The equivalence postulate approach to quantum mechanics aims to formulate quantum mechanics from a fundamental geometrical principle. Underlying the formulation there exists a basic cocycle condition which is invariant under D -dimensional Mobius transformations with respect to the Euclidean or Minkowski metrics. The invariance under global Mobius transformations implies that spatial space is compact. Furthermore, it implies energy quantisation and undefinability of quantum trajectories without assuming any prior interpretation of the wave function. The approach may be viewed as conventional quantum mechanics with the caveat that spatial space is compact, as dictated by the Möbius symmetry, with the classical limit corresponding to the decompactification limit. Correspondingly, there exists a finite length scale in the formalism and consequently an intrinsic regularisation scheme. Evidence for the compactness of space may exist in the cosmic microwave background radiation.
The equivalence postulate approach to quantum mechanics aims to formulate quantum mechanics from a fundamental geometrical principle. Underlying the formulation there exists a basic cocycle condition which is invariant under D -dimensional Mobius transformations with respect to the Euclidean or Minkowski metrics. The invariance under global Mobius transformations implies that spatial space is compact. Furthermore, it implies energy quantisation and undefinability of quantum trajectories without assuming any prior interpretation of the wave function. The approach may be viewed as conventional quantum mechanics with the caveat that spatial space is compact, as dictated by the Möbius symmetry, with the classical limit corresponding to the decompactification limit. Correspondingly, there exists a finite length scale in the formalism and consequently an intrinsic regularisation scheme. Evidence for the compactness of space may exist in the cosmic microwave background radiation.
The equivalence postulate of quantum mechanics offers an axiomatic approach to quantum field theories and quantum gravity. The equivalence hypothesis can be viewed as adaptation of the classical Hamilton-Jacobi formalism to quantum mechanics. The construction reveals two key identities that underlie the formalism in Euclidean or Minkowski spaces. The first is a cocycle condition, which is invariant under D -dimensional Möbius transformations with Euclidean or Minkowski metrics. The second is a quadratic identity which is a representation of the D -dimensional quantum Hamilton-Jacobi equation. In this approach, the solutions of the associated Schrödinger equation are used to solve the nonlinear quantum Hamilton-Jacobi equation. A basic property of the construction is that the two solutions of the corresponding Schrödinger equation must be retained. The quantum potential, which arises in the formalism, can be interpreted as a curvature term. The author proposes that the quantum potential, which is always nontrivial and is an intrinsic energy term characterising a particle, can be interpreted as dark energy. Numerical estimates of its magnitude show that it is extremely suppressed. In the multiparticle case the quantum potential, as well as the mass, is cumulative.
We consider the two main theorems in the derivation of the Quantum Hamilton--Jacobi Equation from the Equivalence Postulate (EP) of quantum mechanics. The first one concerns a basic cocycle condition, which holds in any dimension with Euclidean or Minkowski metrics and implies a global conformal symmetry underlying the Quantum Hamilton--Jacobi Equation. In one dimension such a condition fixes the Schwarzian equation. The second theorem concerns energy quantization which follows rigorously from consistency of the equivalence postulate.
The removal of the peculiar degeneration arising in the classical concepts of rest frame and time parametrization is at the heart of the recently formulated equivalence principle (EP). The latter, stating that all physical systems can be connected by a coordinate transformation to the free one with vanishing energy, univocally leads to the quantum stationary HJ equation (QSHJE). This is a third order nonlinear differential equation which provides a trajectory representation of quantum mechanics (QM). The trajectories depend on the Planck length through hidden variables which arise as initial conditions. The formulation has manifest p-q duality, a consequence of the involutive nature of the Legendre transformation and of its recently observed relation with second order linear differential equations. This reflects in an intrinsic ψ D -ψ duality between linearly independent solutions of the Schrödinger equation. Unlike Bohm's theory, there is a nontrivial action even for bound states and no pilot waveguide is present. A basic property of the formulation is that no use of any axiomatic interpretation of the wave function is made. For example, tunneling is a direct consequence of the quantum potential which differs from the Bohmian one and plays the role of particle's self-energy. Furthermore, the QSHJE is defined only if the ratio ψ D /ψ is a local homeomorphism of the extended real line into itself. This is an important feature as the L 2 (ℝ) condition, which in the Copenhagen formulation is a consequence of the axiomatic interpretation of the wave function, directly follows as a basic theorem which only uses the geometrical gluing conditions of ψ D /ψ at q=±∞ as implied by the EP. As a result, the EP itself implies a dynamical equation that does not require any further assumption and reproduces both tunneling and energy quantization. Several features of the formulation show how the Copenhagen interpretation hides the underlying nature of QM. Finally, the nonstationary higher dimensional quantum HJ equation and the relativistic extension are derived.
When compared to quantum mechanics, classical mechanics is often depicted in a specific metaphysical flavour: spatio-temporal realism or a Newtonian "background" is presented as an intrinsic fundamental classical presumption. However, the Hamiltonian formulation of classical analytical mechanics is based on abstract generalized coordinates and momenta: It is a mathematical rather than a philosophical framework. If the metaphysical assumptions ascribed to classical mechanics are dropped, then there exists a presentation in which little of the purported difference between quantum and classical mechanics remains. This presentation allows to derive the mathematics of relativistic quantum mechanics on the basis of a purely classical Hamiltonian phase space picture. It is shown that a spatio-temporal description is not a condition for but a consequence of objectivity. It requires no postulates. This is achieved by evading spatial notions and assuming nothing but time translation invariance.
Implementation of the equivalence principle, stating that all physical systems can be connected by a coordinate transformation (trivializing map) to the free system with vanishing energy, uniquely leads to the quantum analogue of the Hamilton–Jacobi (HJ) equation. A property of the formalism is manifest p–q duality, a consequence of the involutive nature of the Legendre transformation and of the recently observed relation with second–order differential equations. This makes clear the intrinsic ψD –ψ duality between linearly independent solutions of the Schrödinger equation. Being a third–order non–linear differential equation, the Quantum Stationary HJ Equation (QSHJE) gives rise to initial conditions which cannot be seen in the Schrödinger equation. Solutions of the QSHJE provide a trajectory representation of quantum mechanics, depending on the Planck length. Unlike Bohm’s theory, there is a non– trivial action even for bound states and no wave guide is present. A fundamental aspect of the formulation is that no use of any axiomatic interpretation of the wave function is made. The QSHJE is defined only if the ratio ψD /ψ is a local homeomorphism of the extended real line into itself. In this respect the quantization of the energy is strictly related to the S duality of the Schwarzian derivative. As a result, the equivalence principle, according to which all the potentials are equivalent under coordinate transformations, by itself implies a dynamical equation that does not need any assumption and reproduces both the tunnel effect and energy quantization. Finally, we derive the quantum HJ equation also in the higher dimensional case with time dependent potentials and in the relativistic case. Contents
Abstract Penrose’s Spin Geometry Theorem is extended further, from SU(2) and E(3) (Euclidean) to E(1, 3) (Poincaré) invariant elementary quantum mechanical systems. The Lorentzian spatial distance between any two non-parallel timelike straight lines of Minkowski space, considered to be the centre-of-mass world lines of E(1, 3)-invariant elementary classical mechanical systems with positive rest mass, is expressed in terms of E(1, 3)-invariant basic observables, viz. the 4-momentum and the angular momentum of the systems. An analogous expression for E(1, 3)-invariant elementary quantum mechanical systems in terms of the basic quantum observables in an abstract, algebraic formulation of quantum mechanics is given, and it is shown that, in the classical limit, it reproduces the Lorentzian spatial distance between the timelike straight lines of Minkowski space with asymptotically vanishing uncertainty. Thus, the metric structure of Minkowski space can be recovered from quantum mechanics in the classical limit using only the observables of abstract quantum mechanical systems.