Providing a pedagogical introduction to the essential principles of path integrals and Hamiltonians, this book describes cutting-edge quantum mathematical techniques applicable to a vast range of fields, from quantum mechanics, solid state physics, statistical mechanics, quantum field theory, and superstring theory to financial modeling, polymers, biology, chemistry, and quantum finance. Eschewing use of the Schrödinger equation, the powerful and flexible combination of Hamiltonian operators and path integrals is used to study a range of different quantum and classical random systems, succinctly demonstrating the interplay between a system's path integral, state space, and Hamiltonian. With a practical emphasis on the methodological and mathematical aspects of each derivation, this is a perfect introduction to these versatile mathematical methods, suitable for researchers and graduate students in physics and engineering.

Quantum mechanics is undoubtedly one of the most accurate and important scientific theories in the history of science. The theoretical foundations of quantum mechanics have been discussed in depth in Baaquie (2013e), where the main focus is on the interpretation of the mathematical symbols of quantum mechanics and on its enigmatic superstructure. In contrast, the main focus of this book is on the mathematics of path integral quantum mechanics.
The traditional approach to quantum mechanics has been to study the Schrödinger equation, one of the cornerstones of quantum mechanics, and which is a special case of partial differential equations. Needless to say, the study of the Schrödinger equation continues to be a central task of quantum mechanics, yielding a steady stream of new and valuable results.
Interestingly enough, there are two other formulations of quantum mechanics, namely the operator approach of Heisenberg and the path integral approach of Dirac–Feynman, that provide a mathematical framework which is independent of the Schrödinger equation. In this book, the Schrödinger equation is never directly solved; instead the Hamiltonian operator is analyzed and path integrals for different quantum and classical random systems are studied to gain an understanding of quantum mathematics.
I became aware of path integrals when I was a graduate student, and what intrigued me most was the novelty, flexibility and versatility of their theoretical and mathematical framework. I have spent most of my research years in exploring and employing this framework.
Path integration is a natural generalization of integral calculus and is essentially the integral calculus of infinitely many variables, also called functional integration. There is, however, a fundamental feature of path integration that sets it apart from functional integration, namely the role played by the Hamiltonian in the formalism. All the path integrals discussed in this book have an underlying linear structure that is encoded in the Hamiltonian operator and its linear vector state space. It is this combination of the path integral and its underlying Hamiltonian that provides a powerful and flexible mathematical machinery that can address a vast variety and range of diverse problems. Path integration can also address systems that do not have a Hamiltonian and these systems are not studied. Instead, topics have been chosen that can demonstrate the interplay of the system's path integral, state space, and Hamiltonian.
The Hamiltonian operator and the mathematical formalism of path integration make them eminently suitable for describing quantum indeterminacy as well as classical randomness. In two chapters of the book, namely Chapter 7 on stochastic processes and Chapter 17 on compact degrees of freedom, path integrals are applied to classical stochastic and random systems. The rest of the chapters analyze systems that have quantum indeterminacy.
The range and depth of subjects that come under the sway of path integrals are unified by a common thread, which is the mathematics of path integrals. Problems seemingly unrelated to indeterminacy such as the classification of knots and links or the mathematical properties of manifolds have been solved using path integration. The applications of path integrals are almost as vast as calculus, ranging from finance, polymers, biology, and chemistry to quantum mechanics, solid state physics, statistical mechanics, quantum field theory, superstring theory, and all the way to pure mathematics. The concepts and theoretical underpinnings of quantum mechanics lead to a whole set of new mathematical ideas and have given rise to the subject of quantum mathematics.
The ground-breaking and pioneering book by Feynman and Hibbs (1965) laid the foundation for the study of path integrals in quantum mechanics and is always worth reading. More recent books such as those by Kleinert (1990) and Zinn-Justin (2005) discuss many important aspects of path integration and cover a wide range of applications. Given the complex theoretical and mathematical nature of the subject, no single book can conceivably cover the gamut of worthwhile topics that appear in the study of path integration and there is always a need for books that break new ground. The topics chosen in this book have a minimal overlap with other books on path integrals.
A major field of theoretical physics that is based on path integrals is quantum field theory, which includes the Standard Model of particles and forces. The study of quantum field theory leads to the concept of nonlinear gauge fields and to the concept of renormalization, both of which are beyond the scope this book.
The purpose of the book is to provide a pedagogical introduction to the essential principles of path integrals and of Hamiltonians; for this reason many examples have been worked out in full detail so as to elucidate some of the varied methods and techniques that have proven useful in actually performing path integrations. The emphasis in all the derivations is on the methodological and mathematical aspect of the problem – with matters of interpretation being discussed only in passing. Starting from the simplest examples, the various chapters lay the ground work for analyzing more advanced topics. The book provides an introduction to the foundations of path integral quantum mechanics and is a primer to the techniques and methods employed in the study of quantum finance, as formulated by Baaquie (2004) and Baaquie (2010).

Preface page xv

Acknowledgements xviii

1 Synopsis 1

Part one Fundamental principles 5

2 The mathematical structure of quantum mechanics 7

2.1 The Copenhagen quantum postulate 7

2.2 The superstructure of quantum mechanics 10

2.3 Degree of freedom space F 10

2.4 State space V(F) 11

2.4.1 Hilbert space 14

2.5 Operators O(F) 14

2.6 The process of measurement 18

2.7 The Schrödinger differential equation 19

2.8 Heisenberg operator approach 22

2.9 Dirac–Feynman path integral formulation 23

2.10 Three formulations of quantum mechanics 25

2.11 Quantum entity 26

2.12 Summary 27

3 Operators 30

3.1 Continuous degree of freedom 30

3.2 Basis states for state space 35

3.3 Hermitian operators 36

      3.3.1 Eigenfunctions; completeness 37

      3.3.2 Hamiltonian for a periodic degree of freedom 39

3.4 Position and momentum operators xˆ and pˆ 40

      3.4.1 Momentum operator pˆ 41

3.5 Weyl operators 43

3.6 Quantum numbers; commuting operator 46

3.7 Heisenberg commutation equation 47

3.8 Unitary representation of Heisenberg algebra 48

3.9 Density matrix: pure and mixed states 50

3.10 Self-adjoint operators 51

      3.10.1 Momentum operator on finite interval 52

3.11 Self-adjoint domain 54

      3.11.1 Real eigenvalues 54

3.12 Hamiltonian's self-adjoint extension 55

      3.12.1 Delta function potential 57

3.13 Fermi pseudo-potential 59

3.14 Summary 60

4 The Feynman path integral 61

4.1 Probability amplitude and time evolution 61

4.2 Evolution kernel 63

4.3 Superposition: indeterminate paths 65

4.4 The Dirac–Feynman formula 67

4.5 The Lagrangian 69

      4.5.1 Infinite divisibility of quantum paths 70

4.6 The Feynman path integral 70

4.7 Path integral for evolution kernel 73

4.8 Composition rule for probability amplitudes 76

4.9 Summary 79

5 Hamiltonian mechanics 80

5.1 Canonical equations 80

5.2 Symmetries and conservation laws 82

5.3 Euclidean Lagrangian and Hamiltonian 84

5.4 Phase space path integrals 85

5.5 Poisson bracket 87

5.6 Commutation equations 88

5.7 Dirac bracket and constrained quantization 90

      5.7.1 Dirac bracket for two constraints 91

5.8 Free particle evolution kernel 93

5.9 Hamiltonian and path integral 94

5.10 Coherent states 95

5.11 Coherent state vector 96

5.12 Completeness equation: over-complete 98

5.13 Operators; normal ordering 98

5.14 Path integral for coherent states 99

      5.14.1 Simple harmonic oscillator 101

5.15 Forced harmonic oscillator 102

5.16 Summary 103

6 Path integral quantization 105

6.1 Hamiltonian from Lagrangian 106

6.2 Path integral's classical limit h → 0 109

      6.2.1 Nonclassical paths and free particle 111

6.3 Fermat's principle of least time 112

6.4 Functional differentiation 115

      6.4.1 Chain rule 115

6.5 Equations of motion 116

6.6 Correlation functions 117

6.7 Heisenberg commutation equation 118

      6.7.1 Euclidean commutation equation 121

6.8 Summary 122

Part two Stochastic processes 123

7 Stochastic systems 125

7.1 Classical probability: objective reality 127

      7.1.1 Joint, marginal and conditional probabilities 128

7.2 Review of Gaussian integration 129

7.3 Gaussian white noise 132

      7.3.1 Integrals of white noise 134

7.4 Ito calculus 136

      7.4.1 Stock price 137

7.5 Wilson expansion 138

7.6 Linear Langevin equation 140

      7.6.1 Random paths 142

7.7 Langevin equation with potential 143

      7.7.1 Correlation functions 144

7.8 Nonlinear Langevin equation 145

7.9 Stochastic quantization 148

      7.9.1 Linear Langevin path integral 149

7.10 Fokker–Planck Hamiltonian 151

7.11 Pseudo-Hermitian Fokker–Planck Hamiltonian 153

7.12 Fokker–Planck path integral 156

7.13 Summary 158

Part three Discrete degrees of freedom 159

8 Ising model 161

8.1 Ising degree of freedom and state space 161

      8.1.1 Ising spin's state space V 163

      8.1.2 Bloch sphere 164

8.2 Transfer matrix 165

8.3 Correlators 167

      8.3.1 Periodic lattice 168

8.4 Correlator for periodic boundary conditions 169

      8.4.1 Correlator as vacuum expectation values 171

8.5 Ising model's path integral 171

      8.5.1 Ising partition function 172

      8.5.2 Path integral calculation of Cr 173

8.6 Spin decimation 175

8.7 Ising model on 2×N lattice 176

8.8 Summary 179

9 Ising model: magnetic field 180

9.1 Periodic Ising model in a magnetic field 180

9.2 Ising model's evolution kernel 182

9.3 Magnetization 183

      9.3.1 Correlator 184

9.4 Linear regression 185

9.5 Open chain Ising model in a magnetic field 189

      9.5.1 Open chain magnetization 190

9.6 Block spin renormalization 191

      9.6.1 Block spin renormalization: magnetic field 195

9.7 Summary 196

10 Fermions 198

10.1 Fermionic variables 199

10.2 Fermion integration 200

10.3 Fermion Hilbert space 201

      10.3.1 Fermionic completeness equation 203

      10.3.2 Fermionic momentum operator 204

10.4 Antifermion state space 204

10.5 Fermion and antifermion Hilbert space 206

10.6 Real and complex fermions: Gaussian integration 207

      10.6.1 Complex Gaussian fermion 209

10.7 Fermionic operators 211

10.8 Fermionic path integral 211

10.9 Fermion–antifermion Hamiltonian 214

      10.9.1 Orthogonality and completeness 216

10.10 Fermion–antifermion Lagrangian 217

10.11 Fermionic transition probability amplitude 219

10.12 Quark confinement 220

10.13 Summary 222

Part four Quadratic path integrals 223

11 Simple harmonic oscillator 225

11.1 Oscillator Hamiltonian 226

11.2 The propagator 226

      11.2.1 Finite time propagator 227

11.3 Infinite time oscillator 230

11.4 Harmonic oscillator's evolution kernel 230

11.5 Normalization 233

11.6 Generating functional for the oscillator 234

      11.6.1 Classical solution with source 234

      11.6.2 Source free classical solution 236

11.7 Harmonic oscillator's conditional probability 239

11.8 Free particle path integral 240

11.9 Finite lattice path integral 241

      11.9.1 Coordinate and momentum basis 243

11.10 Lattice free energy 243

11.11 Lattice propagator 245

11.12 Lattice transfer matrix and propagator 246

11.13 Eigenfunctions from evolution kernel 249

11.14 Summary 250

12 Gaussian path integrals 251

12.1 Exponential operators 252

12.2 Periodic path integral 253

12.3 Oscillator normalization 254

12.4 Evolution kernel for indeterminate final position 256

12.5 Free degree of freedom: constant external source 260

12.6 Evolution kernel for indeterminate positions 261

12.7 Simple harmonic oscillator: Fourier expansion 264

12.8 Evolution kernel for a magnetic field 267

12.9 Summary 270

Part five Action with acceleration 271

13 Acceleration Lagrangian 273

13.1 Lagrangian 273

13.2 Quadratic potential: the classical solution 275

13.3 Propagator: path integral 277

13.4 Dirac constraints and acceleration Hamiltonian 280

13.5 Phase space path integral and Hamiltonian operator 283

13.6 Acceleration path integral 286

13.7 Change of path integral boundary conditions 289

13.8 Evolution kernel 291

13.9 Summary 293

14 Pseudo-Hermitian Euclidean Hamiltonian 294

14.1 Pseudo-Hermitian Hamiltonian; similarity transformation 295

14.2 Equivalent Hermitian Hamiltonian HO 297

14.3 The matrix elements of e−τQ 298

14.4 e−τQ and similarity transformations 301

14.5 Eigenfunctions of oscillator Hamiltonian HO 304

14.6 Eigenfunctions of H and H† 305

      14.6.1 Dual energy eigenstates 307

14.7 Vacuum state; eQ/2 309

14.8 Vacuum state and classical action 312

14.9 Excited states of H 313

      14.9.1 Energy ω1 eigenstate Ψ10(x, v) 314

      14.9.2 Energy ω2 eigenstate Ψ01(x, v) 315

14.10 Complex ω1,ω2 317

14.11 State space V of Euclidean Hamiltonian 318

      14.11.1 Operators acting on V 320

      14.11.2 Heisenberg operator equations 322

14.12 Propagator: operators 323

14.13 Propagator: state space 324

14.14 Many degrees of freedom 327

14.15 Summary 329

15 Non-Hermitian Hamiltonian: Jordan blocks 330

15.1 Hamiltonian: equal frequency limit 331

15.2 Propagator and states for equal frequency 331

15.3 State vectors for equal frequency 334

      15.3.1 State vector |ψ1(τ)⧽ 334

      15.3.2 State vector |ψ2(τ)⧽ 335

15.4 Completeness equation for 2 × 2 block 336

15.5 Equal frequency propagator 337

15.6 Hamiltonian: Jordan block structure 339

15.7 2×2 Jordan block 340

      15.7.1 Hamiltonian 342

      15.7.2 Schrödinger equation for Jordan block 343

      15.7.3 Time evolution 344

15.8 Jordan block propagator 344

15.9 Summary 347

Part six Nonlinear path integrals 349

16 The quartic potential: instantons 351

16.1 Semi-classical approximation 352

16.2 A one-dimensional integral 353

16.3 Instantons in quantum mechanics 355

16.4 Instanton zero mode 362

16.5 Instanton zero mode: Faddeev–Popov analysis 364

      16.5.1 Instanton coefficient N 368

16.6 Multi-instantons 370

16.7 Instanton transition amplitude 371

      16.7.1 Lowest energy states 372

16.8 Instanton correlation function 373

16.9 The dilute gas approximation 374

16.10 Ising model and the double well potential 376

16.11 Nonlocal Ising model 377

16.12 Spontaneous symmetry breaking 380

      16.12.1 Infinite well 381

      16.12.2 Double well 381

16.13 Restoration of symmetry 381

16.14 Multiple wells 383

16.15 Summary 383

17 Compact degrees of freedom 385

17.1 Degree of freedom: a circle 386

      17.1.1 Poisson summation formula 387

      17.1.2 The S1 Lagrangian 388

17.2 Multiple classical solutions 388

      17.2.1 Large radius limit 391

17.3 Degree of freedom: a sphere 391

17.4 Lagrangian for the rigid rotor 393

17.5 Cancellation of divergence 395

17.6 Conformation of DNA 397

17.7 DNA extension 399

17.8 DNA persistence length 401

17.9 Summary 403

18 Conclusions 405

References 409

Index 413

Belal E. Baaquie is a Professor of Physics at the National University of Singapore, specializing in quantum field theory, quantum mathematics and quantum finance. He is the author of Quantum Finance (2004), Interest Rates and Coupon Bonds in Quantum Finance (2009), The Theoretical Foundations of Quantum Mechanics (2013) and co-author of Exploring Integrated Science (2010).

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