Chapter 1: Introduction to Richard Feynman

Richard Feynman, an eminent American theoretical physicist, is renowned globally for his groundbreaking contributions to physics. His work encompasses the development of a novel formulation of quantum mechanics utilizing path integrals, the theory of quantum electrodynamics (including his renowned Feynman diagrams), insights into the superfluidity of supercooled liquid helium, and foundational advancements in quantum computing and nanotechnology. Feynman's excellence was recognized with the Nobel Prize in Physics in 1965.

In the biography by American-Canadian physicist Lawrence Krauss, Feynman is quoted as saying: "There is pleasure in recognizing old things from a new viewpoint." — Richard Feynman

This article, inspired by Sakurai's work, aims to delve into Feynman's innovative approach to quantum theory via path integrals.

Chapter 2: Quantum Propagators

Propagators in quantum mechanics and quantum field theory are essential functions that determine the probability amplitude for a particle's transition from one location to another over a specified time interval. The square of this amplitude gives the transition probability.

Let's denote the time-dependent position operator in the Heisenberg picture as X(t). Using the elegant bra-ket notation developed by Dirac, we can consider the initial and final vector states, which are eigenstates of the operator X(t):

text{Transition between states}

The corresponding propagator for this transition is represented as:

text{Propagator corresponding to the transition}

This function, the propagator, acts as a transition amplitude.

Section 2.1: Completeness of Position Kets

The certainty that any measurement of a system's position at a fixed time must yield a result equal to 1 implies that position kets form a complete set:

text{The position kets at a fixed time form a complete set}

Here, the unity on the right side represents the identity operator. This relationship can be utilized to express the propagator as a series of integrals, thereby allowing us to view a transition as a combination of intermediary transitions. The simplest case can be expressed as:

text{The use of the above relation to re-express the propagator}

By leveraging this compositional property, we can divide the time interval associated with a transition into equal segments, allowing us to express the transition amplitude as:

text{Transition amplitude as a composition of amplitudes}

We are effectively summing all potential paths with fixed endpoints, as illustrated in the accompanying diagram.

In classical mechanics, a particle follows a distinct path, determined by its associated Lagrangian L(x, dx/dt). The path can be derived by extremizing the action.

Chapter 3: Feynman’s Space-Time Approach

In classical physics, a particle's motion is described by a specific path; however, in quantum mechanics, all potential paths must be accounted for. The challenge is to demonstrate how quantum mechanics aligns with classical mechanics as h approaches 0.

Paul Dirac made a cryptic remark in a 1933 paper that intrigued Feynman. Using Sakurai's notation, Dirac stated:

text{Dirac's mysterious remark}

Feynman pondered the significance of Dirac's statement, prompting him to develop his influential space-time approach to quantum mechanics.

Following Sakurai, the action for a small segment of a path can be expressed as:

text{Action corresponding to a small segment of the path}

For any given path, we derive the corresponding amplitude by multiplying such expressions. The propagator for the transition can thus be expressed as a sum over all possible paths:

text{The propagator for the transition obtained by summing paths}

For small h, the dominant term in this sum corresponds to a path that remains relatively unchanged with slight deformations, thereby satisfying the classical path equation.

Feynman articulated Dirac's equation as a proportionality:

text{Feynman's proportionality of Dirac's assumption}

James Gleick recounts a conversation between Feynman and Dirac, highlighting the impact of Dirac's earlier remark on Feynman's groundbreaking discoveries.

To simplify, by setting V=0 and using a linear approximation, one can quickly derive the prefactor:

text{The prefactor evaluation}

The full transition amplitude is obtained by integrating over all intermediate states:

text{Transition amplitude in Feynman's formulation}

This represents Feynman's path integral, encompassing all possible trajectories. Unlike other quantum mechanics formulations, the probability amplitude here is not merely a linear superposition of quantum states; it is a superposition of entire alternative histories in spacetime.

It's important to note that these trajectories are not restricted by special relativity rules; any path is permissible as long as the endpoints remain fixed.

Chapter 4: Proving the Equivalence of Feynman's and Schrödinger's Approaches

Now, we will demonstrate that Feynman's methodology aligns with Schrödinger's wave mechanics. We begin with:

text{Total amplitude expressed as a composition of two steps}

Assuming a small time difference, we can rename the final position and time, expanding the exponential and amplitude in powers. Collecting the terms and conducting trivial Gaussian integrals leads us to:

text{Schrödinger equation for Feynman's propagator}

Thus, we conclude that Feynman's propagator and Schrödinger's propagator represent the same entities.

Thank you for reading! Your constructive feedback is always appreciated. For more intriguing content on physics and related topics, feel free to visit my LinkedIn, personal website at www.marcotavora.me, and GitHub.