Planck length in classical and quantum Hamiltonian formulations of general relativity

The physical meaning of the Planck length (\(\ell _{P}\)) is investigated in the framework of the unconstrained synchronous variational formulation of classical general relativity (GR). This theoretical setting permits the establishment of manifestly-covariant Lagrangian and Hamiltonian theories for the Einstein field equations of the continuum gravitational field. It is shown that such a formulation is distinguished by the existence of a novel variational contribution expressed by an infinite series summation of suitable 4-scalar terms in which the coupling coefficients are even powers of the Planck length. However, the requirement of realization of a classical GR Hamiltonian theory places stringent constraints on the admissible Planck-length power terms to be retained. In fact, excluding the trivial gauge constant, it is proved that only the \(O\left( \ell _{P}^{0}\right) \) contribution of the series is ultimately permitted, namely the unique one which is independent of \(\ell _{P}\). Therefore, the Planck length is effectively not allowed to appear at the classical level for consistency with the Hamiltonian principle. This places important consequences on the mathematical establishment of the corresponding canonical quantum gravity theory, which is then found to be correct through \(O\left( \ell _{P}^{2}\right) \). Additional implications concern the physical significance of related quantum momenta and their meaning in the semi-classical limit, as well as the role of the Planck length in the same quantum-gravity realm.