It’s perhaps a striking realisation in modern theoretical physics that the question that now drives new theories is “how are things the same?”. Noether’s theorem simply states that if there is a continuous symmetry in nature then there is a conserved quantity. If the laws of physics are the same in London as they are anywhere else in the universe then there should be a conserved quantity related to continuous movement from one place to another. This quantity is observed as momentum. If the laws of physics are the same now as they where then dinosaurs walked the earth then there is a conserved quantity; energy. But then Einstein has something to point out. He realised that what is time for one observer may be space for another space and time are the same they are spacetime. And so what appears as energy to one observer may be momentum to another or as it is more famously known E= mc2. Essentially the famous equation says that even when something is stationary in space it is still travelling through time and hence has energy proportional to its mass. Einstein’s other great insight was to understand that the laws of physics in a gravitational field are equivalent to those in an accelerating space ship. From this equivalence principle he understood that gravity must be the curvature of spacetime and that this curvature must be equal to the energy-momentum distribution throughout the universe.
From the combination of the two great theories of the early 20th century quantum mechanics and Einstein’s relativity came (relativistic) quantum field theory (QFT). We understand all the forces with the exception of gravity through QFTs. But what is a QFT? Essentially it is way of understanding the world as continuous set mathematical fields which themselves only give a probability of what can actually be measured. By a mathematical field I simply mean that there is a number (or set of numbers) at every point in spacetime (note spacetime as it’s a relativistic theory).
To understand, deeply, what a QFT is we need to understand first what a classical field is and then in apply a generalisation of principles of quantum mechanics. A classical field description of reality is one where matter is consciously spread throughout spacetime so the field itself is essentially the matter density. There are no particles in this classical description. But in quantum mechanics we understand that everything is particles that cannot be created or destroyed. Before we discuss how these two pictures can be married into one and the same picture we need to discus a peculiarity of the classical field. The field can be negative! It is a essential feature of matter fields that they can be negative or positive.
These negative and positive field values can be interpreted as matter and anti-matter. So it makes sense that in QFT we are going to have particles and anti-particles. If we add a positive field value to a negative field value (which remember is just a numbers in spacetime) they are going to cancel. So a particle will annihilate with an anti particle. So we have to give up the idea that particles cannot be created or destroyed as we move from quantum mechanics to QFT. This has an important implication. When we look at the field on small scales we become uncertain to the energy and momentum of the field (see Heisenberg’s uncertainty principle) and hence we actually become uncertain in the number of particles that exist on that scale. The smaller we go the more and more uncertain we get to what particles exist. Let’s say we have an electron and we measure where it is. At a large(for an electron) scale say that of an atom we could be certain that we are just looking at an electron. But as we zoom in closer we might start to see an two electrons and a positron (note 2 electrons + 1 positron = 1 electron). If we keep zooming we see more and more electrons and positrons until we just have a blur of particles and antiparticles. This blur is then actually closer to our classical field theory picture! Think about it this way if we have more and more particles in a smaller and smaller space we essentially must have shorter and shorter distances between the particles until all we see is a continuous density distribution. But now think about larger scales where we are looking at many many particles such as a gas. We are now less concerned with the exact position of each particle and instead see it as a distribution of particles. This kind of description of reality is known as statistical or thermal physics. We see then that there is a symmetry of scales that take us from statistical physics right down to QFT.
Next post i will discus grand unification and supersymmetry