A compact representation is obtained for the quantum statistical sum of indistinguishable nonrelativistic spin-1/2 fermions in the form of Feynman path integrals which can be used as the basis to develop a fundamentally exact method of computer modeling for systems of strongly interacting electrons at nonzero temperature. A basis of symmetrized wave functions is constructed using Young symmetry operators. An exact permutation symmetrization procedure leads to an avalanche-like multiplication in the number of diagrams of linked Feynman integrals of the order of N !. The partition function can be simplified without introducing any approximations and this is performed numerically by computer by direct sorting of diagrams. The control tables obtained, containing combinatorial weights of diagrams, direct the Markov random walk process in virtual trajectory space which is achieved numerically by computer. The equilibrium characteristics of the quantum system are calculated by averaging. This approach is an expansion of the Monte Carlo-Metropolis method to systems of quantum indistinguishable particles with spin. Demonstration numerical calculations using this method were made for the simplest exchange systems, for a hydrogen molecule, a Be + ion, and a Li atom. The ground state of the hydrogen molecule is reproduced with a statistical error of 0.2%. Exchange-correlation effects lead to nontrivial structural changes in the thermally excited electron shells of ions in a state of strong plasma compression.A compact representation is obtained for the quantum statistical sum of indistinguishable nonrelativistic spin-1/2 fermions in the form of Feynman path integrals which can be used as the basis to develop a fundamentally exact method of computer modeling for systems of strongly interacting electrons at nonzero temperature. A basis of symmetrized wave functions is constructed using Young symmetry operators. An exact permutation symmetrization procedure leads to an avalanche-like multiplication in the number of diagrams of linked Feynman integrals of the order of N !. The partition function can be simplified without introducing any approximations and this is performed numerically by computer by direct sorting of diagrams. The control tables obtained, containing combinatorial weights of diagrams, direct the Markov random walk process in virtual trajectory space which is achieved numerically by computer. The equilibrium characteristics of the quantum system are calculated by averaging. This approach is an expansion of the Monte Carlo-Metropolis method to systems of quantum indistinguishable particles with spin. Demonstration numerical calculations using this method were made for the simplest exchange systems, for a hydrogen molecule, a Be + ion, and a Li atom. The ground state of the hydrogen molecule is reproduced with a statistical error of 0.2%. Exchange-correlation effects lead to nontrivial structural changes in the thermally excited electron shells of ions in a state of strong plasma compression.

Journal of Experimental and Theoretical PhysicsJul 1, 200010 citations

The Feynman integral path a Henstock integral: a survey and open problems

R. Gaitán, María Guadalupe Morales

The Feynman path integral is defined over the space $\mathbb{R}^T$ of all possible paths; it has been a powerful tool to develop Quantum Mechanics. The absolute value of Feynman's integrand is not integrable, then Lebesgue integration theory could not be used by Feynman. However, it exists formally as a Henstock integral (which does not require the measure concept) and is a suitable alternative to the ordinary integrals that normally appear in path integrals. Feynman proved the equivalence of his theory with the traditional formulation of Quantum Mechanics, since his path integral satisfies Schrödinger's equation. On the other hand, Feynman's path integral is related to the diagrams of Feynman. For the application of this integral in Feynman's diagrams it is necessary to exchange the integral $\int_{\mathbb{R}^T}$ and the series. We discuss the impossibility to exchange the integral and the sum, considering integral of Henstock and the version of Dominated Convergence Theorem. Even it has not been proved through the several mathematical formalisms that have been used.The Feynman path integral is defined over the space $\mathbb{R}^T$ of all possible paths; it has been a powerful tool to develop Quantum Mechanics. The absolute value of Feynman's integrand is not integrable, then Lebesgue integration theory could not be used by Feynman. However, it exists formally as a Henstock integral (which does not require the measure concept) and is a suitable alternative to the ordinary integrals that normally appear in path integrals. Feynman proved the equivalence of his theory with the traditional formulation of Quantum Mechanics, since his path integral satisfies Schrödinger's equation. On the other hand, Feynman's path integral is related to the diagrams of Feynman. For the application of this integral in Feynman's diagrams it is necessary to exchange the integral $\int_{\mathbb{R}^T}$ and the series. We discuss the impossibility to exchange the integral and the sum, considering integral of Henstock and the version of Dominated Convergence Theorem. Even it has not been proved through the several mathematical formalisms that have been used.

arXiv:2002.12691Feb 28, 20200 citations

The Feynman path integral is defined over the space

\mathbb{R}^T

of all possible paths; it has been a powerful tool to develop Quantum Mechanics. The absolute value of Feynman's integrand is not integrable, then Lebesgue integration theory could not be used by Feynman. However, it exists formally as a Henstock integral (which does not require the measure concept) and is a suitable alternative to the ordinary integrals that normally appear in path integrals. Feynman proved the equivalence of his theory with the traditional formulation of Quantum Mechanics, since his path integral satisfies Schrödinger's equation. On the other hand, Feynman's path integral is related to the diagrams of Feynman. For the application of this integral in Feynman's diagrams it is necessary to exchange the integral

\int_{\mathbb{R}^T}

and the series. We discuss the impossibility to exchange the integral and the sum, considering integral of Henstock and the version of Dominated Convergence Theorem. Even it has not been proved through the several mathematical formalisms that have been used.