Derivation of an expression for a Hamiltonian functional integral in theories with first and second class constraints

A Hamiltonian functional integral in theories with first and second class constraints (both stationary and nonstationary) is derived in a simple and consistent manner. The gauge conditions need not be in involution and may contain the time explicitly. In contrast to other studies, much attention is devoted to the proof of the Hamiltonicity of the theory on the physical submanifold $\Gamma^*$ of the phase space $\Gamma$. It is shown that $\Delta^{-1}$, where $\Delta$ is the Faddeev–Popov determinant, is simply the volume element of the subspace $\Bar\Gamma=\Gamma\backslash\Gamma^*$ in noncanonical coordinates determined by the constraints and gauge conditions. It is proved that the expression for the functional integral is invariant under finite transformations of the gauge conditions.