Aspects of Quantum Field Theory in Curved Spacetime (London Mathematical Society Student Texts Book 17) 1st Edition by Stephen A. Fulling | (PDF) Free Download

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The theory of quantum fields on curved spacetimes has attracted great attention since the discovery, by Stephen Hawking, of black-hole evaporation. It remains an important subject for the understanding of such contemporary topics as inflationary cosmology, quantum gravity and superstring theory. This book provides, for mathematicians, an introduction to this field of physics in a language and from a viewpoint which such a reader should find congenial. Physicists should also gain from reading this book a sound grasp of various aspects of the theory, some of which have not been particularly emphasised in the existing review literature. The topics covered include normal-mode expansions for a general elliptic operator, Fock space, the Casimir effect, the ‘Klein’ paradox, particle definition and particle creation in expanding universes, asymptotic expansion of Green’s functions and heat kernels, and renormalisation of the stress tensor. The style is pedagogic rather than formal; some knowledge of general relativity and differential geometry is assumed, but the author does supply background material on functional analysis and quantum field theory as required. The book arose from a course taught to graduate students and could be used for self-study or for advanced courses in relativity and quantum field theory.

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⭐Here is a fine example of the cross-fertilization between mathematics and physics. An exceptionally complete Index of ten pages. Next, a fifteen-page bibliography, which includes a number of favorite textbooks: Partial Differential Equations, by Garabedian, for instance. Keep in mind that the author maintains an academic webpage where you can freely access the bibliography and Chapter eight of this text (twenty-five page account, “Some Geometrical Apparatus” that is one highlight of this text). On to words of wisdom:(1) “A natural definition of particles does not exist, it seems reasonable, therefore, to consider taking the field itself as the basic object.” (page 92) and “Casimir effects are really of two types–boundary effects and quantization, or discretization, effects. Neither of these has anything directly to do with topology.” (page 108).(2) “Any commutation or anticommutation relations consistent with the dynamics can define a possible model.”(page 126) and “The physical implications of the dynamics must be sought in terms of field observables. This forces us again to the problem of defining a meaningful energy-momentum tensor in curved-spacetime.” (page 158).(3) Finally, “The central problem of fundamental theoretical physics today is reconciling general relativity with quantum theory.” (page 217).On to technicalities:(1) A twenty-page summary of quantum mechanics introduces the book (No path integrals). An elementary discussion of dimensional analysis is of interest (pages 18-19). Concluding: “the field is not a wave function giving probabilities.”(2) Next, mathematics of differential operators. One will need a copy of Titchmarsh (1962) as reference.(3) Third, quantization of scalar field. An exceptional discussion of twenty-five pages.(4) Next: “Green Function Zoo” (page 75). Glance at page 87: indicative of the mathematical manipulations. Lucid.(5) Stress-Energy Tensors will be effected with a “negative signature” as “my feeling is that energies and energy densities are naturally positive and it is silly for them to change sign when an index is raised or lowered.” (page 95).I am sold on that discussion ! Pages 109-115 make some of the best reading with “history and contact with the real world.” Initial five chapters (115 pages) laid the groundwork for the remainder of the book:(6) A list of examples of “spaces that are not globally hyperbolic” is presented (pages 121-125). As mentioned, the chapter of geometrical apparatus (covariant derivatives,curvature, parallel transport) is exceptional. Fulling highlights John Synge’s ‘world-function’ (page 176, Bryce DeWitt also highlights Synge’s world function).(7) An introductory account of renormalization occupies chapter nine. Note the Table (page 189) being quite useful.The book concludes with a two-part appendix–Klein Paradox, from electrodynamics following which black-hole analogue. We learn: “The virtual particles of quantum field theory cannot always be reasoned about as if they were real classical particles.” (page 283).(8) Concluding my review: An excellent text (with problems and hints) for advanced students. A mixture of quantum mechanics, field theory, general relativity, attention to certain mathematical niceties that are wanting in elementary treatments. Fulling writes: “a competent physicist must know some genuine spectral theory” and “there are interesting mathematical points still to be settled, this book will have served its purpose if its very deficiencies recruit a new generation of investigators to finish the job.”Highly recommended for enrichment.

⭐This is a very good book. It contains many details of physical significances and applicability of quantum gravity.

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