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E-grāmata: Local Zeta Regularization And The Scalar Casimir Effect: A General Approach Based On Integral Kernels

(Univ Degli Studi Di Milano, Italy), (Univ Degli Studi Di Milano, Italy)
  • Formāts: 276 pages
  • Izdošanas datums: 06-Oct-2017
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • Valoda: eng
  • ISBN-13: 9789813225015
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Local Zeta Regularization And The Scalar Casimir Effect: A General Approach Based On Integral KernelsPalielināt
  • Formāts: 276 pages
  • Izdošanas datums: 06-Oct-2017
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • Valoda: eng
  • ISBN-13: 9789813225015
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Zeta regularization is a method to treat the divergent quantities appearing in several areas of mathematics and physics, say Fermi and Pizzocchero, and these are managed by introducing a complex parameter, with the role of a regular, and defining their renormalized versions in terms of the analytic continuation with respect to the regulator. Their topics include the zeta-regularized stress-energy vacuum expectation value in terms of integral kernels, total energy and forces on the boundary, a massless field between parallel hyperplanes, a massive field constrained by perpendicular hyperplanes, and a scalar field with a harmonic background potential. Annotation ©2018 Ringgold, Inc., Portland, OR (protoview.com)

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.
Preface v
Acknowledgements xi
General theory 1(76)
1 Zeta regularization for a scalar field
3(12)
1.1 Conventions and notations
4(1)
1.2 The general setting
5(3)
1.3 Zeta regularization and the stress-energy tensor
8(4)
1.4 More about the stress-energy tensor and its VEV
12(3)
1.4.1 Staticity features of the stress-energy VEV
12(1)
1.4.2 Conformal and non-conformal parts of the stress-energy VEV
12(1)
1.4.3 A few remarks on the total energy and pressure on the boundary
13(2)
2 The zeta-regularized stress-energy VEV in terms of integral kernels
15(40)
2.1 Basics on integral kernels
15(2)
2.2 The operator A
17(2)
2.3 The Green function
19(1)
2.4 The Dirichlet kernel
20(2)
2.4.1 Basic facts
20(1)
2.4.2 Some additional remarks
21(1)
2.5 The regularized propagator and the stress-energy VEV: connections with the Dirichlet kernel
22(4)
2.6 The heat kernel, the cylinder kernel and some variations
26(11)
2.6.1 Basic facts
26(3)
2.6.2 The modified cylinder kernel
29(1)
2.6.3 The case of a non-negative A
30(1)
2.6.4 Connections between the cylinder kernel and a (d + 1)-dimensional Green function
31(3)
2.6.5 Behaviour of the heat and cylinder kernels for small and large t
34(3)
2.7 The Dirichlet kernel as Mellin transform of the heat or cylinder kernel
37(3)
2.8 Analytic continuation of Mellin transforms
40(10)
2.8.1 Continuation via asymptotic expansions
40(4)
2.8.2 Continuation via integration by parts
44(2)
2.8.3 Continuation via complex integration
46(4)
2.9 Product configurations: factorization of the heat kernel
50(1)
2.10 Slab configurations: reduction to a lower-dimensional problem
51(4)
3 Total energy and forces on the boundary
55(12)
3.1 The total energy
55(5)
3.1.1 The reduced energy for a slab configuration
58(2)
3.2 The pressure on the boundary
60(2)
3.2.1 An explicit expression for the regularized pressure
61(1)
3.3 An equivalent characterization of boundary forces
62(2)
3.4 Integrated forces on the boundary
64(1)
3.5 A comment on some previous "anomalies"
65(2)
4 Some variations of the previous schemes
67(10)
4.1 The basic Hilbert space when 0 is a proper eigenvalue of A; the case of Neumann and periodic boundary conditions
67(2)
4.2 The case where 0 belongs to the continuous spectrum of A
69(5)
4.3 Variations involving the spatial domain
74(3)
Applications 77(132)
5 A massless field on the segment
79(18)
5.1 Introducing the problem for arbitrary boundary conditions
80(1)
5.2 The cylinder and Dirichlet kernels
81(2)
5.3 The stress-energy tensor
83(1)
5.4 The total energy
84(1)
5.5 The boundary forces
85(1)
5.6 Dirichlet boundary conditions
85(3)
5.7 Dirichlet-Neumann boundary conditions
88(1)
5.8 Neumann boundary conditions
89(1)
5.9 Periodic boundary conditions
90(2)
5.10 Comparison with the previous literature
92(5)
6 A massless field between parallel hyperplanes
97(16)
6.1 Introducing the problem for arbitrary boundary conditions
99(1)
6.2 The reduced Dirichlet and cylinder kernels
100(2)
6.3 The stress-energy tensor
102(1)
6.4 The reduced energy
103(1)
6.5 The boundary forces
104(1)
6.6 Dirichlet boundary conditions
105(2)
6.7 Dirichlet-Neumann boundary conditions
107(2)
6.8 Neumann boundary conditions
109(1)
6.9 Periodic boundary conditions
110(3)
7 A massive field constrained by perpendicular hyperplanes
113(24)
7.1 Introducing the problem for arbitrary boundary conditions
114(1)
7.2 The reduced heat kernel
114(2)
7.3 The reduced Dirichlet kernel
116(1)
7.4 The d-dimensional Dirichlet kernel
117(2)
7.5 The stress-energy tensor
119(2)
7.6 The boundary forces
121(3)
7.7 Introducing two examples
124(1)
7.8 A half-space in spatial dimension d = 3
124(3)
7.9 The rectangular wedge
127(5)
7.10 The previous examples in the zero mass limit
132(5)
8 A massless field in a three-dimensional wedge
137(16)
8.1 Introducing the problem for arbitrary boundary conditions
138(2)
8.2 The Dirichlet kernel
140(1)
8.3 The stress-energy tensor
140(1)
8.4 The boundary forces
141(1)
8.5 Some remarks
142(1)
8.6 Dirichlet boundary conditions
143(3)
8.7 Dirichlet-Neumann boundary conditions
146(1)
8.8 Neumann boundary conditions
147(2)
8.9 Periodic boundary conditions (the cosmic string)
149(4)
9 A scalar field with a harmonic background potential
153(28)
9.1 Introducing the problem
154(1)
9.2 The heat kernel
155(1)
9.3 Rescaled spherical coordinates
156(2)
9.4 Regularity properties of some coefficients
158(1)
9.5 Analytic continuation of the regularized stress-energy tensor
159(3)
9.6 Asymptotics for small values of the radial coordinate
162(1)
9.7 Asymptotics for large values of the radial coordinate
163(2)
9.8 The total energy
165(3)
9.8.1 The bulk energy
165(2)
9.8.2 The boundary energy
167(1)
9.8.3 A remark on energy anomalies
168(1)
9.9 A harmonic potential in spatial dimension d = 2
168(4)
9.10 A harmonic potential in spatial dimension d = 3
172(9)
10 A massless field inside a rectangular box
181(50)
10.1 Introducing the problem
183(1)
10.2 The heat kernel
184(3)
10.2.1 First representation: large t expansion
185(1)
10.2.2 Second representation: small t expansion
186(1)
10.3 The Dirichlet kernel
187(5)
10.3.1 Series representation and analytic continuation of Ds(>)
188(1)
10.3.2 Series representation and analytic continuation of Ds(<)
188(3)
10.3.3 Conclusions for the Dirichlet kernel
191(1)
10.4 The stress-energy tensor
192(1)
10.5 The pressure on the boundary
193(2)
10.6 The total energy
195(2)
10.7 The total force on a side of the box
197(2)
10.8 On the convergence of the previous series representations
199(2)
10.9 Scaling considerations
201(1)
10.10 A rectangular box in spatial dimension d = 2
202(7)
Appendix A The "improved" stress-energy tensor 209(6)
Appendix B On the regularity of some integral kernels 215(10)
Appendix C A contour integral representation for Mellin transforms 225(2)
Appendix D Some identities for the Dirichlet kernel in a slab configuration 227(4)
Appendix E Derivation of some results on boundary forces 231(4)
E.1 The regularized pressure in the case of Dirichlet boundary conditions
231(2)
E.2 A variational identity
233(2)
Appendix F An explicit expression for the renormalized Dirichlet kernel of half-integer order 235(4)
Bibliography 239(12)
Index 251
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