17.7 Feynman Quantum Path Integrals I

This paragraph introduces the topic of Feynman's quantum path integral formulation of quantum mechanics, which is considered a challenging and subtle subject. The lecturer emphasizes its importance and fascination, stating that it provides a new perspective on quantum mechanics and is foundational for field theory and quantum chromodynamics (QCD), the advanced theory of elementary particles. The paragraph also touches on the dissatisfaction with the standard approach to quantum mechanics and Feynman's endeavor to connect it more closely with classical mechanics, focusing on the concept of the classical trajectory and its generalization in quantum mechanics.

The lecturer delves into Feynman's exploration of the relationship between classical mechanics and quantum mechanics, discussing the concept of the classical trajectory and its uniqueness in classical physics. The paragraph discusses the statistical nature of quantum mechanics, where multiple paths are possible, each with its own probability, contrasting with the single, unique path in classical mechanics. Feynman's approach to quantum mechanics involves considering all possible paths a particle can take, integrating over these paths to find the wave function at a point in space-time, using a function known as the propagator or Green's function. The paragraph also mentions Dirac's influence on Feynman's thought process, suggesting the consideration of Hamilton's principle, a fundamental concept in physics.

This paragraph focuses on Hamilton's principle, which states that the motion of a particle along the classical trajectory is such that the action is at an extremum. The action is defined as a functional of the path, involving an integral of the Lagrangian, which is the difference between the kinetic and potential energy of the particle. The lecturer explains that Hamilton's principle has been used to derive various fundamental theories in physics, including Newton's laws and Einstein's general theory of relativity. The paragraph also discusses the concept of a functional in the context of Hamilton's principle and the constraints of the system, which require that all paths begin and end at specific points in space-time.

The lecturer applies Hamilton's principle to quantum mechanics, considering a free particle and its action. The action for a free particle is derived and related to the Green's function, which is a solution to the wave equation. The paragraph explains how the Green's function, or propagator, is related to the action along a path from point A to point B. It also discusses how the wave function at a point in space-time is the result of summing over all possible paths connecting that point to every other point in space, with each path weighted by its action. The paragraph concludes by highlighting the importance of Planck's constant in determining the probability of each path, with paths near the classical trajectory being the most probable due to their minimal action.

This paragraph discusses the challenge of calculating quantum mechanics using path integrals. The lecturer explains that the Green's function for a quantum system can be expressed as a sum over all possible paths, each weighted by the exponential of the classical action divided by Planck's constant. The paragraph details the process of using a complete set of eigenstates to find the Green's function and then using this function to determine the wave function of the system. The lecturer also touches on the difficulty of the calculations involved and the need to relate the quantum mechanical Green's function to the classical action.

The final paragraph introduces Wick rotation, a technique used to evaluate the Green's function for imaginary values of time. This rotation transforms the oscillatory exponentials in the Green's function into decaying exponentials, simplifying the calculation. The lecturer explains that at very long imaginary times, only the ground state contributes significantly to the sum, allowing the wave function to be determined from the limit of the Green's function as imaginary time approaches infinity. The paragraph concludes by emphasizing the importance of this method in calculating quantum mechanical wave functions and the significance of the ground state in this context.