Path integral over probability functional

In summary, a path integral over probability functional is a mathematical technique used in quantum physics to calculate the probability of a particle moving from one position to another by integrating over all possible paths and taking into account their associated probabilities. It is also used in quantum mechanics and quantum field theory to calculate the probability amplitudes for particles and quantum fields to evolve between states. This concept is also applied in other fields such as statistical mechanics, finance, economics, computer science, and machine learning. Feynman diagrams visually represent the path integral over probability functional and its applications.
  • #1
Jezuz
31
0
Hi. Can anyone tell me how to solve the path integral

[tex] \int D F \exp \left\{ - \frac{1}{2} \int_{t'}^{t} d \tau \int_{t'}^{\tau} ds F(\tau) A^{-1}(\tau - s) F(s) + i \int_{t'}^{t} d\tau F(\tau) \xi(\tau) \right\} [/tex]

In case my Latex doesn't work the integral is over all possible forces F over the functional

\exp \left\{ - \frac{1}{2} \int_{ t' } ^{ t } d \tau \int_{ t' } ^{ \tau } ds F( \tau ) A^{-1} ( \tau - s ) F( \tau) + i \int_{ t' } ^{t} d \tau F( \tau ) \xi ( \tau ) \right\}

I have tried to solve it by making the discrete Fourier transform of the functions F, A^{-1} and \xi but I run into some trouble when doing that.

/Jezuz
 
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  • #2


Hi Jezuz,

The path integral you are trying to solve is a common problem in quantum field theory, known as the Gaussian path integral. It is used to calculate the probability amplitude for a particle to travel from one point to another in a given time interval, taking into account all possible paths that the particle can take.

To solve this integral, you will need to use techniques from functional analysis and quantum mechanics. First, let's break down the integral into smaller parts:

1. The first term in the exponential, \frac{1}{2} \int_{t'}^{t} d \tau \int_{t'}^{\tau} ds F(\tau) A^{-1}(\tau - s) F(s), represents the action of the particle. This is a functional of the path F, which describes the trajectory of the particle. A^{-1} is the inverse of the propagator, which tells us how the particle evolves in time.

2. The second term, i \int_{t'}^{t} d\tau F(\tau) \xi(\tau), represents the external force acting on the particle, given by the function \xi(\tau).

To solve this integral, we need to find the functional derivative of the action with respect to the path F. This will give us the equation of motion for the particle. We can then use this equation to find the most probable path that the particle will take.

To do this, we can use the method of steepest descent, where we find the path that minimizes the action. This is equivalent to finding the classical path in classical mechanics. The solution to the integral will then be given by the value of the action at the minimum.

I would recommend consulting a textbook on quantum field theory or seeking help from a colleague who has experience with functional analysis. It can be a complex problem, but with the right approach and understanding, you will be able to solve it. Good luck!
 

1. What is a path integral over probability functional?

A path integral over probability functional is a mathematical technique used in quantum physics to calculate the probability of a particle moving from one position to another. It involves integrating over all possible paths that the particle could take between the two positions and taking into account the various probabilities associated with each path.

2. How is a path integral over probability functional used in quantum mechanics?

In quantum mechanics, the path integral over probability functional is used to calculate the probability amplitude for a particle to move from one state to another. This technique takes into account all possible paths that the particle could take and assigns a probability amplitude to each path, resulting in a final probability for the particle to end up in a particular state.

3. What is the significance of a path integral over probability functional in quantum field theory?

In quantum field theory, the path integral over probability functional is used to calculate the probability amplitude for a quantum field to evolve from one state to another. This is important in understanding the behavior of quantum fields and their interactions with particles.

4. How does a path integral over probability functional relate to Feynman diagrams?

Feynman diagrams are visual representations of the path integral over probability functional. Each line in a Feynman diagram represents a possible path that a particle can take, and the probability amplitude associated with that path is represented by the height of the line. The sum of all Feynman diagrams gives the total probability amplitude for a particle's behavior.

5. Are there any applications of path integral over probability functional outside of quantum physics?

Yes, the concept of path integral over probability functional has been applied in various fields such as statistical mechanics, finance, and economics. It has also been used in computer science for solving optimization problems and in machine learning algorithms.

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