Quantum Mechanics Without Wavefunctions

We present a self-contained formulation of spin-free nonrelativistic quantum mechanics that makes no use of wavefunctions or complex amplitudes of any kind. Quantum states are represented as ensembles of real-valued quantum trajectories, obtained by extremizing an action and satisfying energy conservation. The theory applies for arbitrary configuration spaces and system dimensionalities. Various beneficial ramifications - theoretical, computational, and interpretational - are discussed.


I. INTRODUCTION
For nearly a century, quantum mechanics has presented philosophical and interpretational conundrums that remain as controversial as ever. Far from disappearing into the realm of esoteric academic debate, recent experimental advances, e.g. in entanglement, decoherence, and quantum computing, have brought such questions to the forefront of topical interest. The various competing viewpoints-Copenhagen, 1  The aim of this paper is to show that for non-relativistic spin-free quantum mechanics, the above can be realized (A different approach is described in Ref. 6.) In particular, a quantum state can be represented as an ensemble of real-valued quantum trajectories, satisfying a self-contained partial differential equation (PDE). This was shown in a recent paper 7 by one of the authors (Poirier) for both the one-dimensional (1D) TISE and TDSE. It transpired that similar work had already been done for the TISE in 1D, 8 the TDSE in 3D, 9 and in greater generality. 10 In the current paper, we simplify, unify, and generalize these previous constructions, presenting quantum trajectory PDEs for arbitrary configuration spaces and system dimensionalities. The goal here is to present an alternative, standalone reformulation of quantum mechanics, that neither relies on the TDSE nor makes any mention of any external constructs such as Ψ, and which, in addition, is likely to provide far-reaching benefits for numerical calculations, e.g. of accurate quantum scattering dynamics for chemically reactive molecular systems.
The relevant PDEs can be derived from an action principle. We discuss symmetries of the action principle, associated conserved quantities, and other properties, such as a heretofore unexpected Hamiltonian structure in the case of 1D time-independent quantum mechanics (TIQM), which also serves as the basis of an accurate many-D numerical scheme. We single out a specific choice of Lagrangian or gauge, and show why this choice may be regarded as physically preferred. Likewise, we single out a specific choice of trajectory-labeling coordi-nate(s), in terms of which the resultant PDE exhibits no explicit coordinate dependences.
We present simple analytical solutions, and discuss some initial numerical results that appear very promising for molecular and chemical physics applications. The reformulation of quantum mechanics in terms of trajectory ensembles, in addition to shedding light on complex theoretical issues, evidently also provides important practical benefits.

II. THE 1D TIME-INDEPENDENT CASE
As shown in Ref. 7, 1D TIQM states can be represented uniquely with a single trajectory, x(t). The theory is one of a broad class of dynamical laws, for which the Lagrangian and energy are of the form Equations (1) and (2) are natural generalizations of the well-known classical forms. The quantum correction, Q, is similar to the potential, in that it appears with opposite signs in L and E, but is actually related to the TISE kinetic energy operator. 4 It has a universal "kinematic" form, i.e. no explicit x dependence.
As in classical theory, the quantum trajectories, x(t), are obtained via extremization of the action, S = L dt. Since L is autonomous, the resultant x(t) solutions exhibit timetranslation invariance and energy conservation for any choice of Q. However, for general Q, the conserved energy for Eq. (1) obtained via Noether's theorem does not take the form of Eq. (2). Requiring this equivalence imposes rather special conditions on Q, e.g., that Q must be invariant under time rescaling. The simplest, well-behaved, nontrivial Q giving the form of Eq. (2) is Hereh is, in principle, an arbitrary positive constant. However, with the usual identification ofh as Planck's constant, Eq. (3) is equivalent to the quantum potential of Bohmian mechanics 2-4 for the 1D Cartesian TISE. 7 This is surprising, given that the expression Eq. (3), being universal and kinematic, is determined entirely by the trajectory, x(t). In particular, no reference to the wavefunction, Ψ, or the TISE itself, is used in this derivation.
Our quantum trajectories are nevertheless Bohmian trajectories, although our formulation and interpretation are not at all that of Bohmian mechanics, because no Ψ is involved.
Action extremization applied to Eqs. (1) and (3) yields the following fourth-order autonomous ODE, describing 1D TIQM quantum trajectories: Any fourth-order ODE can be rewritten as a set of four coupled, first-order ODE's.
Remarkably, Eq. (4) can be rewritten: which are Hamilton's equations for a 2D system, for the Hamiltonian In the above equations, (x, p) are the "classical" dimension phase space variables, and (r, s) correspond to an additional, "quantum" dimension, essentially describing quantum interference. Note that Eq. (7) reduces to the classical Hamiltonian when r = 0 and s = p.
In terms of the time derivatives of x(t), Substitution of Eq. (8) into Eq. (7) then reveals H to be the conserved Noether energy of Eq. (2). For the free particle case [∂V /∂x = 0], p is a second conserved quantity, in involution with H (and also derivable from Noether's theorem). The importance of p is difficult to overstate; it represents the "particle momentum," analogous to the well-known "particle energy," E = H. 3,4 Yet remarkably, p has barely been considered 8 in the previous literature, which generally regards s = mẋ as particle momentum. Note that p is conserved for all free particle TIQM states, including those exhibiting interference, whereas s is conserved only for plane wave states-i.e., the classical special case for which p = s. to an approximate simulation scheme that has also proven to be remarkably accurate (i.e., to two or three digits). 11

III. THE 1D TIME-DEPENDENT CASE
For the case of 1D time-dependent quantum mechanics (TDQM), any self-contained formulation must involve a PDE, rather than an ODE. It is no longer possible to exactly represent a quantum state as a single trajectory, x(t), but rather as a one-parameter ensemble of trajectories, x(C, t), where the real-valued, space-like coordinate C labels individual trajectories. The equation of motion should be a PDE involving C and t derivatives, preferably derived from a field-theoretic action principle.
In Refs. 7, 9, and 10, C was chosen as the initial trajectory value [x(C, 0)=x 0 =C]. The resultant PDE is complicated, exhibits explicit x 0 dependence through the initial probability density, ρ 0 (x 0 )=ρ(x 0 , 0), and bears little resemblance to Eq. (4). In addition, Q is expressed in terms of C rather than t derivatives of x. The PDE can be simplified by a better choice of the trajectory parameter, C, which in general can be taken to be any monotonic function of x 0 (regardless of the initial wavefunction). A crucial idea of the current paper is that C should be chosen so as to uniformize the probability density. In particular, since then C takes values from 0 to 1 (for normalized wavepackets), and ρ C (C) = 1.
Working with Eq. (9) (or any uniformizing choice of C), and writing x ′ = ∂x/∂C,ẋ = ∂x/∂t etc., the PDE of Ref. 7 simplifies very substantially to Equation (10) c.f. Eqs. (1) and (3). This action is invariant under translations of both coordinates t and C. By Noether's theorem this gives rise to two conservation laws, which are easily found to be, respectively, ∂ ∂t The first corresponds to conservation of energy. In the free particle case, there is also a momentum conservation law, arising from x-translation symmetry: We now consider Gaussian wavepacket evolution under the free particle (V = 0) and harmonic oscillator (V = 1 2 mω 2 x 2 ) potentials. The respective x(C, t) solutions are and (15) where x 0 , p 0 , t 0 , a are real wavefunction parameters, and 0 ≤ C ≤ 1. Note these solutions diverge as C → 0 or 1. In general, x(C, t) must diverge at the C endpoints; modulo this requirement, any solution of Eq. (10) (or its arbitrary-C generalization) can be used to reconstruct a normalized solution Ψ(x, t) of the 1D TDSE.
We have successfully numerically integrated Eq. (10) for an Eckart potential and initial Gaussian wavepacket, using the Stormer-Verlet and other methods. Initial results compare favorably with those obtained using standard Ψ-based methods, but further improvements in efficiency and accuracy are planned. In any case, these calculations represent a milestone achievement, as the first successful synthetic Bohmian quantum trajectory calculations ever achieved for a system with substantial reflection interference-a much-sought goal eluding chemical dynamics researchers for over a decade, and a major hurdle preventing exact quantum wavepacket calculations for large molecular systems with few reaction pathways. 4,12 IV. THE MANY-D TIME-DEPENDENT CASE The 1D analysis generalizes to many-D. The single variable x is replaced with the ndimensional configuration space vector x, with C likewise replaced with C, so that x(C, t) represents an n-parameter family of trajectories. One option 9 is to take C = x 0 , though we wish to consider more general choices for which C and x 0 are related via any invertible coordinate transformation.
In analogy with the 1D case, it can be shown that for an arbitrary choice of the parametrization C, where J is the Jacobi matrix, J i j = ∂x i /∂C j . As in the 1D case, we mostly work with a uniformizing C for which ρ C (C) = 1. The resulting perturbed Newton equation, i.e. the many-D generalization of Eq. (10), can be written in various different forms, the most compact being Here K = J −1 denotes the inverse Jacobi matrix. The Einstein summation convention is used, albeit with some mismatched indices as we are currently assuming the Euclidean metric on x space.
Equation (18) is the variational PDE for the action Equation (19) is the many-D generalization of Eq. (11), which preserves the L = T −V −Q

V. CONCLUDING REMARKS
We have developed a self-contained, trajectory-based formulation of spin-free nonrelativistic TDQM, achieving all goals as outlined in the Introduction. Further developments are underway. Theoretical progress will require a correct treatment of spin, relativity, particle indistinguishability, and second quantization-with a promising start having been made by Holland and others. 3,9,10 The invariance of Eq. (21) under volume-preserving diffeomorphisms suggests a connection with gravity, though the physical significance of C is not yet clear.
Numerically, the prospect of stable, synthetic quantum trajectory calculations for many-D molecular applications will be fully explored, as the benefits here could prove profound. 4,7,11 Our formalism offers flexibility for restricting action extremization to trajectory ensembles of a desired form (e.g., reduced dimensions), thereby providing useful variational approximations. Our exact TDQM equations are PDEs, not single-trajectory ODEs-the entire ensemble must be determined at once. But they provide the great advantage of making no reference to any external fields, such as densities or wavefunctions. Alternatively, the many-D Hamiltonian ODE approach is approximate, but evidently quite accurate. 11 Regarding interpretation, we draw no definitive conclusions here. However, it is clearly of great significance that the form of Q can be expressed in terms of x and its C derivativesimplying the key idea that the interaction of nearby trajectories, rather than particles, is the