A lower bound is placed on the fermionic determinant of Euclidean quantum electrodynamics in three dimensions in the presence of a smooth, finite--flux, static, unidirectional magnetic field $\mathbf{ B}(\mathbf{ r})=(0,0,B(\mathbf{ r}))$, where $B(\mathbf{ r})\geq 0$ or $B(\mathbf{ r})\leq 0$ and $\mathbf r$ is a point in the $xy\mbox{ -plane}$. Bounds are also obtained for the induced spin for $2+1$ dimensional QED in the presence of $\mathbf{ B}(\mathbf{ r})$. An upper bound is placed on the fermionic determinant of Euclidean QED in four dimensions in the presence of a strong, static, directionally-varying, square-integrable magnetic field $\mathbf{ B}(\mathbf{ r})$ on $\R^3 $.
Categories: hep-th