The Generalized Euler Identity: A Unified Rotation Between the Four Dynamical Universality Classes
Abstract
Euler’s identity e^{iθ} = cos θ + i sin θ expresses rotation in the complex plane. We extend this identity by replacing the imaginary unit i with an operator J satisfying J^2 = −I, where J acts on a doubled dynamical state encoding the four fundamental PDE universality classes: elliptic, hyperbolic, parabolic (diffusive), and unitary (Schrodinger-type). This gives the Generalized Euler Identity e^{θJ} = cos θ I + sin θ J , which performs Wick rotation through physical law itself. This formalism reveals a natural holofractal cyclicity among the canonical dynamical regimes.
Fields of Study & Subjects
Advanced Mathematical Theories and Applications
Algebraic and Geometric Analysis
Nonlinear Waves and Solitons
Publication details
- Journal / source
- Zenodo (CERN European Organization for Nuclear Research)
- Publication date
- 15 January 2026
- Language
- EN
- Document type
- Preprint