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Assure time field

perhaps one of the greatest advancements in physics occurred during the 20th century with the unification of special relativity and quantum mechanics this incredible theory is known as quantum field theory and has made all of modern particle physics possible from quantum electrodynamics to quantum chromodynamics to the standard model and beyond but what exactly is a quantum field to understand that we should start by talking about classical fields one common example of this is an electric field if we have some charged particle called a source then the charge creates an electric field extending all the way to infinity this field is closely related to how strong of a force a second charge will feel from the source as we get farther and farther away from the source the strength of the force that a second charge will feel will get weaker and weaker and so the value of the electric field gets smaller and smaller however this isn't the only example of how fields are used in physics many fluids can also be modeled as fields even though they're not technically continuous where the value of the field might give for example the density at a certain point in words all a field refers to is a continuous object which has some value at each point in space and time so a quantum field is simply a field which obeys the rules of quantum mechanics now this concept might seem a bit strange how can a field be quantum when quantum exactly means cut up into discrete pieces also how does a theory of fields give us the study of particle physics to answer this let's start by trying to understand a classical field theory the simplest of which being a string so say we set up our system in a way where the string is forced to stay in a circular loop now the string in principle has some elasticity to it so we can stretch it and compress it we also could imagine segmenting this string into teeny tiny little pieces where each one of these pieces will only feel a pull from the pieces right next to it now if we treat each of these tiny pieces as rigid bodies we can see that it is beginning to look like a bunch of small masses connected together by springs also known as harmonic oscillators specifically it looks like an infinite number of these harmonic oscillators infinitesimally close together in fact this turns out to be a great way to mathematically describe field theories so instead of just jumping straight into the case of the string let's instead look at the more intuitive case of a bunch of masses connected by identical springs in a ring here's the setup we'll take our ring of harmonic oscillators displace a single one let go and see what happens let's start with just a few then really crank up the number of oscillators to see what we might expect from a true field theory in these simulations the springs which are compressed will be red and the springs which are extended will be blue the more compressed the more red and the more extended the more blue when the springs are relaxed they'll be gray okay let's see what happens when we start with a small number of masses there aren't a whole lot of patterns to see once we release the mass everything just starts oscillating back and forth in a messy way however we can immediately see something interesting it looks like it takes some time for these oscillations to actually reach the other end of the ring in other words it looks like there's some maximum speed that the forces can travel through this ring [Music] when we get above about 100 masses i'll make a few changes due to the fact that it gets very hard to see what's going on with individual masses and springs first instead of drawing all the springs and masses i'll only show the colors of the compressed or extended springs second instead of starting out by displacing only a single mass we'll start by compressing all of the springs in a certain region and extending all of the springs in the other this makes more sense to do anyway since our end goal is to describe the continuous string and it doesn't really make sense to displace a single piece of a continuous object so for about a hundred masses nothing's really too much different from the cases with fewer masses however watch what happens when we increase to 500 masses it looks like two copies of the initial compression and tension regions start to zoom around the circle in opposite directions while everything else seems somewhat stationary but it isn't perfect so they start to develop tails of oscillating masses in their wakes when we increase the number of masses even more these tails take longer to develop and stay smaller than the case of 500 masses by 5000 the tails almost don't even show up at all this pattern of course continues until we have a true continuum in which case we don't develop the tails at all and we just get these two copies of the initial state moving around the ring at a set speed so already we see how something which looks sort of particle-like could arise from a field theory if we start with pieces of our field in a certain state these pieces will keep their form and move around like we might expect a particle to however these aren't truly particles they're really just regions of high and low density in the string in other words these are sound waves which propagate around the string but we aren't done there's nothing quantum about this field theory so how do we upgrade this classical string to a quantum field theory well it's easy instead of using classical harmonic oscillators like a mass on a spring we use a quantum harmonic oscillator now there are many interesting differences between classical and quantum harmonic oscillators but the one that's really important here is that the quantum oscillator can't have any energy that we want in fact the energy can only come in integer multiples of a constant determined by the properties of the oscillator in essence the energy of a quantum harmonic oscillator comes in chunks and every oscillator can have only a whole number of these chunks of energy in it so let's set up our system in the same way as the classical system just take a ring of quantum harmonic oscillators and start increasing the number of them to see what happens we'll prepare the state so that there's a single chunk of energy in a single oscillator now these oscillators can still interact and pass this chunk of energy between each other however due to the nature of quantum mechanics after we start the system running we can't exactly say where this chunk of energy is we can only determine the probability of finding this piece of energy in each particular oscillator i'll represent this by how red each oscillator is the more red the higher the probability of finding the unit of energy in that oscillator okay let's look at how this works like in the classical case when the number of oscillators is small things look pretty random [Music] now like we saw in the classical case around n equals 500 we see a big change we get two main regions where the probability is highest which propagate around the circle however after a while we again see the probabilities start to smear out over the full ring as we increase the number of oscillators even more this high probability region gets even smaller and holds together for longer when we reach n equals 5000 we really start to see how we can get particles from this theory the two very red regions are very small and also quite stable for the entirety of the simulation we don't see these regions smear out nearly as much as we did for the case of fewer oscillators as we might expect when we take n to infinity the non-zero probabilities are localized in two points which travel at a fixed speed around the ring as it turns out the speed is exactly the speed of sound we found in the classical case so it seems that instead of having sound waves like in the classical case we have sound particles called phonons in the quantum theory the two parts just come from the fact that there's a 50 chance that the phonon is traveling clockwise and a 50 chance that it is traveling counterclockwise so why do we consider these actual particles whereas in the classical theory we did not well remember in the quantum case we have to have discrete units of energy in specific oscillators in the case that we only have a single unit of energy in the system there's no other option than have it localized at a single site even though multiple oscillators may have some probability to be the home of this unit of energy in this sense each unit of energy behaves very much like we would expect from a particle they always have the same properties and they're localized in a specific place in the classical case we don't have any restrictions on what the energy in the system should be or how we configure that energy so we don't have a similar interpretation all right let's recap a bit we can visualize a field theory as a collection of a bunch of harmonic oscillators really close together the only difference between the classical field theory and the quantum field theory is how these harmonic oscillators behave in the quantum case the energy of each oscillator is quantized so we can only put energy into the system in set units when we place a single unit of energy into the system at a specific point the unit of energy behaves very much like a particle the possible sites where we could find it are localized points which zoom around the ring at the speed of sound this is why a theory of quantum fields gives rise to physics involving particles now before i wrap up this video i want to say that i've put all of the code that i used to create these animations in a github repository as well as some documents which explain the math behind the code so if you're interested and want to play around with it a bit i'll link it in the description for you to check out while its interpretation as a bunch of quantum harmonic oscillators very close together is very straightforward quantum field theory remains an incredibly mysterious area of research due to its mathematical complexity in fact even one of the seven one million dollar millennium prize problems is really a quantum field theory problem nevertheless despite these mathematical difficulties theory has been arguably the most successful of any physical theory to date giving us the full standard model of particle physics as well as finding applications in other fields such as condensed matter physics and nuclear physics quantum field theory is an absolutely beautiful theory both in terms of physics and mathematics and has completely shifted the way we understand the universe around us