I am studying textbook by Schulman od path integration. In first chapters of this book he uses Lagrangian form of the path integral and arrives at the conclusion that when a particle is moving in external gauge field one needs to use midpoint rule for evaluation of $\vec A (\vec x)$ in path integrals. I'd like to compare that to what is done in Weinberg (quantum fields). He explains there that choice of the evaluation point is equivalent to choosing ordering of operators in quantum counterpart of classical function. I followed steps made there and found that we can construct path integrals evaluated using the endpoint rule provided that we take classical Hamiltonian as $$ H_{eff}= \frac{(\vec p - q \vec A)^2 + iq \vec \nabla A}{2m} + q V (\vec x). $$ If we perform Gaussian integration in Hamiltonian version of path integral we get Lagrangian formulation with Lagrangian $$ L_{eff} = \frac{m}{2} \dot{\vec{x}}^2 + q \vec A \cdot \dot{\vec x} - \frac{iq}{2m} \vec{\nabla} \cdot \vec A - q V (\vec x) .$$ It can be verfied straightforwardly that with this choice of Lagrangian usual path integral formula approximated with saddle point method produces correct results in the sense that the wave-function satisfies Shrodinger equation with correct Hamiltonian operator. At first it seems that everything is in order, but actually not quite! The problem is that the action obtained with these $L$ or $H$ does not transform "as it should" under gauge transformations. It seems that something important is lost here.
Question: Can one really treat explanations in Weinberg seriously? What is the exact relation between evaluation point in the path integral and ordering ambiguities in quantization? Is there a way to preserve gauge covariance obtained by the procedure presented in Weinberg (with endpoint rule)?
