Feynman Diagrams

May 11, 2018
Among physicists, there is a popular story told and retold about a pair of lectures that were given at a conference in 1948 that demonstrates the brilliance of Richard Feynman. How much of it is true and how much has been embellished to build up the legend is unknown, but it is still an entertaining and information anecdote.

Since the development of quantum mechanics in the 1920s and 1930s there had been an unsolved problem regarding the proper treatment of particle interactions. The laws of quantum mechanics had been very successful at describing the properties of a single particle in an external field, or multiple non-interacting or weakly interacting particles, but they failed to describe two identical particles scattering from each other or similar reactions. At this particular conference, a young physicist named Julian Schwinger claimed to have solved this problem, and the great minds of physics gathered to hear his presentation of the solution. For several hours (the exact duration seems to lengthen with each retelling) Schwinger filled blackboards with equations and lectured to his audience on every detail of the calculation. Every space was filled with symbols and formulae, and not even the greatest experts in attendance could follow everything at first viewing. In the end Schwinger gave a prediction for certain atomic properties that were already measured by the experimentalists, and proved this his method gave the correct results. The audience was impressed and lauded his achievement appropriately.

Next up to give a presentation was another young physicist named Richard Feynman. He gave a very brief lecture on the same problem, but instead of pages of calculations he just drew a couple of simple stick figures on the board and a couple of equations. After a very short calculation, Feynman arrived at the same result that Schwinger had just presented a short time earlier. His methods were dismissed by the experts, and his correct predictions were dismissed as either coincidence or worse. No one took Feynman's method seriously at that conference.

Seventy years later, Feynman diagrams are the taught to all physics students, are used in all particle physics research, and even appear routinely in popular physics books and articles. Only the very dedicated experts in mathematical physics can reproduce Schwinger's methods in full detail, and that brilliant calculation has been primarily forgotten by the scientific community.

And so I thought it would be worth explaining exactly what Feynman diagrams are and what the represent. As usual in this series of articles, I will attempt to keep it accessible to the general audience at the possible cost of losing some mathematical rigor. Those who are interested in the more precise details of such calculations are directed to any number of excellent introductory textbooks in quantum field theory and particle physics. 

In my previous article, I explained how Feynman developed the idea of path integrals as a method of understanding quantum mechanics. In this method, the probability that a physical system in state A will eventually be in state B is calculated by summing over all "paths" through the space of possible states between them. It is a very powerful method, but also very difficult to calculate in practice for any realistic system.

However there is an alternative to calculating the full path integral, and that is to consider a series of successive approximations. In particle physics, these approximations to the path integral are referred to as Feynman diagrams. 

Suppose that I want to study how an electron interacts with a proton. Then I can start with a state in which there exists a single electron and a single proton, with some given energies and momenta. The most likely evolution of this state, at least on short time scales, is that both particles will travel forward in time and not interact with each other at all. This is one "path" between the initial and final states of the two particles, and will form our first approximation.

The next most likely interaction will be that the electron emits a single photon which is then absorbed by the proton (or the reverse process, but in Feynman diagrams these two process are considered equivalent for reasons too technical to give here). This can be represented by drawing each particle's path as a straight line, and the exchanged photon connects them. 

In this diagram, as in all Feynman diagrams, the horizontal axis represents time. The initial state is drawn vertically on the left - in this case two particles - and the final state is drawn vertically on the right - in this case still two particles. If you were to mask this diagram so that only a small vertical slit was visible, then moving the mask from left to right would show an animation of the particle reaction, as the two particles move closer together before exchanging a photon and moving apart again. This is another "path" between the two states.

However we might need a more precise calculation of the probability that these two particles will interact, and so we must consider slightly more complicated "paths". One such path has the electron emit two photons at different times, each of which is absorbed by the proton. Another such path has one of the two photons re-absorbed by the electron itself. These two paths can be represented in the following two Feynman diagrams:


These diagrams actually represent a larger set of paths between the initial and final states, because only the sum of the energies of the two photons is known, while the individual energy of each photon is otherwise arbitrary. This fact has serious consequences for the practical calculations of such paths, but those are best left for another article.

And of course the series of Feynman diagrams continues with increasingly complicated diagrams representing all possible combinations of particles and interactions, with each added interaction making the effect of the related "path" slightly smaller. In principle the full calculation of any particle reaction would require an infinite series of such diagrams, but in practice it is rare to require more than a few of the most significant terms in the series.

As an aside, I spent many hours debating with myself whether to give more details of how these diagrams are used in actual calculations. The calculation of probabilities from the diagrams is just as beautiful as the diagrams themselves, but it does require a level of mathematics that I prefer to avoid in these popular articles. So instead I will give just a very coarse outline. In short, for each physical theory one has a set of mathematical functions - one for each line and vertex that can be added to the diagrams. Each diagram is then a product of all the functions that appear in the diagram - each time a photon is created or absorbed, multiply by the charge of the electron and for each line divide by the magnitude of energy-momentum vector. Add the mathematical term corresponding to each diagram, and then square the final result to get the probability that the initial state leads to the final state. In practice the calculations are more complicated than this, but that is the general concept of calculating probabilities using Feynman diagrams.

In the end, this is why the Feynman diagram method has become the standard method of performing calculations in particle physics and quantum field theory. It is no more precise than any other method, but the use of simple diagrams makes it quite easy to both remember all the "paths" and to communicate them to other researchers. A complex mathematical equation can be full of typos and is difficult to understand quickly, but a set of stick figure diagrams is easy to keep track of and to understand at a glance.

And that is perhaps the greatest aspect of Feynman's brilliance as a scientist. He was able to put aside complexities and reduce the calculation to an intuitive set of diagrams that anyone could comprehend easily. It is the underlying physics that matters more than the mathematical details. He was truly a magician among the mathematicians...

Path Integrals

May 11, 2018
In honour of the 100th anniversary of the birth of Richard Feynman, I will today present a very basic overview of one of the great ideas of modern physics which was developed and popularized by Feynman. And the first part of that theory is the path integral formulation of quantum mechanics.

By now quantum mechanics is firmly established as a confirmed and proven theory of nature. Even popular society has come to embrace some of the stranger aspects of quantum theory (though unfortunately they ...

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Happy Birthday Mr. Feynman

May 11, 2018

I do not normally celebrate celebrity birthdays and anniversaries, since the focus on this series of articles is on the principles of science and the laws of nature. However I do feel that today's anniversary warrants an exception to this general rule. It was one century ago today that Richard Feynman entered the world and would go on to change physics forever. 

Although never reaching the same level of popular recognition as Einstein or Newton or Hawking, in many ways Richard Feynman was grea...

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Dominion Astrophysical Centenary

May 6, 2018
Today marks the anniversary of one of the biggest events in local astronomy history, and probably in the history of science in Canada. It was exactly one century ago today that the Plaskett telescope at the Dominion Astrophysical Observatory on Little Saanich Mountain in my hometown of Victoria,British Columbia, first collected light from the stars and became the second largest telescope in the world.

For those who haven't had a chance to see the telescope, it is an amazing piece of equipment ...
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Of Bitcoins & Quantum Computing

April 6, 2018
There has been a lot of discussion in the media over the last few months on the rise of bitcoins and cryptocurrencies, and what it means for society and the economy. With uncertainty in the world due to Brexit and Trump and countless other political crises, individuals and organizations have been increasingly turning to the decentralized currency that is controlled by the masses, and as a result what was once given away for free a mere decade ago is now selling for tens of thousands of dollar...
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The Future of Social Media

March 25, 2018
Today's article will be a departure from my usual scientific fare, and is a copy of an article that I recently wrote for a futurist website that I also contribute to. This is more of an editorial than a scientific review, and some users may wish to skip over it.

This has been a bad week for Facebook. By now I suspect most people are aware of the latest scandal, in which the user data of tens of millions of users was harvested by a British company that then used the information to influence vot...
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Hawking Radiation

March 14, 2018
With the recent passing of legendary theoretical physicist Stephen Hawking, I have received a number of requests to explain some of his pioneering research work in a popular format. To be honest it is simply not possible to do better than Hawking's own work in communicating his research to the masses. I would strongly encourage those who are interested to read some of his many popular science books to get a true understanding of his genius.

However for those who are still reading, I will make ...
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Stephen Hawking

March 14, 2018
There is sad news this morning with the announcement that the legendary theoretical physicist and science popularized, Stephen Hawking, has passed away at the age of 76.

He was a rare figure in the scientific community in that he made significant contributions to research, and yet he was also a famous celebrity outside of the theoretical physics community due to his popular books and willingness to bring modern physics theories to the masses. 

As an academic, he was noted for being one of the f...
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A New Test Of General Relativity

February 24, 2018
One of the great unsolved problems in modern physics is the nature of gravity. Since Einstein first published the general theory of relativity over a century ago, it has proven to be a very accurate model of the solar system and the cosmos. Repeated experiments have confirmed its predictions in the form of planetary orbits, gravitational lensing, and high precision measurements of time and frequency on the Earth and in orbit. So far no deviations from the predictions of general relativity hav...
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Surreal Numbers

February 23, 2018
I am always amazed at how simple some of the most interesting ideas and research in modern mathematics truly is. There are problems in mathematics that can be explained to a small child and yet the greatest minds of the past centuries have been unable to solve. Mathematics is one of the few fields of study where anyone can understand topics that the leading experts are still trying to solve. One such topic is the surreal numbers.

Everyone remembers as a child learning the integers, or counting...
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About Me

Dr. Chris Bird I am a theoretical physicist & mathematician, with training in electronics, programming, robotics, and a number of other related fields.


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