is there a way to generalize the electromagnetic field strength tensor to general, specifically non-integer dimensions? As context: I am currently working on a calculation in the high energy QFT framework, using dimensional regularization. However, I have classical electromagnetic field strength tensors, $F_{\mu\nu}$, and I have absolutely no idea how to generalize them to $4-2\epsilon$ dimensions in a unique, mathematically consistent way. Is it even possible?
My assumption is, like the situation with $\gamma^5$ matrices, one has to abandon certain assumptions (in the $\gamma^5$ case it would be either abandoning the cyclicity of Dirac traces, or the anti-commutativity of $\gamma^5$). However, I am not sure about the assumptions about $F_{\mu\nu}$, since defining equations like the Maxwell equations are not that clear in general dimensions either.
I have read paper working in similar situations, yet the integrals are performed using rather in hand-wavy ways.
Edit: To provide a little more context, I have a constant external magnetic field along an a chosen axis, and $F_{\mu\nu}$ manifests itself in form of a Schwinger's phase, i.e. $$\Phi_\Sigma(x,y,z)=-\frac{e_f|B|}{2}\left(x_\mu \hat{F}_c^{\mu\nu}y_\nu+y_\mu \hat{F}_c^{\mu\nu}z_\nu+z_\mu \hat{F}_c^{\mu\nu}x_v\right)$$ where $\hat{F}_c^{\mu\nu}=\frac{F_c^{\mu\nu}}{|B|}$ is the unitless and normalized classical field strength tensor. Since $B$ is along an axis and constant, contraction with a vector produces another vector that is perpendicular to the magnetic field.
