AR-2019-2020

can extract the information contained in the other four wedges of the Lorentzian sector from the expression valid in the Euclidean sector. In this work, they provided the four different analytic continuations (see Tab.1) of the Euclidean polar coordinates such that one can reach all the four wedges in the Lorentzian sector from the Euclidean Rindler spacetime. The procedure is based on a simple unifying principle, viz, that the analytic continuation should map Euclidean squared distance σ 2 E to ( σ 2 M + i ), with a positive, infinitesimal, imaginary part in the Lorentzian sector. The authors explicitly demonstrate that this procedure leads to the correct expressions for the propagators in the Lorentzian sector, even when the two events are in two different wedges. Case Euclidean → Lorentzian σ 2 E → σ 2 RR ( r, θ ) → ( ρ, iτe − i ) ρ 2 + ρ 2 − 2 ρρ cosh( τ − τ ) + i 0 + ( r θ ) → ( ρ , iτ e − i ) RF ( r, θ ) → ( ρ R , iτ R ) − ρ 2 F + ρ 2 R − 2 ρ F ρ R sinh( τ F − τ R ) + i 0 + ( r , θ ) → ( iρ F , iτ F + π 2 + ) FF ( r < , θ ) → ( − e i iρ < , iτ + π 2 ) − ρ 2 < − ρ 2 > + 2 ρ < ρ > cosh( τ − τ ) + i 0 + ( r > , θ ) → ( iρ > , iτ − π 2 ) Table 1: Recipe for analytic continuation: Here R , L and F denotes the right, left and future Rindler wedges, respectively. The notation RR denotes the case when both points at which the propagator is evaluated lies in R , while RF denotes the case when one point is in the right and other in the future wedge, and so on. ( r, θ ) are the Euclidean polar coordinates and ( ρ, τ ) the corresponding Rindler coordinates, with the subscripts indicating the appropriate wedges. Complex time route to quantum backreaction When the degrees of freedom of a system can be naturally divided into two subsystems, say C and q , apart from the classical limit (viz. the → 0 limit), one can also study another useful limit. This corresponds to the limit in which one subsystem, say C , is effectively classical, while the other is quantum mechanical. In the study of such systems, quantum backreaction refers to the correction to classical dynamics of the subsystem C due to the feedback from the quantum excitations of q . One approach towards studying backreaction equation, that is often discussed in the literature, uses an effective action S eff [ C ] for the system C obtained by ‘integrating out’ the quantum degree of freedom q . To obtain the dynamical equation that describes the backreaction on the system C , we may demand that δ Re[ S eff ] /δC = 0 for the effective classical ‘trajectory’ C ( t ). Unfortunately, there are some severe issues in this approach: (i) the backreaction equation is non-causal, and (ii) the dynamics of C obtained by this approach does not seem to completely incorporate the effects of particle production. The origin of these undesired features is the presence of matrix element of operators acting on the Hilbert space of q -subsystem evaluated between the ‘in-vacuum’ and the ‘out-vacuum’. The ‘in-in’ approach, where the ‘in-out’ matrix elements are replaced by ‘in-in’ expectation values, is devoid of the issues (i) and (ii). The main drawback concerning the ‘in-in’ prescription is that the manner in which one has to postulate – rather than derive– the backreaction equation is ad hoc. In this work, Karthik Rajeev illustrated how the ‘in-in’ prescription could be given a path integral basis. For this purpose, he considered a C − q system described by the following Lagrangian: L = m ( C ) 2 ˙ q 2 − ω 2 ( C ) q 2 + M ˙ C 2 2 − V ( C ) . (4) He then formulated a path integral based effective action S T eff for time evolution along a general complex-time contour T . When the evolution is along the T 1 of Fig.3, the corresponding back reaction equation, obtained by varying the effective action S T 1 eff [ C ], matches exactly with that of the ‘in-out’ formalism. On the other hand, for the choice of time-contour T 2 of Fig.4, the backreaction