Definition of Feynmann integral ?

[tpm]

Definition of Feynman integral ??

Using the definition of integral as a limit of certain sums...couldn't we define an integral over paths $$x(t)$$ as the limit whenever 'epsilon' tends to 0 of the Sum:

$$\sum _{i} F[x(t)] (L[x(t)+i \delta (t-t')]-L[x(t)+(i-1)\delta (t-t')])$$

Where F is a functional and $$L[x(t)]=\int_{a}^{b}dtx(t)$$

Or something like that..in the sense that Riemann Path integrals are just the limit of a Riemann sum over an space of paths X(t)

Or in terms of 'hyperreal' function so $$x(t)+i \epsilon \delta (t-t')$$

for i=1,2,3,4,5,6,7,8,9,... and 'epsilon' is a infinitesimal quantity.

 

[eljose79]

Yes, you are correct. The Feynman integral, also known as the path integral, can be defined as a limit of sums over infinitesimal paths. This is similar to how the Riemann integral is defined as a limit of Riemann sums over infinitesimal intervals.

The main difference is that in the Feynman integral, the paths are not necessarily straight lines, but can be any path between two points. This allows for a more general and powerful way of integrating over functions, especially in quantum mechanics where the paths represent possible trajectories of particles.

In addition, the use of hyperreal numbers in the definition allows for a more rigorous and flexible approach to the integral, as it takes into account infinitesimal quantities that may be ignored in traditional calculus.

Overall, the Feynman integral is an important tool in theoretical physics, and its definition as a limit of sums over paths is a fundamental concept in understanding its applications.

 

[denian]

The Feynman integral is a mathematical tool used in quantum field theory to calculate the probability of a particle's movement through space and time. It was developed by physicist Richard Feynman in the 1940s and is based on the concept of a path integral, which is a generalization of the concept of an integral in calculus.

The Feynman integral is defined as the limit of a sum over an infinite number of paths, where each path represents a possible trajectory of the particle. The sum is multiplied by a functional, which assigns a numerical value to each path based on physical properties such as energy and momentum.

The limit is taken as the infinitesimal parameter 'epsilon' tends to 0, making the paths increasingly dense and allowing for a more accurate calculation of the particle's behavior. This approach allows for a more comprehensive understanding of quantum systems, as it takes into account all possible paths that the particle could take, rather than just a single trajectory.

In summary, the Feynman integral is a mathematical tool used to calculate the probability of a particle's movement in quantum field theory, taking into account all possible paths the particle could take. It is a powerful tool that has greatly contributed to our understanding of the quantum world.