About the Feynman Path Integral and Principle of Least Time

In summary, the shortest path does not dominate in the path integral, but the method of stationary phase shows that the largest contribution is from the paths that satisfy the classical equations of motion, also known as Fermat's principle.
  • #1
EsbMtrx
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I don't entirely get why we usually say that only the shortest path contributes in the path integral. If you calculate the volume of nth fresnel zones which is the locus where the path length is between n-1 and n wavelengths from the shortest path in 3 dimensions, they are the same I believe. So I figured each path length is contributed by the same number of paths and the shortest one wouldn't dominate. My question pertains to Fermat's principle.

ps: this is my first thread here so I apologize in advance if I did anything wrong
 
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  • #2
It is not correct that only the shortest path contributes in the path integral. All paths contribute. However, you can use the method of stationary phase to approximate the path integral, and then you find that the largest contribution is from the paths that satisfy the classical equations of motion.
 
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1. What is the Feynman Path Integral?

The Feynman Path Integral is a mathematical formulation used in quantum mechanics to calculate the probability of a particle taking a particular path between two points in space and time. It is based on the principle of superposition, where all possible paths are considered and their contributions are added together to determine the most likely path.

2. What is the Principle of Least Time?

The Principle of Least Time, also known as Fermat's Principle, states that light will travel between two points in the least amount of time possible. This principle is used in optics and is related to the Feynman Path Integral as it considers all possible paths that light can take and determines the most probable path.

3. How is the Feynman Path Integral used in quantum mechanics?

In quantum mechanics, the Feynman Path Integral is used to calculate the probability of a particle's position and momentum at a given time. It takes into account all possible paths that the particle could take and their associated amplitudes, which are then added together to determine the most likely path.

4. What are the limitations of the Feynman Path Integral?

The Feynman Path Integral is a powerful tool in quantum mechanics, but it has its limitations. It is most effective in systems with a small number of particles and becomes increasingly difficult to calculate as the number of particles increases. It also does not take into account the effects of gravity.

5. How does the Feynman Path Integral relate to other principles in physics?

The Feynman Path Integral is closely related to other principles in physics, such as the Principle of Least Action and the Principle of Least Entropy. These principles all involve finding the most probable path or state of a system by considering all possible paths or states and their associated probabilities.

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