Complex Time in Relativistic Quantum Physics: Analytic Continuation, Wick Rotation, and Physical Implications

The concept of time plays a central role in both relativity and quantum mechanics, yet its interpretation becomes increasingly subtle in relativistic quantum theory. This paper examines the theoretical significance and physical implications of complex time, with particular emphasis on imaginary time introduced through analytic continuation and Wick rotation. By analyzing foundational relativistic wave equations—the Klein-Gordon and Dirac equations—the study explores how complex time alters wave propagation, probability amplitudes, and mathematical structure. The imaginary-time formalism is further examined within Feynman’s path integral approach, highlighting its role in simplifying quantum-field-theoretic calculations and establishing connections with statistical mechanics. Applications to quantum tunneling, vacuum fluctuations, and cosmology, including the Hartle-Hawking no-boundary proposal, are also discussed. The analysis demonstrates that complex time is not merely a computational tool but a meaningful extension that provides deeper insight into non-perturbative quantum phenomena and the structure of relativistic quantum systems.