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Local Zeta Regularization And The Scalar Casimir Effect A General Approach Based On Integral Kernels

Author: Pizzocchero Livio
Publisher: World Scientific
ISBN: 9813225017
Size: 27.67 MB
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Zeta regularization is a method to treat the divergent quantities appearing in several areas of mathematical physics and, in particular, in quantum field theory; it is based on the fascinating idea that a finite value can be ascribed to a formally divergent expression via analytic continuation with respect to a complex regulating parameter. This book provides a thorough overview of zeta regularization for the vacuum expectation values of the most relevant observables of a quantized, neutral scalar field in Minkowski spacetime; the field can be confined to a spatial domain, with suitable boundary conditions, and an external potential is possibly present. Zeta regularization is performed in this framework for both local and global observables, like the stress-energy tensor and the total energy; the analysis of their vacuum expectation values accounts for the Casimir physics of the system. The analytic continuation process required in this setting by zeta regularization is deeply linked to some integral kernels; these are determined by the fundamental elliptic operator appearing in the evolution equation for the quantum field. The book provides a systematic illustration of these connections, devised as a toolbox for explicit computations in specific configurations; many examples are presented. A comprehensive account is given of the existing literature on this subject, including the previous work of the authors. The book will be useful to anyone interested in a mathematically sound description of quantum vacuum effects, from graduate students to scientists working in this area. Contents: General Theory: Zeta Regularization for a Scalar FieldThe Zeta Regularized Stress-Energy VEV in Terms of Integral KernelsTotal Energy and Forces on the BoundarySome Variations of the Previous SchemesApplications: A Massless Field on the SegmentA Massless Field Between Parallel HyperplanesA Massive Field Constrained by Perpendicular HyperplanesA Massless Field in a Three-Dimensional WedgeA Scalar Field with a Harmonic Background PotentialA Massless Field Inside a Rectangular BoxAppendices: The "Improved" Stress-Energy TensorOn the Regularity of Some Integral KernelsA Contour Integral Representation for Mellin TransformsSome Identities for the Dirichlet Kernel in a Slab ConfigurationDerivation of Some Results on Boundary ForcesAn Explicit Expression for the Renormalized Dirichlet Kernel of Half-Integer Order Readership: Graduate students and researchers including academics in theoretical physics. Keywords: Quantum Field Theory;Zeta Regularization;Casimir Effect;Stress-Energy TensorReview: Key Features: Zeta regularization is used in a systematic way for both local and global aspects related to the vacuum state of a quantized field, marking a difference with respect to the existing literature, in which local aspects (say, the stress-energy tensor) do not receive the full attention they would deserve (especially, in the presence of boundary conditions)Explicit computations are carried out for several configurations, applying in a uniform way the general algorithmsGive a more intuitive approach to the subject by implementing the regularization using canonical quantization in a Lorentzian framework

Advances In Quantum Mechanics

Author: Alessandro Michelangeli
Publisher: Springer
ISBN: 3319589040
Size: 59.71 MB
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This volume collects recent contributions on the contemporary trends in the mathematics of quantum mechanics, and more specifically in mathematical problems arising in quantum many-body dynamics, quantum graph theory, cold atoms, unitary gases, with particular emphasis on the developments of the specific mathematical tools needed, including: linear and non-linear Schrödinger equations, topological invariants, non-commutative geometry, resonances and operator extension theory, among others. Most of contributors are international leading experts or respected young researchers in mathematical physics, PDE, and operator theory. All their material is the fruit of recent studies that have already become a reference in the community. Offering a unified perspective of the mathematics of quantum mechanics, it is a valuable resource for researchers in the field.

Analytic Aspects Of Quantum Fields

Author: Andrei A. Bytsenko
Publisher: World Scientific
ISBN: 9812383646
Size: 15.13 MB
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One of the aims of this book is to explain in a basic manner the seemingly difficult issues of mathematical structure using some specific examples as a guide. In each of the cases considered, a comprehensible physical problem is approached, to which the corresponding mathematical scheme is applied, its usefulness being duly demonstrated. The authors try to fill the gap that always exists between the physics of quantum field theories and the mathematical methods best suited for its formulation, which are increasingly demanding on the mathematical ability of the physicist.

Zeta Regularization Techniques With Applications

Author: E Elizalde
Publisher: World Scientific
ISBN: 9814502987
Size: 54.25 MB
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This book is the result of several years of work by the authors on different aspects of zeta functions and related topics. The aim is twofold. On one hand, a considerable number of useful formulas, essential for dealing with the different aspects of zeta-function regularization (analytic continuation, asymptotic expansions), many of which appear here, in book format, for the first time are presented. On the other hand, the authors show explicitly how to make use of such formulas and techniques in practical applications to physical problems of very different nature. Virtually all types of zeta functions are dealt with in the book. Contents:Riemann and Related Zeta FunctionsZeta-Function Regularization of Sums Over Known SpectrumZeta Function Regularization GeneralizedThe Casimir Effect in Flat SpacetimeHeat-Kernel and Zeta-Function Regularization Techniques in the Theory of Quantum Fields on Hyperbolic SpaceQuantum Properties of Extended ObjectsFinite-Temperature Effects in Quantum Field TheoryString Theory at Non-Zero TemperatureCovariant Effective in 2D GravityAppendix: Calculation of Heat-Kernel CoefficientsIndex Readership: High energy physicists. keywords:Zeta-Function Regularization;Heat Kernel Expansion;Selberg Trace Formula;Casimir Effect;Finite Temperature Effects;Strings at Finite Temperature;Compact Hyperbolic Manifold;p-Branes;2-Dimensional Quantum Gravity

General Principles Of Quantum Field Theory

Author: N.N. Bogolubov
Publisher: Springer Science & Business Media
ISBN: 9400904916
Size: 39.92 MB
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The majority of the "memorable" results of relativistic quantum theory were obtained within the framework of the local quantum field approach. The explanation of the basic principles of the local theory and its mathematical structure has left its mark on all modern activity in this area. Originally, the axiomatic approach arose from attempts to give a mathematical meaning to the quantum field theory of strong interactions (of Yukawa type). The fields in such a theory are realized by operators in Hilbert space with a positive Poincare-invariant scalar product. This "classical" part of the axiomatic approach attained its modern form as far back as the sixties. * It has retained its importance even to this day, in spite of the fact that nowadays the main prospects for the description of the electro-weak and strong interactions are in connection with the theory of gauge fields. In fact, from the point of view of the quark model, the theory of strong interactions of Wightman type was obtained by restricting attention to just the "physical" local operators (such as hadronic fields consisting of ''fundamental'' quark fields) acting in a Hilbert space of physical states. In principle, there are enough such "physical" fields for a description of hadronic physics, although this means that one must reject the traditional local Lagrangian formalism. (The connection is restored in the approximation of low-energy "phe nomenological" Lagrangians.

Lessons From Nanoelectronics

Author: Supriyo Datta
Publisher: World Scientific Publishing Company
ISBN: 9814483907
Size: 23.10 MB
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Everyone is familiar with the amazing performance of a modern smartphone, powered by a billion-plus nanotransistors, each having an active region that is barely a few hundred atoms in length. These lecture notes are about a less appreciated by-product of the microelectronics revolution, namely the deeper understanding of current flow, and device operation that it has enabled, which forms the basis for a new approach to transport problems. The book assumes very little background beyond linear algebra and differential equations, and is intended to be accessible to anyone in any branch of science or engineering. Readers are encouraged to visit the website http://nanohub.org/groups/lnebook to access revisions, corrections, video lectures, tutorials, quizzes and also to join a Q&A forum based on questions from readers.

The Casimir Effect

Author: K A Milton
Publisher: World Scientific
ISBN: 9814492507
Size: 79.62 MB
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In its simplest manifestation, the Casimir effect is a quantum force of attraction between two parallel uncharged conducting plates. More generally, it refers to the interaction — which may be either attractive or repulsive — between material bodies due to quantum fluctuations in whatever fields are relevant. It is a local version of the van der Waals force between molecules. Its sweep ranges from perhaps its being the origin of the cosmological constant to its being responsible for the confinement of quarks. This monograph develops the theory of such forces, based primarily on physically transparent Green's function techniques, and makes applications from quarks to the cosmos, as well as observable consequences in condensed matter systems. It is aimed at graduate students and researchers in theoretical physics, quantum field theory, and applied mathematics. Contents:Introduction to the Casimir EffectCasimir Force Between Parallel PlatesCasimir Force Between Parallel DielectricsCasimir Effect with Perfect SphericalThe Casimir Effect of a Dielectric Ball: The Equivalence of the Casimir Effect and van der Waals ForcesApplication to Hadronic Physics: Zero-Point Energy in the Bag ModelCasimir Effect in Cylindrical GeometriesCasimir Effect in Two Dimensions: The Maxwell-Chern-Simons Casimir EffectCasimir Effect on a D-dimensional SphereCosmological Implications of the Casimir EffectLocal EffectsSonoluminescene and the Dynamical Casimir EffectRadiative Corrections to the Casimir EffectConclusions and OutlookAppendices:Relation of Contour Integral Method to Green's Function ApproachCasimir Effect for a Closed String Readership: High-energy, condensed-matter and nuclear physicists. Keywords:Van Der Waals Forces;Quantum Field Theory;Quantum Fluctuations;Green's Functions;Finite-Temperature Field Theory;Sonoluminescence;Cosmological Constant;Boundary Conditions;Dimensional Dependence;Energy-Momentum TensorReviews:“The Casimir effect is about a lot more than a force between two metals plates, and Milton's book offers a great opportunity to read about it and learn the techniques by which it can be calculated … the book is very comprehensive, clearly written and filled with wonderful physics.”CERN Courier “Milton's attractively slim book will guide the serious beginner around the shoals, and its analytical rigor should also attract experienced theorists. It is appealing also in its unified approach, based mainly on Green's function methods that Milton and others developed in collaboration with Julian Schwinger.”Physics Today “Milton's monograph provides an outstanding review of the present state of the theory and applications of Casmir effects to diverse problems … Because of its clarity and advanced treatment, this book will likely stimulate the next stage in that development by opening a number of impenetrable fields to advanced students.”American Journal of Physics “This carefully written book gives a survey of the experimental and theoretical situation concerning the Casimir effect … All technical points are presented clearly and thoroughly. The book belongs on the shelf of anyone who wishes to make practical calculations involving the Casimir effect. The book contains an extensive bibliography. There are occasional historical notes as well. A graduate student or researcher would gain a good perspective of the role of the Casimir effect in physics from a reading.”Mathematical Reviews “Milton also tries to survey the field while emphasizing the Green's function technique in concrete calculations. In the past decades, he contributed in so many ways to our present understanding of the effect that the book more or less recounts his personal journey through the Casimir world.”Zentralblatt MATH

The Casimir Effect And Its Applications

Author: Vladimir Mikhaĭlovich Mostepanenko
Publisher: Oxford University Press
ISBN: 9780198539988
Size: 55.58 MB
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Of value to the general scientific public, this is the first book in the world scientific literature devoted to the Casimir effect. This topic has important applications in the fields of elementary particle physics, statistical physics, quantum field theory, gravitation and cosmology.