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arXiv:2606.17622

Quantum mechanics in configuration space in context

Arwa Bukhari, Margherita Moro, Max Davies, Alastair Wilson, Almut Beige

quant-phSubmitted Jun 16, 2026• Announced Jun 17, 2026• 0 citations
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Abstract

To enhance the way in which wave-particle duality is implemented in the modelling of quantum mechanical systems, Bukhari et al. [New J. Phys. 27, 084501 (2025)] recently introduced an alternative approach to quantum mechanics, namely quantum mechanics in configuration space. This formalism is based on a physically motivated quantisation of Newtonian mechanics and promotes the classical position-velocity states (x,v) to pairwise distinguishable quantum states. The resulting |x,v> states form the basis of the Hilbert space of individual quantum mechanical particles and evolve along classical trajectories. In this paper, we consider the modelling of a mechanical particle in free space and put quantum mechanics in configuration space into context. It is shown that this formalism increases the continuity between quantum and classical mechanics by avoiding a conceptual inconsistency associated with the definition of momentum in canonical quantisation. In addition, we emphasise that standard quantum mechanics and quantum mechanics in configuration space are based on two distinct formulations of classical mechanics.

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11 pages, 3 figures, 1 table

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Among the equivalent formulations of quantum mechanics, Heisenberg's matrix mechanics and quantum theory, based on the Schrödinger wave equation, are the most technically advanced.Since the advent of quantum mechanics, these two schemes have provided the mathematical tools and the primary basis for the description of quantum phenomena.The path integral method developed by Dirac and Feynman has important advantages in the quantisation of gauge theories.We discuss here the formulation of quantum mechanics in phase space, known as deformation quantisation.Deformation quantisation uses the Wigner-Weyl association rule (Weyl, 1927(Weyl, , 1929(Weyl, , 1931;;Wigner, 1932) to establish a one-to-one correspondence between the functions in the phase space and the operators in the Hilbert space.Wigner's function appears as the Weyl symbol of the density matrix.A consistent dynamical description of the systems with the help of the Wigner function leads to deformation quantization.A useful formulation of the Wigner-Weyl association rule was proposed by Groenewold (1946) andStratonovich (1957).Groenewold introduced a non-commutative associative -product (star-product) of the functions in the phase space (Groenewold, 1946).The evolution of the quantum systems is determined by the antisymmetric part of the -product (Groenewold, 1946;Moyal, 1949), known as the Moyal bracket.The Moyal bracket represents the quantum deformation of the Poisson bracket.Deformation quantisation preserves many features of classical Hamiltonian dynamics.The formulation of deformation quantisation is based on the Wigner function and the Moyal bracket (i.e., the -product).The Wigner-Weyl association rule is necessary to prove the equivalence of deformation quantisation and the standard formalisms of quantum mechanics.Extensive literature has reported on the formulation of quantum mechanics in the phase space and the -product.We refer the reader to excellent reviews by Bayen et al. 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InTech eBooks•Feb 22, 2012•0 citations
To enhance the way in which wave-particle duality is implemented in the modelling of quantum mechanical systems, Bukhari et al. [New J. Phys. 27, 084501 (2025)] recently introduced an alternative approach to quantum mechanics, namely quantum mechanics in configuration space. This formalism is based on a physically motivated quantisation of Newtonian mechanics and promotes the classical position-velocity states (x,v) to pairwise distinguishable quantum states. The resulting |x,v> states form the basis of the Hilbert space of individual quantum mechanical particles and evolve along classical trajectories. In this paper, we consider the modelling of a mechanical particle in free space and put quantum mechanics in configuration space into context. It is shown that this formalism increases the continuity between quantum and classical mechanics by avoiding a conceptual inconsistency associated with the definition of momentum in canonical quantisation. In addition, we emphasise that standard quantum mechanics and quantum mechanics in configuration space are based on two distinct formulations of classical mechanics.
Abstract To enhance the consistency between the quantum descriptions of waves and particles, we quantise mechanical point particles in this paper in the same physically-motivated way as we previously quantised light in quantum electrodynamics (Bennett et al 2016 Eur. J. Phys. 37 014001). To identify the relevant Hilbert space, we notice that mechanical particles can occupy any position x while moving at any velocity v. Afterwards, we promote the classical states (x, v) to pairwise orthogonal quantum states | x , v ⟩ and demand that these evolve according to Newton’s equations of motion. The resulting quantum theory is mass-independent, when Newton’s equations of motion are mass-independent, as one would expect. The basic formulation of quantum mechanics emerges from quantum mechanics in configuration space as a semi-classical approximation when a fixed mass is imposed and several other adjustments are made.
Quantum mechanics is empirically unrivalled, yet the origin of its distinctive non-classical features remains a foundational question. This paper proves, within the unchanged standard quantum formalism, that these features—collectively termed “quantumness”—arise if and only if certain configurations of the system are indistinguishable in principle. Quantumness is defined as any deviation from classical probability distributions, i.e., probabilities that require coherent amplitude addition rather than incoherent probability addition. Three theorems are established using only the standard Hilbert-space formalism and unitary evolution: 1. Indistinguishability of configurations is necessary and sufficient for quantumness.2. Indistinguishability of identical particles is neither necessary nor sufficient.3. An equivalent restatement in precise operational terms: quantumness occurs exactly when a variable lacks physical definition (distinguishability in principle by macroscopic outcomes). These results are presented and interpreted within the Logical Interpretation of quantum mechanics, a minimalist framework that adds no new postulates and applies classical logic strictly to physically defined facts. The interpretation provides the conceptual tools—particularly the precise notion of “indistinguishability in principle”—that make the theorems sharp and unified. This work unifies the origin of quantumness and serves as the conceptual prequel to the full Logical Interpretation developed in the author’s companion paper.
Within the traditional Hilbert space formalism of quantum me-chanics, it is not possible to describe a particle as possessing, simul-taneously, a sharp position value and a sharp momentum value. Is it possible, though, to describe a particle as possessing just a sharp position value (or just a sharp momentum value)? Some, such as Teller [21], have thought that the answer to this question is No — that the status of individual continuous quantities is very different in quantum mechanics than in classical mechanics. On the contrary, I shall show that the same subtle issues arise with respect to contin-uous quantities in classical and quantum mechanics; and that it is, after all, possible to describe a particle as possessing a sharp position value without altering the standard formalism of quantum mechanics. 1
In physics, experiments ultimately inform us about what constitutes a good theoretical model of any physical concept: physical space should be no exception. The best picture of physical space in Newtonian physics is given by the configuration space of a free particle (or the center of mass of a closed system of particles). This configuration space (as well as phase space) can be constructed as a representation space for the relativity symmetry. From the corresponding quantum symmetry, we illustrate the construction of a quantum configuration space, similar to that of quantum phase space, and recover the classical picture as an approximation through a contraction of the (relativity) symmetry and its representations. The quantum Hilbert space reduces into a sum of one-dimensional representations for the observable algebra, with the only admissible states given by coherent states and position eigenstates for the phase and configuration space pictures, respectively. This analysis, founded firmly on known physics, provides a quantum picture of physical space beyond that of a finite-dimensional manifold and provides a crucial first link for any theoretical model of quantum space-time at levels beyond simple quantum mechanics. It also suggests looking at quantum physics from a different perspective.
An interpretation and re-formulation of modern physics which removes the presumption of the space-time continuum, and bases physical theory on a small number of rational and empirical principles. After briefly describing the philosophical principles underlying the theory, we rigorously construct a discrete model of quantum mechanics. Kets are names, or labels which categorise states of matter but do not directly describe them. The principle of superposition is seen as a definitional truism in the categorisation of states. This resolves the measurement problem of quantum mechanics by attributing the collapse of the wave function to information. The probability interpretation has a natural meaning in which the configuration of interacting particles plays the role of a hidden variable. The model supports a form of relativistic quantum field theory which does not depend on quantisation or second quantisation from classical mechanics. Continuous laws of wave mechanics are found in a discrete metaphysic which does not involve waves. Classical law is the expected behaviour of many elementary particles. The constraints on the theory are sufficient to establish that particles are point-like entities with specific properties found in nature. Newton's first law and conservation of momentum are established from the principle of homogeneity. Maxwell's equations are derived from the simple interaction in which a Dirac particle emits or absorbs a photon. Feynman rules are calculated for the discrete theory and differ from the standard rules by the removal of the ultraviolet divergence and the use of proper loop integrals. They give finite and unambiguous results.
Among the equivalent formulations of quantum mechanics, Heisenberg's matrix mechanics and quantum theory, based on the Schrödinger wave equation, are the most technically advanced.Since the advent of quantum mechanics, these two schemes have provided the mathematical tools and the primary basis for the description of quantum phenomena.The path integral method developed by Dirac and Feynman has important advantages in the quantisation of gauge theories.We discuss here the formulation of quantum mechanics in phase space, known as deformation quantisation.Deformation quantisation uses the Wigner-Weyl association rule (Weyl, 1927(Weyl, , 1929(Weyl, , 1931;;Wigner, 1932) to establish a one-to-one correspondence between the functions in the phase space and the operators in the Hilbert space.Wigner's function appears as the Weyl symbol of the density matrix.A consistent dynamical description of the systems with the help of the Wigner function leads to deformation quantization.A useful formulation of the Wigner-Weyl association rule was proposed by Groenewold (1946) andStratonovich (1957).Groenewold introduced a non-commutative associative -product (star-product) of the functions in the phase space (Groenewold, 1946).The evolution of the quantum systems is determined by the antisymmetric part of the -product (Groenewold, 1946;Moyal, 1949), known as the Moyal bracket.The Moyal bracket represents the quantum deformation of the Poisson bracket.Deformation quantisation preserves many features of classical Hamiltonian dynamics.The formulation of deformation quantisation is based on the Wigner function and the Moyal bracket (i.e., the -product).The Wigner-Weyl association rule is necessary to prove the equivalence of deformation quantisation and the standard formalisms of quantum mechanics.Extensive literature has reported on the formulation of quantum mechanics in the phase space and the -product.We refer the reader to excellent reviews by Bayen et al. (1978a, 1978b), Carruthers & Zachariasen (1983), Balazs & Jennings (1984), Hillery et al. (1983), Karasev & Maslov (1991), and Osborn & Molzahn (1995), where one may find additional references.Wigner's function, as a fundamental object of deformation quantisation, has numerous applications in many-body physics, kinetic theory, collision theory, and quantum chemistry.Transport models, originally created to simulate chemical reactions, have been modified and are widely used to describe heavy-ion collisions.www.intechopen.comSome Applications of Quantum Mechanics 68 Deformation quantisation does not have many recognised successful applications in quantum theory.Recently, attempts have been made to use specific properties of the formalism to investigate semiclassical expansion (Osborn & Molzahn, 1995;McQuarrie et al., 1998) and to calculate the determinants of one-loop operators in field theory (Pletnev & Banin, 1999;Banin et al., 2001) and the high-order corrections to the Bohr-Sommerfeld quantisation rule (Gracia-Saz, 2004;Cargo et al., 2005).Potential applications of deformation quantisation in transport models are presented in this review.Transport models for heavy-ion physics are designed for phenomenological descriptions of the complex dynamics of nuclear collisions.Several types of advanced transport models are based on the Boltzmann-Uhlenbeck-Ueling equations (BUU) (Blaettel et al., 1993), (relativistic) quantum molecular dynamics (QMD/RQMD) (Sorge et al., 1989;Aichelin, 1991;Faessler, 1992), or antisymmetrised molecular dynamics (AQMD) (Feldmeier & Schnack, 1997).These approaches have the correct classical limit and they contain special plug-ins and quantum-mechanical attributes, such as Pauli blocking for binary collisions of fermions.Numerical solutions are implemented through the distribution of test particles (BUU) or centroids of wave packets (QMD, AQMD) for classical trajectories in the phase space.For AQMD, the wave packets are antisymmetrised in their parameters.The transport models provide a solid basis for a phenomenological description of a variety of complex nuclear collisions phenomena.However, quantum coherence effects and non-localities are beyond the scope of these models.The internal consistencies of the approximations in the models remain a subject of debate, promoting further developments (see, for example, papers by Kohler (1995) andFeldmeier &Schnack (1997) and references therein).The most striking feature of the transport models is the depiction of the trajectories in the phase space, which are test particles or centroids of wave packets.The evolution of the system of classical particles can be calculated using standard programs to solve first-order ordinary differential equations (ODEs).At the same time, the evolution of wave functions of many-body systems is a field-theoretic problem with an infinite number of degrees of freedom that cannot be solved either analytically or numerically.Any simulation of many-body quantum dynamics must be based on a concept of trajectories, which is the only attribute allowing access to an approximate description of complex quantum systems.The concept of phase-space trajectories arises naturally in the formalism of deformation quantisation through the Wigner transformation of the operators of the canonical coordinates and momenta in the Heisenberg representation.These trajectories satisfy the quantum version of Hamilton's equations (Osborn & Molzahn, 1995;Krivoruchenko & Faessler, 2006b) and are the characteristics by which the time-dependent Weyl's symbols for the other operators can be determined (Krivoruchenko & Faessler, 2006b;Krivoruchenko et al., 2006c Krivoruchenko et al., , 2007)).In the classical limit, quantum characteristics reduce to classical trajectories.Knowledge of the quantum phase flow, i.e., the quantum trajectories, is equivalent to a complete knowledge of the quantum dynamics.In this chapter, we provide an introduction to deformation quantisation and demonstrate the usefulness of the formalism in solving the evolution problem for many-body systems in terms of semiclassical expansion.We show that, in any fixed order of expansion over the Planck's constant, the evolution problem can be reduced to a statistical-mechanics problem of calculating an ensemble of quantum characteristics in the phase space and their Jacobi fields.In comparison with the corresponding rules of classical statistical mechanics, the rules for computing the probabilities and time-dependent averages of observables are www.intechopen.com