Path Integrals and Anomalies in Curved Space
Path integrals provide a powerful method for describing quantum phenomena. This book introduces the quantum mechanics of particles that move in curved space by employing path integrals and then using them to compute anomalies in quantum field theories. The authors start by deriving path integrals for particles moving in curved space and their supersymmetric generalizations. They then discuss the regularization schemes essential to constructing and computing these path integrals. This topic is used to introduce regularization and renormalization in quantum field theories in a wider context. These methods are then applied to discuss and calculate anomalies in quantum field theory. Such anomalies provide enormous constraints in the search for physical theories of elementary particles, quantum gravity and string theories. An advanced text for researchers and graduate students of quantum field theory and string theory, the first part is also a stand-alone introduction to path integrals in quantum mechanics.
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anticommuting antisymmetric approach bosonic boundary conditions calculation chiral anomaly classical action coeﬃcients complete compute consistent anomaly contribute coordinate corresponding counterterm coupled covariant covariant anomaly curvature deﬁned denote derivatives diﬀerent dimensional regularization dimensions Dirac matrices discuss divergences eﬀective action Einstein equation evaluate external extra term factor fermions Feynman graphs field ﬁnal ﬁnd ﬁnite ﬁrst ﬁxed follows gauge anomalies gauge ﬁelds gauge invariance ghosts given Grassmann variables gravitational anomaly gravitino Hamiltonian Hence inﬁnite interactions Jacobian linear loops Lorentz invariant Lorentz transformation Majorana Majorana fermions matrix mode regularization momenta nonlinear sigma models obtain one-loop operator path integral phase-space propagators quantum ﬁeld theories quantum mechanical regularization scheme regulator representation result scalar self-dual spinor supergravity supersymmetric symmetry tensor trace anomalies transition amplitude transition element two-loop vanishes vector vertex vertices vielbein Weyl ordering Weyl-ordered Yang–Mills yields