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Bridged: Gaps between mathematical methods of understanding nature

Read time: 8 mins
28 May 2021
Bridged: Gaps between mathematical methods of understanding nature

A Feynman diagram in quantum electrodynamics, a quantum field theory. [Image Credits: Wikimedia Commons / CC BY-SA 4.0]

In the first two decades of the twentieth century, Albert Einstein developed the special theory of relativity, which unifies the previously separate entities ‘space’ and ‘time’. Later, he developed the general theory of relativity, which provides a mathematical and conceptual framework of gravitation. In the next decade, theoretical physicists had developed a mathematical framework that describes the world of small things –– electrons, protons, atoms, molecules –– called ‘quantum mechanics’. Despite tremendous successes of special relativity and quantum mechanics in describing a plethora of natural phenomena, they remained incompatible in mathematical terms. In the next few years emerged the field of ‘quantum field theory’, a general mathematical framework used to describe multiple physical theories that unify the salient features of special relativity and quantum mechanics.

Over the next few decades, theoretical physicists developed multiple quantum field theories with the help of advanced mathematics. These theories successfully plugged the mathematical loopholes in physical theories and predicted the existence of new particles in nature –– the ‘neutrons’ which along with protons make up the nuclei of atoms, ‘positrons’ or positively charged electrons, and the list continued. Meanwhile, experimental physicists were busy hunting for these particles, and surprisingly, finding them. Due to the interplay of theoretical and experimental physics, one discovery led to another, and ‘particle physics’ became the cool thing to be investigating. The ‘Conseil Européen pour la Recherche Nucléaire’ (CERN), or European Council for Nuclear Research, came into existence, and so did numerous other particle physics detectors around the world. In 2012, CERN finally discovered the ‘Higgs particle’ –– the final building block of the now-famous ‘Standard Model of Particle Physics’, a model that successfully accommodates multiple quantum field theories into one mathematical framework.

The Standard Model of Particle Physics comes with multiple limitations –– for example, it cannot explain the well-established experimental fact that neutrinos, a particle that travels close to the speed of light, possesses mass. Gravitation, the phenomenon that Einstein explained with his general theory of relativity, does not feature in the Standard Model either. In the last few decades, theoretical physicists have worked extensively on alternate theories of the Universe that provides a quantum field theory for gravitation, the most famous among them being ‘string theory’. In the search for such theories, one aspect got largely forgotten in the last 50 years –– that quantum field theories encounter infinities.

While trying to calculate the probabilities of events using quantum field theories, physicists encountered mathematical infinities that made no sense. After all, infinities do not correspond to anything tangible in nature. While a group of physicists concluded that quantum field theories were inadequate in describing nature, another group devised an alternative explanation. According to the second group, while some terms in the equations had positive infinities, others had negative infinities of the same magnitude. These infinities cancelled, they said –– there was no logical fallacy. The infinities arose because of the limitation of the mathematical method they used to calculate probabilities of physical events. However, they were not ready to divorce from the method. Called the ‘Feynman diagrams’, the mathematical method helped physicists picturise physical processes. An alternative mathematical method, called the ‘Bootstrap approach’, was neater. It had no pesky infinities nor cancellation of positive and negative infinities. However, the mathematical steps were complex, and physicists could not clearly understand the probabilities they calculated via physical processes. The Bootstrap approach quickly fell out of favour and was soon forgotten.

In a new study, two researchers from the Indian Institute of Science (IISc), Bengaluru, have used the ‘Bootstrap approach’ of quantum field theories to explain ‘Feynman diagrams’. Professor Aninda Sinha and PhD scholar Ahmadullah Zahed carried out the research in the Centre for High Energy Physics of IISc. Published in the journal Physical Review Letters, the study was supported by the Department of Science and Technology, Government of India.

Aninda explains why physicists did not pursue the Bootstrap approach after its formulation. “They did not know how exactly to use the complicated equations that arise from the Bootstrap approach,” he says. But the scenario has changed with the advent of better computers and improved computational algorithms. “Since 2008, new numerical methods have led to new insights into using the Bootstrap equations,” says Aninda. “From 2015, my collaborators and I have been trying to make sense of the correct way to take the Bootstrap approach,” he adds. When they investigated the technical difficulties of this approach, they found that it did not consider an important factor –– certain symmetries of nature.

Symmetries are very common in nature. When a particular object is subjected to a specific change, for example, rotation –– but remains similar to before the change, it is symmetric under that change. Mathematical equations describing physical theories also exhibit symmetries, that is, they remain unchanged when subject to mathematical operations. The mathematician Amelie Emmy Noether discovered that a particular physical quantity remains conserved whenever there are symmetries in physical laws. For example, the conservation of mass-energy is related to symmetries of the mathematical equations describing nature with respect to time. The Feynman diagrams also exhibit a special kind of symmetry, the ‘crossing symmetry’, which has interesting consequences. For example, physical processes involving a couple of electrons and a couple of positrons have the same probabilities even when an electron exchanges with a positron.

The crossing symmetry is an inherent property of the Feynman diagram approach of quantum field theories. However, it is a restriction that needs to be imposed mathematically on the Bootstrap method. Aninda and Ahmadullah did just that. In doing so, the calculations of the Bootstrap approach became simpler. Their equations, which relate the probabilities of different processes, started looking similar to the Feynman diagram method. They calculated some of these mathematical steps with pen on paper. For others, they used an advanced analytical software meant for automating such complicated calculations.

“We had to reinterpret conceptually as well as mathematically an older work from the 1970s as well as connect it up with current attempts over the last two years by other groups. It was quite a challenge!” shares Aninda.

They credit their work to calculations documented in a 1972 study by two physicists Auberson and Khuri. When they came across this paper, they found that it was hardly ever cited by other researchers. “No one knew about this paper, which is evidence that there are hidden treasures in the past,” remarks Aninda. Two or three other groups, one involving S. M. Roy, an Indian physicist at the Tata Institute of Fundamental Physics, Mumbai, had followed up on Auberson and Khuri. However, these efforts remained largely forgotten. Delays caused in the pre-email communication also contributed to a lack of coherent communication. Today, Aninda, Ahmadullah, and their colleagues from various institutions spread across India are perennially connected online.

The magic of complex variables enables us to see Feynman diagrams emerge “locally”. [Image Credits: Ahmadullah Zahed, an author of the study.]

The study has provided a bridge between the seemingly different approaches to quantum field theories. But, there is more to it. The Feynman diagram approach is also useful in predicting things that happen around us –– atoms do not crumble, radioactive elements decay with time, particles collide to give rise to other particles. Nobel Laureate Kenneth Wilson and his collaborators used it to study physical quantities like specific heat, the amount of heat a kilogram of water requires to be heated per unit degree Celsius rise in temperature. He had shown that it is possible to calculate how the specific heat changes with temperature, specifically at temperatures above which water cannot be liquified even after applying tremendous pressure.

In an earlier work conducted in 2017, the researchers, in collaboration with Professor Rajesh Gopakumar, director of the International Centre for Theoretical Sciences (ICTS), Bengaluru, had used the Bootstrap approach to study the dependence of specific heat on temperature. “There were a couple of mathematical gaps in the broad scheme of calculation we had proposed in 2017,” says Rajesh. By invoking the crossing symmetries, they have now fixed those gaps, and it has opened up a whole new range of questions both theoretically and experimentally. This study has also been published in the journal Physical Review Letters.

Changes between different phases of matter, like solids and liquids, happen primarily in two ways, explains Rajesh. How physical quantities like temperature, pressure change as the transition occurs –– determine the kind of transition. In one type, the changes are steady, while in the other, sudden. Hence, studying these physical properties becomes essential for understanding the properties of the transition. The Bootstrap approach makes mathematical predictions of these physical properties for materials in which the phase transitions are continuous. It is now up to the experimentalists to verify these predictions. Given the tremendous progress of methods in experimental physics, it might take only a few years, opines Rajesh.

“Our study shows how ideas inspired by a theory of particle physics and gravitation can play a role in explaining ordinary phenomena. It is a remarkable example of how one field of physics can influence another,” he signs off.

This article has been run past the researchers, whose work is covered, to ensure accuracy

Editor's Note: The aticle was edited to include a couple of hyperlinks.