Quantum Transformations

We show that the stationary quantum Hamilton–Jacobi equation of non–relativistic 1D systems, underlying Bohmian mechanics, takes the classical form with ∂q replaced by ∂q̂ where dq̂ = dq √ 1−β . The β term essentially coincides with the quantum potential that, like V − E, turns out to be proportional to a curvature arising in projective geometry. In agreement with the recently formulated equivalence principle, these “quantum transformations” indicate that the classical and quantum potentials deform space geometry. One of the main aspects of contemporary theoretical research concerns the quantization of gravity. Despite many efforts and results, such as those of superstring theory, the understanding of the problem is still incomplete. While Einstein’s general relativity, based on a simple principle, describes gravity in a purely geometrical framework, foundations of quantum mechanics rely on a set of axioms which apparently seem unrelated to any geometrical principle. It is then natural to think that the difficulties which arise in considering quantization of gravity merit a better understanding of the possible relationship between the foundations of general relativity and quantum mechanics. Recently we proposed that quantum mechanics may in fact arise from an equivalence principle [1][2]. While the original formulation considered the case of non–relativistic one–dimensional stationary systems with Hamiltonian of the form H = p/2m+ V (q), which is also the case we consider in the present Letter, it will be shown in [3][4] that the principle actually implies the higher dimensional time–dependent Schrödinger equation. In this Letter we show that the Quantum Stationary Hamilton–Jacobi Equation (QSHJE), that we derived from the equivalence principle [1][2], maps to the classical form under “quantum transformations” whose structure is strictly related to the quantum potential. This indicates that the classical and quantum potentials deform space geometry. We will also show that both the quantum potential and V −E are proportional to curvatures arising in projective geometry. These aspects, together with the investigation of p–q duality, related to the properties of the Legendre transformation, constitute the main results of the present Letter. The solution S0 of the QSHJE derived in [1][2] is the quantum version of the Hamiltonian characteristic function (also called reduced action). In this respect the theory is consistently defined in terms of trajectories [5][2][3]. Although reminiscent of Bohmian mechanics [6][7], the formulation we consider has some differences which will be further considered in [3]. In particular, as noticed also by Floyd [5], while in Bohm theory one identifies ψ = Re i h̄ S with the Schrödinger wave function, one can see that in the 1D stationary case the natural identification is ψ = R(Ae i h̄ S0 + Be i h̄ 0). While in Bohm theory the state described by a real wave function corresponds to S0 = 0, this is never the case in the approach we consider. Furthermore, we note that the Schwarzian derivative {S0, q} is not defined for S0 = cnst. As a consequence, while in Bohm theory the states described by a real wave function unavoidably have a vanishing conjugate momentum, this is never the case in the proposed formulation. While in Bohmian mechanics there is the issue of recovering the classical limit for states with real wave function, e.g. for the harmonic oscillator in which S0 = 0 [7], this limit is rather natural in the formulation