Path Integral Quantization of Time Field Theory: A Natural Framework from Time Field Fluctuations to the Unification of Optics and Quantum Measurement

The canonical quantization of Time Field Theory (TFT) has been completed in previous work, demonstrating the compatibility between TFT classical field theory and standard quantum mechanics. However, canonical quantization is essentially an operatorization procedure, which bears a structural discrepancy with the geometrical physical picture of TFT — the time field, Fermat's principle, and flux conservation. This paper proposes and systematically elaborates the path integral quantization scheme for TFT, arguing that it is not only a technically superior choice but also the inevitable mathematical expression of TFT's physical logic at the quantum level. The core arguments are as follows: quantum fluctuations of the time field directly correspond to the "all possible paths" in the Feynman path integral; Fermat's principle is the semiclassical approximation of the path integral in the classical limit; wave function collapse is a natural consequence of the establishment of the metric reference (the emergence of the observer), which imposes new boundary conditions on the path space. This framework provides a unified TFT interpretation for wave optical phenomena such as diffraction and interference, offers a clear physical picture for the quantum measurement problem, and paves the way for the deep integration of TFT and quantum field theory.