Differential Geometry, Analysis and Physics Jeffrey M. Lee c 2000 Jeffrey Marc lee (cid:141) ii Contents 0.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 1 Preliminaries and Local Theory 1 1.1 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Chain Rule, Product rule and Taylor’s Theorem . . . . . . . . . 11 1.3 Local theory of maps . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Differentiable Manifolds 15 2.1 Rough Ideas I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Topological Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Differentiable Manifolds and Differentiable Maps . . . . . . . . . 17 2.4 Pseudo-Groups and Models Spaces . . . . . . . . . . . . . . . . . 22 2.5 Smooth Maps and Diffeomorphisms . . . . . . . . . . . . . . . . 27 2.6 Coverings and Discrete groups . . . . . . . . . . . . . . . . . . . 30 2.6.1 Covering spaces and the fundamental group . . . . . . . . 30 2.6.2 Discrete Group Actions . . . . . . . . . . . . . . . . . . . 36 2.7 Grassmannian manifolds . . . . . . . . . . . . . . . . . . . . . . . 39 2.8 Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.9 Manifolds with boundary. . . . . . . . . . . . . . . . . . . . . . . 43 3 The Tangent Structure 47 3.1 Rough Ideas II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4 The Tangent Map . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.5 The Tangent and Cotangent Bundles . . . . . . . . . . . . . . . . 55 3.5.1 Tangent Bundle. . . . . . . . . . . . . . . . . . . . . . . . 55 3.5.2 The Cotangent Bundle . . . . . . . . . . . . . . . . . . . . 57 3.6 Important Special Situations. . . . . . . . . . . . . . . . . . . . . 59 4 Submanifold, Immersion and Submersion. 63 4.1 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Submanifolds of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3 Regular and Critical Points and Values. . . . . . . . . . . . . . . 66 4.4 Immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 iii iv CONTENTS 4.5 Immersed Submanifolds and Initial Submanifolds . . . . . . . . . 71 4.6 Submersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.7 Morse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.8 Problem set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5 Lie Groups I 81 5.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . 81 5.2 Lie Group Homomorphisms . . . . . . . . . . . . . . . . . . . . . 84 6 Fiber Bundles and Vector Bundles I 87 6.1 Transitions Maps and Structure . . . . . . . . . . . . . . . . . . . 94 6.2 Useful ways to think about vector bundles . . . . . . . . . . . . . 94 6.3 Sections of a Vector Bundle . . . . . . . . . . . . . . . . . . . . . 97 6.4 Sheaves,Germs and Jets . . . . . . . . . . . . . . . . . . . . . . . 98 6.5 Jets and Jet bundles . . . . . . . . . . . . . . . . . . . . . . . . . 102 7 Vector Fields and 1-Forms 105 7.1 Definition of vector fields and 1-forms . . . . . . . . . . . . . . . 105 7.2 Pull back and push forward of functions and 1-forms . . . . . . . 106 7.3 Frame Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.4 Lie Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 7.5 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.6 Action by pullback and push-forward . . . . . . . . . . . . . . . . 112 7.7 Flows and Vector Fields . . . . . . . . . . . . . . . . . . . . . . . 114 7.8 Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.9 Time Dependent Fields . . . . . . . . . . . . . . . . . . . . . . . 123 8 Lie Groups II 125 8.1 Spinors and rotation . . . . . . . . . . . . . . . . . . . . . . . . . 133 9 Multilinear Bundles and Tensors Fields 137 9.1 Multilinear Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . 137 9.1.1 Contraction of tensors . . . . . . . . . . . . . . . . . . . . 141 9.1.2 Alternating Multilinear Algebra. . . . . . . . . . . . . . . 142 9.1.3 Orientation on vector spaces . . . . . . . . . . . . . . . . 146 9.2 Multilinear Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 147 9.3 Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 9.4 Tensor Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . 149 10 Differential forms 153 10.1 Pullback of a differential form. . . . . . . . . . . . . . . . . . . . 155 10.2 Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 156 10.3 Maxwell’s equations. . . . . . . . . . . . . . . . . . . . . . . . . 159 10.4 Lie derivative, interior product and exterior derivative. . . . . . . 161 10.5 Time Dependent Fields (Part II) . . . . . . . . . . . . . . . . . . 163 10.6 Vector valued and algebra valued forms. . . . . . . . . . . . . . . 163 CONTENTS v 10.7 Global Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . 165 10.8 Orientation of manifolds with boundary . . . . . . . . . . . . . . 167 10.9 Integration of Differential Forms. . . . . . . . . . . . . . . . . . . 168 10.10Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 10.11Vector Bundle Valued Forms. . . . . . . . . . . . . . . . . . . . . 172 11 Distributions and Frobenius’ Theorem 175 11.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 11.2 Integrability of Regular Distributions . . . . . . . . . . . . . . . 175 11.3 The local version Frobenius’ theorem . . . . . . . . . . . . . . . . 177 11.4 Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 11.5 The Global Frobenius Theorem . . . . . . . . . . . . . . . . . . . 183 11.6 Singular Distributions . . . . . . . . . . . . . . . . . . . . . . . . 185 12 Connections on Vector Bundles 189 12.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 12.2 Local Frame Fields and Connection Forms . . . . . . . . . . . . . 191 12.3 Parallel Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 193 12.4 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 13 Riemannian and semi-Riemannian Manifolds 201 13.1 The Linear Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 201 13.1.1 Scalar Products . . . . . . . . . . . . . . . . . . . . . . . 201 13.1.2 Natural Extensions and the Star Operator . . . . . . . . . 203 13.2 Surface Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 13.3 Riemannian and semi-Riemannian Metrics . . . . . . . . . . . . . 214 13.4 The Riemannian case (positive definite metric) . . . . . . . . . . 220 13.5 Levi-Civita Connection. . . . . . . . . . . . . . . . . . . . . . . . 221 13.6 Covariant differentiation of vector fields along maps. . . . . . . . 228 13.7 Covariant differentiation of tensor fields . . . . . . . . . . . . . . 229 13.8 Comparing the Differential Operators . . . . . . . . . . . . . . . 230 14 Formalisms for Calculation 233 14.1 Tensor Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 14.2 Covariant Exterior Calculus, Bundle-Valued Forms . . . . . . . . 234 15 Topology 235 15.1 Attaching Spaces and Quotient Topology . . . . . . . . . . . . . 235 15.2 Topological Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 15.3 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 15.4 Cell Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 16 Algebraic Topology 245 16.1 Axioms for a Homology Theory . . . . . . . . . . . . . . . . . . . 245 16.2 Simplicial Homology . . . . . . . . . . . . . . . . . . . . . . . . . 246 16.3 Singular Homology . . . . . . . . . . . . . . . . . . . . . . . . . . 246 vi CONTENTS 16.4 Cellular Homology . . . . . . . . . . . . . . . . . . . . . . . . . . 246 16.5 Universal Coefficient theorem . . . . . . . . . . . . . . . . . . . . 246 16.6 Axioms for a Cohomology Theory . . . . . . . . . . . . . . . . . 246 16.7 DeRham Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 246 16.8 Topology of Vector Bundles . . . . . . . . . . . . . . . . . . . . . 246 16.9 deRham Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 248 16.10The Meyer Vietoris Sequence . . . . . . . . . . . . . . . . . . . . 252 16.11Sheaf Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 253 16.12Characteristic Classes . . . . . . . . . . . . . . . . . . . . . . . . 253 17 Lie Groups and Lie Algebras 255 17.1 Lie Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 17.2 Classical complex Lie algebras. . . . . . . . . . . . . . . . . . . . 257 17.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . 258 17.3 The Adjoint Representation . . . . . . . . . . . . . . . . . . . . . 259 17.4 The Universal Enveloping Algebra . . . . . . . . . . . . . . . . . 261 17.5 The Adjoint Representation of a Lie group. . . . . . . . . . . . . 265 18 Group Actions and Homogenous Spaces 271 18.1 Our Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 18.1.1 Left actions . . . . . . . . . . . . . . . . . . . . . . . . . . 272 18.1.2 Right actions . . . . . . . . . . . . . . . . . . . . . . . . . 273 18.1.3 Equivariance . . . . . . . . . . . . . . . . . . . . . . . . . 273 18.1.4 The action of Diff(M) and map-related vector fields. . . 274 18.1.5 Lie derivative for equivariant bundles. . . . . . . . . . . . 274 18.2 Homogeneous Spaces. . . . . . . . . . . . . . . . . . . . . . . . . 275 19 Fiber Bundles and Connections 279 19.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 19.2 Principal and Associated Bundles . . . . . . . . . . . . . . . . . . 282 20 Analysis on Manifolds 285 20.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 20.1.1 Star Operator II . . . . . . . . . . . . . . . . . . . . . . . 285 20.1.2 Divergence, Gradient, Curl . . . . . . . . . . . . . . . . . 286 20.2 The Laplace Operator . . . . . . . . . . . . . . . . . . . . . . . . 286 20.3 Spectral Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 289 20.4 Hodge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 20.5 Dirac Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 20.5.1 Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . 291 20.5.2 The Clifford group and Spinor group . . . . . . . . . . . . 296 20.6 The Structure of Clifford Algebras . . . . . . . . . . . . . . . . . 296 20.6.1 Gamma Matrices . . . . . . . . . . . . . . . . . . . . . . . 297 20.7 Clifford Algebra Structure and Representation . . . . . . . . . . 298 20.7.1 Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . 298 20.7.2 Hyperbolic Spaces And Witt Decomposition. . . . . . . . 299 CONTENTS vii 20.7.3 Witt’s Decomposition and Clifford Algebras . . . . . . . . 300 20.7.4 The Chirality operator . . . . . . . . . . . . . . . . . . . 301 20.7.5 Spin Bundles and Spin-c Bundles . . . . . . . . . . . . . . 302 20.7.6 Harmonic Spinors . . . . . . . . . . . . . . . . . . . . . . 302 21 Complex Manifolds 303 21.1 Some complex linear algebra . . . . . . . . . . . . . . . . . . . . 303 21.2 Complex structure . . . . . . . . . . . . . . . . . . . . . . . . . . 306 21.3 Complex Tangent Structures . . . . . . . . . . . . . . . . . . . . 309 21.4 The holomorphic tangent map. . . . . . . . . . . . . . . . . . . . 310 21.5 Dual spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 21.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 21.7 The holomorphic inverse and implicit functions theorems. . . . . 312 22 Classical Mechanics 315 22.1 Particle motion and Lagrangian Systems . . . . . . . . . . . . . . 315 22.1.1 Basic Variational Formalism for a Lagrangian . . . . . . . 316 22.1.2 Two examples of a Lagrangian . . . . . . . . . . . . . . . 319 22.2 Symmetry, Conservation and Noether’s Theorem . . . . . . . . . 319 22.2.1 Lagrangians with symmetries. . . . . . . . . . . . . . . . . 321 22.2.2 Lie Groups and Left Invariants Lagrangians . . . . . . . . 322 22.3 The Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . 322 23 Symplectic Geometry 325 23.1 Symplectic Linear Algebra . . . . . . . . . . . . . . . . . . . . . . 325 23.2 Canonical Form (Linear case) . . . . . . . . . . . . . . . . . . . . 327 23.3 Symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 327 23.4 Complex Structure and K¨ahler Manifolds . . . . . . . . . . . . . 329 23.5 Symplectic musical isomorphisms . . . . . . . . . . . . . . . . . 332 23.6 Darboux’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 332 23.7 Poisson Brackets and Hamiltonian vector fields . . . . . . . . . . 334 23.8 Configuration space and Phase space . . . . . . . . . . . . . . . 337 23.9 Transfer of symplectic structure to the Tangent bundle . . . . . . 338 23.10Coadjoint Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 23.11The Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 23.11.1The configuration in R3N . . . . . . . . . . . . . . . . . . 342 23.11.2Modelling the rigid body on SO(3) . . . . . . . . . . . . . 342 23.11.3The trivial bundle picture . . . . . . . . . . . . . . . . . . 343 23.12The momentum map and Hamiltonian actions. . . . . . . . . . . 343 24 Poisson Geometry 347 24.1 Poisson Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 347 viii CONTENTS 25 Quantization 351 25.1 Operators on a Hilbert Space . . . . . . . . . . . . . . . . . . . . 351 25.2 C*-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 25.2.1 Matrix Algebras . . . . . . . . . . . . . . . . . . . . . . . 354 25.3 Jordan-Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 354 26 Appendices 357 26.1 A. Primer for Manifold Theory . . . . . . . . . . . . . . . . . . . 357 26.1.1 Fixing a problem . . . . . . . . . . . . . . . . . . . . . . . 360 26.2 B. Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . 361 26.2.1 Separation Axioms . . . . . . . . . . . . . . . . . . . . . . 363 26.2.2 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 364 26.3 C. Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . 365 26.3.1 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . 367 26.3.2 Orthonormal sets . . . . . . . . . . . . . . . . . . . . . . . 368 26.4 D. Overview of Classical Physics . . . . . . . . . . . . . . . . . . 368 26.4.1 Units of measurement . . . . . . . . . . . . . . . . . . . . 368 26.4.2 Newton’s equations. . . . . . . . . . . . . . . . . . . . . . 369 26.4.3 Classical particle motion in a conservative field . . . . . . 370 26.4.4 Some simple mechanical systems . . . . . . . . . . . . . . 375 26.4.5 The Basic Ideas of Relativity . . . . . . . . . . . . . . . . 380 26.4.6 Variational Analysis of Classical Field Theory . . . . . . . 385 26.4.7 Symmetry and Noether’s theorem for field theory . . . . . 386 26.4.8 Electricity and Magnetism . . . . . . . . . . . . . . . . . . 388 26.4.9 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . 390 26.5 E. Calculus on Banach Spaces . . . . . . . . . . . . . . . . . . . . 390 26.6 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 26.7 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 26.8 Chain Rule, Product rule and Taylor’s Theorem . . . . . . . . . 400 26.9 Local theory of maps . . . . . . . . . . . . . . . . . . . . . . . . . 405 26.9.1 Linear case. . . . . . . . . . . . . . . . . . . . . . . . . . 411 26.9.2 Local (nonlinear) case. . . . . . . . . . . . . . . . . . 412 26.10The Tangent Bundle of an Open Subset of a Banach Space . . . 413 26.11Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 26.11.1Existence and uniqueness for differential equations . . . . 417 26.11.2Differential equations depending on a parameter. . . . . . 418 26.12Multilinear Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . 418 26.12.1Smooth Banach Vector Bundles . . . . . . . . . . . . . . . 435 26.12.2Formulary . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 26.13Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 26.14Group action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 26.15Notation and font usage guide . . . . . . . . . . . . . . . . . . . . 445 27 Bibliography 453 0.1. PREFACE ix 0.1 Preface InthisbookIpresentdifferentialgeometryandrelatedmathematicaltopicswith the help of examples from physics. It is well known that there is something strikingly mathematical about the physical universe as it is conceived of in the physical sciences. The convergence of physics with mathematics, especially differential geometry, topology and global analysis is even more pronounced in the newer quantum theories such as gauge field theory and string theory. The amount of mathematical sophistication required for a good understanding of modern physics is astounding. On the other hand, the philosophy of this book is that mathematics itself is illuminated by physics and physical thinking. The ideal of a truth that transcends all interpretation is perhaps unattain- able. Even the two most impressively objective realities, the physical and the mathematical, are still only approachable through, and are ultimately insepa- rable from, our normative and linguistic background. And yet it is exactly the tendency of these two sciences to point beyond themselves to something tran- scendentally real that so inspires us. Whenever we interpret something real, whether physical or mathematical, there will be those aspects which arise as mere artifacts of our current descriptive scheme and those aspects that seem to beobjectiverealitieswhicharerevealedequallywellthroughanyofamultitude of equivalent descriptive schemes-“cognitive inertial frames” as it were. This theme is played out even within geometry itself where a viewpoint or interpre- tive scheme translates to the notion of a coordinate system on a differentiable manifold. A physicist has no trouble believing that a vector field is something beyond its representation in any particular coordinate system since the vector field it- selfissomethingphysical. Itisthewaythatthevariouscoordinatedescriptions relate to each other (covariance) that manifests to the understanding the pres- ence of an invariant physical reality. This seems to be very much how human perceptionworksanditisinterestingthatthelanguageoftensorshasshownup in the cognitive science literature. On the other hand, there is a similar idea as to what should count as a geometric reality. According to Felix Klein the task of geometry is “givenamanifoldandagroupoftransformationsofthemanifold, to study the manifold configurations with respect to those features which are not altered by the transformations of the group” -Felix Klein 1893 The geometric is then that which is invariant under the action of the group. As a simple example we may consider the set of points on a plane. We may imposeoneofaninfinitenumberofrectangularcoordinatesystemsontheplane. If, in one such coordinate system (x,y), two points P and Q have coordinates (x(P),y(P)) and (x(Q),y(Q)) respectively, then while the differences ∆x = x(P) x(Q) and ∆y =y(P) y(Q) are very much dependent on the choice of − − these rectangular coordinates, the quantity (∆x)2+(∆y)2 is not so dependent. x CONTENTS If (X,Y) are any other set of rectangular coordinates then we have (∆x)2 + (∆y)2 = (∆X)2+(∆Y)2. Thus we have the intuition that there is something morerealaboutthatlaterquantity. Similarly,thereexistsdistinguishedsystems forassigningthreespatialcoordinates(x,y,z)andasingletemporalcoordinate t to any simple event in the physical world as conceived of in relativity theory. Thesearecalledinertialcoordinatesystems. Nowaccordingtospecialrelativity the invariant relational quantity that exists between any two events is (∆x)2+ (∆y)2+(∆z)2 (∆t)2. We see that there is a similarity between the physical − notion of the objective event and the abstract notion of geometric point. And yet the minus sign presents some conceptual challenges. Whiletheinvarianceunderagroupactionapproachtogeometryispowerful itisbecomingcleartomanyresearchersthattheloosernotionsofgroupoidand pseudogroup has a significant role to play. Since physical thinking and geometric thinking are so similar, and even at times identical, it should not seem strange that we not only understand the physical through mathematical thinking but conversely we gain better mathe- matical understanding by a kind of physical thinking. Seeing differential geom- etry applied to physics actually helps one understand geometric mathematics better. Physics even inspires purely mathematical questions for research. An example of this is the various mathematical topics that center around the no- tion of quantization. There are interesting mathematical questions that arise whenonestartsthinkingabouttheconnectionsbetweenaquantumsystemand its classical analogue. In some sense, the study of the Laplace operator on a differentiablemanifoldanditsspectrumisa“quantizedversion”ofthestudyof thegeodesicflowandthewholeRiemannianapparatus; curvature, volume, and so forth. This is not the definitive interpretation of what a quantized geometry should be and there are many areas of mathematical research that seem to be related to the physical notions of quantum verses classical. It comes as a sur- prisetosomethattheuncertaintyprincipleisacompletelymathematicalnotion within the purview of harmonic analysis. Given a specific context in harmonic analysis or spectral theory, one may actually prove the uncertainty principle. Physicalintuitionmayhelpevenifoneisstudyinga“toyphysicalsystem”that doesn’texistinnatureoronlyexistsasanapproximation(e.g. anonrelativistic quantummechanicalsystem). Attheveryleast,physicalthinkinginspiresgood mathematics. I have purposely allowed some redundancy to occur in the presentation be- cause I believe that important ideas should be repeated. Finally we mention that for those readers who have not seen any physics for a while we put a short and extremely incomplete overview of physics in an appendix. The only purpose of this appendix is to provide a sort of warm up which might serve to jog the readers memory of a few forgotten bits of undergraduate level physics.