Boundary versus Bulk Dynamics of Extended Objects and the Fractal Structure of Quantum Spacetime

List of Figures......Page 3
List of Tables......Page 9
List of Definitions......Page 11
List of Propositions......Page 13
List of Notations......Page 17
Introduction......Page 23
Boundary Dynamics......Page 43
``Shadow'' Dynamics......Page 44
Preliminaries......Page 53
Area Quantization Scheme: Original Formulation......Page 56
The Basic Action......Page 58
Reparametrized Schild Formulation......Page 68
Hamilton--Jacobi Theory......Page 72
Hamilton--Jacobi Equation: Ogielski Formulation......Page 73
Hamilton-Jacobi Equation: Reparametrized Formulation......Page 76
Classical Area Effective Formulation......Page 80
Reparametrized Canonical Formulation......Page 84
Covariant Schild Action......Page 88
Quantum Propagator......Page 91
Functional Wave Equation......Page 96
Integrating the Functional Wave Equation......Page 102
Integrating the Path--Integral......Page 107
Summary, Comments, Highlights and More......Page 109
Reparametrized Schild Action......Page 117
p-brane Hamilton Jacobi Theory......Page 122
Equivalence with Nambu--Goto Dynamics......Page 125
Functional Schrödinger Equation and p-brane......Page 129
p-brane Quantum Propagator......Page 130
String Functional Solutions......Page 135
Plane Wave Solution......Page 136
Gaussian Loop Wave--Packet......Page 138
Preliminaries......Page 141
The String......Page 142
The p-brane......Page 150
The Shape Uncertainty Principle......Page 153
The Hausdorf Dimension of a Quantum String......Page 158
Classical--to--Fractal Geometric Transition......Page 161
Couplings and Limits......Page 165
Classical Limit......Page 166
Overview......Page 173
Loop and String States......Page 174
Nonstandard Functional Quantization......Page 187
Boundary Shadow Dynamics and M-Theory......Page 193
Covariant ``Speculation''......Page 196
Superconductivity and Quantum Spacetime......Page 198
Conclusion......Page 200
Lagrangian, Hamiltonian and Coefficients!......Page 211
bold0mu mumu Raw and bold0mu mumu Raw Variation of the Action for a p-brane......Page 212
Momentum Variation in Reparametrized Hamiltonian Formalism......Page 215
Hamilton Principle with one Free Boundary......Page 216
Functional Integration of the bold0mu mumu Raw( bold0mu mumu Raw) Fields......Page 219
Functional Integration of the bold0mu mumu Raw- ( bold0mu mumu Raw ) Fields......Page 221
Functional Integration of the bold0mu mumu PPRawPPPP- ( bold0mu mumu Raw ) Fields......Page 222
Saddle Point Evaluation......Page 223
Equations Satisfied by the Kernel......Page 224
Kernel Functional Wave Equation......Page 225
Holographic Coordinates: Functional Derivatives......Page 226
Functional Derivatives of the Classical Action......Page 230
Solutions for the  and  Kernel Ansatz Parameters......Page 232
Infinitesimals and Infinities......Page 233
Some Examples......Page 239
Ultra Euclidean Space......Page 241
Functional Spaces......Page 245
Diffusion Process on  E { e n } ( Rd  R)......Page 251
Overview......Page 261
Functional and Holographic Derivatives......Page 262