What are the fundamental mathematics behind path integrals in QM?

[pivoxa15]

What is the 'fundamental' maths behind path integrals in QM?

 

[PRB147]

Path integral gets another name, functional integral,
For detail, see
http://www.physicstoday.org/pt/vol-54/iss-8/p48.html

 

[denian]

The fundamental mathematics behind path integrals in quantum mechanics (QM) is rooted in the principles of calculus, linear algebra, and complex analysis. Path integrals are a mathematical tool used to calculate the probability amplitude of a particle's trajectory in quantum systems.

In QM, the state of a particle is described by a wave function, which evolves over time according to the Schrödinger equation. Path integrals provide a way to visualize and calculate the evolution of a particle's wave function by summing over all possible paths that the particle could take.

This involves using the mathematical concept of integration to sum up the contributions of all possible paths, taking into account the phase factors associated with each path. This approach is based on the principle of superposition, which states that the wave function of a system is a combination of all possible states.

To perform path integrals, we use the mathematical tools of functional analysis and measure theory. Functional analysis is used to define the mathematical objects called functionals, which are used to describe the probability amplitudes of a particle's trajectory. Measure theory is used to define the measure of these functionals, which allows us to calculate the probability amplitudes.

Furthermore, path integrals involve the use of complex numbers and complex analysis. This is because the wave function in QM is a complex-valued function, and the path integral formula involves taking the exponential of a complex number.

In summary, the fundamental mathematics behind path integrals in QM involves the use of calculus, linear algebra, complex analysis, functional analysis, and measure theory. These mathematical tools provide a rigorous framework for understanding the behavior of quantum systems and calculating their probabilities.