Abstract

Complex Spacetime and the Schrödinger Equation

Author(s): Bhushan Poojary

In this paper, we extend the formulation of the Schrödinger equation into the complex spacetime framework, introducing a novel approach that bridges quantum mechanics, electromagnetism, and general relativity. By incorporating complex derivatives and applying the Cauchy-Riemann conditions, we reveal that the imaginary components of spacetime contribute to quantum fluctuations, while the real components govern classical behavior. This framework naturally connects the imaginary curvature of spacetime to electromagnetic field dynamics, suggesting that electromagnetic phenomena could emerge from the geometry of complex spacetime. We demonstrate that the imaginary Ricci tensor, derived from the curvature of the imaginary spacetime dimensions, aligns with the mathematical structure of Maxwell's equations in curved spacetime. This connection implies a geometric origin of electromagnetism, where quantum fluctuations arise from the imaginary curvature of spacetime. Furthermore, the preservation of standard quantum commutation relations within this framework suggests consistency with established quantum mechanical principles. This approach provides a potential pathway for unifying quantum mechanics and electromagnetism within a single geometric framework. The results presented offer new insights into the fundamental nature of spacetime and open avenues for exploring the deeper connections between quantum field theory, gravity, and the complex structure of the universe.