Mathematician, astronomer and physicist C.F. Gauss famously asserted that: "Mathematics is the Queen of the Sciences, and Arithmetic the Queen of Mathematics. She often condescends to render service to Astronomy and other natural sciences, but under all circumstances the first place is her due." (1)
The sheer complexity of the calculations involved in string theory for example, lead physics titan Edward Witten to describe them as a bit of 21st century physics that somehow dropped into the 20th century. Witten's own work in string theory was revolutionary and led Witten to mathematical results so profound that he become the first physicist to be awarded the Fields medal for mathematics in 1990.
With this in mind, this blog entry considers some of the most profound, mysterious and powerful diagram in physics, diagrams which seem to transcend their mathematical origins and function at meta-levels in terms of their efficiency and the value of their insights.
The first and most iconic example of such diagrams is the Feynman diagram, named after the American physicist Richard Feynman (1918-88). Feynman was the eccentric 'genius's genius' with a legendary reputation for creative problem solving and the ability to teach the complexities of quantum physics to students and non-physicists.
Figure 2: One of the earliest published Feynman diagrams (Sightings, Sept. 2003) depicting electron-electron scattering by means of a virtual photon (labelled here as virtual quantum and depicted as a wavy line). The diagram presents a quantum-mechanical view of how particles with like charge repel one another.
Feynman first created his diagrams as mnemonic doodles to keep track of the long and complex calculations of QED, or Quantum Electrodynamics, the study of electromagnetism at the quantum-mechanical level. However he quickly realised they could be developed into a method of organising calculations with such efficiency that they avoid having to write out sheets of dense mathematical notation.
Feynman diagrams are powerful tools because they provide a transparent picture for particle interactions in spacetime. They usually represent sub-atomic events in two dimensions: space on the horizontal axis and time on the vertical axis (figure 2). Straight lines represent fermions, such as electrons, and wavy lines represent bosons, such as photons (except for the Higgs boson which uses a dashed line, and gluons which use loops).
which he decorated with hand painted Feynman diagrams.
At the quantum level particles interact in every way available to them, and so an exact description of the scattering process involves summing up a large number of diagrams, each with their own mathematical formula for the likelihood they will occur.
Pioneer of data visualization and expert on information graphics Edward Tufte, had 120 Feynman diagrams constructed in stainless steel (see figure 4). His wall mounted constructs represent all 120 different ways that a 6-photon scattering event can be depicted.
Wall mounted installation of stainless steel with shadows, 530 x 230 x 10 cm (Installation view at Fermilab)
Feynman introduced his ingenious schematic in 1948, but by the 1980's their limitations were starting to become apparent, and Feynman himself went on to prove that the diagrams were only approximations that involved an enormous amount of redundancy that arose from their reliance on involving virtual particles (see figure 2).
Feynman diagrams were designed to describe all the possible results of subatomic particle collisions, but even a seemingly simple event like two gluons colliding to produce four less energetic gluons, involves some 220 diagrams. Such collisions occur billions of times a second during experiments carried out using modern day particle accelerators.
In the mid-2000s patterns began to emerge from events recorded in particle accelerators that repeatedly hinted at an unknown, underlying, coherent mathematical structure. A new set of formulas were proposed by the physicists Ruth Britto, Freddy Cachazo, Bo Feng and Edward Witten, known as the BCFW recursion relations after their discoverers. The formulas dispense with familiar variables of Feynman diagrams such as position and time, and involves an entirely new diagrammatic system first developed in the 1970's by Roger Penrose, named twistor diagrams.
Figure 5: Twistor diagrams depicting an interaction between six gluons that can be used to derive a simple formula for the 6-gluon scattering amplitude.
According to Andrew Hodges:
"Twistor diagrams for scattering amplitudes have been explored since the early 1970s, when Roger Penrose first wrote them down. But the ideas underlying them suddenly received quite new attention at the end of 2003, when Ed Witten's twistor string model brought together twistor geometry, string theory and scattering amplitudes for pure gauge fields." (2)
After over a decade of research with his collaborators, Arkani-Hamed showed how twistor diagrams could be pieced together to create a timeless, multidimensional object known as an 'Amplituhedron' (figure 6).
The Amplituhedron has been described as an intricate, multi-faceted, higher dimensional jewel at the heart of quantum mechanics, a meta-level Feynman diagram completely new to mathematics.
figure 7: Arkani-Hamed's hand drawn diagram of the amplituhedron representing an 8-gluon particle interaction.
This theoretical object enables simplified calculation of particle interactions with such astounding efficiency that according to Jacob Bourjaily, "you can easily do, on paper, computations that were infeasible even with a computer before." (4)
The amplituhedron is a geometrical representation of real particle data. The scattering amplitude can be derived from its volume and details of a particular scattering process determine its dimensionality and facets.
Figure 7 shows the amplituhedron representing an 8-gluon particle interaction, an event that would require almost 500 pages of algebra using Feynman diagrams.
From Feynman diagrams to twistor diagrams and the discovery of the enigmatic amplituhedron, diagrams remain a powerful, albeit mysterious tool in theoretical physics. They permit information to be stored and shared with high fidelity, but they also mobilise and shape new knowledge by allowing intuition and rational thought to play a role in the creative process.
For his 'Momentum' series (2010-2013), Spanish photographer Alejandro Guijarro traveled to several international academic institutions that specialize in quantum mechanics: CERN, Stanford, Berkeley and Oxford. In a form of documentation, Guijarro measured and photographed blackboards that he found in lecture theatres, meeting rooms and offices, then printed the images at a 1:1 scale.
The series highlights the transitive nature of diagrams at work during the creation and transmission of knowledge. It presents the process as a physically involved gestural performance, as various trains of thought are followed and erased to leave a blurred palimpsest.
'Momentum' is reminiscent of Marcel Duchamp’s project 'Unhappy Readymade', discussed in this previous Blog: The Diagrams of Geometry part II- A soggy book of diagrams as a wedding present from Marcel Duchamp.
Both projects present us with a token of something lost - information and knowledge made manifest through the substrates ink, paper, chalk and board only to be subject to entropy.
In the case of 'Unhappy Readymade' it's the wind and rain which add entropy, in the case of Guijarro’s 'Momentum' it's the hand of the professor, janitor or the student armed with a blackboard eraser that return the arena of ideas to a tabula rasa.
1) C.F. Gauss quoted in Gauss zum Gedächtniss (1856) by Wolfgang Sartorius von Waltershausen
2) Andrew Hodges, Online at: http://www.twistordiagrams.org.uk/papers/
3) Arkani-Hamed, quoted in 'A Jewel at the Heart of Quantum Mechanics' by Natalie Wolchova, online
4) Jacob Bourjaily, quoted in 'A Jewel at the Heart of Quantum Mechanics' by Natalie Wolchova, online