Path Integral Troubleshooting: Dealing with Delta Distributions in the Exponent

In summary, the conversation discusses difficulties in solving a functional integral involving a Dirac delta distribution. The integral resembles a Gaussian integral, but the presence of the delta distributions complicates the solution. There is confusion about the nature of the variable X, whether it is a function or a parameter. The conversation also raises a question about the presence of an integral in the exponent in the path integral representation.
  • #1
mhill
189
1
I am having troubles to solve the functional integral:

[tex] \int D( X) e^{i(\dot X)^{2}+ a\delta (X-1)+ b\delta (X-3) [/tex]

if a and b were 0 the integral is just a Gaussian integral but i do not know how to deal with the Delta distribution inside some may help ??
 
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  • #2
I have difficulty understanding what X is. Is it some variable, or is it a function, like X(x). When you write

[tex]
\int\mathcal{D}X
[/tex]

it looks like X is a function, for example [itex]X:\mathbb{R}\to\mathbb{R}[/itex] or [itex]X:[-L,L]\to\mathbb{R}[/itex] or something similar. But when you write

[tex]
\delta(X-3)
[/tex]

it looks like X is just some parameter, like [itex]X\in\mathbb{R}[/itex].

Or is the number 3 a constant function [itex]\mathbb{R}\to\mathbb{R}[/itex], [itex]3(x)=3[/itex], and the delta function an infinite dimensional delta function, like [itex]\delta^{\mathbb{R}}[/itex]?
 
  • #3
if that is in path integral repreesetation, shouldn't there be an integral in the exponent.?,
[tex]e^{S(q)}[/tex] where S(q) is the action which is an integration over relevant time period of the Lagrangian of system.
 

1. What is a path integral in physics?

A path integral is a mathematical concept used in physics to calculate the probability of a particle's movement from one point to another. It is a summation of all possible paths the particle could take, and the path with the highest probability is the one the particle is most likely to follow.

2. How is a path integral different from other types of integrals?

A path integral is different from other types of integrals because it takes into account all possible paths a particle could take, rather than just one specific path. It also involves complex numbers and is often used in quantum mechanics to describe the behavior of particles at the microscopic level.

3. What are some practical applications of path integrals?

Path integrals have many practical applications in physics, including in quantum field theory, statistical mechanics, and fluid dynamics. They are also used in financial mathematics to calculate the risk of investments.

4. How do I solve a path integral?

Solving a path integral can be a complex task, and it often involves using advanced mathematical techniques such as contour integration and the Feynman-Kac formula. It also requires a solid understanding of the underlying physical principles and concepts.

5. What are some common challenges when working with path integrals?

One common challenge when working with path integrals is dealing with infinite or highly complex integrals. Another challenge is understanding the physical meaning behind the mathematical calculations. Additionally, path integrals can be difficult to visualize, as they involve summing over an infinite number of paths.

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