Foundations of Physics

, Volume 45, Issue 3, pp 295–332 | Cite as

Quantum Mechanics and the Principle of Least Radix Economy

  • Vladimir Garcia-MoralesEmail author


A new variational method, the principle of least radix economy, is formulated. The mathematical and physical relevance of the radix economy, also called digit capacity, is established, showing how physical laws can be derived from this concept in a unified way. The principle reinterprets and generalizes the principle of least action yielding two classes of physical solutions: least action paths and quantum wavefunctions. A new physical foundation of the Hilbert space of quantum mechanics is then accomplished and it is used to derive the Schrödinger and Dirac equations and the breaking of the commutativity of spacetime geometry. The formulation provides an explanation of how determinism and random statistical behavior coexist in spacetime and a framework is developed that allows dynamical processes to be formulated in terms of chains of digits. These methods lead to a new (pre-geometrical) foundation for Lorentz transformations and special relativity. The Parker-Rhodes combinatorial hierarchy is encompassed within our approach and this leads to an estimate of the interaction strength of the electromagnetic and gravitational forces that agrees with the experimental values to an error of less than one thousandth. Finally, it is shown how the principle of least-radix economy naturally gives rise to Boltzmann’s principle of classical statistical thermodynamics. A new expression for a general (path-dependent) nonequilibrium entropy is proposed satisfying the Second Law of Thermodynamics.


Quantum mechanics Variational methods Entropy  Discrete physics 



I thank Prof. José Antonio Manzanares for his many helpful remarks and Prof. José María Isidro San Juan for his comments on a previous version of this manuscript. Past support from the Technische Universität München - Institute for Advanced Study (funded by the German Excellence Initiative) through a three-years Carl von Linde Junior Fellowship (when this research was initiated) is also gratefully acknowledged.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institute for Advanced StudyTechnische Universität MünchenGarchingGermany
  2. 2.Nonequilibrium Chemical Physics - Physics DepartmentTechnische Universität MünchenGarchingGermany

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