Abstract
In the present paper on the one hand we apply the central limit theorem to the solution of the sign problem of a path integral of two-interacting particles in potential and give an expression for the sign solved propagator (SSP) derived from that solution and on the other hand we perform the angular decomposition of the path integrals of the 2D and 3D Helium atoms. Finally, we combine those two results and derive the SSPs of the 2D and 3D Helium atoms.
Highlights
The central limit theorem has an enormous variety of applications in probability, statistics and various other areas of mathematics and mathematical physics [1] [2] [3]
We apply the central limit theorem to the solution of the sign problem [4] and the extraction of the sign solved propagator (SSP) [5] [6]
According to our methods in the case of path integrals which involve inner or outer products in the interaction term or the potential, after the angular decomposition, we evaluate the infinite product from n = 1 to n = N as N → ∞ and integrate the n= N +1 factor considering the parameter ε as a one form
Summary
The central limit theorem has an enormous variety of applications in probability, statistics and various other areas of mathematics and mathematical physics [1] [2] [3]. We develop further those methods and consider double phase space path integrals of interacting particles. We study the path integral of the 2D and 3D Helium atoms We perform their angular decomposition, and we derive their sign solved propagators (SSP). We consider the angular decomposition of the 3D Helium path integral and in Section 4 of the 2D one and apply the results of Section 2 to extract the SSPs. In Section 5, we give our conclusions.
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