Rewriting Quantum Reality: A New Path Beyond Wavefunction Collapse

Author: Denis Avetisyan

A novel variational approach to quantum mechanics proposes that randomness isn’t inherent, but emerges from fundamental limits at the smallest scales.

This work introduces a time-symmetric formulation leveraging Fisher information and primal-dual optimization to derive Schrödinger dynamics and objective boundary conditions.

The persistent challenge of interpreting quantum mechanics stems from the seemingly arbitrary imposition of dynamics and the measurement problem. This paper, ‘A Time-Symmetric Variational Formulation of Quantum Mechanics with Emergent Schrödinger Dynamics and Objective Boundary Randomness’, presents a novel framework where Schrödinger dynamics and Bohmian trajectories emerge as optimal solutions to a time-symmetric variational principle. Crucially, this formulation generates the Born rule intrinsically and posits that fundamental randomness arises not from wavefunction collapse, but from scale limits enforced at boundary conditions. Could this approach offer a pathway towards a more fundamental understanding of quantum reality, bridging the gap between deterministic and probabilistic descriptions?

The Illusion of Wavefunctions: Beyond Computational Limits

The conventional framework of quantum mechanics, while remarkably successful, often presents significant challenges due to its reliance on wavefunctions. Describing a quantum system necessitates solving the $\text{Schrödinger equation}$ for these wavefunctions, a process that scales exponentially with the number of particles – quickly becoming computationally intractable for even moderately complex systems. Furthermore, the wavefunction itself, a complex-valued function representing the probability amplitude, doesn’t directly reveal the underlying physical dynamics. Interpreting its behavior requires additional calculations to extract observable quantities, obscuring the intuitive connection between the quantum state and the system’s evolution. This computational burden and lack of direct physical insight have motivated the search for alternative formulations that can offer both efficiency and a more transparent understanding of quantum phenomena.

Quantum systems, traditionally described by the often-abstract mathematical construct of a wavefunction, can alternatively be understood through a ‘Hydrodynamic Description’. This approach recasts the behavior of quantum particles not as waves, but as the flow of a quantum fluid, defined by its density and current. Analogous to how fluid dynamics governs the movement of water or air, this formulation represents the quantum state using fields that describe the probability density of finding a particle and the associated flow of probability – essentially, how quickly and in what direction the particle is likely to move. By framing quantum mechanics in this more intuitive, fluid-like manner, researchers aim to unlock new computational strategies and potentially reveal deeper connections between quantum phenomena and the classical world, offering a fresh perspective on the fundamental laws governing reality.

By recasting quantum problems through the lens of hydrodynamic variables – density and current – researchers are effectively transforming traditionally complex calculations into optimization challenges. This approach moves away from directly solving the $\text{Schrödinger equation}$ , instead focusing on finding the density and current distributions that minimize a specific cost function. This reformulation is particularly powerful because optimization algorithms, well-developed in fields like machine learning, can then be applied to efficiently approximate solutions, even for systems where exact analytical solutions are intractable. Consequently, this hydrodynamic formulation offers a pathway to tackle increasingly complex quantum systems and explore phenomena previously beyond the reach of conventional computational methods, potentially accelerating advancements in materials science, quantum chemistry, and beyond.

From Quantum Systems to Optimization Landscapes

The primal problem in formulating quantum systems as optimization tasks centers on directly minimizing or maximizing a functional dependent on hydrodynamic variables. These variables, typically representing quantities like density and velocity, are subject to constraints dictated by the underlying physical laws governing the system – such as the conservation of probability or energy. The objective function to be optimized is therefore constructed from these variables, and the solution represents the state of the quantum system that best satisfies the defined physical constraints. Formally, this can be expressed as minimizing $F[ρ, v]$ subject to constraints $C(ρ, v) = 0$ , where ρ represents density, $v$ represents velocity, and $C$ represents the set of physical constraints.

The introduction of the Dual Problem utilizes Lagrange multipliers to address continuity constraints inherent in formulating quantum problems as optimization tasks. This technique transforms inequality and equality constraints – defining permissible solutions – into penalty terms added directly to the original objective function. Specifically, for a constrained optimization problem minimizing $f(x)$ subject to $g_i(x) = 0$ and $h_j(x) \leq 0$ , the Lagrangian function $L(x, \lambda, \nu) = f(x) + \sum_{i} \lambda_i g_i(x) + \sum_{j} \nu_j h_j(x)$ is formed, where $\lambda_i$ and $\nu_j$ are Lagrange multipliers. Optimizing the Lagrangian with respect to both $x$ and the multipliers effectively incorporates the constraints into a single, unconstrained optimization problem, allowing for a more tractable solution process.

The Karush-Kuhn-Tucker (KKT) conditions represent a set of necessary conditions for a solution to be optimal in a constrained optimization problem. These conditions consist of equations and inequalities that must be satisfied at the optimal point. Specifically, they involve the gradient of the Lagrangian function-derived from the objective function and constraints-being equal to zero, primal feasibility (satisfying the original constraints), dual feasibility (non-negativity of Lagrange multipliers), and complementary slackness-stating that for each constraint, either the constraint is satisfied as an equality or the corresponding Lagrange multiplier is zero. Satisfying the KKT conditions does not guarantee a global optimum, but any local optimum must satisfy them; therefore, algorithms solving the dual problem often seek solutions that fulfill these conditions to identify potential optimal solutions. $\nabla_x L(x, \lambda) = 0$ , $g(x) \le 0$ , $\lambda \ge 0$ , and $\lambda_i g_i(x) = 0$ represent simplified forms of these conditions.

The Quantum Hamilton-Jacobi Equation: A Bridge to Classicality

The Karush-Kuhn-Tucker (KKT) conditions, arising from constrained optimization of the wavefunction’s phase, directly yield the Quantum Hamilton-Jacobi Equation (QHJE). This equation describes the time evolution of the phase $S$ of the wavefunction ψ, where $\psi = \exp(iS/\hbar)$ . Derivation involves treating the wavefunction’s phase as a functional variable subject to constraints related to the probability density. The QHJE, therefore, represents a fundamental equation governing the dynamics of quantum systems when viewed through the lens of optimal control and variational principles, linking classical Hamiltonian mechanics to quantum evolution via the phase.

The incorporation of $Fisher Information$ into the $Fisher Regularized Action$ serves to resolve ambiguities and ensure a well-posed problem when deriving equations of motion. This regularization approach quantifies the cost associated with constraining the probability density to a narrow, classical state; specifically, the action cost scales inversely with the square of the Gaussian approximation’s width, $σ²$ . A smaller $σ²$ indicates a more localized, classical state, and therefore incurs a higher action cost, reflecting the energetic penalty for reducing quantum uncertainty. This inverse relationship directly influences the derived quantum potential and ensures mathematical consistency in the variational principle.

Implementation of a variational principle for regularization yields a specific quantum potential form. This potential is expressed as $ℏ²/4mσ²(1-x²/2σ²)$ , where $ℏ$ is the reduced Planck constant, m is mass, x represents position, and $σ²$ is the variance. Derivation involves minimizing the Fisher Regularized Action, which incorporates Fisher Information as a regularization term, thereby establishing a well-defined framework for obtaining the equations of motion and subsequently, the quantum potential. The resulting potential directly relates the variance of the wavefunction to the quantum mechanical effects on the system.

The Illusion of Randomness: A Limit on Localization

Objective randomness, rather than stemming from fundamental unpredictability within a system, emerges as a consequence of the limitations inherent in defining a precise localization. The Maximum-Relative-Entropy Principle establishes this by suggesting that, given incomplete information, the probability distribution with the highest entropy – and thus the most uncertainty – best represents the system’s state. This principle doesn’t imply a lack of determinism at a deeper level; instead, it reveals that randomness arises from the observer’s inability to fully specify the initial conditions due to practical or fundamental limits on measurement precision. Consequently, what appears random isn’t a property of the system itself, but a reflection of the observer’s limited knowledge and the constraints imposed by the act of observation – a probabilistic description arising from a lack of complete localization rather than intrinsic chaos.

The nature of a system isn’t fixed, but rather shifts depending on the observer’s scale of examination – a principle known as scale dependence. This implies that the very ontology – the fundamental categories of being – changes as one zooms in or out. What appears deterministic at a macroscopic level, governed by classical physics, may dissolve into probabilistic behavior at the quantum scale, and vice versa. This isn’t merely a matter of differing levels of description; the underlying reality itself is scale-dependent, suggesting that properties aren’t intrinsic to the system but emerge from the interaction between the system and the observational framework at a given resolution. Consequently, understanding a system requires acknowledging that its characteristics are relational, defined not by what it is absolutely, but by how it manifests at a specific scale of analysis.

The perceived randomness inherent in quantum systems isn’t a fundamental property, but rather a consequence of the finite resolution of any measurement apparatus. Analyzing a Gaussian wave packet – a standard solution to the Schrödinger equation – reveals this principle; as the standard deviation σ of the packet approaches zero – representing an attempt at increasingly precise localization – the total action associated with the system diverges. This mathematical divergence indicates a physical limit to how precisely a system can be defined, effectively establishing a fundamental scale below which meaningful measurement becomes impossible. Consequently, what appears as randomness isn’t an intrinsic unpredictability of the universe, but the inevitable outcome of attempting to observe a system with insufficient resolution, forcing a probabilistic interpretation due to the limitations of the observational process itself.

The presented work, grounded in a time-symmetric variational principle, echoes a profound humility regarding the limits of theoretical constructs. It posits that inherent randomness, dictated by the Fisher information and a fundamental scale limit, isn’t a flaw in the description but a foundational aspect of reality. This aligns with the sentiment expressed by Paul Dirac: “I have not the slightest idea of what I am doing.” The article’s exploration of emergent Schrödinger dynamics, arising from a primal-dual optimization, suggests that even our most successful equations may be approximations, valid only within a specific domain, much like a classical description breaking down at the event horizon of a black hole. The boundaries of applicability, whether defined by scale or singularity, reveal the inherent provisional nature of physical law and human intuition.

What Lies Beyond?

The presented variational formulation, while offering an intriguing path toward objective randomness and circumventing the conventional wavefunction collapse postulate, does not eliminate the fundamental challenges inherent in interpreting quantum phenomena. The reliance on a primal-dual optimization scheme, linked to the Fisher information metric, merely shifts the locus of inquiry. The emergence of Schrödinger dynamics, while elegant, begs the question of its ultimate justification; is it a truly fundamental property, or an artifact of the chosen scale limit? Further investigation must address the physical interpretation of the boundary conditions imposed by the Fisher information, and whether these conditions can be derived from more foundational principles.

The connection to Bohmian mechanics, while noted, remains largely unexplored. A rigorous demonstration of equivalence, or divergence, under varying conditions could illuminate the strengths and weaknesses of both approaches. More importantly, the scale dependence inherent in this formalism introduces a potential for non-locality that demands careful scrutiny. Any attempt to reconcile this with relativistic causality will prove a formidable undertaking.

Ultimately, the paper serves as a reminder that any attempt to formulate a complete theory of quantum mechanics is, perhaps, an exercise in self-deception. The very act of imposing a variational principle, or selecting a particular metric, represents a subjective choice, a construction built upon assumptions that may themselves be illusory. Gravitational collapse forms event horizons with well-defined curvature metrics; similarly, any theoretical edifice may vanish beyond the limits of its own applicability. Singularity is not a physical object in the conventional sense; it marks the limit of classical theory applicability.

Original article: https://arxiv.org/pdf/2512.22320.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

The Illusion of Wavefunctions: Beyond Computational Limits

From Quantum Systems to Optimization Landscapes

The Quantum Hamilton-Jacobi Equation: A Bridge to Classicality

The Illusion of Randomness: A Limit on Localization

What Lies Beyond?

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