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Germán V. RODRIGO, PhD

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Universal opening of four-loop scattering amplitudes to trees

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Originaly published 2018 in the IFIC's website

IFIC Celebrates the Feynman's Centenial


The year 2018 marks the 100th anniversary of the birth of one of the most prolific physicists of the 20th century: Richard P. Feynman. His major contributions encompassed, in addition to particle physics, the physics of superfluid supercooled liquid helium. He was also a pioneer in introducing nanotechnology and quantum computing as concepts. To commemorate his life and scientific legacy, a conference was held in Hungary, funded by the COST Action PARTICLEFACE led by IFIC, whose objective has also been to present the latest techniques in quantum field theory used to predict the behavior of subatomic particles in collisions at very high energies, such as those occurring at the LHC, from the diagrams that bear his name.

Richard P. Feynman

Richard Phillips Feynman was born in New York on May 11, 1918, and received the Nobel Prize in Physics in 1965, along with Julian Schwinger and Shinichiro Tomonaga, for their contributions to the development of Quantum Electrodynamics (QED), the quantum field theory that describes electromagnetic interactions. Feynman first introduced the diagrams that bear his name at a conference in the Pocono Mountains of Pennsylvania in the spring of 1948, precisely to describe QED. Feynman's presentation, however, did not, because of its novelty, received much attention from the audience. In contrast, at the same conference, Schwinger gave a day-long presentation of his version of the theory. Feynman and Schwinger did not use the same language and found it very difficult to understand each other, yet they arrived at similar results, which served to confirm the validity of QED. Tomonaga's version would come soon after, and it was somewhat simpler than Schwinger's version.

Feynman is also known for many other relevant contributions such as the complete development of the path integral formalism in quantum mechanics (based on earlier work by P.A.M. Dirac), the physics of superfluidity of supercooled liquid helium, the quantization of gravity and Yang-Mills theories, the V-A form describing the electroweak interactions responsible for radioactive processes, the parton model, and for pioneering the introduction of nanotechnology and quantum computing as concepts. He is also well known for his public work, particularly after having participated in the Rogers Commission that investigated the Space Shuttle Challenger accident (1986), and for his pedagogical and outreach work. He also wrote a couple of well-known semi-autobiographical books ("Surely You're Joking, Mr. Feynman!" and "What Do You Care What Other People Think?").

Feynman's diagrams

But undoubtedly the diagrams that bear his name are his most famous contribution since they represent graphically and simply the quantum processes that subatomic particles undergo when they interact with each other, and not only in QED but also in any quantum field theory. These diagrams are formed by a series of vertices interconnected by lines. Each line symbolizes the propagation of a particle from one point in space-time to another, each vertex the interaction between several particles at a point. There are two types of diagrams, those in which the interconnected lines form closed circuits, called loops, and those in which they do not, called tree diagrams because of their similarity to the branches of a tree.


Feynman diagrams at two loops
Fig.1: Feynman diagrams representing the decay of the Higgs boson to a pair of photons.


The main advantage of Feynman diagrams lies in the ease of translating them into mathematical expressions with which physicists can predict what will be the behavior of subatomic particles in collision or decay processes at very high energies, for example, what is the probability of producing a Higgs boson at the LHC, and what is the probability of that Higgs boson decaying to two photons rather than to two top quarks. Each theory allows for different Feynman diagrams, and the translation of those diagrams into mathematical formulas also depends on the theory. Thus, Feynman diagrams are the most direct way to obtain theoretical predictions with which to compare experimental data and discern which theory best fits experiment. In addition, Feynman diagrams define a whole series of new mathematical functions with very interesting properties. Also from the mathematical point of view, Feynman diagrams introduce innovative aspects.

The theoretical field related to the applications of Feynman diagrams has undergone a spectacular development in recent years, motivated by the need for accurate theoretical predictions and sophisticated analysis tools to analyze the LHC data. When physicists analyze a collision process at the LHC, the tree-like diagrams that describe it represent only a rough approximation. To improve the accuracy of the theory it is necessary to consider loop diagrams as well, since they are those that incorporate the quantum vacuum fluctuations that contribute to the same process. The greater the number of loops that are incorporated into the theoretical calculations the greater the accuracy of the theory, but also the greater the number of possible quantum fluctuations and the complexity of the mathematical expressions derived from them. Each new loop introduces new mathematical functions, and new challenges for theoretical physicists.

On the shores of Lake Balaton

To celebrate the centenary of the birth of Richard P. Feynman, the COST Action PARTICLEFACE led by IFIC organized a conference in Balatonfüred, on the shores of Lake Balaton in Hungary, where the latest developments in the field were also presented and discussed. The choice of Balatonfüred was not accidental and is justified for several reasons. In June 1972, the first of a series of international conferences on neutrino physics took place in Balatonfüred, organized by the Hungarian physicist György Marx and attended by 139 participants including Richard P. Feynman, Tsung-Dao Lee (Nobel Prize in Physics 1957), three researchers who have subsequently received the Nobel Prize, and the now professor at the University of Zürich Zoltan Kunszt.

The place also has an interesting history behind it. Since 1926, Balatonfüred was frequented by the Indian poet, musician and artist Rabindranath Tagore. Tagore, the first non-European to win the Nobel Prize in Literature in 1913, planted a tree on the site. That first tree was followed by others that were planted by other Indian personalities on their visit to Tagore, giving rise to what is known today as Tagore Park. Later, Salvatore Quasimodo (Nobel Prize in Literature 1959) also planted a tree, thus establishing the tradition that if any Nobel Prize winner visits Balatonfüred a tree must planted. This was the case of Feynman in 1972, whose tree is accompanied today by trees planted by Bruno Pontecorvo, Nevill Francis Mott (Nobel Prize in Physics 1977), Eugene Wigner (Nobel Prize in Physics 1963) and Rudolf Mössbauer (Nobel Prize in Physics 1961), all of them identified with their corresponding commemorative plaque.

Kunszt explained how the 1972 conference was only possible when more than 10 years after the 1956 revolution, crushed by the Soviet Union, Hungary began to open up to scientific collaboration with the outside world, allowing researchers to visit other countries and organize international conferences. This is a clear example of the diplomatic value of science in general. Feynman, who confirmed his attendance at Balatonfüred at the last minute, gave a talk on "What neutrinos can tell us about partons". The parton model had been proposed by Feynman in 1969 and describes hadrons (e.g. protons and neutrons) as entities consisting of point-like sub-subatomic particles called partons that at very high energies behave as if they were free moving in the direction of the hadron of which they are part. Feynman's conclusion at Balatonfüred was that the partons are quarks (today we know that they are also gluons). Kunszt pointed out that the parton model was not accepted by the community until the publication in 1977 of the article by Guido Altarelli and Giorgio Parisi "Asymptotic Freedom in Parton Language", which predicts from the theory known as Quantum Chromodynamics what the density of these partons within hadrons is as a function of energy.

Professor Kunszt also emphasized the article published in Acta Physica Polonica in 1963 entitled "Quantum Theory of Gravitation", which is based on tape recordings of a lecture given in Jablonna, Poland. In this paper of only 14 pages and 12 additional pages reflecting the subsequent discussion, Feynman introduces three important novel concepts: what are known as Fadeev-Popov ghost particles (1967), the now called Feynman tree theorem that establishes a correspondence between loop-type diagrams and tree-type diagrams with additional particle emission, and the so-called Berezin integrals (1965) for fermions, an extension of the path integral formalism developed by Feynman himself.

In this article Feynman recognizes that accidentally and at the suggestion of Murray Gell-Mann (Nobel in Physics 1969) he decided to apply the tree theorem to a Yang-Mills theory with zero mass because of its analogy with gravitation. According to Feynman, such a theory cannot exist since it would immediately produce radiation outside the nucleus. Although this theory would not represent a practical case, it is simpler than gravity. Well, this theory is now known as Quantum Chromodynamics and describes with great precision the strong interactions and which are the most relevant in hadronic accelerators like the LHC. We know today that the theory works because the strong interactions become even more intense at large distances, not letting the potential radiation of the nucleus to escape.

Yang-Mills theory with zero mass cannot exist since it would immediately produce radiation outside the nucleus.


Many of the recent theoretical advances in the field are also indirectly related to the Feynman tree theorem. This theorem has also served as inspiration for the loop-tree duality theorem and its applications, proposed and developed by a group of IFIC researchers. Loop-tree duality, like Feynman's tree theorem, reduces the many-loop problem to the computation of tree-like diagrams, and presents some interesting advantages over Feynman's original tree theorem.

The conference concluded with a visit to Tagore Park and its trees, including the tree planted by Feynman in 1972.



Feynman tree

This tree was planted on June 13, 1972, by Nobel Prize-winning physicist Richard Feynman




   INVITED TALK 9-10 MAY 2022
   6th Red LHC



   INVITED TALK 15-17 SEP 2021
   Matter To The Deepest


   SEMINAR 5 JUN 2020
   Universität Wien


   INVITED TALK 13-17 JAN 2020
   FCC Week CERN


   INVITED TALK 26-28 JUN 2019
   FCC Week Brussels




   WG/MC MEETING 26-28 FEB 2018
   Kick-off of COST Action PARTICLEFACE

Germán RODRIGO @ 2021