The functional formulation of quantum mechanics and relativistic quantum field theories are widely studied subjects. Nevertheless, certain aspects of this formulation are frequently overlooked, in spite of their relevance from a structural point of view. This book aims to fill up this gap. The non-relativistic regime has been found particularly appropriate for these

Preface vii

1. Correspondence rules. The phase space path integral 1

1.1 Operator formulation of quantum mechanics 1

      1.1.1 Schrodinger picture 2

      1.1.2 Heisenberg picture 4

      1.1.3 The physical system 5

      1.1.4 The propagator 8

1.2 Correspondence rules 9

      1.2.2 The ordering problem 17

1.3 The phase space path integral 20

1.4 One dimensional harmonic oscillator 24

      1.4.1 Computation of the propagator 24

      1.4.2 Eigenvalues and eigenvectors 34

1.5 One dimensional free particle 38

1.6 The stochastic nature of the path integral 40

1.7 Problem 46

2.1 Fictitious source picture 47

2.2 Schwinger action principle 49

2.3 Green functions 51

2.4 Schwinger equations 56

2.5 Integration of Schwinger equations 60

2.6 Schwinger equations and the GWT 64

3. Generating functional of Green functions 75

3.1 Generating functional for the one dimensional harmonic oscillator 75

3.2 More on the computation of the generating functional for the one dimensional harmonic oscillator 85

3.3 The canonical algebra and the functional formalism 89

3.4 The one dimensional free particle 92

3.5 The general case 97

3.6 Approximation methods 101

3.7 Effective action and effective potential 110

4. Constrained systems 117

4.1 Classical dynamics of constrained systems. Hamiltonian formulation 117

4.2 Functional quantization of constrained systems 126

4.3 Problem 135

5. Noncommutative systems 137

5.1 Classical-quantum transition for noncommutative systems 138

5.2 Example: the noncommutative two-dimensional harmonic oscillator 140

5.3 Functional formulation of the quantum dynamics for non­commutative systems 144

5.4 Noncommutative systems and the time slicing definition of the path integral 147

5.5 Problem s 150

Appendix A Some useful integrals 155

Appendix B Preliminaries on functional analysis 165

B.1 Functional differentiation 165

B.2 Functional integrals 168

      .2.1 Functional integrals over real functions 168

      B.2.2 Functional integrals over complex functions 173

      B.2.3 Change of variables in the functional integral 174

      B.2.4 Functional Fourier transform 179

Bibliography 181

Index 185

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