Fractional Calculus Across Physical Scales: From Classical to Quantum Theories in Low- and High-Dimensional Systems

Dear Colleagues,

Fractional calculus, a generalization of classical calculus to non-integer orders, has emerged as a powerful mathematical framework for modeling complex physical systems with memory effects, non-locality, and anomalous dynamics. While its applications in engineering and applied mathematics are well documented, recent advancements highlight its potential in foundational areas of physics, including classical mechanics (e.g., viscoelastic continua and turbulence), classical field theory (e.g., non-local electromagnetism and gravity), quantum mechanics (e.g., open systems with memory kernels), and quantum field theory (e.g., renormalization in fractional spacetimes and symmetries in fractional field theories). This Special Issue seeks to explore innovative mathematical techniques in fractional calculus and their cross-disciplinary applications to advance our understanding of physical phenomena from across dimensional scales—from one-dimensional classical oscillators to quantum gravitational interactions and high-dimensional quantum field theories—and invites contributions that bridge foundational theory, computational methods, and experimental validation.

Key themes include, but are not limited to, the following:

• Fractional Hamiltonian/Lagrangian formulations;
• Fractional dynamical systems in classical mechanics;
• Anomalous wave propagation in fractional continua and field theories;
• Fractional Schrödinger equations, quantum coherence, and decoherence mechanisms in open quantum systems;
• Fractional Schrödinger and Dirac equations in low-dimensional systems;
• Fractional field theories and their role in modeling non-local interactions in electromagnetism, gravity, and cosmology;
• Fractal spaces and calculus in classical and quantum physics;
• Non-local quantum field theories and renormalization in fractional spacetime dimensions;
• Dimensional regularization techniques using fractional operators;
• Numerical schemes for fractional PDEs in higher-dimensional classical/quantum systems;
• Interplay between fractional operators and symmetry principles in theoretical physics.

This Special Issue will provide a platform for cutting-edge research at the intersection of fractional calculus and physics and rigorous mathematical advances and applications that address open challenges, such as the physical meaning of fractional derivatives in quantum gravity or the experimental feasibility of fractional models in condensed matter systems, fostering dialogue between theoretical, computational, and applied communities. Submissions may include theoretical, computational, or applied studies that deepen the dialogue between fractional calculus and physics. We invite researchers worldwide to contribute original articles, reviews, or perspectives within the outlined themes.

Prof. Dr. Ion Vancea
Dr. Cresus F.L. Godinho
Dr. Thiago R. Araujo
Guest Editors

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• fractional calculus
• fractal calculus
• classical and quantum non-local field theory
• fractional quantum mechanics
• dimensional regularization
• anomalous transport
• viscoelastic continua
• fractional renormalization
• fractional quantum field theory
• fractional gravity
• Hamiltonian systems
• memory kernels
• high-dimensional PDEs
• symmetries in fractional systems