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This paper presents a theoretical framework for "polymer quantum mechanics," an alternative quantization scheme derived from loop quantum gravity that utilizes discrete configuration spaces. The authors derive exact energy eigenvalues and wavefunctions for standard systems like the particle in a box, demonstrating how these polymer states converge to conventional Schrödinger results in the continuum limit. This work advances fundamental understanding of quantum formalism but does not address practical AI applications or near-term commercializable technologies.
Polymer quantum mechanics on compact configuration spaces
arXiv:2606.06019v3 Announce Type: replace Abstract: "Polymer quantum mechanics" is the name given to a quantization scheme inspired by loop quantum gravity in which the configuration space of the theory is chosen to have a discrete topology. Polymer quantization yields a representation of the canonical commutation relations that is genuinely distinct from the conventional "Schr\"odinger" representation. In this paper, we summarize the main features of polymer quantum mechanics and investigate in detail the polymer quantization of systems with configuration spaces that are classically compact. We show explicitly how using the standard construction of polymer states leads to a Hilbert space of states defined on a finite graph of points. By way of example, we find the exact energy eigenvalues and eigenfunctions for a particle on a ring and a particle in a box defined on such lattices, and discuss similarities and differences from standard Schr\"odinger quantum mechanics. We also explore the continuum limit of states in these systems, and demonstrate in detail how the exact eigenfunctions in the position representation approach their continuum counterparts.
Abstract:"Polymer quantum mechanics" is the name given to a quantization scheme inspired by loop quantum gravity in which the configuration space of the theory is chosen to have a discrete topology. Polymer quantization yields a representation of the canonical commutation relations that is genuinely distinct from the conventional "Schrödinger" representation. In this paper, we summarize the main features of polymer quantum mechanics and investigate in detail the polymer quantization of systems with configuration spaces that are classically compact. We show explicitly how using the standard construction of polymer states leads to a Hilbert space of states defined on a finite graph of points. By way of example, we find the exact energy eigenvalues and eigenfunctions for a particle on a ring and a particle in a box defined on such lattices, and discuss similarities and differences from standard Schrödinger quantum mechanics. We also explore the continuum limit of states in these systems, and demonstrate in detail how the exact eigenfunctions in the position representation approach their continuum counterparts.
43 pages, 8 figures. v2: Minor edits. References added. v3: Minor edits. Typos corrected, references updated. Version that appears in Universe (more or less)
arXiv:2606.06019 [quant-ph]
(or arXiv:2606.06019v3 [quant-ph] for this version)
https://doi.org/10.3390/universe12080246 DOI(s) linking to related resources
From: David A. Craig [view email] [v1] Thu, 4 Jun 2026 11:09:41 UTC (1,016 KB) [v2] Wed, 24 Jun 2026 10:29:44 UTC (1,023 KB) [v3] Sat, 15 Aug 2026 09:37:33 UTC (1,016 KB)
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