Gauge Theories in Rainbow Space-Time

We construct Maxwell and Yang-Mills theories in the rainbow space-time. We show that the time-dependent Aharonov-Bohm phases for both Abelian and non-Abelian gauge fields are non-zero in the rainbow space-time, unlike in Minkowski space-time, where they are zero. Starting from the Poisson brackets between coordinates and velocities of a massive particle moving in rainbow space-time and interacting with Abelian/non-Abelian gauge fields, we derive Maxwell's/Yang-Mills equations in rainbow space-time using a variant of Feynman's approach. In this approach, one replaces kinetic momentum with conjugate momentum and the gauge field to calculate the Poisson brackets. Finding the time-derivative of these Poisson brackets and exploiting their anti-symmetry property, a second rank anti-symmetric tensor is introduced which is the field strength associated with Maxwell/Yang-Mills theory, respectively. We show the gauge invariance/covariance of these field strengths explicitly and by demanding the gauge invariance, construct Maxwell's/Yang-Mills Lagrangians. We derive the Lorentz force equation for the particle moving in rainbow space-time in the presence of Abelian/non-Abelian gauge fields. Using the gauge field strength in rainbow space-time, expressed in terms of field strength in the Minkowski space-time and rainbow parameters, we explicitly evaluate the Aharonov-Bohm phases for different choices of rainbow functions and also for different possible gauge field configurations. We show that the time-dependent Aharonov-Bohm phases for Abelian as well as non-Abelian fields are, in general, non-zero in the rainbow space-time. The non-vanishing of time-dependent Aharonov-Bohm phase, which distinguishes rainbow space-time from Minkowski space-time, is a clear signal to be measured to validate the rainbow space-time.