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THE EQUIVALENCE POSTULATE OF QUANTUM MECHANICS
Full text linkThe Equivalence Principle (EP), stating that all physical systems are connected by a coordinate transformation to the free one with vanishing energy, univocally leads to the Quantum Stationary HJ Equation (QSHJE). 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 transform and of its recently observed relation with second-order linear differential equations. This reflects in an intrinsic psi^D-psi duality between linearly independent solutions of the Schroedinger equation. Unlike Bohm's theory, there is a non-trivial action even for bound states. No use of any axiomatic interpretation of the wave-function is made. Tunnelling is a direct consequence of the quantum potential which differs from the usual one and plays the role of particle's self-energy. The QSHJE is defined only if the ratio psi^D/psi is a local self-homeomorphism of the extended real line. This is an important feature as the L^2 condition, which in the usual 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 psi^D/psi at q=\pm\infty 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 tunnelling and energy quantization. Several features of the formulation show how the Copenhagen interpretation hides the underlying nature of QM. Finally, the non-stationary higher dimensional quantum HJ equation and the relativistic extension are derived
EQUIVALENCE PRINCIPLE, HIGHER DIMENSIONAL MOBIUS GROUP AND THE HIDDEN ANTISYMMETRIC TENSOR OF QUANTUM MECHANICS
Full text linkWe show that the recently formulated equivalence principle (EP) implies a basic cocycle condition both in Euclidean and Minkowski spaces, which holds in any dimension. This condition, that in one dimension is sufficient to fix the Schwarzian equation, implies a fundamental higher-dimensional Möbius invariance which, in turn, unequivocally fixes the quantum version of the Hamilton-Jacobi equation. This also holds in the relativistic case, so that we obtain both the time-dependent Schrödinger equation and the Klein-Gordon equation in any dimension. We then show that the EP implies that masses are related by maps induced by the coordinate transformations connecting different physical systems. Furthermore, we show that the minimal coupling prescription, and therefore gauge invariance, arises quite naturally in implementing the EP. Finally, we show that there is an antisymmetric 2-tensor which underlies quantum mechanics and sheds new light on the nature of the quantum Hamilton-Jacobi equation
EQUIVALENCE PRINCIPLE: TUNNELLING, QUANTIZED SPECTRA AND TRAJECTORIES FROM THE QUANTUM HJ EQUATION
Full text linkA basic aspect of the recently proposed approach to quantum mechanics is that no use of any axiomatic interpretation of the wave function is made. In particular, the quantum potential turns out to be an intrinsic potential energy of the particle, which, similarly to the relativistic rest energy, is never vanishing. This is related to the tunnel effect, a consequence of the fact that the conjugate momentum field is real even in the classically forbidden regions. The quantum stationary Hamilton-Jacobi equation is defined only if the ratio ψD/ψ of two real linearly independent solutions of the Schrödinger equation, and therefore of the trivializing map, is a local homeomorphism of the extended real line into itself, a consequence of the Möbius symmetry of the Schwarzian derivative. In this respect we prove a basic theorem relating the request of continuity at spatial infinity of ψD/ψ, a consequence of the qq-1 duality of the Schwarzian derivative, to the existence of L2(R) solutions of the corresponding Schrödinger equation. As a result, while in the conventional approach one needs the Schrödinger equation with the L2(R) condition, consequence of the axiomatic interpretation of the wave function, the equivalence principle by itself implies a dynamical equation that does not need any assumption and reproduces both the tunnel effect and energy quantization
EQUIVALENCE PRINCIPLE, PLANCK LENGTH AND QUANTUM HAMILTON-JACOBI EQUATION
Full text linkThe Quantum Stationary HJ Equation (QSHJE) that we derived from the equivalence principle, gives rise to initial conditions which cannot be seen in the Schrödinger equation. Existence of the classical limit leads to a dependence of the integration constant l=l1+il2 on the Planck length. Solutions of the QSHJE provide a trajectory representation of quantum mechanics which, unlike Bohm's theory, has a non-trivial action even for bound states and no wave guide is present. The quantum potential turns out to be an intrinsic potential energy of the particle which, similarly to the relativistic rest energy, is never vanishing
QUANTUM MECHANICS FROM AN EQUIVALENCE PRINCIPLE
Full text linkWe postulate that physical states are equivalent under coordinate transformations. We then implement this equivalence principle first in the case of one-dimensional stationary systems showing that it leads to the quantum analogue of the Hamilton-Jacobi equation which in turn implies the Schrödinger equation. In this context the Planck constant plays the role of covariantizing parameter. The construction is deeply related to the GL(2,C)-symmetry of the second-order differential equation associated to the Legendre transformation which selects, in the case of the quantum analogue of the Hamiltonian characteristic function, self-dual states which guarantee its existence for any physical system. The universal nature of the self-dual states implies the Schrödinger equation in any dimension
A Statistical Interpretation of Space and Classical-Quantum duality
No full textBy defining a prepotential function for the stationary Schrödinger equation we derive an inversion formula for the space variable as a function of the wave--function . The resulting equation is a Legendre transform that relates , the prepotential , and the probability density. We invert the Schrödinger equation to a third--order differential equation for and observe that the inversion procedure implies a -- duality. This phenomenon is related to a modular symmetry due to the superposition of the solutions of the Schrödinger equation. We propose that in quantum mechanics the space coordinate can be interpreted as a macroscopic variable of a statistical system with playing the role of a scaling parameter. We show that the scaling property of the space coordinate with respect to is determined by the ``beta--function''. We propose that the quantization of the inversion formula is a natural way to quantize geometry.By defining a prepotential function for the stationary Schr\"odinger equation we derive an inversion formula for the space variable as a function of the wave--function . The resulting equation is a Legendre transform that relates , the prepotential , and the probability density. We invert the Schr\"odinger equation to a third--order differential equation for and observe that the inversion procedure implies a -- duality. This phenomenon is related to a modular symmetry due to the superposition of the solutions of the Schr\"odinger equation. We propose that in quantum mechanics the space coordinate can be interpreted as a macroscopic variable of a statistical system with playing the role of a scaling parameter. We show that the scaling property of the space coordinate with respect to is determined by the ``beta--function''. We propose that the quantization of the inversion formula is a natural way to quantize geometry.By defining a prepotential function for the stationary Schr\"odinger equation we derive an inversion formula for the space variable as a function of the wave--function . The resulting equation is a Legendre transform that relates , the prepotential , and the probability density. We invert the Schr\"odinger equation to a third--order differential equation for and observe that the inversion procedure implies a -- duality. This phenomenon is related to a modular symmetry due to the superposition of the solutions of the Schr\"odinger equation. We propose that in quantum mechanics the space coordinate can be interpreted as a macroscopic variable of a statistical system with playing the role of a scaling parameter. We show that the scaling property of the space coordinate with respect to is determined by the ``beta--function''. We propose that the quantization of the inversion formula is a natural way to quantize geometry.By defining a prepotential function for the stationary Schrödinger equation we derive an inversion formula for the space variable as a function of the wave-function . The resulting equation is a Legendre transform that relates , the prepotential , and the probability density. We invert the Schrödinger equation to a third-order differential equation for and observe that the inversion procedure implies a - duality. This phenomenon is related to a modular symmetry due to the superposition of the solutions of the Schrödinger equation. We propose that in quantum mechanics the space coordinate can be interpreted as a macroscopic variable of a statistical system with playing the role of a scaling parameter. We show that the scaling property of the space coordinate with respect to is determined by the ``beta-function''. We propose that the quantization of the inversion formula is a natural way to quantize geometry. The formalism is extended to higher dimensions and to the Klein-Gordon equation
THE EQUIVALENCE POSTULATE OF QUANTUM MECHANICS: MAIN THEOREMS
No full textWe 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
QUANTUM TRANSFORMATIONS
Full text linkWe show that the stationary quantum Hamilton-Jacobi equation of nonrelativistic 1D systems, underlying Bohmian mechanics, takes the classical form with ∂q replaced by ∂q where dq=dq/sqrt(1-β2). The β2 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
THE EQUIVALENCE PRINCIPLE OF QUANTUM MECHANICS: UNIQUENESS THEOREM
Full text linkRecently we showed that the postulated diffeomorphic equivalence of states implies quantum mechanics. This approach takes the canonical variables to be dependent by the relation p=∂qS0 and exploits a basic GL(2,C)-symmetry which underlies the canonical formalism. In particular, we looked for the special transformations leading to the free system with vanishing energy. Furthermore, we saw that while on the one hand the equivalence principle cannot be consistently implemented in classical mechanics, on the other it naturally led to the quantum analogue of the Hamilton-Jacobi equation, thus implying the Schrödinger equation. In this letter we show that actually the principle uniquely leads to this solution. Furthermore, we find the map reducing any system to the free one with vanishing energy and derive the transformations on S0 leaving the wave function invariant. We also express the canonical and Schrödinger equations by means of the brackets recently introduced in the framework of N=2 SYM. These brackets are the analogue of the Poisson brackets with the canonical variables taken as dependent
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