The Equivalence Postulate of Quantum Mechanics, Dark Energy, and the Intrinsic Curvature of Elementary Particles

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 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.

1. Introduction

Understanding the synthesis of quantum mechanics and gravity is an important challenge in theoretical physics. The main effort in this endeavour is in the framework of string theory. The primary advantage of string theory is that it gives rise to the gauge and matter ingredients of elementary particle physics and predicts the number of degrees of freedom needed to obtain a consistent theory. String theory therefore enables the construction of quasirealistic models and the development of a phenomenological approach to quantum gravity. The state of the art in this regard is the heterotic string models in the free fermionic formulation, which reproduce the matter content of the minimal supersymmetric standard model and preserve its unification picture [1–5]. Despite its phenomenological success string theory does not provide a framework for a rigorous formulation of quantum gravity from fundamental principles.

Important characteristics of string theories are its various perturbative and nonperturbative dualities. Indeed, one of the interesting approaches to formulating string theory aims to promote 𝑇-duality [6] to a manifest property of the formalism [7]. 𝑇-duality can be viewed heuristically as phase-space duality in compact space.

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 [8–15]. The equivalence postulate of quantum mechanics hypothesises that all physical systems are equivalent under coordinate transformations. In particular, there should always exist a coordinate transformation connecting a physical system with a nontrivial potential 𝑉 and energy 𝐸, to the one with 𝑉−𝐸=0. Conversely, any allowed physical state should arise by a coordinate transformation from the state with 𝑉−𝐸=0. Thus, nontrivial states arise from the inhomogeneous term stemming from the transformation of the trivial state under coordinate transformations. It is then 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. More precisely, 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. The quantum modification removes this state from the space of allowed solutions and enables the consistency of the equivalence postulate, as well as definability of phase-space duality, for all physical states.

In this paper, I argue that another feature of the equivalence postulate approach is the existence of dark energy, which arises as an intrinsic property of elementary particles. This property of elementary particles arises from the quantum potential, which is never vanishing and corresponds to an intrinsic curvature associated with elementary particles.

2. Equivalence Postulate of Quantum Mechanics

In this section I review the equivalence postulate approach to quantum mechanics. It is important to emphasise that the equivalence postulate formulation of quantum mechanics does not entail an interpretation or a modification, but rather a mathematically rigorous derivation of quantum mechanics from a fundamental postulate. In this respect the equivalence postulate provides an axiomatic approach to quantum mechanics and quantum field theories. The formalism has some reminiscences of Bohm’s approach to quantum mechanics in the sense that both approaches may be regarded as a modification of the classical Hamilton-Jacobi formalism. However, aside from this superficial similarity the two approaches are fundamentally distinct.

A view of the equivalence postulate approach is obtained by an analogy with the classical Hamilton-Jacobi formalism. The classical Hamilton equations of motion are given by 𝑝̇=−𝜕𝐻,̇𝜕𝑞𝑞=𝜕𝐻,𝜕𝑝 (1) where 𝑝 and 𝑞 are the phase space variables and 𝐻 is the Hamiltonian. In classical mechanics the solution to the mechanical problem is obtained by performing canonical transformations to a new set of phase space variables 𝑞⟶𝑄𝑞,𝑝,𝑡,𝑝⟶𝑃𝑞,𝑝,𝑡 (2) such that the new Hamiltonian and Hamilton equations of motion are trivial 𝐻(𝑞,𝑝)→𝐾(𝑄,𝑃)≡0. Consequently, the transformed phase space coordinates are constants of the motion. The solution to this problem in classical mechanics is given by the classical Hamilton-Jacobi equation (CHJE) which for systems that conserve the total energy gives the classical stationary Hamilton-Jacobi equation (CSHJE) 12𝑚𝜕𝑆𝜕𝑞²+𝑉𝑞=𝐸, (3) where 𝑆(𝑞) is a generating function. For the purpose of this paper it is sufficient to focus here on the stationary case. In performing these trivialising transformations the phase space variables are taken as independent variables. The functional dependence between them is only extracted after solving the CSHJE equation (3) from the relation 𝑞𝑝=𝜕𝑆.𝜕𝑞 (4) Since the transformations (2) are invertible we also have the inverse transformation 𝑄⟶𝑞(𝑄,𝑃,𝑡),𝑃⟶𝑝(𝑄,𝑃,𝑡). (5) However, the intrinsic property of quantum mechanics is that the phase space variables are not independent; that is, they satisfy the commutation relation: 𝑞,𝑝=𝑖ℏ. (6) We therefore relax the condition that the phase space variable are independent in the application of the trivialising transformation; and, hence, that the transformations are canonical. We further assume the functional relation between the phase-space variables via the generating function (4). We can ask the following question: is it possible in classical mechanics to start with a system with a nontrivial Hamiltonian, that is with a given nonvanishing 𝑊(𝑞), and connect via some coordinate transformations to any other system. This demand dictates that the Hamilton-Jacobi equation retains its form under the coordinate transformations. From the form of (3) we see that this is not possible in classical mechanics. The reason is that in classical mechanics the trivial state, with 𝑉(𝑞)−𝐸≡0, is a fixed state under the transformations. Insisting that all physical states, including the trivial one, are connected by coordinate transformations necessitates the modification of the CSHJE. The modification is given by the quantum stationary Hamilton-Jacobi equation (QSHJE) 1𝑞2𝑚𝜕𝑆𝜕𝑞²+𝑊𝑞+𝑄𝑞=0, (7) where 𝑊(𝑞)=𝑉(𝑞)−𝐸. From the form of (7) it is seen that the combination 𝑊(𝑞)+𝑄(𝑞) transforms as a quadratic differential under coordinate transformations, whereas each of the functions 𝑊(𝑞) and 𝑄(𝑞) transform, as a quadratic differential up to an additive term; that is, under 𝑞→𝑞(𝑞) we have: 𝑊𝑞⟶𝑊𝑞=𝜕𝑞𝜕𝑞²𝑊𝑞+𝑞,𝑄𝑞⟶𝑄𝑞=𝑞;𝜕𝑞𝜕𝑞²𝑄𝑞−𝑞,𝑊𝑞𝑞⟶𝑊𝑞+𝑄𝑞=𝑞;+𝑄𝜕𝑞𝜕𝑞²𝑊𝑞+𝑄𝑞. (8) All physical states with a nontrivial 𝑊(𝑞) then arise from the inhomogeneous part in the transformation of the trivial state 𝑊⁰(𝑞⁰)≡0; that is, 𝑊(𝑞)=(𝑞⁰;𝑞).

Considering the transformation 𝑞^𝑎→𝑞^𝑏→𝑞^𝑐 versus 𝑞^𝑎→𝑞^𝑐 gives rise to the cocycle condition on the inhomogeneous term 𝑞^𝑎;𝑞^𝑐=𝜕𝑞^𝑏𝜕𝑞^𝑐2𝑞^𝑎;𝑞^𝑏−𝑞^𝑐;𝑞^𝑏. (9) The cocycle condition (9) underlies the equivalence postulate and embodies its underlying symmetries. In particular, it is invariant under the Möbius transformations 𝛾𝑞^𝑎;𝑞^𝑏=𝑞^𝑎;𝑞^𝑏, (10) where 𝛾𝑞=𝐴𝑞+𝐵𝐶𝑞+𝐷 (11) and ^𝐴𝐵_𝐶𝐷∈GL(2,𝐶). In one dimension the cocycle condition (9) uniquely defines the Schwarzian derivative up to a constant and a coboundary term. Specifically, one obtains (𝑞^𝑎;𝑞^𝑏)=−𝛽{𝑞^𝑎,𝑞^𝑏}/4𝑚, where {𝑓,𝑞}=𝑓^′′′/𝑓^′−3(𝑓^′′/𝑓^′)²/2 denotes the Schwarzian derivative and 𝛽 is a constant with the dimension of an action. We further have that {𝛾(𝑞),𝑞}≡0.

The one-dimensional stationary case is instructive to reveal the symmetry properties that underlie quantum mechanics in the equivalence postulate formalism. In one dimension the unique solution of the problem is given in terms of the Schwarzian identity 𝜕𝑆𝑞𝜕𝑞²=𝛽²2e^{(2𝑖/𝛽)𝑆},𝑞−𝑆,𝑞 (12) which embodies the equivalence postulate and leads to the Schrödinger equation. Making the identification 𝑊𝑞𝑞𝛽=𝑉−𝐸=−²e4𝑚^{(𝑖2𝑆0)/𝛽},𝑄𝑞=𝛽,𝑞₂𝑆4𝑚⁰,𝑞, (13)(14) we have that 𝑆₀ is the solution of the quantum stationary Hamilton-Jacobi equation (QSHJE) 12𝑚𝜕𝑆₀𝜕𝑞²𝑞ℏ+𝑉−𝐸+²𝑆4𝑚⁰,𝑞=0, (15) From (13) and the properties of the Schwarzian derivative, we deduce that 𝑆₀, the solution of the QSHJE equation (15), is given by (see also [16–21]): e^{(2𝑖/𝛽)𝑆0}=𝛾(𝑤)=𝐴𝑤+𝐵𝐶𝑤+𝐷=e^𝑖𝛼𝑤+𝑖ℓ,𝑤−𝑖ℓ (16) where ℓ=ℓ₁+𝑖ℓ₂;{𝛼,ℓ₁,ℓ₂}∈𝑅. Here 𝑤=𝜓^𝐷/𝜓 and 𝜓^𝐷 and 𝜓 are two linearly independent solutions of a second order differential equation given by −𝛽²𝜕2𝑚𝜕𝑞²²+𝑉𝑞−𝐸𝜓𝑞=0, (17) That is, 𝜓^𝐷 and 𝜓 are the two solutions of the Schrödinger equation and we can identify 𝛽≡ℏ. We can note the relation between the Shrödinger equation in an alternative way. Inserting the solution 𝜓=𝑅e^{(𝑖/ℏ)𝑆0} (18) into the Schrödinger equation produces the two equations 𝜕_𝑞𝑆₀²𝑞ℏ+𝑉−𝐸−²𝜕²_𝑞𝑅()𝜕2𝑚𝑅=0,_𝑞𝑅²𝜕_𝑞𝑆₀=0. (19)(20) The continuity equation, (20), gives 𝑅=1/𝑆^′₀ and consequently 𝑄𝑞ℏ=−²𝜕2𝑚^2𝑞𝑅𝑅=ℏ²𝑆4𝑚⁰,𝑞 (21) and (19) corresponds to (15).

The steps taken in deriving the quantum Hamilton-Jacobi equation (15) from the Schrödinger equation are reminiscent of its derivation in the framework of Bohmian quantum mechanics. However, there is a crucial difference. While in Bohmian mechanics one identifies the solution (18) with the wave function, and hence 𝑅² with the probability density, it is noted that the equivalence postulate necessitates that both solutions 𝜓 and 𝜓^𝐷 are kept in the formalism. This can be seen from the properties of the Schwarzian derivative that show that the trivialising transformation is given by 𝑞⟶𝑞⁰𝜓≡𝛾^𝐷𝜓, (22) that is, up to a Möbius transformation, 𝑞⁰ is given by the ratio of the two solutions of the corresponding Schrödinger equation. Hence, in general the wave function in the equivalence postulate approach is given by Ψ𝑞=𝑅𝑞𝐴e^{(𝑖/ℏ)𝑆0}+𝐵e^{−(𝑖/ℏ)𝑆0}. (23) Furthermore, the equivalence postulate necessitates that 𝑆₀(𝑞)≠𝐴𝑞+𝐵; that is, S₀(𝑞) cannot be a linear function of 𝑞. Strictly speaking the condition that the Schwarzian derivative is well defined only necessitates the weaker condition 𝑆₀(𝑞)≠constant. However, the condition that the quantum potential is always nonvanishing leads to the stronger constraint 𝑆₀(𝑞)≠𝐴𝑞+𝐵, which also follows from arguments concerning phase-space duality [8–14]. In the time-dependent case the equivalence postulate implies that the wave function should always take the form Ψ𝑞,𝑡=𝑅𝑞,𝑡𝐴e^{(𝑖/ℏ)𝑆(𝑞,𝑡)}+𝐵e^{−(𝑖/ℏ)𝑆(𝑞,𝑡)}, (24) where 𝑆=𝑆₀−𝐸𝑡 in the stationary case. As discussed previously, consistency of the equivalence postulate of quantum mechanics dictates the necessity of employing the two solutions of the Schrödinger equation. This condition is well known in relativistic quantum mechanics and signals the departure from the single particle interpretation in nonrelativistic quantum mechanics to the multiparticle representation of quantum field theories. However, in nonrelativistic Bohmian mechanics, and in particular in the case of bound states, only the solution with the positive exponent is kept, which implies that in those cases in Bohmian mechanics 𝑆₀=constant. The equivalence postulate approach, on the other hand, necessitates that both solutions are kept in the formalism and that 𝑆₀≠0 always. Consistency of the equivalence postulate further implies that the trivialising map, 𝑞→𝑞=𝜓^𝐷/𝜓, is continuous on the extended real line [8–14]. It is then seen that this condition is synonymous with the requirement that the physical solution of the corresponding Schrödinger equation admits a square integrable solution and selects the correct physical eigenstates for the bound states. In the relativistic case the inclusion of the “negative energy” states reveals the existence of antiparticles. The implications of incorporating the two solutions in the nonrelativistic case require more detailed scrutiny. It is noted that this decomposition of the wave function has been employed successfully in studies of molecular dynamics, [22, 23] and is referred to there as the bipolar decomposition.

The equivalence postulate formalism extends to the higher dimensional case both with respect to the Euclidean and Minkowski metrics [15]. The key to these extensions is the generalisations of the cocycle condition equation (9) and of the quadratic identity equation (12). Denote the transformations between two sets of coordinate systems by 𝑞⟶𝑞^𝑣=𝑣𝑞 (25) and the conjugate momenta by the generating function 𝑆₀(𝑞), 𝑝_𝑘=𝜕𝑆₀𝜕𝑞_𝑘. (26) Under the transformations (25) we have 𝑆^𝑣₀(𝑞^𝑣)=𝑆₀(𝑞); hence 𝑝_𝑘⟶𝑝^𝑣_𝑘=^𝐷_𝑖=1𝐽_𝑘𝑖𝑝_𝑖, (27) where 𝐽 is the Jacobian matrix 𝐽_𝑘𝑖=𝜕𝑞_𝑖𝜕𝑞^𝑣_𝑗. (28) Introducing the notation 𝑝^𝑣=∑∣𝑝^𝑘𝑝^𝑣_𝑘²∑_𝑘𝑝²_𝑘=𝑝^𝑡𝐽^𝑡𝐽𝑝𝑝^𝑡𝑝, (29) the cocycle condition takes the form 𝑞^𝑎;𝑞^𝑐=𝑝^𝑐∣𝑝^𝑏𝑞^𝑎;𝑞^𝑏−𝑞^𝑐;𝑞^𝑏, (30) which captures the symmetries that underlie quantum mechanics. It is shown that the cocycle condition (30) is invariant under 𝐷-dimensional Möbius transformations, which include dilatations, rotations, translations, and reflections in the unit sphere [15]. The quadratic identity, (12), is generalised by the basic identity 𝛼²∇𝑆₀²=Δ𝑅e^𝛼𝑆0𝑅e^𝛼𝑆0−Δ𝑅𝑅−𝛼𝑅²∇⋅𝑅²∇𝑆₀, (31) which holds for any constant 𝛼 and any functions 𝑅 and 𝑆₀. Then, if 𝑅 satisfies the continuity equation ∇⋅𝑅²∇𝑆₀=0, (32) and setting 𝛼=𝑖/ℏ, we have 12𝑚∇𝑆₀²ℏ=−²Δ2𝑚𝑅e^{(𝑖/ℏ)𝑆0}𝑅e^{(𝑖/ℏ)𝑆0}+ℏ²2𝑚Δ𝑅𝑅. (33) In complete analogy with the one-dimensional case we make identifications, 𝑊𝑞𝑞ℏ=𝑉−𝐸=²Δ2𝑚𝑅e^{(𝑖/ℏ)𝑆0}𝑅e^{(𝑖/ℏ)𝑆0},𝑄𝑞ℏ=−²2𝑚Δ𝑅𝑅. (34)(35) Equation (34) implies the 𝐷-dimensional Schrödinger equation −ℏ²𝑞2𝑚Δ+𝑉Ψ=𝐸Ψ, (36) and the general solution Ψ=𝑅𝑞𝐴e^{(𝑖/ℏ)𝑆0}+𝐵e^{−(𝑖/ℏ)𝑆0}, (37) is mandated by consistency of the equivalence postulate.

The equivalence postulate formalism generalises to the relativistic case as well. In this case, setting 𝑞≡(𝑐𝑡,𝑞₁,…,𝑞_𝐷−1), with 𝑞^𝑣=𝑣(𝑞) a general transformation of the coordinates, we have 𝑝^𝑣=𝜂∣𝑝^𝜇𝜌𝑝^𝑣_𝜇𝑝^𝑣_𝜌𝜂^𝜇𝜌𝑝_𝜇𝑝_𝜈=𝑝^𝑡𝐽𝜂𝐽^𝑡𝑝𝑝^𝑡,𝜂𝑝 (38) and 𝐽 is the Jacobian matrix 𝐽^𝜇_𝜌=𝜕𝑞^𝜇𝜕𝑞^𝑣𝜌. (39) Furthermore, we obtain the cocycle condition 𝑞^𝑎;𝑞^𝑐=𝑝^𝑐∣𝑝^𝑏𝑞^𝑎;𝑞^𝑏−𝑞^𝑐;𝑞^𝑏, (40) is invariant under 𝐷-dimensional Möbius transformations with respect to Minkowski metric. The quadratic identity in this case takes the form, 𝛼²(𝜕𝑆)²=□𝑅e^𝛼𝑆𝑅e^𝛼𝑆−□𝑅𝑅−𝛼𝑅²𝜕⋅𝑅²𝜕𝑆, (41) which holds for any constant 𝛼 and any functions 𝑅 and 𝑆. Then, if 𝑅 satisfies the continuity equation 𝜕(𝑅²⋅𝜕𝑆)=0, and setting 𝛼=𝑖/ℏ, we have (𝜕𝑆)²=−ℏ²□𝑅e^{(𝑖/ℏ)𝑆}𝑅e^{(𝑖/ℏ)𝑆}+ℏ²□𝑅𝑅. (42) Setting 𝑊𝑞=𝑚𝑐²=−ℏ²□𝑅e^{(𝑖/ℏ)𝑆}𝑅e^{(𝑖/ℏ)𝑆},𝑄𝑞=ℏ²□𝑅𝑅 (43) reproduces the relativistic Klein-Gordon equation ℏ²□+𝑚𝑐²Ψ𝑞=0 (44) with the general solution Ψ=𝑅𝑞𝐴e^{(𝑖/ℏ)𝑆}+𝐵e^{−(𝑖/ℏ)𝑆}. (45) The equivalence postulate formalism also incorporates gauge interactions via the generalisation of the quadratic identities equations (12), (31), and (41), and of the cocycle conditions equations (9), (30), and (40). In the nonrelativistic stationary case the quadratic identity takes the form 𝛼²∇𝑆₀+𝑒𝐴²=(∇+𝑒𝛼𝐴)²𝑅e^𝛼𝑆0𝑅e^𝛼𝑆0−Δ𝑅𝑅−𝛼𝑅²∇⋅𝑅²∇𝑆₀+𝑒𝐴, (46) and the cocycle condition equation (30) retains its form with 𝑝^𝑣𝑝=𝑝^𝑣+𝑒𝐴^𝑣𝑡𝑝^𝑣+𝑒𝐴^𝑣𝑝+𝑒𝐴^𝑡𝑝+𝑒𝐴, (47) whereas in the relativistic case they take the form 𝛼²(𝜕𝑆+𝑒𝐴)²=𝐷²𝑅𝑒^𝛼S𝑅𝑒^𝛼𝑆−𝑅𝑅−𝛼𝑅²𝜕⋅𝑅²(𝜕𝑆+𝑒𝐴), (48) where 𝐷_𝜇=𝜕_𝜇+𝛼𝑒𝐴_𝜇, (49) and the cocycle identity (30) retains its form with 𝑝^𝑏=𝑝∣𝑝^𝑏+𝑒𝐴^𝑏2𝑝+𝑒𝐴²=𝑝+𝑒𝐴^𝑡𝐽𝜂𝐽^𝑡𝑝+𝑒𝐴𝑝+𝑒𝐴^𝑡𝜂,𝑝+𝑒𝐴 (50) and 𝐽 is the Jacobian matrix 𝐽^𝜇_𝜈=𝜕𝑞^𝜇𝜕𝑞^𝑏𝜈. (51) We note the symmetry structure that underlies the formalism. Seeking further generalisation of this approach simply entails that this robust symmetry structure is retained. While the formalism has some reminiscences of Bohmian quantum mechanics, it is clearly distinct from it as no physical interpretation has so far been assigned to the wave function. Furthermore, consistency of the equivalence postulate formalism requires that the physical solutions are square integrable and selects the same eigenstates that in conventional quantum mechanics arise due to the probability interpretation of the wave function. In the framework of the equivalence postulate, the Schrödinger equation, and its solutions, is merely a device to find solutions of the nonlinear quantum Hamilton-Jacobi equation. To investigate the physical picture of the quantum potential in the equivalence postulate formalism we have to turn to its interpretation as a curvature term.

3. The Quantum Potential as a Curvature Term

The quantum stationary Hamilton-Jacobi equation (QSHJE), (7) can be viewed as a deformation of the classical stationary Hamilton-Jacobi equation (CSHJE), (3), by a “conformal factor.” Noting that 𝑆₀,𝑞=−𝜕_𝑞𝑆₀²𝑞,𝑆₀, (52) we have that the QSHJE (7) is equivalent to 12𝑚𝜕𝑆₀𝜕𝑞²1−ℏ²𝑈𝑆₀+𝑉𝑞−𝐸=0, (53) where 𝑈(𝑆₀) is the canonical potential 𝑈𝑆₀=12𝑞,𝑆₀, (54) that arises in the framework of the 𝑝−𝑞 duality [8–14]. Equation (53) can be written in the form 12𝑚𝜕𝑆₀𝜕𝑞²+𝑉𝑞−𝐸=0, (55) where 𝜕𝑞𝜕𝑞²=1−ℏ²𝑈𝑆₀, (56) or equivalently 𝑑𝑞=𝑑𝑞1−𝛽²𝑞, (57) with 𝛽²(𝑞)=ℏ²𝑈(𝑆₀)=ℏ²{𝑞,𝑆₀}/2. Integrating (56) yields 𝑞=^𝑞𝑑𝑥1−𝛽²(𝑥). (58) It follows that lim_ℏ→0𝑞=𝑞. (59) In the case of the trivial state 𝑊⁰(𝑞⁰)≡0, (53) becomes 12𝑚𝜕𝑆⁰₀𝜕𝑞⁰²1−ℏ²𝑈𝑆₀=0, (60) whose solution is 𝑈ℏ𝑞2𝑖ln𝛾⁰=1ℏ², (61) where 𝛾𝑞⁰=𝐴𝑞⁰+𝐵𝐶𝑞⁰,+𝐷 (62) with 𝐴𝐷−𝐵𝐶≠0. Equation (53) shows that the quantum potential can be regarded as a deformation of the space geometry. From (46) and (48) we note that in this respect the quantum potential is distinct from the interactions that arise from the gauge potentials, that can be viewed as a shift of the momentum. The quantum potential has a universal character, which is independent of gauge charges. Furthermore, Flanders showed that the Schwarzian derivative can be interpreted as a curvature of an equivalence problem for curves in 𝐏¹ [24]. For that purpose, introduce a frame for 𝐏¹, that is a pair 𝐱,𝐲 of points in affine space 𝐀² such that [𝐱,𝐲]=1, where [𝐱,𝐲]=𝐱^𝑡01−10𝐲=𝑥₁𝑦₂−𝑥₂𝑦₁ (63) is the area function, which has the 𝑆𝐿(2,𝐑)-symmetry [𝐱,𝐲]=[𝐱,𝐲], (64) where 𝐱=𝑅𝐱 and 𝐲=𝑅𝐲 with 𝑅∈𝑆𝐿(2,𝐑). Considering the moving frame 𝑠→{𝐱(𝑠),𝐲(𝑠)} and differentiating [𝐱,𝐲]=1 yield the structure equations 𝐱^′=𝑎𝐱+𝑏𝐲,𝐲^′=𝑐𝐱−𝑎𝐲, (65) where 𝑎,𝑏, and 𝑐 depend on 𝑠. Given a map 𝜙=𝜙(𝑠) from a domain to 𝐏¹, one can choose a moving frame 𝐱(𝑠),𝐲(𝑠) in such way that 𝜙(𝑠) is represented by 𝐱(𝑠). This map can be seen as a curve in 𝐏¹. Two mappings 𝜙 and 𝜓 are said to be equivalent if 𝜓=𝜋∘𝜙, with 𝜋 a projective transformation on 𝐏¹.

Flanders considered two extreme situations. The first case corresponds to 𝑏(𝑠)=0, for all 𝑠. In this case 𝜙 is constant. Taking the derivative of 𝜆𝐱, for some 𝜆(𝑠)≠0, we have by (64) that (𝜆𝐱)^′=(𝜆^′+𝑎𝜆)𝐱. Choosing 𝜆∝exp[−∫^𝑠_𝑠0𝑑𝑡𝑎(𝑡)]≠0, we have (𝜆𝐱)^′=0, so that 𝜆𝐱 is a constant representative of 𝜙.

The other case is for 𝑏 never vanishing. There are only two inequivalent situations. The first one is when 𝑏 is either complex or positive. It turns out that it is always possible to choose the following “natural moving frame” for 𝜙 [24]: 𝐱^′=𝐲,𝐲^′=−𝑘𝐱. (66) In the other case, corresponding to 𝑏 real and negative, the natural moving frame for 𝜙 is 𝐱^′=−𝐲,𝐲^′=𝑘𝐱. (67)

A characteristic property of the natural moving frame is that it is determined up to a sign with 𝑘 an invariant. Thus, for example, suppose that for a given 𝜙 there is, besides (66), the natural moving frame 𝐱^′₁=𝐲₁, 𝐲^′₁=−𝑘₁𝐱₁. Since both 𝐱 and 𝐱₁ are representatives of 𝜙, we have 𝐱=𝜆𝐱₁, so that 𝐲=𝐱^′=𝜆^′𝐱₁+𝜆𝐲₁ and 1=[𝐱,𝐲]=𝜆². Therefore, 𝐱₁=±𝐱, 𝐲₁=±𝐲, and 𝑘₁=𝑘 [24].

Let us now review the derivation of Flanders formula for 𝑘. Consider 𝑠→𝐳(𝑠) to be an affine representative of 𝜙 and let 𝐱(𝑠),𝐲(𝑠) be a natural frame. Then 𝐳=𝜆𝐱 where 𝜆(𝑠) is never vanishing. Now note that, since 𝐳^′=𝜆^′𝐱+𝜆𝐲, we have that 𝜆 can be written in terms of the area function [𝐳,𝐳^′]=𝜆². Computing the relevant area functions, one can check that 𝑘 has the following expression 2𝑘=𝐳,𝐳^′′′+3𝐳^′,𝐳^′′𝐳,𝐳^′−32[𝐳,𝐳^′′][𝐳,𝐳^′]². (68)

Given a function 𝑧(𝑠), this can be seen as the nonhomogeneous coordinate of a point in 𝐏¹. Therefore, we can associate with 𝑧 the map 𝜙 defined by 𝑠→(1,𝑧(𝑠))=𝐳(𝑠). In this case we have [𝐳,𝐳^′]=𝑧^′, [𝐳,𝐳^′′]=𝑧^′′, [𝐳,𝐳^′′′]=𝑧^′′′, [𝐳^′,𝐳^′′]=0, and the curvature becomes [24] 1𝑘=2{𝑧,𝑠}. (69)

For an arbitrary physical state with potential function 𝑊 we have ℏ𝑊=−²𝑒4𝑚^{(2𝑖/ℏ)𝑆0}ℏ,𝑞=−²𝑘2𝑚^𝑊, (70) and similarly for the quantum potential ℏ𝑄=²𝑆4𝑚⁰=ℏ,𝑞²𝑘2𝑚^𝑄, (71) where 𝑘_𝑊 is the curvature associated with the map 𝑞⟶1,𝑒^{(2𝑖/ℏ)𝑆0(𝑞)}, (72) while the curvature 𝑘_𝑄 is associated with the map 𝑞⟶1,𝑆₀𝑞. (73) The function defining the map (72) coincides with the trivialising map, whereas the Schwarzian identity (12) can be now seen as difference of curvatures 𝜕_𝑞𝑆₀²=ℏ²𝑘_𝑊−ℏ²𝑘_𝑄. (74) The QSHJE (7) can be written in the form 12𝑚𝜕𝑆₀𝑞𝜕𝑞²𝑞+ℏ+𝑊²𝑘2𝑚^𝑄=0. (75) We therefore note that in the one-dimensional QSHJE the quantum potential is interpreted as a curvature and is an intrinsically quantum characteristic of the particle. In higher dimension the curvature term takes the form ℏ𝑄=−²2𝑚Δ𝑅𝑅 (76) and the corresponding form in the relativistic case. The continuity condition, (32), implies that 𝑅²𝜕_𝑖𝑆₀=𝜖^{𝑖2⋯𝑖𝐷}_𝑖𝜕_𝑖2𝐹_{𝑖3⋯𝑖𝐷}, (77) where 𝐹 is a (𝐷−2) form. The 3𝐷 case 𝑅²𝜕_𝑖𝑆₀ is the curl of a vector, denoted by 𝐵, 𝑅²∇𝑆₀=∇×𝐵. (78) Hence, the QSHJE, (7), takes the form (∇×𝐵)²=ℏ²𝑅³Δ𝑅+2𝑚𝐸𝑅⁴. (79) In the time-dependent relativistic case 𝐹 is a (𝐷−1)-form. We have 𝑅²𝜕_𝜇𝑆−𝑒𝐴_𝜇=𝜖^{𝜎1⋯𝜎𝐷}_𝜇𝜕_𝜎1𝐹_{𝜎2⋯𝜎𝐷}=𝜕^𝜈𝐵_𝜇𝜈, (80) that is 𝑅²=𝜕^𝜇𝑆−𝑒𝐴^𝜇(𝜕𝑆−𝑒𝐴)²𝜕^𝜈𝐵_𝜇𝜈. (81) In terms of 𝐵 and 𝑅 the RQHJE, which correspond to the real part of (46) with 𝛼=𝑖/ℏ, takes the form 𝜕^𝜈𝐵_𝜇𝜈𝜕_𝜎𝐵^𝜇𝜎+𝑅⁴𝑚²𝑐²−ℏ²𝑅³□𝑅=0. (82) Thus, in the higher dimensional case the quantum potential corresponds to the curvature of the function 𝑅(𝑞). We can investigate the properties of the curvature term by studying the free one-dimensional particle, with 𝑊⁰(𝑞⁰)≡0. In this case the Schrödinger equation takes the form 𝜕²𝜕𝑞²𝜓=0, (83) with the two linearly independent solutions being 𝜓^𝐷=𝑞⁰ and 𝜓=const. As discussed previously an essential tenant of the equivalence postulate formalism is that both solutions of the Schrödinger equation must be retained. The linear combination of 𝜓 and 𝜓^𝐷, 𝜓^𝐷+𝑖ℓ𝜓, appears in the solution for 𝑆₀, (16). In the case of the state 𝑊⁰(𝑞⁰)≡0, with 𝜓^𝐷=𝑞⁰ and 𝜓=1, this combination reads 𝑞⁰+𝑖ℓ₀, and ℓ₀≡ℓ should have the dimension of length. On the other hand, from the fact that by (22) the trivialising coordinate 𝑞⁰ is given by the ratio 𝜓^𝐷/𝜓 up to a Möbius transformation, it follows that ℓ and 𝑤=𝜓^𝐷/𝜓 have the dimension of length for any state. The reduced action 𝑆⁰₀ corresponding to the trivial state 𝑊⁰ is e^{(2𝑖/ℏ)𝑆00}=e^𝑖𝛼𝑞⁰+𝑖ℓ₀𝑞⁰−𝑖ℓ₀, (84) and the conjugate momentum 𝑝₀=𝜕_𝑞0𝑆⁰₀ has the form 𝑝₀ℏℓ=±⁰+ℓ₀2𝑞⁰−𝑖ℓ₀². (85) It follows that 𝑝₀ vanishes only for 𝑞⁰→±∞ and is maximised at 𝑞⁰=−Imℓ₀𝑝₀−Imℓ₀=ℏReℓ₀. (86) Since Reℓ₀≠0, 𝑝₀ is always finite, and ℓ₀ provides an ultraviolet cutoff. This property extends to any state [8–14]. It follows that the equivalence postulate implies an ultraviolet cutoff on the conjugate momentum. We can contemplate the potential form of this ultraviolet cutoff by studying the classical limit, lim_ℏ→0𝑝₀=0. (87) From (85) it follows that Imℓ₀ can be absorbed by a shift of 𝑞⁰. Hence, in the limit equation (87) we can set Imℓ₀=0 and distinguish the cases 𝑞⁰=0 and 𝑞⁰≠0. Let us define 𝛾 by Reℓ₀∼_ℏ→0ℏ^𝛾. (88) Then 𝑝₀∼_ℏ→0ℏ^𝛾+1,𝑞₀ℏ≠0,_1−𝛾,𝑞₀=0, (89) and by (87) −1<𝛾<1. (90) A constant length having powers of ℏ can be constructed by means of the Compton length, 𝜆_𝑐=ℏ/(𝑚𝑐), and the Planck length, 𝜆_𝑝=ℏ𝐺/𝑐³. It is also noted that a constant length which is independent of ℏ is provided by 𝜆_𝑒=𝑒²/𝑚𝑐² where 𝑒 is the electric charge. Thus ℓ₀ can be considered as a suitable function of 𝜆_𝑐, 𝜆_𝑝, and 𝜆_𝑒 satisfying the constraint (90). Of the three constants it is noted that 𝜆_𝑝 satisfies the condition (90), whereas 𝜆_𝑐 and 𝜆_𝑒 do not. Therefore, we can set Reℓ₀=𝜆_𝑝=ℏ𝐺𝑐³. (91) With this choice of Reℓ₀, by (86), the maximum of |𝑝₀| is given by 𝑝₀−Imℓ₀=ℏ𝑐³𝐺. (92) The quantum potential associated with the state 𝑊⁰ is given by 𝑄⁰=ℏ²𝑆4𝑚⁰⁰,𝑞⁰ℏ=−²Reℓ₀²12𝑚𝑞⁰−𝑖ℓ₀⁴. (93) We can make naive estimates of this potential. Taking 𝑚∼100GeV; Reℓ₀=𝜆_𝑝≈10⁻³⁵m; and 𝑞⁰ as the size of the observable universe 𝑞⁰∼93Ly gives |𝑄|∼10⁻²⁰²eV. A tiny amount of energy indeed! It is noted that the sign of 𝑄 is negative in the one-dimensional case. However, there is no reason to expect that this will be the case in higher dimensions as the curvature of the function 𝑅(𝑞) is, in principle, undetermined. It should be emphasised that this is a naive analysis, reflecting the appearance of a length scale in the formalism associated with the intrinsic properties of particle and providing an internal energy component similar to its mass. In this estimate I have taken the particle mass to be 100GeV and 𝑞 to be given by the size of the observable universe, and the quantum potential is that corresponding to the state 𝑊⁰. Taking 𝑞∼1m yields |𝑄|∼10⁻⁹⁶eV. In this respect a relevant question is what is the effective 𝑞 that one should take. A consistency condition of the equivalence postulate formalism dictates that the trivialising transformation is continuous on the extended real line [8–14], that is, at 𝑞=±∞. The question therefore is what is the effective 𝑞=±∞ from the elementary particle perspective. A more detailed analysis can take into account relativistic particles by considering the relativistic quantum Hamilton-Jacobi equation and by considering particles with constant, but finite, 𝑊 functions. It is noted that several authors considered the possibility of interpreting the quantum potential in Bohmian mechanics as the source of dark energy [25, 26], as well as detailed cosmological scenarios [27].

4. The Multiparticle Case

So far the single particle case was discussed. It is interesting to study the question of the multiparticle case and whether the quantum potential is additive in this case. Furthermore, while the single particle case can reveal basic properties of the mathematical formalism, the physical case involves the multi-particle state, as it requires at least an observer and an observee. In the case of two free particles of energy 𝐸 and masses 𝑚₁ and 𝑚₂, the QSHJE reads 12𝑚₁∇₁𝑆₀²+12𝑚₂∇₂𝑆₀²−ℏ−𝐸₂2𝑚₁Δ₁𝑅𝑅−ℏ²2𝑚₂Δ₂𝑅𝑅=0. (94) The continuity equation is 1𝑚₁∇₁⋅𝑅²∇₁𝑆₀+1𝑚₂∇₂⋅𝑅²∇₂𝑆₀=0. (95) Next, we set 𝑟=𝑟₁−𝑟₂,𝑟_c.m.=𝑚₁𝑟₁+𝑚₂𝑟₂𝑚₁+𝑚₂𝑚,𝑚=¹𝑚₂𝑚₁+𝑚₂, (96) where 𝑟₁ and 𝑟₂ are the vectors of the two particles. With respect to the new variables (94) and (95) have the form 12𝑚₁+𝑚₂∇_𝑟c.m.𝑆₀²+12𝑚∇𝑆₀²−ℏ−𝐸₂2𝑚₁+𝑚₂Δ_𝑟c.m.𝑅𝑅−ℏ²2𝑚Δ𝑅𝑅1=0,𝑚₁+𝑚₂∇_𝑟c.m.⋅𝑅²∇_𝑟c.m.𝑆₀+1𝑚∇⋅𝑅²∇𝑆₀=0, (97) where ∇(∇_𝑟c.m.) and Δ(Δ_𝑟c.m.) are the gradient and Laplacian with respect to the components of the vector 𝑟(𝑟_c.m.). These equations can be decomposed into the equations for the centre of mass 𝑟_c.m. and those for the relative motion. Reference [28] focussed on the equations for the relative motion and argued that the quantum potential is at the origin of the gravitational interactions. Estimating the energy due to gravitational interaction between two protons at a distance of 1𝑓𝑚=10⁻¹⁵m gives |𝐸|∼10⁻³¹eV, whereas estimating it from the quantum potential yields |𝐸|∼10⁻³³eV, which is not too far off. While this is an intriguing proposition, it seems that the constructive approach to incorporate gravity into the formalism is to extend the cocycle and quadratic identities to curved backgrounds, which is left for future work. Instead I focus here on the equations for motion of the centre of mass, which take the form 12𝑚₁+𝑚₂∇_𝑟c.m.𝑆₀²−ℏ𝐸−²2𝑚₁+𝑚₂Δ_𝑟c.m.𝑅𝑅1=0,𝑚₁+𝑚₂∇_𝑟c.m.⋅𝑅²∇_𝑟c.m.𝑆₀=0. (98) The main point to note is that the effective mass in the relative motion is the reduced mass, whereas the effective mass in the centre of mass motion is additive, which is what we expect physically. Hence, for large number of particles the mass and potential energies are summed. In the equivalence postulate framework the mass and quantum energy of elementary particles are correlated and are different attributes of the same particle. Finally, the same arguments could have been presented by using the relativistic form of the equations. However, nonrelativistic treatment is viable given that the majority of matter in the universe is expected to be nonrelativistic.

5. Conclusions

Contemporary observations at the smallest and highest length scales are well accounted for by the standard model of particle physics and by general relativity, respectively. Yet the synthesis of these two basic theories remains elusive. 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. A somewhat heuristic view of the equivalence postulate procedure is the following. In the classical Hamilton-Jacobi formalism one seeks a transformation from an a nontrivial Hamiltonian to a trivial Hamiltonian. The transformations are canonical and treat the phase space variables as independent variables. This ensures that the classical equations of motion are preserved. It ensures that the classical path is selected. The key property of quantum mechanics is that the phase space variables are not independent but satisfy the basic quantum mechanical relation 𝑞,𝑝=𝑖ℏ. (99) Therefore, we can view the equivalence postulate procedure as implementing the Hamilton-Jacobi trivialisation algorithm, but taking the phase space variables to be related by the generating function 𝑆(𝑞), via relation (4). The solution to this problem is not given by the CSHJE, (3), but rather by the QSHJE, (7). Covariance of the QSHJE then implies that all physical systems labelled by the potential function 𝑊(𝑞) are connected by the coordinate transformations. This view is in line with the picture that quantum mechanically all possible paths are allowed, and we may imagine that different paths correspond to different 𝑊(𝑞) labels. The equivalence postulate formalism is encoded in two basic identities. The first is the cocycle condition, (9), which embodies the symmetry properties underlying quantum mechanics in this formalism. The second is the quadratic identity, (12), which is the manifestation of the quantum Hamilton Jacobi equation and is compatible with the equivalence hypothesis. Both of these key ingredients admit suitable generalisations in higher dimensions with Euclidean or Minkowski metrics. A key role in this approach is played by the various dualities [8–14, 29], which are implemented as Legendre transformations. These Legendre dualities represent a complementary facet of the underlying physical principles. In this respect they show that the different physics variables used to characterise the physical systems are related by the dualities and none is superior to the other.

In view of the observational supremacy of the standard model and general relativity in their respective domains, we may question how the equivalence postulate formalism is manifested observationally. Given that quantum gravity effects are likely to be notoriously small, we may expect them to generate deviations in the most extreme regimes of contemporary observations. Indeed, recently the OPERA collaboration announced intriguing results on the superluminal propagation of neutrinos [30]. While the fate of the results of this experiment is in doubt [31], it is clear that the neutrino sector will provide experimental data that will continue to probe the validity of the standard model. In the context of the equivalence postulate formalism it was shown in [32–36] that the quantum potential in this formalism leads to dispersion relations that modify the classical relativistic energy-momentum relation. Such modifications of the relativistic energy-momentum relation are expected in different approaches to quantum gravity and the relevant question is whether they are sufficiently large to be observable. In this paper I proposed that the quantum potential that arises in the equivalence postulate formulation can be regarded as an intrinsic curvature term, which characterises elementary particles, and may be interpreted as dark energy.

Acknowledgments

The author would like to thank the theoretical Physics Department at Oxford University for hospitality. This work was supported in part by the STFC (PP/D000416/1).