This book presents an in-depth review of the recent interdisciplinary studies that have shed light on the mechanisms of the creation, energy transfer, and measurement of the dressed photon. A dressed photon is a type of photon that results from the interaction between light and matter in a confined space, typically on the scale of nanometers. It has been applied to nano-fabrication technologies, energy conversion technologies, and silicon light-emitting devices. Despite its extensive applications in various fields, the dressed photons off-shell nature has posed challenges in describing it through conventional optical scientific theories. The book explains how, through a mathematical viewpoint, the underlying spatiotemporal vortex dynamics connect the dressed photon, the dark energy field, and the structure of the universe. The canonical equations of motion, which describe the time evolution of dynamical systems with Hamiltonian structure, play a key role in understanding these physical phenomena. In particular, the covariant form of equations of motion, where vortex tensors explicitly appear, corresponds to the transformed canonical equations of motion in the Eulerian representation.  This newly augmented view of the equations of motion presents a deeper understanding of the interconnectedness of different physical phenomena, which can enlighten graduate students, junior scientists, and industry engineers engaged or interested in this field.