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Counting-Based Effective Dimension and Discrete Regularizations
Ivan Horváth1,2, Peter Markoš3, Robert Mendris4
1Nuclear Physics Institute CAS, 25068 Řež, Czech Republic.
Entropy (Basel, Switzerland)
|March 29, 2023
Summary
This study introduces the effective counting dimension (ECD), a new fractal dimension for quantum structures. The ECD is scheme-independent, providing a robust measure for probabilistic descriptions in physics.
Area of Science:
- Theoretical Physics
- Number Theory
- Complex Systems
Background:
- Fractal-like structures are prevalent in nature, geometrically characterized by dimensions like Minkowski and Hausdorff.
- Quantum mechanics describes structure via probability distributions, posing challenges for traditional geometric dimensions.
Purpose of the Study:
- To establish a robust, measure-based fractal dimension for quantum structures.
- To introduce and validate the effective counting dimension (ECD) for probabilistic descriptions in physics.
Main Methods:
- Utilizing effective number theory to construct counting-based schemes for probability distributions.
- Developing and analyzing the properties of the effective counting dimension (ECD).
Main Results:
- The effective counting dimension (ECD) is demonstrated to be scheme-independent.
- ECD is shown to be a well-defined measure-based dimension analogous to the Minkowski dimension.
- The ECD provides a theoretical foundation for recent findings in quantum chromodynamics and Anderson models.
Conclusions:
- The effective counting dimension (ECD) offers a robust method for characterizing fractal structures in quantum systems.
- This work validates the use of ECD in understanding effective spatial dimensions in complex physical models.
- The study provides a framework for assessing regularization reliability in physical theories.
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