Path Integrals for Pedestrians

The following sections are included:

• Dedication
• Preface
• Contents

The following sections are included:

• Quantum mechanics and summing up amplitudes
• Double slit experiment
• Infinite slits experiment and paths correspondence

The following sections are included:

• Time slicing: From infinitesimal to finite time intervals
• Re-derivation of the Feynman path integrals via the Trotter formula
• Continuous paths but nowhere differentiable
• Commutation relations
• Free particle
• Quadratic potentials and harmonic oscillator
• Perturbation theory via path integrals

The path integral is not only the best tool for perturbation theory but also a very powerful instrument to develop the so-called non-perturbative techniques. One of these is the semiclassical approximation. In this chapter we will briefly review it as it is done in most textbooks of quantum mechanics and then present a version involving path integrals…

Weyl, Wigner and Mojal [Weyl (1927); Wigner (1932); Moyal (1949)] developed a formalism for quantum mechanics which is entirely based on commuting functions in phase space and not on operators. The correspondence between the operators and the functions is called “symbol map” and we are now going to define it…

The following sections are included:

• The work of Koopman-von Neumann (KvN) on the operatorial version of classical mechanics
• Path Integrals for classical mechanics (CPI) from the KvN formalism
• Cartan calculus via the CPI
• Geometric quantization
  • Dequantization in the q and p-polarizations and supertime
  • Generating functionals and Dyson-Schwinger equations
  • Warnings on the dequantization rules
• Superposition in classical mechanics

The following sections are included:

• Appendix A Asynchronous variation of the action
• Appendix B The equation for the function f (t₂, t₁) introduced in Section 2.6
• Appendix C Variational calculus in the discrete formalism
• Appendix D Brief review of Grassmann variables
• Appendix E Dimensional analysis of θ and
• Appendix F Schrödinger and Heisenberg picture in θ and
• Appendix G Classical path integral in the momentum representation
• Appendix H Classical path integral via the Trotter formula
• Appendix I Ordering problems in the classical path integral

The following sections is included:

• Bibliography