Abstract
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 underD-dimensional Möbius transformations with Euclidean or Minkowski metrics. The second is a quadratic identity which is a representation of theD-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.
Highlights
Understanding the synthesis of quantum mechanics and gravity is an important challenge in theoretical physics
While in Bohmian mechanics one identifies the solution (18) with the wave function, and R2 with the probability density, it is noted that the equivalence postulate necessitates that both solutions ψ and ψD are kept in the formalism
By hypothesising that all physical systems labelled by a potential function W(q) are connected by coordinate transformations, the equivalence postulate of quantum mechanics offers an axiomatic starting point for formulating quantum field theories and quantum gravity
Summary
Understanding the synthesis of quantum mechanics and gravity is an important challenge in theoretical physics. A formalism that aims to promote phase-space duality to a level of a fundamental principle was followed in the context of the equivalence postulate approach to quantum mechanics [8,9,10,11,12,13,14,15]. It is seen that both the definability of the phase-space duality, and consistency of the equivalence postulate for all physical states require the modification of classical mechanics by quantum mechanics. The phase-space duality, as well as the equivalence postulate, is ill defined in classical mechanics for the trivial state, for which Hamilton’s generating function S0 is a constant or a linear function of the coordinate. This property of elementary particles arises from the quantum potential, which is never vanishing and corresponds to an intrinsic curvature associated with elementary particles
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