Introduction to Quantum Effects in Gravity

The standard reference for quantum fields in curved spacetimes is Birrell & Davies which is totally incomprehensible for beginners at both conceptual and computational levels - just my 6 months experience with it. The present book is the only one I know that takes beginners to a level no other book can. It is conceptually clear and tailored to beginners in the field which already had courses in Complex analysis (you will need to have some idea what analytic continuation is), Quantum Mechanics, General Relativity and a little Quantum Field Theory (only free fields).

The first part of the book is devoted to specifying a relevant ground/vacuum state of a scalar field and calculations using Bogoljubov coefficients. The usual Unruh effect is given a detailed carefull consideration, Hawking radiation is derived by analogy and Casimir effect is briefly considered too. Most importantly, the book gives a very clear treatment of quantization of a scalar field in expanding universe. Later that is applied to the inflationary spacetime (de Sitter). The vaccum state of the scalar field is specified (Bunch-Davies) and the spectrum of the primordial field fluctuations calculated - that is a hot topic in Cosmology since a derivative of that spectrum is observed in CMB and Large scale structure.

The second part of the book deals with actions and path integral quantization of a system. The effective action of a quantum sub-system interacting with a classical sub-system (the background) is defined and shown how to use it to calculate matrix elements of operators. The effective action of a quantum scalar field in a gravitational classical background is considered. In order to calculate it, it's transformed to Euclidean action and then renormalized (thrown away infinities) using zeta function and heat kernel technique then analytically continued back to Lorentzian action. A whole chapter is devoted to showing how the zeta function renormalization (by analytical continuation of zeta function) throws away infinities that can be viewed as being absorbed in the bare coupling constants of the action. At the end it is shown how to use the renormalized effective action to calculate matrix elements of the energy-momentum tensor (in 2D).

The book is self-contained and derives everything from scratch (path integrals included). It can be read in about 2 months if you read 5 pages/day and solve the problems. The style is consize and straight to the point explaining the correct conceptual understanding. There are numerous problems interspersed around the text with solutions at the end of the book. There are nice appendices on functionals, functional derivatives, distributions (like delta function) and Green's functions which are pretty usefull.

It is amazing and exciting that in such a short time the reader will gain some basic understanding of advanced topics like effective action, zeta function, heat kernel, conformal anomaly, primordial quantum fluctuations etc. An example of textbook writing at the finest level ...