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The Equivalence Postulate of Quantum Mechanics

Alon E. Faraggi, Marco Matone

Published Jan 1, 1998• 70 citations
OpenAlex

Abstract

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

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The Hamilton-Jacobi (HJ) equation for a one-dimensional system is obtained by considering the canonical transformation (q,p) → (Q,P ) so that the old Hamiltonian H maps to a trivialized one, that is H˜ = 0. The old and new momenta are expressed in terms of the generating function of such a transformation, the Hamilton’s principal function p = ∂Scl ∂q , P = cnst = − ∂Scl ∂Q |Q=cnst that satisfies the classical HJ equation H ( q,p = ∂S ∂q , t ) + ∂S ∂t = 0.

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Energy quantisation and time parameterisation

Alon E. Faraggi, Marco Matone

We show that if space is compact, then trajectories cannot be defined in the framework of quantum Hamilton--Jacobi equation. The starting point is the simple observation that when the energy is quantized it is not possible to make variations with respect to the energy, and the time parameterisation t-t_0=\partial_E S_0, implied by Jacobi's theorem and that leads to group velocity, is ill defined. It should be stressed that this follows directly form the quantum HJ equation without any axiomatic assumption concerning the standard formulation of quantum mechanics. This provides a stringent connection between the quantum HJ equation and the Copenhagen interpretation. Together with tunneling and the energy quantization theorem for confining potentials, formulated in the framework of quantum HJ equation, it leads to the main features of the axioms of quantum mechanics from a unique geometrical principle. Similarly to the case of the classical HJ equation, this fixes its quantum analog by requiring that there exist point transformations, rather than canonical ones, leading to the trivial hamiltonian. This is equivalent to a basic cocycle condition on the states. Such a cocycle condition can be implemented on compact spaces, so that continuous energy spectra are allowed only as a limiting case. Remarkably, a compact space would also imply that the Dirac and von Neumann formulations of quantum mechanics essentially coincide. We suggest that there is a definition of time parameterisation leading to trajectories in the context of the quantum HJ equation having the probabilistic interpretation of the Copenhagen School.

•Aug 16, 2016•7 citations
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
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.
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 show that if space is compact, then trajectories cannot be defined in the framework of the quantum Hamilton–Jacobi (HJ) equation. The starting point is the simple observation that when the energy is quantised it is not possible to make variations with respect to the energy, and the time parameterisation t−t0=∂ES0t-t_0=\partial _E{\mathcal {S}}_0t−t0​=∂E​S0​ , implied by Jacobi’s theorem, which leads to the group velocity, is ill defined. It should be stressed that this follows directly from the quantum HJ equation without any axiomatic assumption concerning the standard formulation of quantum mechanics. This provides a stringent connection between the quantum HJ equation and the Copenhagen interpretation. Together with tunnelling and the energy quantisation theorem for confining potentials, formulated in the framework of quantum HJ equation, it leads to the main features of the axioms of quantum mechanics from a unique geometrical principle. Similar to the case of the classical HJ equation, this fixes its quantum analog by requiring that there exist point transformations, rather than canonical ones, leading to the trivial hamiltonian. This is equivalent to a basic cocycle condition on the states. Such a cocycle condition can be implemented on compact spaces, so that continuous energy spectra are allowed only as a limiting case. Remarkably, a compact space would also imply that the Dirac and von Neumann formulations of quantum mechanics essentially coincide. We suggest that there is a definition of time parameterisation leading to trajectories in the context of the quantum HJ equation having the probabilistic interpretation of the Copenhagen School.
We observe that Schrodinger's equation may be written as two real coupled Hamilton-Jacobi (HJ)-like equations, each involving a quantum potential. Developing our established programme of representing the state through exact free-standing deterministic trajectory models, it is shown how quantum evolution may be treated as the autonomous propagation of two coupled congruences. The wavefunction at a point is derived from two action functions, each generated by a single trajectory. The model shows that conservation as expressed through a continuity equation is not a necessary component of a trajectory theory of state; probability is determined by the difference in the actions, and we prove that it is not generated by the densities of the two HJ congruences. The theory also illustrates how time-reversal symmetry may be implemented by transformations that disobey the conventional transformation (T) of displacement components (scalars) and velocity (reversal); rather, the time-reversal transform of one flow is the T-reversal of the other. We prove a theorem whereby an integral curve of the linear superposition of two vectors is derived algebraically from the integral curves of one of the constituent vectors labelled by integral curves associated with the other constituent. A corollary establishes relations between displacement functions in diverse trajectory models, in particular where the functions obey different symmetry transformations. This is illustrated by showing that a (T-obeying) de Broglie-Bohm trajectory is a sequence of points on the (non-T) HJ trajectories, and vice versa.
We show that if space is compact, then trajectories cannot be defined in the framework of quantum Hamilton--Jacobi equation. The starting point is the simple observation that when the energy is quantized it is not possible to make variations with respect to the energy, and the time parameterisation t-t_0=\partial_E S_0, implied by Jacobi's theorem and that leads to group velocity, is ill defined. It should be stressed that this follows directly form the quantum HJ equation without any axiomatic assumption concerning the standard formulation of quantum mechanics. This provides a stringent connection between the quantum HJ equation and the Copenhagen interpretation. Together with tunneling and the energy quantization theorem for confining potentials, formulated in the framework of quantum HJ equation, it leads to the main features of the axioms of quantum mechanics from a unique geometrical principle. Similarly to the case of the classical HJ equation, this fixes its quantum analog by requiring that there exist point transformations, rather than canonical ones, leading to the trivial hamiltonian. This is equivalent to a basic cocycle condition on the states. Such a cocycle condition can be implemented on compact spaces, so that continuous energy spectra are allowed only as a limiting case. Remarkably, a compact space would also imply that the Dirac and von Neumann formulations of quantum mechanics essentially coincide. We suggest that there is a definition of time parameterisation leading to trajectories in the context of the quantum HJ equation having the probabilistic interpretation of the Copenhagen School.