The worldline approach to Quantum Field Theory (QFT) allows to efficiently compute several quantities, such as one-loop effective actions, scattering amplitudes and anomalies, which are linked to particle path integrals on the circle. A helpful tool in the worldline formalism on the circle, are string-inspired (SI) Feynman rules, which correspond to a specific way of factoring out a zero mode. In flat space this is known to generate no difficulties. In curved space, it was shown how to correctly achieve the zero mode factorization by applying BRST techniques to fix a shift symmetry. Using special coordinate systems, such as Riemann Normal Coordinates, implies the appearance of a non-linear map-originally introduced by Friedan-which must be taken care of in order to obtain the correct results. In particular, employing SI Feynman rules, the map introduces further interactions in the worldline path integrals. In the present paper, we compute in closed form Friedan's map for RNC coordinates in maximally symmetric spaces, and test the path integral model by computing trace anomalies. Our findings match known results. * In the worldline approach to Quantum Field Theory (QFT), particle path integrals are used as a versatile computational tool. The method was introduced by Feynman who, already in the 1950, proposed a particle model representation for the dressed scalar propagator in scalar Quantum Electrodynamics [1]. However, it was only in the late 80's that the method started to be taken seriously as an alternative approach to conventional second-quantized methods. Initially it was used as a tool to compute chiral anomalies [2,3,4] and trace anomalies [5,6], and later it was introduced by Bern and Kosower [7], and Strassler [8], as a proper method to compute QFT effective actions and generic QFT Feynman diagrams-see [9] for a comprehensive review of the early stages of the method. Since then, several applications and new implementations of the worldline formalism have been considered. In the realm of perturbative QFT some examples are: the computation of multiloop effective actions [10], Bern-Kosower rules for dressed propagators [11,12], the worldline formalism in curved spacetime [13,14,15,16], higher-spin field theory approaches [17,18,19,20], the spinning particle approach to Yang Mills theories [21,22], as well as applications to noncommutative QFT [23,24], to the Standard Model and Grand Unified theories [25,26], and to QFT on manifolds with boundary [27,28].The extension of the worldline formalism to the computation of effective actions and Feynman diagrams for QFT in curved space time required to tackle some technical issues which, during several years, had resulted in numerous controversial statements and errors. The main issue boils down to the fact that, when the metric is non-flat, the associated particle models are characterized by non-linear sigma models which, in the perturbative path integral approach about the flat space metric, give rise to an infinite set of vertices with double-derivative interacti...
We present a numerical method to evaluate worldline (WL) path integrals defined on a curved Euclidean space, sampled with Monte Carlo (MC) techniques. In particular, we adopt an algorithm known as YLOOPS with a slight modification due to the introduction of a quadratic term which has the function of stabilizing and speeding up the convergence. Our method, as the perturbative counterparts, treats the non-trivial measure and deviation of the kinetic term from flat, as interaction terms. Moreover, the numerical discretization adopted in the present WLMC is realized with respect to the proper time of the associated bosonic point-particle, hence such procedure may be seen as an analogue of the time-slicing (TS) discretization already introduced to construct quantum path integrals in curved space. As a result, a TS counter-term is taken into account during the computation. The method is tested against existing analytic calculations of the heat kernel for a free bosonic point-particle in a D-dimensional maximally symmetric space.
We compute the perturbative short-time expansion for the transition amplitude of a particle in curved space time, by employing dimensional regularization (DR) to treat the divergences which occur in some Feynman diagrams. The present work generalizes known results where DR was applied to the computation of one-loop effective actions, which in the worldline approach are linked to particle path integrals on the circle, i.e. with periodic boundary conditions. The main motivation of the present work comes from revived interest in particle transition amplitudes in curved space-times, due to their use in the recently proposed worldline quantum field theory (in curved space-time).
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