Quantum Inhomogeneous Field Theory: Unruh-Like Effects and Bubble Wall Friction

In this paper, we study a free scalar field in a specific (1+1)-dimensional curved spacetime. By introducing an algebraic state that is locally Hadamard, we derive the renormalized Wightman function and explicitly calculate the covariantly conserved quantum energy-momentum tensor up to a relevant order. From this result, we show that the Hadamard renormalization scheme, which has been effective in traditional quantum field theory in curved spacetime, is also applicable in the quantum inhomogeneous field theory. As applications of this framework, we show the existence of an Unruh-like effect for an observer slightly out of the right asymptotic region, as well as the vanishing of quantum frictional effect in the leading order ($e^{- bx}$) on the bubble wall expansion during the electroweak phase transition in the early universe.