What happens when two electrons collide? In classical physics, the answer is very simple, the equations that govern the course of this subatomic billiards shot are intuitive and easily solvable, but in the realms of Quantum field theory, the phenomenon is fairly complicated. In quantum physics, the momentum exchange between the two electrons can happen in an infinite number of ways. The electron may emit a photon that is absorbed by the other electron, the photon may split to give an electron-positron pair that annihilates to give back the photon, the electron may emit a photon that it reabsorbs at a later instant, the possibilities are endless. All the intermediate states that lead to the same final result are not mere possibilities but actually do happen. To perfectly calculate the scattering of two electrons, we need to add up the probability contributions of all the possibilities and in an attempt to do so, we encounter infinities. Richard Feynman, one of the most brilliant minds of the twentieth century, came up with a way to deal with the infinities, called the Feynman diagrams. He devised a very simple set of rules through which we can pictorially represent the behaviour of subatomic particles and map all the possibilities that contribute significantly to the probability of an event. Feynman diagrams are extremely useful in quantum field theory, and thus, in the whole of theoretical physics.
A Feynman diagram is made up of lines and vertices. The lines represent particles and the vertices represent the interactions between them. For example, electrons are represented by a straight line with an arrow, the positron is represented in a similar way with the direction of the arrow reversed and photons are represented by squiggly lines. There are 6 basic interactions possible among these 3 particles. The electrons as well as positrons may emit or absorb a photon, the photon may split into an electron-positron pair, and an electron and a positron may annihilate to give off a photon. Using these basic interactions, we can account for all the possible ways in which an electrodynamic quantum system moves from an initial state to a final state. Similarly, there are well-defined rules for constructing Feynman diagrams for all elementary particles, using which we can account for all the possible ways in which a generic quantum system moves from an initial state to a final state.
Feynman diagrams allow us to take into account every important interaction using an absurdly simple set of rules. The resemblance to reality is questionable, so much so that all the particles that don’t enter or exit the system are called virtual particles.
Explain like I’m a Geek
Subatomic particles act in strange ways. They pop in and out of existence, observing them changes their behaviour, uncertainty is fundamentally ingrained in their very nature; all in all, they don’t go along with the intuitive understanding of the world we develop through our macro-level experiences. Our intuition has shaped mathematics to a large extent and that makes the mathematical models of the quantum mechanical (read non intuitive) world fairly cumbersome.
To handle the unwieldy equations, physicists have to come up with clever tools that can minimise the required computations without compromising on the accuracy of the results. One of the most popular examples of such tools in theoretical physics are the Feynman diagrams.
A Feynman diagram is a two-dimensional representation of the interaction between subatomic particles in which the axes represent space and time. Straight lines are used to depict fermions—fundamental matter particles with half-integer values of intrinsic angular momentum, such as electrons and wavy lines are used for bosons—force carrying particles with integer values of spin, such as photons. Within the canonical formulation of quantum field theory, a Feynman diagram represents a term in the Wick’s expansion of the perturbative S-matrix (scattering matrix). Feynman diagrams are extremely useful in Quantum field theory, especially in QED (Quantum electrodynamics).
QED had two major problems, one of which was infinities popping up as soon as the calculations were pushed out of the simplest approximations. The force-carrying virtual photons could borrow any amount of energy, even infinite energy, as long as they paid it back quickly enough. This resulted in the equations being plagued by infinities and rendered useless. Also, the mathematics was notoriously complicated, it was an algebraic nightmare of distinct terms to track and evaluate. In principle, a quantum event can have infinite intermediate states to precisely calculate the probability of an event, we need to account for all of them. Fortunately, all the possible paths don’t contribute equally to the probability of the event and to add to our convenience, the more complex a path is, the less it contributes to the probability.
Every Feynman diagram corresponds to a term in the series and represents the probability amplitude that the term contributes to the probability of the event and the terms weigh in according to the number of virtual particles involved. The weightage of the term representing the interaction involving two virtual particles is ten-thousandth of the weightage of the term involving just one and as we keep adding virtual particles, the weightage keeps dropping by a factor of ten thousand. Although, in principle, the full calculations extend to include an infinite number of separate terms, in practice any given calculation can be truncated after a few that contribute significantly to the probability. This is known as a perturbative calculation. Perturbation is essentially a way to solve an unsolvable equation by taking a solvable approximation and tweaking it until the results are suitably accurate.The perturbative approach combined with Feynman Diagrams proved to be an extremely powerful tool in quantum field theory.
In spite of all of the simplifications, calculations in QED were still extraordinarily difficult in practice. Most of the integrals for the two photon diagrams blew up to infinity, rather than providing a finite answer, just as physicists had been finding with their non-diagrammatic calculations for two decades. So Feynman next showed how some of the troublesome infinities could be removed, the step physicists dubbed “renormalization” by using a combination of calculational tricks, some of his own design and others borrowed. The order of operations was important: Feynman started with the diagrams as a mnemonic aid in order to write down the relevant integrals, and only later altered these integrals, one at a time, to remove the infinities. By using the diagrams to organize the calculational problem, Feynman had thus solved a long-standing puzzle that had stymied the world’s best theoretical physicists for years.