Quantum Electrodynamics and Feynman Diagrams

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quantum

electrodynamics

when two electrons approach each other they repel this explains why we do not fall from a chair when we sit why we can exert a force on an object and why air exerts friction on a feather Apart From gravity and radioactivity on a nuclear scale, almost all phenomena in the universe can be explained by how electrons behave at the beginning. One would be tempted to describe this repulsion by a force, the electromagnetic force, but electrons are not small marbles that would obey classical mechanics. They are

quantum

objects, particles, and to describe their interactions it is necessary to reconcile electromagnetism with quantum physics.

For this, in the middle of the 20th century, the most precise model ever created in the history of physics was developed, an elegant model that allows the use of simple

diagrams

to calculate with astonishing precision the most fundamental phenomena of physics quantum

electrodynamics

quantum electrodynamics is an example of quantum field theory we consider our universe as a kind of space-time box that contains two fluid fields formed by mathematical objects the electron field and the electromagnetic field, within these fields small packets of energy move that can appear or disappear, called particles, electrons, for example, they are disturbances that propagate as waves within the electron field.

The electromagnetic field also contains disturbances, quanta of energy that can appear or disappear, photons. field of electrons and that of photons are of a different nature, the mathematical objects that compose them are not of the same type, the field of electrons in particular is formed by spinners rather abstract objects that are described by complex numbers to simplify we can imagine a complex a number has a size and a color its phase when an electron propagates in the field the phase of complex numbers rotates with time this is called electric charge electrons have an electric charge that transcribes the fact that their phase changes as they move towards the future apart from electrons this same field can also contain disturbances whose phases rotate in the other direction in a way we could say that this is an electron but that it is moving in the opposite direction towards the past from our point of view the phase of this the particle seems to rotate in the opposite direction we perceive an opposite electric charge this is called antielectron a positron on the other hand the photon field is made up of vectors that are expressed with real numbers they are just ordinary numbers that have no phase so the photons They do not have an electrical charge.

To understand how electrons interact, let's imagine that we place two of them at the beginning of our quantum field to schematically represent the content of the universe. It is convenient to use lines to symbolize the movement of the particles aligned with an arrow towards the future to symbolize a electron with an arrow pointing back in the other direction to symbolize a positron and a wavy line to symbolize a photon to describe how particles evolve quantum electrodynamics allows our two fields to interact using interaction vertices one interaction vertex involves a photon and two particles of the electron type, these can be electrons or positrons depending on their orientation with respect to time, such a vertex can symbolize an electron that emits a photon, an electron that absorbs a photon, a positron that emits a photon or absorbs a photon or even an electron and a positron annihilating into a photon or a photon becoming an electron-positron pair, all of these interactions are allowed as long as for each vertex the overall momentum of the particles in space and time before and after the interaction follows being the same. the electric charge must also be conserved each interaction necessarily has an arrow that enters and another that leaves the vertex by allowing our two electrons to interact with this type of vertices then we can imagine a whole variety of different scenarios in the simplest scenario the two electrons continue In a straight line, in another more interesting scenario, the two electrons exchange a photon that acts as a messenger carrying part of the impulse from the first electron to the second electron.

It is important to keep in mind that the particles behave like waves and can be exchanged in one direction although they carry an impulse. which is oriented differently, that way in some scenarios the exchange will bring the two electrons closer and in other scenarios the exchange will separate them, then we can imagine more complex scenarios involving many interaction points, the electrons can exchange several photons in different places and at different times, sometimes a photon becomes an electron-positron pair that annihilates to form a photon again. We call this a loop. An electron can also emit and then reabsorb a photon as long as the overall momentum and electrical charge are conserved.

In all imaginable scenarios an infinite number of more or less complex possibilities can occur and if we stop the evolution of the field after a certain time to observe the result of each scenario, sometimes we find our two electrons and sometimes more particles, some of which have appeared between the two. initial and final instance each of these scenarios that start from an initial situation and reach a final situation is called a refinement diagram. Feynman

diagrams

transcribe the different possible evolutions of our quantum fields from a given initial situation apart from the particles. initial and final particles that are real particles that can be detected the particles that act as messengers within these diagrams are said to be virtual these are particles that cannot be detected and can exhibit some rather strange properties they only serve as intermediaries to describe how our two electrons interact mathematically at a distance each scenario each Feynman diagram corresponds to a very rigorous equation and virtual particles are just a way to intuitively interpret certain parts of the equations that said that although they are only intermediaries resulting from our mathematical model it is essential to consider these virtual particles because they account for the interactions of the fields and therefore how the electrons behave.

Let's summarize what we have so far. We describe a space-time that contains two fields that have electrons and that of photons from an initial situation with two electrons. We are interested in all the possible evolutions that allow interactions that link a photon to two particles of the electron type. From this we can create a catalog, a list of all the possible diagrams, there are infinite numbers, some are simple, containing few interactions, and others are more complex, involving many interactions. but this doesn't help us much if we want to predict the actual behavior of electrons, we would like to know which of these scenarios would actually occur if we carried out the experiment.

Is this scenario or another scenario that would occur? The answer to this question is subtle and may seem counterintuitive, but it is precisely what makes quantum theory so powerful: our universe does not follow just one of these scenarios, but evolves at the same time according to all possible scenarios, so that, starting from a given initial situation, all possibilities occur at the same time. parallel time as a superposition of all imaginable scenarios to describe the behavior of electrons it is necessary to take into account all finement diagrams in quantum electrodynamics each diagram corresponds to an equation that allows us to calculate a number for each scenario and amplitude that we can imagine the diagrams as layers and their breadth as a kind of opacity that sometimes adds constructively and other times destructively. different scenarios have different amplitudes, but for the sake of calculations we can usually ignore the more complex scenarios by considering only the first few simpler diagrams and still get reasonably accurate results, it is by performing the sum of all these scenarios with their different amplitudes as if We would superimpose more or less opaque layers that we get the real evolution of the physical system when we perform the experiment in the real world if we throw two electrons and observe them a little later.

The amplitude of each plot allows us to calculate the overall probability that we will observe a specific outcome as a result of our experiment, and in particular, the most likely outcome is that we will observe our two electrons with slightly different values. impulse repelled each other in a way Feynman diagrams are not so much descriptions of real phenomena but rather very powerful tools that allow us to calculate the probability of observing this or that result we have a greater probability of finding our two electrons with outward momentum The most likely result is that they repel each other thanks to the exchange of virtual particles.

At our scale, we have the impression that the two electrons suffer a continuous force while fundamentally this force is only the probabilistic synthesis of all the possible interactions in which the electrons Are exchanged. movement through virtual particles to conclude, quantum electrodynamics is a complex theory but it allows us to predict in an amazing way how electrons, positrons and photons interact, synthesizing all possible scenarios with their different amplitudes. This theory explains and predicts on the fundamental scale all the laws of optics, the behavior of light when it interacts with a material, the Maxwell equations that govern the electric and magnetic fields, and the interactions between electrons from which, on our scale, almost all of the interactions arise. forces historically this model was the first great success of quantum field theory by describing matter as quantum fields interactions as virtual particles and by proposing a very elegant calculation method based on diagrams and amplitudes, quantum electrodynamics has allowed in particular Scientists predict with unprecedented precision the way in which an electron reacts to a magnetic field through virtual photons.

The magnetic field causes the spin of the electron to process and this precession movement is perfectly predicted by quantum electrodynamics with almost 10 figures. significant to this day all theories considered this is the best verified experimental prediction in the history of physics the anomalous magnetic moment of the electron