A Three-Dimensional Time Framework for Quantum Mechanics, Relativity, Feynman Path Integral and Fundamental Interactions

We propose a fundamental extension of spacetime in which time is described as a three-dimensional vector $\mathbf{T}=(t_1,t_2,t_3)$ alongside ordinary three-dimensional space. A generalized invariant interval\[ds^2 = c^2 d\mathbf{T}\cdot d\mathbf{T} - d\mathbf{x}\cdot d\mathbf{x}\]is introduced, yielding an underlying $(3,3)$ signature geometry. Within this framework, quantum dynamics isformulated through a multi-time Schr\"odinger equation\[i\hbar\,\partial_{t_a}\psi = \hat{H}_a \psi,\]together with consistency conditions that select a single physical evolution parameter$\tau=\mathbf{n}\cdot\mathbf{T}$. Relativistic wave equations are obtained by constructing Klein--Gordon andDirac operators on the extended manifold, while gravitation is described by a six-dimensionalEinstein--Hilbert action. A generalized path-integral formulation is also presented, in which particlehistories evolve in the extended time-space manifold. The formalism reduces to standard quantum mechanicsand general relativity under an appropriate dimensional reduction, while predicting small deviations athigh energies. This work establishes a unified mathematical foundation for exploring multiple timedimensions across quantum theory, relativity, string-inspired models, and fundamental interactions.