Ten physical applications of spectral zeta functions

Cover ......Page 1
Lecture Notes in Physics: Monographs 35 ......Page 2
Ten Physical Applications of Spectral Zeta Functions ......Page 4
Preface ......Page 8
Contents ......Page 10
1.1.1 The Riemann zeta function ......Page 14
1.1.2 The Hurwitz zeta function ......Page 16
1.1.3 The Epstein zeta function ......Page 17
1.2.1 The zeta function of a differential operator ......Page 19
1.2.2 Regularization of the vacuum energy ......Page 20
1.2.3 Regularization of one-loop graphs ......Page 21
1.3.1 Some explicit examples ......Page 22
1.3.2 Comparison with other regularization methods ......Page 24
1.3.3 A word of warning ......Page 27
1.4.1 Calculation of heat-kernel coefficients ......Page 28
1.4.2 Determinant of the Laplacian ......Page 30
1.4.3 Proper definition of the zeta function of a partial differential operator ......Page 32
2.1 A simple recurrence for the higher derivatives of the Hurwitz zeta function ......Page 34
2.2 The zeta-function regularization theorem ......Page 41
2.2.1 The theorem (special form) ......Page 43
2.3 Immediate application of the theorem ......Page 51
2.4 Expressions for multi-series on combinations ihvolving arbitrary constants and exponents ......Page 54
3.1 Dealing with the non-polynomical term AER ......Page 64
3.1.1 Proof of asymptotlcity of the series ......Page 66
3.1.2 The remainder term and the Poisson resummation formula ......Page 71
3.2 Numerical estimates of the remainder ......Page 73
3.3 Application: summation of the string partition function for different ranges of the temperature ......Page 77
4 ANALYTICAL AND NUMERICAL STUDY OF INHOMOGENEOUS EPSTEIN AND EPSTEIN-HURWITZ ZETA FUNCTIONS ......Page 86
4.1 Explicit analytical continuation of inhomogeneous Epstein zeta functions ......Page 87
4.1.1 The particular case of the basic one-dimensional Epstein-Hurwitz series ......Page 89
4.1.2 The homogeneous case: Chowla-Selberg\'s formula ......Page 92
4.1.3 Derivation of the general formula ......Page 93
4.2 Numerical analysis of the inhomogeneous generalized Epsteln-Hurwltz zeta function ......Page 97
4.2.1 Asymptotic expansions of the function and its derivatives with respect to the variable and parameters ......Page 98
5.1.1 The classical Casimir effect ......Page 110
5.1.2 Connection with the van der Waals forces and the London theory ......Page 111
5.1.3 The specific contribution of Casimir and Polder: retarded van der Waals forces ......Page 112
5.1.4 The Lifschitz theory ......Page 114
5.2.2 The experiment of Tabor and Wintert\'on: transition from normal to retarded van der Waals forces ......Page 116
5.2.3 The experiment of Sabiskl and Anderson ......Page 117
5.2.4 A contemporary experiment ......Page 118
5.3.1 The local formulation of the Casimir effect ......Page 119
5.3.2 The mystery of the Casimlr effect ......Page 120
5.3.3 The concept of the vacuum energy ......Page 121
5.3.4 The explicit, regularized definition of the Casimlr energy ......Page 123
5.3.5 Definition of the Casimir energy density and its relation with the vacuum energy ......Page 124
5.4 A very simple computation of the Casimir effect ......Page 126
5.4.1 The Caslmir effect for a free massless scalar field in S 1 x R d and in T 2 x R 2 spacetlmes ......Page 127
5.4.2 The case of a massless scalar field between p perpendicular pairs of parallel walls with Dirlchlet boundary conditions ......Page 130
5.4.3 Massless scalar field with periodic and Neumann boundary conditions, and electromagnetic field ......Page 135
6 FOUR PHYSICAL APPLICATIONS OF THE INHOMOGENEOUS GENERALIZED EPSTEIN-HURWITZ ZETA FUNCTIONS ......Page 142
6.1 Application: the Caslmir energy over Riemann surfaces ......Page 143
6.2 Application: Kaluza-Kleln model with spherical compactificatlon ......Page 149
6.3 Critical behavior of a field theory at non-zero temperature ......Page 156
6.4 Application to quantlzlng through the Wheeler-De Witt equation ......Page 159
6.4.1 Explicit zeta-functlon calculation of the essential determinant and extrema of the potential ......Page 160
6.4.2 An alternative treatment by means of Eisensteln series ......Page 166
7.1 Relation between the generalized Pauli-Villars and the covariant regularizations ......Page 170
7.2 The Casimir energy corresponding to a piecewlse uniform string ......Page 176
7.2.1 The zero temperature theory ......Page 179
7.2.2 Regularized Casimir energy and numerical results ......Page 182
7.2.3 The finite temperature theory ......Page 186
8 APPLICATIONS TO GRAVITY, STRINGS AND P-BRANES ......Page 192
8.1 Application to spontaneous compactificatlon in two-dlmenslonal quantum gravity ......Page 193
8.2 Application to the study of the stability of the rigid membrane ......Page 196
8.2.1 Calculation of the potential ......Page 197
8.2.2 The limit of large spacetime dimensionality ......Page 198
8.2.3 A saddle point analysis ......Page 200
8.2.4 Explicit expressions for the zeta-functlon regularization of the effective potential ......Page 201
8.2.5 Discussion of the general case ......Page 203
9.1 General considerations ......Page 206
9.2 The one-loop effective potential for the self-interacting theory ......Page 208
9.3 The one-loop topological mass ......Page 212
9.4 Renormalization of the theory ......Page 215
9.5 Symmetry breaking mechanism for a massless scalar field ......Page 217
References ......Page 222
Index ......Page 234