Feynman diagrams for the many-body perturbational theory are enumerated by solving the system of Hedin’s equations in zero dimension. We extend the treatment of Molinari [Phys. Rev. B 71, 113102 (2005)] and give a complete solution of the enumeration problem in terms of Whittaker functions. An important relation between the generating function of the electron propagator and anomalous dimension in quantum field theory of massless fermions and mesons in four dimensions (Yukawa theory) is found. The Hopf algebra of undecorated rooted trees yields the anomalous field dimension in terms of the solution of the same differential equation. Its relation to the mathematical problem of combinatorics of chord diagrams is discussed; asymptotic expansions of the counting numbers are obtained.
Feynman diagrams for the many-body perturbational theory are enumerated by solving the system of Hedin’s equations in zero dimension. We extend the treatment of Molinari [Phys. Rev. B 71, 113102 (2005)] and give a complete solution of the enumeration problem in terms of Whittaker functions. An important relation between the generating function of the electron propagator and anomalous dimension in quantum field theory of massless fermions and mesons in four dimensions (Yukawa theory) is found. The Hopf algebra of undecorated rooted trees yields the anomalous field dimension in terms of the solution of the same differential equation. Its relation to the mathematical problem of combinatorics of chord diagrams is discussed; asymptotic expansions of the counting numbers are obtained.
Hedin's equations are solved perturbatively in zero dimension to count Feynman graphs for self-energy, polarization, propagator, effective potential and vertex function in a many-body theory of fermions with two-body interaction. Counting numbers are also obtained in the GW approximation.
In many ways the solution to the Hedin equations represents an exact solution to the many body problem. However, for most systems of practical interest, the solution to the Hedin equations is rendered nearly numerically intractable because the Hedin equations are of functional derivative form. Integral equations, on the other hand, are much more numerically tractable, than functional derivative equations, as they can often be solved iteratively. In this work we present a systematic set of integral equations (with no functional derivatives) - Hedin approximations I, II, III, IV etc. - whose solutions converge to the solutions of the exact Hedin equations. The Hedin approximations are well suited to iterative numerical solutions (which we also describe). Furthermore Hedin approximation I is just the GW approximation (as such this work may be viewed as a systematic improvement of the GW approximation). We present a systematic study of the Hedin equations for zero dimensional field theory (which, in particular, is a method to enumerate Feynman diagrams for field theories in arbitrary dimensions) and show better and better convergence to the solutions of the Hedin equations for higher and higher Hedin approximations, with Hedin approximations I, II and III being explicitly studied. We, in particular, show that the higher Hedin approximations capture more and and more Feynman diagrams for the self energy. We also show that already Hedin approximation II captures more diagrams than the state of the art diagrammatic vertex corrections approach. Furthermore Hedin approximation III is a near perfect match to the exact solutions of the Hedin equations, at least in the zero dimensional case, and enumerates a large number of Feynman diagrams.
This chapter discusses a very popular method of studying critical phenomena analytically: the so-called ϵ-expansion. It describes the renormalisation group calculation of the critical exponents in the complete Landau theory, near d = 4. The chapter outlines the strategy of the calculation of 0, to prepare the reader for the algebraic horrors to follow. R. P. Feynman developed useful shorthand for the terms, which is to write down diagrams indicating how different fields are paired up. There is one-to-one correspondence between each Feynman diagram or Feynman graph and a term in the perturbation series. This correspondence is not unique: there are many ways to write down diagrams for the terms, and people have their own preferences. Once one has figured out this correspondence, then one only has to be able to draw all the appropriate diagrams to see what one should compute.
The computational complexity of time-dependent perturbation theory is well known to be largely combinatorial whatever the chosen expansion method and family of parameters (combinatorial sequences, Goldstone and other Feynman-type diagrams, etc.). We show that a very efficient perturbative expansion, both for theoretical and numerical purposes, can be obtained through an original parametrization by trees and generalized iterated integrals. We emphasize above all the simplicity and naturality of the new approach that links perturbation theory with classical and recent results in enumerative and algebraic combinatorics. These tools are applied to the adiabatic approximation and the effective Hamiltonian. We prove perturbatively and nonperturbatively the convergence of Morita’s generalization of the Gell–Mann and Low wave function. We show that summing all the terms associated with the same tree leads to an utter simplification where the sum is simpler than any of its terms. Finally, we recover the Rayleigh-Schrödinger time-independent equation for the wave operator and we give an explicit nonrecursive expression for the term corresponding to an arbitrary tree.
I review the formalism, Feynman rules, and combinatorics that constrain a field to propagate ``classically", strictly in tree diagrams, either by itself, or interacting with other, purely quantum fields.
The perturbation theory is reorganized by virtue of the linear terms that introduce the constraints via Lagrange multipliers, generalizing and giving results that cannot be obtained with the standard procedures which start at the quadratic terms.
I apply the formalism to a theory of an O(N)-symmetric quantum field interacting with a ``classical" scalar field via cubic interactions in six spacetime dimensions. Using the renormalization group, I examine the effective potential, symmetry breaking with radiative corrections, the fixed points in d=6−ε dimensions, and compare with other works.
Other possible generalizations and applications of the formalism are also discussed.
Hedin's equations provide an elegant route to compute the exact one-body Green's function (or propagator) via the self-consistent iteration of a set of nonlinear equations. Its first-order approximation, known as GW, corresponds to a resummation of ring diagrams and has shown to be extremely successful in physics and chemistry. Systematic improvement is possible, although challenging, via the introduction of vertex corrections. Considering anomalous propagators and an external pairing potential, we derive a self-consistent set of closed equations equivalent to the famous Hedin equations but having as a first-order approximation, the particle-particle (pp) T-matrix approximation, where one performs a resummation of the ladder diagrams. This pp version of Hedin's equations offers a way to go systematically beyond the T-matrix approximation by accounting for low-order pp vertex corrections.