Automatic Generation of Vacuum Amplitude Many-body Perturbation Series

Received (received date) Revised (revised date) An algorithm and a computer program in Fortran 95 are presented which enumerate the Hugenholtz diagram representation of the many-body perturbation series for the ground state energy with a two-body interaction. The output is in a form suitable for post-processing such as automatic code generation. The results of a particular application, generation of L A T E X code to draw the diagrams, is shown. 1. Many-Body Perturbation Theory Many problems of interest in quantum mechanics have no known analytic solution, and some approximation is necessary to obtain a solution. A particularly useful approximation method is perturbation theory, which consists of finding a way of separating the Hamiltonian into a part which a solvable, either numerically or analytically , and a part which is small (See e.g. 1 for a general treatment). One may then start from the solution of the solvable part and build in successive orders of the perturbation in a power series approach to obtain a closer approximation to the solution of the full problem, providing the series converges. In many-body quantum mechanics, the perturbation series is convenently resp-resented in diagrammatic form, based on the work of Feynman 2 , Goldstone 3 and Hugenholtz 4. Diagrams may be prescriptively expressed in algebraic form which give a complete representation of the perturbation series. One may construct such series of diagrams and expressions for different observables in Quantum Mechanics. This paper is concerned with the ground state expectation value of the Hamiltonian operator that is, the ground state energy, or vacuum amplitude for a many-fermion system in the special, but common case that the unperturbed problem is given by the solution of the Hartree-Fock equations and the system is governed by a two-body interaction. This situation is widely dealt with in textbooks 5,6. Each order of correction in perturbation theory for the ground state energy involves, in general, Automatic Generation of Vacuum Amplitude many diagrams and it is important to be able to enumerate them, order by order, if one is to perform a calculation 7. The perturbation series for the ground state energy can be given by diagrams in the Hugenholtz vein constructed according to the following rules (see 5 for a more detailed explanation): For an n th order diagram, n dots (vertices) are drawn in a column. These vertices are connected by directed lines subject to the conditions 1. Each …