Researchers at the CNRS, École polytechnique, and Université Claude Bernard Lyon 1 have developed a new approach to calculating quantum dynamics on Lie groups, overcoming longstanding challenges in handling noncommutative momentum spaces and compact directions within these complex mathematical structures. The work, detailed in a recent paper identified as CPHT-RR020.072026, builds path integrals, tools for determining transition amplitudes, by generalizing the familiar “sum over winding numbers” typically used for calculations on a circle to Lie groups. This advancement allows the team to compute semiclassical approximations of propagators and partition functions for Euler-Arnold systems, achieving accuracy up to and including two-loop order. The research continues a study initiated in a previous paper, aiming to better understand quantum systems with inherent symmetries found in areas ranging from rigid body dynamics to condensed matter physics.

Quantum Dynamics on Lie Groups: Path Integral Construction

Recent work, detailed in a preprint identified as CPHT-RR020.072026, addresses a longstanding challenge in quantum dynamics on Lie groups: properly accounting for noncommutative momentum space and the presence of compact directions. Researchers Mathieu Beauvillain, Blagoje Oblak, and Marios Petropoulos have constructed path integrals, essential tools for calculating transition amplitudes, specifically designed to overcome these hurdles, extending the applicability of quantum mechanics to a broader range of group-based systems. The team’s approach builds upon earlier studies, notably their own work initiated in a prior publication [1], and leverages a decompactification of the group onto its Lie algebra, a technique analogous to methods used for path integrals on a circle. A key innovation lies in how they handle compactness, achieving this through a sum over winding numbers in maximal tori, effectively generalizing the familiar summation technique typically employed for circular path integrals.

These Euler-Arnold systems, described as nonabelian generalizations of free particles, are central to the research; their classical dynamics reduce to geodesic motion on a Lie group endowed with an invariant metric. The researchers distinguish between left and right Haar measures on a Lie group, acknowledging the presence of modular functions in their formulas. The paper clarifies the method’s ability to accurately represent quantum behavior in these complex systems. This refined path integral construction promises more accurate modeling of diverse physical phenomena, from rigid bodies to quantum liquids.

Researchers for Theoretical Physics (CPHT), affiliated with CNRS and École Polytechnique, are refining methods for describing quantum dynamics on Lie groups, complex mathematical spaces increasingly relevant to theoretical physics. A key advancement lies in generalizing a familiar technique used for simpler systems. While path integrals on a circle traditionally rely on a “sum over winding numbers,” the CPHT team extends this concept to Lie groups, a sophisticated mathematical development. This allows for a more nuanced treatment of compactness within the group structure, overcoming limitations present in earlier approaches. Beyond the theoretical framework, the team has achieved concrete results.

This advancement isn’t merely a mathematical curiosity; it directly impacts the ability to model complex systems. The approach diverges from conventional heat kernel computations by leveraging the inherent Lie group structure, yielding a detailed list of vertices to be used at all loop orders. The study distinguishes itself from prior work in loop quantum gravity by providing a complete framework that accounts for both unimodular and non-unimodular groups, carefully differentiating between left and right Haar measures.

This isn’t merely an abstract refinement of existing theory; it’s a practical advancement enabling more accurate calculations within quantum systems exhibiting group symmetries. By focusing on these systems, the researchers provide a concrete application of their theoretical development, demonstrating its utility in calculating physical properties with increased precision.

CPHT-RR020.072026 ultimately paves the way for a rigorous definition of path integrals on group manifolds. This isn’t merely an extension of existing methods, but a significant leap in the mathematical framework underpinning quantum dynamics on manifolds. Beyond the theoretical framework, the team delivered concrete results. The ability to model these systems to this degree suggests a pathway toward more accurate predictions in diverse areas of physics, from classical hydrodynamics to quantum liquids.

The ability to accurately model quantum systems evolving on Lie groups, complex mathematical spaces with inherent symmetry, has received a significant boost from recent work detailed in CPHT-RR020.072026. A key advancement lies in how the team handles compactness within these groups. This extension allows for a more comprehensive and accurate representation of quantum states, particularly when dealing with systems exhibiting cyclical or rotational behavior. The work, the authors state, promises more accurate modeling of diverse physical systems and furthering the development of quantum simulations.

👉 More information
🗞 Quantum Mechanics on Lie Groups: II. Path Integrals
✍️ Mathieu Beauvillain, Blagoje Oblak and Marios Petropoulos
🧠 ArXiv: https://arxiv.org/abs/2607.16029

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