i \nabla_R  \cdot \nabla_R \times   \langle \psi_n \mid \nabla_R \mid  \psi_n \rangle$$

since the vector identity of the triple vector product of $\mathbb{A}$,

$$\nabla \cdot \nabla \times  \mathbb{A} = 0$$

reduces the divergence of $\mathbb{B}$ to,

$$\nabla_R \cdot \mathbb{B} = i \nabla_R \cdot  \nabla_R \quad \times   \langle \psi_n \mid \nabla_R \mid  \psi_n \rangle = 0$$

so the magnetic flux,

This proves that no isolated magnetic flux could exist in the context of the inflationary period of the Big Bang and since an isolated magnetic flux is the definition of a magnetic monopole:

Therefore the principle of Revolution of Matter predicts that no magnetic monopoles could evolve during the Big Bang.

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