Path Integrals Reimagined: A New Framework for Quantization

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Author: Denis Avetisyan

Researchers have developed a novel approach to geometric quantization by directly constructing a prequantum groupoid from the space of paths, offering a unifying framework for diverse quantum mechanical concepts.

This work extends geometric quantization to general parasymplectic spaces using diffeology, period groups, and momentum maps.

Traditional approaches to geometric quantization struggle with generalizing beyond simply connected spaces and incorporating the full complexity of dynamical systems. This paper, ‘Geometric Quantization by Paths Part II: The General Case’, addresses this limitation by constructing a prequantum groupoid directly from the space of paths over arbitrary parasymplectic diffeological spaces. The resulting framework identifies a ‘quantum system’ embodied by a groupoid whose automorphisms correspond precisely to the symmetries of the classical dynamical system. Does this path-based construction offer a novel and geometrically natural route towards a fully consistent theory of quantum mechanics?

Beyond Classical Limitations: A Geometric Foundation for Quantization

Conventional quantization procedures, designed to bridge the gap between classical and quantum mechanics, frequently encounter difficulties when confronted with geometries that deviate from simple, well-behaved forms. These methods often rely on assumptions about the smoothness and commutativity of phase space, which break down in systems exhibiting complex topological features or non-commutative algebraic structures. Consequently, applying these techniques to systems with irrational periods, singularities, or intrinsically non-commutative dynamics often leads to inconsistencies or requires the introduction of ad hoc modifications. The limitations stem from a fundamental reliance on classical notions of phase space, hindering the development of a truly general quantization scheme capable of consistently describing a broader range of physical systems, and necessitating the exploration of alternative geometric frameworks.

The conventional framework of symplectic geometry, while successful in describing many classical systems, reveals inherent limitations when attempting to fully encompass the subtleties of quantum mechanics. This geometry, defined by the preservation of area in phase space, struggles to adequately represent the non-commutativity and inherent uncertainty that define quantum behavior. Attempts to directly quantize systems solely through symplectic methods often encounter inconsistencies, particularly when dealing with complex potentials or systems exhibiting chaotic dynamics. The geometry’s reliance on well-defined trajectories and conserved quantities breaks down in regimes where quantum effects dominate, necessitating ad-hoc modifications or the introduction of topological considerations to achieve consistent quantization. Ultimately, while serving as a valuable approximation in certain contexts, a purely symplectic approach proves insufficient to capture the full richness and predictive power demanded by a complete description of quantum phenomena, hinting at the need for a more generalized geometric structure.

A novel approach to quantization centers on parasymplectic geometry, offering a more robust foundation for constructing quantum systems than traditional methods. Unlike classical symplectic geometry, which struggles with systems exhibiting irrational periods or singularities, parasymplectic geometry inherently accommodates these complexities without the need for supplementary topological constructions. This framework allows for a consistent quantization procedure applicable to a broader range of physical systems, effectively extending beyond the limitations of classical phase space. By relaxing the constraints of symplectic structure, parasymplectic geometry provides a more fundamental and versatile platform for bridging the gap between classical and quantum mechanics, potentially unlocking new avenues for understanding complex quantum phenomena and consistently describing systems previously considered intractable.

Constructing the Prequantum Groupoid: A Path-Based Formulation

The Prequantum Groupoid functions as the foundational mathematical object for representing a quantum system, effectively encoding its prequantum geometric properties. This groupoid is not merely a descriptive tool; it is the system, with its elements representing the possible configurations and transformations. Specifically, the structure of the groupoid – its objects, morphisms, composition laws, and identities – directly corresponds to the geometric features of the quantum system before quantization. This allows for the treatment of quantum mechanics as a consequence of geometric principles, where the groupoid’s structure dictates the system’s dynamics and observable quantities. \text{Prequantum Groupoid} \rightarrow \text{Quantum System}

The prequantum groupoid is built upon the space of paths, defined as mappings from an interval [0,1] to the system’s configuration space. Utilizing this path space allows the groupoid to inherently capture the system’s dynamics by representing all possible trajectories. Furthermore, the topology of the configuration space is directly incorporated; paths that can be continuously deformed into one another are considered equivalent, effectively encoding the connectivity and global structure of the system’s possible states. This path-based construction provides a framework where relationships between different points in the configuration space are determined not by direct adjacency, but by the existence of continuous paths connecting them, a crucial aspect in prequantum mechanics as it moves beyond simple point-particle descriptions.

The Chain Homotopy Operator, denoted ∂, plays a fundamental role in constructing the prequantum groupoid by establishing a relationship between differential forms and paths within the system. Specifically, ∂ acts on forms, mapping them to paths, and conversely, relates paths back to forms, allowing for the definition of composition rules and ensuring that the geometric structure remains consistent under variations. This operator effectively encodes the notion of boundaries and coboundaries, crucial for defining the algebraic structure of the groupoid and verifying that the prequantum structure satisfies necessary geometric constraints, such as the closure of the groupoid under composition and the existence of inverses.

Diffeology addresses the challenges presented by path spaces which, in prequantum constructions, are frequently infinite-dimensional and may contain singularities due to the nature of generalized functions and non-smooth paths. Traditional differential geometry requires well-defined manifolds with smooth structures; however, diffeology provides a weaker, more flexible framework based on plots – smooth maps from ℝ to the path space. This allows for the consistent definition of derivatives and tangent spaces even on spaces lacking a conventional smooth manifold structure. The use of plots circumvents the need for atlases and transition functions, simplifying calculations and permitting the inclusion of paths exhibiting discontinuities or non-differentiability at isolated points, which are crucial for representing certain quantum phenomena and ensuring a geometrically sound foundation for the prequantum groupoid.

Encoding Quantum Properties: Periods, Symmetry, and Topological Invariants

The quantum phase space within this framework is formally defined by the Total Group of Periods, which integrates three distinct period types: spherical, toric, and surfacic. Spherical periods relate to the S^2 sphere and represent the simplest form of cyclic behavior. Toric periods, associated with tori T^n, introduce multi-dimensional cyclic variables. Surfacic periods, arising from Riemann surfaces, represent more complex, non-trivial cycles and are crucial for describing systems with higher-dimensional phase spaces. The combination of these periods provides a complete basis for describing all possible quantum states and their evolution within the defined system, effectively mapping the geometric properties of these periods onto the quantum realm.

The Moment Map establishes a correspondence between paths in the quantum phase space and elements of the system’s symmetry group, effectively identifying conserved quantities. Mathematically, it is a continuous function \mu : M \rightarrow \mathfrak{g} , where M represents the phase space manifold and \mathfrak{g} is the Lie algebra of the symmetry group. Crucially, the critical points of the Moment Map correspond to constant values of these conserved quantities, directly linking quantum states to classical mechanical invariants like energy and angular momentum. This connection allows for the application of classical techniques – such as Hamiltonian mechanics and Noether’s theorem – to analyze and understand the behavior of the quantum system, and provides a rigorous framework for quantization procedures.

The fundamental group, denoted as \pi_1(X), characterizes the topology of the space X and directly impacts the calculation of surfacic periods. These periods, which represent integrals over surfaces, are sensitive to the presence of non-contractible loops within the space, as identified by the fundamental group. Specifically, non-trivial fundamental groups indicate the existence of loops that cannot be continuously shrunk to a point, leading to multi-valuedness in the surfacic periods. This multi-valuedness manifests as non-commutativity in the associated quantum mechanical operators, as the order of operations becomes significant due to the path dependence introduced by the non-contractible loops. Therefore, the structure of the fundamental group dictates the commutation relations and fundamentally shapes the quantum behavior within this framework.

The Diffeomorphism Group, consisting of smooth, invertible mappings, guarantees the geometric invariance of the quantum system. This group acts on the space of possible configurations, ensuring that physical observables remain consistent under continuous deformations. Formally, a diffeomorphism φ maps a region of space without tearing or folding, preserving local geometric properties like angles and distances. Consequently, the Diffeomorphism Group establishes a framework where the system’s behavior is independent of coordinate choices, a fundamental requirement for a physically meaningful model. Maintaining diffeomorphism invariance is crucial for eliminating spurious solutions and ensuring the robustness of the quantum description under geometric transformations.

Beyond Dirac: A Geometrically Robust Foundation for Quantum Theory

Conventional Dirac quantization, while successful in many contexts, relies on specific geometric constraints that limit its applicability to more complex physical systems. This work presents a significant departure, establishing a framework that transcends those limitations by prioritizing the intrinsic geometric properties of the system itself. Rather than imposing constraints a priori, the approach leverages the structure of prequantum groupoids to naturally accommodate systems with intricate geometries and symmetries – including those possessing irrational periods or singularities. This geometric robustness not only broadens the scope of quantizable systems but also offers a more consistent and potentially more accurate description of quantum phenomena, particularly in scenarios where traditional methods falter or require artificial topological constructions. The result is a generalization of Dirac quantization, providing a powerful new tool for exploring the foundations of quantum mechanics and tackling previously inaccessible problems in theoretical physics.

The foundational Path Integral formulation of quantum mechanics, traditionally introduced as a postulate, arises unexpectedly from the intrinsic structure of the Prequantum Groupoid. This mathematical object, built upon the geometry of a system’s configuration space, inherently encodes the rules for summing over all possible paths a quantum particle can take. The groupoid’s composition law directly translates into the weighting factors within the Path Integral \in t \mathcal{D}[q(t)] e^{iS[q(t)]} , where S[q(t)] represents the action. Consequently, the familiar principles of quantum field theory are not merely imposed, but emerge as a natural consequence of this geometric framework, offering a deeper understanding of how quantum amplitudes are calculated and revealing a previously unseen connection between geometry and quantum dynamics.

A deeper understanding of quantum mechanics arises when attention shifts from merely calculating probabilities to examining the geometric structure of the space encompassing all possible paths a quantum system can take. This approach reveals that the very foundations of the theory are deeply intertwined with the shape and properties of this path space, rather than existing as an abstract mathematical formalism. By treating paths not simply as trajectories, but as geometric objects with intrinsic characteristics, researchers can uncover relationships previously hidden within the standard formulation. This geometric perspective allows for a more intuitive grasp of quantum phenomena, potentially resolving long-standing conceptual difficulties and paving the way for novel approaches to quantization, particularly in scenarios involving complex geometries or non-conventional symmetries, where traditional methods often falter.

The presented framework transcends limitations inherent in conventional quantization techniques by providing a robust pathway for systems characterized by intricate geometries and symmetries. Unlike standard methods which often struggle with, or require ad-hoc solutions for, systems possessing irrational periods or singularities, this approach consistently quantizes these complex scenarios without necessitating the introduction of auxiliary topological structures. Recent results demonstrate the efficacy of this method, offering a fundamentally geometric approach to quantization that successfully navigates challenges previously considered insurmountable, thereby expanding the scope of quantifiable physical systems and potentially revealing new insights into the nature of quantum reality.

The presented work meticulously constructs a prequantum groupoid from the space of paths, a decidedly non-trivial undertaking demanding mathematical rigor. This approach, extending geometric quantization to general parasymplectic spaces, resonates with James Clerk Maxwell’s assertion: “The true voyage of discovery… never ends.” The pursuit of a comprehensive framework, unifying concepts like period groups and momentum maps within quantum mechanics, isn’t merely about finding a functional method; it’s about establishing a logically sound, scalable foundation. The elegance lies not in the complexity of the calculations, but in the demonstrable correctness of the underlying principles, ensuring the theory holds true beyond specific examples and limitations.

What Lies Ahead?

The construction of a prequantum groupoid directly from path space, as demonstrated, offers a certain… elegance. Yet, the devil, as always, resides in the details – specifically, the translation of this geometric structure into genuinely predictive quantum phenomena. The current formalism handles the machinery, but the critical question of extracting measurable quantities – probabilities, energies – remains largely a matter of further, rigorous derivation. To claim unification is premature; demonstrating equivalence with established methods, and crucially, surpassing them in predictive power, is the only true validation.

A significant limitation lies in the inherent complexity of dealing with general parasymplectic spaces. While the framework allows for such generality, the computational cost of actually performing calculations in these spaces may prove prohibitive. The search for simplifying assumptions, or for classes of parasymplectic manifolds that admit tractable solutions, will be paramount. Moreover, the connection to path integrals, while suggestive, demands a more precise mathematical formulation. Simply being a path integral is insufficient; it must yield results consistent with established quantum theory.

In the chaos of data, only mathematical discipline endures. The next step isn’t merely expanding the formalism, but subjecting it to the unforgiving scrutiny of concrete problems. Until this geometric quantization yields not just equations, but predictions that surpass existing models, it remains a beautiful, but ultimately incomplete, symphony of symbols.

Original article: https://arxiv.org/pdf/2512.24627.pdf

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

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2026-01-04 01:00