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In this paper, we consider a compact five dimensional spacetime with thestructure $\mathcal{M}^{1,3}\times S^{1}$. Generally speaking, motion on such astructure will break Lorentz invariance, allowing for causal bulk signals topropagate superluminally. Based on recent articles, we calculate the anomalousfunction of a gauge invariant but Lorentz-violating term in the $4D$ QEDeffective action by using path integral. Finally, we find that the compact bulkscalar propagator and path integral duality are consistent, this result bringsa new perspective: the behavior of breaking Lorentz invariance caused bydimensional compactness can be seen as path integral duality.

Anomalous function of a Lorentz-violating QED effective action and the relation between compact bulk scalar propagator and path integral duality