Enhancing wave-particle duality

Technical Summary: Enhancing Wave-Particle Duality via Configuration Space Quantisation

Problem Statement
The paper addresses a fundamental inconsistency between the quantum descriptions of waves and particles, specifically regarding the quantisation of mechanical point particles. In standard quantum mechanics (basic QM), the quantisation of mechanical particles relies on Hamiltonian formalism, identifying position $\hat{x}$ and momentum $\hat{p}$ as canonically conjugate operators. However, this approach creates a disconnect with classical Newtonian mechanics, where dynamics are naturally described by position $x$ and velocity $v$.

The authors highlight several specific issues in basic QM:

1. Inconsistency with Classical Trajectories: Ehrenfest's theorem, which requires quantum expectation values to follow classical equations of motion, holds strictly only for free particles or harmonic oscillators, or for very narrow wave packets. In general potentials, basic QM does not guarantee that expectation values evolve according to Newton's laws.
2. Causality and Localization: Basic QM implies that a particle initially localized in position will immediately spread to infinity (superluminal spreading) if the Hamiltonian is bounded from below, violating causality in a relativistic sense.
3. Momentum vs. Velocity: Basic QM conflates momentum and velocity by defining momentum as $p=mv$ with a fixed mass. This leads to inconsistencies in scenarios like free fall, where spatial translational symmetry implies momentum conservation, yet the velocity (and thus $mv$) increases linearly due to acceleration.
4. Wave-Particle Duality: The quantisation of mechanical particles differs significantly from the quantisation of light (photons) in quantum electrodynamics (QED). Photons are quantised based on their propagation, whereas mechanical particles in basic QM are quantised via canonical conjugation, leading to a lack of unified treatment for wave-particle duality.

Methodology
The authors propose a "physically-motivated" quantisation scheme for mechanical point particles that mirrors the approach used for light in QED. The methodology proceeds as follows:

1. Identification of Configuration Space: Instead of starting with Hamiltonian phase space $(x, p)$, the authors start with Newtonian mechanics, where the state of a particle is defined by its position $x$ and velocity $v$. They define the configuration space as the set of all possible classical states $(x, v)$.
2. Promotion to Quantum States: The classical distinguishable states $(x, v)$ are promoted to pairwise orthogonal quantum basis states $|x, v\rangle$. These states form a complete basis for an extended Hilbert space $\mathcal{H}$, which is significantly larger than the Hilbert space of basic QM.
3. Definition of Observables:
   • Position ($\hat{x}$) and Velocity ($\hat{v}$): Defined as diagonal operators in the $|x, v\rangle$ basis.
   • Momentum ($\hat{p}$): Defined as the generator of spatial translations, analogous to basic QM but acting on the extended space.
   • Acceleratum ($\hat{a}$): A new operator defined as the generator of velocity changes (analogous to how $\hat{p}$ generates position changes).
4. Derivation of the Dynamical Hamiltonian ($\hat{H}_{dyn}$): The authors demand that the "most classical" quantum states $|x(t), v(t)\rangle$ evolve according to Newton's equations of motion ($\dot{x}=v, \dot{v}=f(x)$). By enforcing this condition on the Schrödinger equation, they derive a dynamical Hamiltonian:
   $$\hat{H}_{dyn} = \hat{v}\hat{p} + f(\hat{x})\hat{a}$$
   where $f(x) = -\frac{1}{m}\frac{\partial V}{\partial x}$. This Hamiltonian is unbounded from below (having both positive and negative eigenvalues), distinguishing it from the standard energy observable.
5. Recovery of Basic QM: The paper demonstrates that basic QM emerges as a semi-classical approximation of this new formalism. This occurs when:
   • The Hilbert space is restricted to subspaces where $|p/v| = m$ (fixed mass).
   • The velocity operator is replaced by $\hat{v} = \hat{p}/m$.
   • The dynamical Hamiltonian is replaced by the standard Hamiltonian $\hat{H}_{BQM} = \frac{\hat{p}^2}{2m} + V(\hat{x})$.

Key Contributions and Results

• Enhanced Wave-Particle Duality: The formalism treats mechanical point particles and photons on an equal footing. Both are described by states in a configuration space where the dynamics are governed by a similar structure. The only difference is that photons are restricted to a single speed $c$, while mechanical particles can have any velocity.
• Consistency with Classical Mechanics: By construction, the expectation values of position and velocity in this framework evolve exactly according to Newton's equations of motion for any initial state, satisfying the correspondence principle more robustly than basic QM.
• Resolution of Momentum-Velocity Ambiguity: The paper establishes that position $\hat{x}$ and velocity $\hat{v}$ are commuting observables ($[\hat{x}, \hat{v}] = 0$), allowing a particle to be simultaneously localized in position and velocity. Conversely, momentum $\hat{p}$ and velocity $\hat{v}$ are independent commuting observables. This resolves the paradox in free fall where momentum is conserved (due to translational symmetry) while velocity changes.
• Uncertainty Relations: The formalism retains Heisenberg's uncertainty principle but redefines the conjugate pairs. The canonical pairs are $(\hat{x}, \hat{p})$ and $(\hat{v}, \hat{a})$, satisfying $[\hat{x}, \hat{p}] = i\hbar$ and $[\hat{v}, \hat{a}] = i\hbar$. However, $[\hat{x}, \hat{v}] = 0$ and $[\hat{p}, \hat{v}] = 0$. This implies that while position and momentum cannot be known simultaneously with arbitrary precision, position and velocity can.
• Causality and Localization: Unlike basic QM, where localized wave packets spread instantaneously, the dynamical Hamiltonian in this framework allows for localized wave packets to remain localized over time, avoiding superluminal spreading issues associated with bounded-from-below Hamiltonians.

Significance and Claims
The authors claim that this formulation enhances wave-particle duality by providing a unified quantisation scheme for both light and matter. The paper argues that basic QM is not a fundamental theory but a semi-classical approximation of "QM in configuration space" that arises when specific constraints (fixed mass, specific Hamiltonian form) are applied.

The significance of this work lies in:

1. Theoretical Consistency: It offers a framework where quantum dynamics are strictly consistent with classical Newtonian trajectories for expectation values, addressing limitations of Ehrenfest's theorem in standard QM.
2. Unified Description: It bridges the gap between the quantisation of electromagnetic fields (photons) and mechanical particles, treating them as excitations in a configuration space with similar dynamical generators.
3. Potential Applications: The authors suggest this approach may provide new insights into quantum transport, non-Hermitian quantum physics (where Hamiltonians with complex eigenvalues are used), quantum chaos, and relativistic quantum information. They note that the formalism naturally accommodates Hamiltonians with positive and negative eigenvalues, which are common in phenomenological models in condensed matter physics (e.g., Hubbard models) but are often viewed as problematic in standard energy-based interpretations.

The paper concludes that while basic QM has been successful, the configuration space formulation offers a more fundamental and consistent description of mechanical point particles, clarifying the distinct roles of momentum and velocity and restoring a stronger link between quantum and classical mechanics.