The Formation of Classical Defects After a Slow Quantum Phase Transition

Classical defects (monopoles, vortices, etc.) are a characteristic consequence of many phase transitions of quantum fields. We show a model in which the onset of classical probability distributions, for the long-wavelength modes at early times, allows the identification of line-zeroes of the field with vortex separation. We obtain a refined version of Kibble's causal results for defect separation, but from a completely different approach. It is apparent that vortices are not created from thermal fluctuations in the Ginzburg regime.Comment: 10 pages, RevTex file. No figures. To appear in Phys. Lett.

An explicit open image theorem for products of elliptic curves

Let $K$ be a number field and $E_1, \ldots, E_n$ be elliptic curves over $K$, pairwise non-isogenous over $\overline{K}$ and without complex multiplication over $\overline{K}$. We study the image of the adelic representation of the absolute Galois group of $K$ naturally attached to $E_1 \times \cdots \times E_n$. The main result is an explicit bound for the index of this image in $\left\{ (x_1,\ldots,x_n) \in \operatorname{GL}_2(\hat{\mathbb{Z}})^n \bigm\vert \det x_i = \det x_j \;\; \forall i,j \right\}$.Comment: 18 pages. v2: improved expositio