Integral in an infinte dimensional space

[eljose79]

let,s suppose we have to perform an integral into a infinite dimensional space,then we would use the Montecarlo,s Method as it is known to be independent of the dimension of the integral, but my problem is still the same..¿how do you define a point into a infinite dimensional space?...how would you perform this integral?..

This is useful in Quantum Field Theory where such integrals appear.

 

[Feynman]

Hi,
In this case you should know what is the property of this space,then construct a knew measure of integration like Feynman path integral in quantum mechanics

 

[M98Ranger]

Performing integrals in an infinite dimensional space is a challenging task, as it is difficult to define a point in such a space. However, the Monte Carlo method is a useful tool in this situation as it is independent of the dimension of the integral. This method involves generating random points in the infinite dimensional space and using these points to approximate the integral.

To define a point in an infinite dimensional space, we can use a coordinate system that assigns a value to each dimension. For example, in a 3-dimensional space, we have x, y, and z coordinates. In an infinite dimensional space, we can use an infinite number of coordinates, such as x1, x2, x3,...xn. This allows us to define a point in the space as a set of values for each coordinate.

To perform the integral, we can use the Monte Carlo method to generate random points in the infinite dimensional space. These points are then used to calculate the function at each point and average the results. By increasing the number of random points, we can improve the accuracy of the integral approximation.

In the context of Quantum Field Theory, this approach is particularly useful as it allows us to perform integrals in spaces with an infinite number of dimensions, such as the space of all possible configurations of a quantum field. The Monte Carlo method is a powerful tool in this field as it allows us to handle these complex integrals and obtain meaningful results.