Path Integral of Bianchi I models in Loop Quantum Cosmology

A path integral formulation of the Bianchi I models containing a massless scalar field in loop quantum cosmology is constructed. Following the strategy used in the homogenous and isotropic case, the calculation is extended to the simplest non-isotropic models according to the $\bar{\mu }$ and ${\bar{\mu }}^{\mathrm{\prime }}$ scheme. It is proved from the path integral angle that the quantum dynamic lacks the full invariance with respect to fiducial cell scaling in the $\bar{\mu }$ scheme, but it does not in the ${\bar{\mu }}^{\mathrm{\prime }}$ scheme. The investigation affirms the equivalence of the canonical approach and the path integral approach in loop quantum cosmology.