Some infinite series related to Feynman diagrams

On Some Infinite Series Related to the Twin Primes

It is shown that all positive integers can be divided into numbers that can lead to a pair of twin primes by a simple algebraic operation and numbers that cannot. The paper devises a formula for finding the numbers in each category, and shows that the numbers in the second category can be arranged in infinite groups and super-groups with an inner symmetry, a precise interval length and a well-d...

Simple method to make asymptotic series of Feynman diagrams converge.

We show that, for two nontrivial lambda phi(4) problems (the anharmonic oscillator and the Landau-Ginzburg hierarchical model), improved perturbative series can be obtained by cutting off the large field contributions. The modified series converge to values exponentially close to the exact ones. For lambda larger than some critical value, the method outperforms Padé's approximants and Borel sum...

On one dimensional digamma and polygamma series related to the evaluation of Feynman diagrams

We consider summations over digamma and polygamma functions, often with summands of the form (±1)nψ(n + p/q)/nr and (±1)nψ(m)(n + p/q)/nr, where m, p, q, and r are positive integers. We develop novel general integral representations and present explicit examples. Special cases of the sums reduce to known linear Euler sums. The sums of interest find application in quantum field theory, including...

On Perturbation Theory, Dyson Series, and Feynman Diagrams

To study the transitions (scattering, making new particles, decays, etc.) caused by V̂ we want to use a fixed basis of Ĥ0 eigenstates, but we want to keep the transitions separate from wave-packet spreading and other effects due to Schrödinger phases e−iEt of the Ĥ0 itself. The picture of QM which separates these effects is the interaction picture. In the Schrödinger picture, the operators are t...

On Perturbation Theory, Dyson Series, and Feynman Diagrams