# Functional Equation & Feynman Path Integral Solution

In summary, the conversation discusses the relationship between functional equations and functional derivatives, specifically in the context of the Feynman Path integral. It is noted that there may be difficulty in defining such a relationship due to the path integral not being a traditional integral. The possibility of the Dyson-Schwinger equation being used as a method to evaluate propagators without relying on path integrals is also mentioned."
Is there any Functional equation In functional derivatives so the Feynman Path integral is its solution?.. i mean given:

$$A[\Phi]=\int \bold D[\Phi]e^{iS/\hbar}$$

Then A (functional) satisfies:

$$G( \delta , \delta ^{2} , B[\phi] )A[\Phi]=0$$

where B is a known functional and "delta" here is the functional derivative.

I think there would be some difficulty in defining such a thing since the path integral isn't technically an integral at all, since it's defined over a space with no clearly defineable measure.

If you take this A to be the vacuum to vacuum transition amplitude then this equation exists and as far as I know it is known as Dyson-Schwinger equation. You can find the derivation in section 6.4 of Ryder's "Quantum Field Theory".

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"Hellfire" is the Dyson-Schwinger equation a method to evaluate propagators (Non-perturbative) without recalling to Path Integrals??

## 1. What is a functional equation?

A functional equation is an equation in which the unknown quantities are functions rather than specific values. It relates the values of the unknown function to the values of its input variables.

## 2. What is the Feynman path integral solution?

The Feynman path integral solution is a mathematical technique used to solve certain types of functional equations, particularly in the field of quantum mechanics. It involves summing over all possible paths that a particle could take between two points in space and time, and using this information to calculate the probability of the particle's behavior.

## 3. How is the Feynman path integral solution related to quantum mechanics?

The Feynman path integral solution is closely related to quantum mechanics because it allows us to calculate the quantum mechanical amplitude for a particle to travel from one point to another. This amplitude can then be used to determine probabilities for the behavior of quantum systems.

## 4. What is the significance of the Feynman path integral solution?

The Feynman path integral solution is significant because it provides a powerful mathematical tool for solving certain types of functional equations and performing calculations in quantum mechanics. It has been used to make predictions and explain various phenomena in physics, and has applications in other fields such as finance and biology.

## 5. Are there any limitations to the use of the Feynman path integral solution?

Yes, there are some limitations to the use of the Feynman path integral solution. It is not applicable to all types of functional equations, and can be challenging to use for complex systems. Additionally, it may not provide exact solutions in all cases and may require approximations. However, it remains a valuable and widely used tool in solving functional equations in quantum mechanics and other fields.

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