A recap of the Article – Bohm and Feynman Path Integrals – author – Marius Oltean, U Waterloo.

Feynman’s Path Integral

Feynman’s approach to quantum mechanics involves the concept of path integrals, where the probability amplitude for a particle to move from one point to another is computed by summing over all possible paths that the particle can take between these points. Each path contributes an amplitude, which is an integral of the classical action. This method provides a reformulation of the Schrödinger equation and is particularly useful in quantum field theory.

  • Mathematical Tool: Feynman’s paths are not real paths along which particles move. They are mathematical constructs used to calculate the evolution of the wavefunction.
  • Sum Over Histories: The method involves summing the contributions from all possible paths, weighted by the exponential of the classical action.
  • Wavefunction Propagation: The Feynman path integral provides a way to compute the propagator, which describes how the wavefunction evolves over time.

Bohm’s Path Integral

In contrast, Bohm’s approach, also known as the de Broglie-Bohm theory or Bohmian mechanics, posits that particles have definite trajectories guided by the wavefunction. This theory incorporates both the wave and particle nature of quantum entities.

  • Real Trajectories: In Bohm’s theory, particles follow actual paths determined by a guiding equation derived from the wavefunction.
  • Quantum Potential: The particle’s motion is influenced by a quantum potential, which is derived from the wavefunction and affects the trajectory.
  • Single Path: Unlike Feynman’s sum over all paths, Bohm’s formulation involves integrating the quantum Lagrangian along a single, real path—the actual trajectory of the particle.

Distinguishing Between Feynman’s and Bohm’s Paths

  1. Conceptual Basis:
    • Feynman: Paths are mathematical constructs without physical reality.
    • Bohm: Paths are real trajectories that particles actually follow.
  2. Mathematical Formulation:
    • Feynman: Involves summing over an infinite number of possible paths.
    • Bohm: Involves a single path determined by the guiding equation and quantum potential.
  3. Interpretation of Paths:
    • Feynman: The paths represent all possible histories of the particle.
    • Bohm: The path represents the actual history of the particle as it moves according to the guiding equation.
  4. Wavefunction Propagation:
    • Feynman: Uses the path integral to compute the propagator and describe the evolution of the wavefunction.
    • Bohm: Uses the de Broglie-Bohm trajectory to directly integrate the quantum Lagrangian for the wavefunction’s evolution.
  5. Physical Reality:
    • Feynman: No single path is real; all paths contribute probabilistically.
    • Bohm: The particle’s path is real and unique, influenced by the quantum potential.

The article demonstrates that while these two approaches originate from different conceptual bases, they can be connected. The Feynman method of summing over all paths can be derived from the de Broglie-Bohm theory by considering the contributions of all possible trajectories​