## Feynman Diagrams

Posted by on Friday, May 11, 2018 Under: Particle Physics

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...

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...

In : Particle Physics