ABSTRACT
A clear mathematical theory of time remains one of the most difficult challenges of science, which seems highly intriguing. In this work, we assume that time is the main independent attribute of nature and therefore may serve as the foundation of a comprehensive field theory. Furthermore, we assume that division algebras with the Euclidean norm are essential mathematical tools of time and the physical world in general. We use a four-dimensional normed division algebra of quaternions to describe time mathematically, as originally envisioned by Hamilton. We systematically define basic quaternion concepts related to time, such as the quaternion time interval, scalar measured time, the arrow of time, vector velocity, and quaternion frequency. We apply quaternion time concepts to the optical Doppler effect and demonstrate that our approach predicts known experimental results. Furthermore, we show that the quaternion solution of the Doppler effect enhances the relativity theory by resolving the notorious twin paradox. We identify quaternion frequency with the traditional concept of energy. We assume that quaternion energy, which is generally dependent on time and external interactions, can be used to describe dynamic properties of matter. In conclusion, we suggest that a state of matter can be represented by the eight-dimensional octonion configuration space, consisting of a quaternion time interval and a time dependent quaternion frequency. Therefore, it appears that the application of normed division algebras for the study of time and nature is highly logical, credible, and compelling.