AR-2019-2020

. Quantum Theory and Gravity Exploring the Rindler vacuum and the Euclidean plane In flat spacetime, two inequivalent vacuum states which arise naturally are the Rindler vacuum |R and the Minkowski vacuum |M . One can then build standard QFT based on these two vacua and study their inter-relationship. In particular, one can study Minkowski and Rindler Feynmann- propagators G M ( x 2 , x 1 ) and G R ( x 2 , x 1 ), respectively, defined by the standard procedure. It is well known that the Minkowski propagator G M can be thought of as a ‘thermalised’ version of the Rindler propagator G R in the following sense: G M ( iτ ) = ∞ n = −∞ G R iτ + i 2 πn g − 1 , (1) where τ denotes the Rindler time coordinate and g is the acceleration parameter, which is taken to be unity henceforth. There is, however, another intriguing relationship between G M and G R , which has received very little attention in the literature. It turns out that, for events ( x 1 , x 2 ) in the right-Rindler wedge, Candelas and Raine (1976) showed there was a curious relation between G R and G M given by: G R ( x 1 , x 2 ) = G M [ σ ( x 1 , x 2 )] − ∞ −∞ dλ G M [ σ ( x 1 , x ( r ) 2 ( τ 2 − λ ))] π 2 + ( λ − τ 1 ) 2 , (2) where σ 2 ( x, y ) is the square of the invariant distance between the two events, and the event x ( r ) 2 ( τ ) is defined through the relation x ( r ) 2 ( τ ) = x 2 ( τ ± iπ ). Geometrically, one can interpret x ( r ) 2 ( τ ) as the ‘reflection’ of x 2 ( τ ) about the origin of the x − t plane, as shown in Fig.1. The original derivation of Eq.2 makes use of the fact that G R and G M are the Feynman propaga- tors in the two vacua |R and |M , respectively. It was not clear whether the same relation holds for a much wider class of functions, and if so, what are the essential ingredients which go into this relation. In this work, two functions { F R ( τ ) , F M ( τ ) } , such that F M is the periodic sum of F R in the sense of Eq.(1) are considered. When both these functions are even, Karthik Rajeev and T. Padmanabhan showed that an integral transformation exists, which express F R in terms of F M and hence, can be interpreted as the ‘inversion’ of the thermal sum. This transformation is given by: F R ( z ) = C du ( iπ ) u ( u 2 − z 2 ) F M ( u ) , (3) where the contour is C shown in Fig.2. Further, they showed that for real values of z , this integral transformation reduced to a relation between F R and F M , which has exactly the form of Eq.2. This result has the physical consequence that Feynman propagators of appropriate vacua in a general spacetime with a bifurcate killing horizon, whose explicit expressions may even be unknown, simultaneously satisfy equations analogous to Eq.1 and Eq.2. Euclidean quantum field theory serves a useful mathematical tool to calculate important phys- ical quantities in the real-world Lorentzian quantum field theories. In this work, Karthik and Padmanabhan clarified an important issue related to the Euclidean to Lorentzian continuation of points in Minkowski spacetime, namely, that the analytic continuation of the Euclidean polar coordinates — which involves replacing t E → − it and τ E → − iτ in x = ρ cos τ E , t E = ρ sin τ E — would lead us only to the events in the right-Rindler wedge. The question arises as to how one