The equations that describe the world

Ever since mathematics has been introduced into the description of physics we have striven to describe reality in terms of equations. One of the arguably most know equations is Newton's law that the acceleration of an object is given by the ratio of the force acting upon this object divided by its mass. These equations should not be taken as everlasting truths. For this law of Newton we know that its not fully satisfied if we try to describe a quantum object or if the speed of the object is close to the one of light. However, this makes the equation nonetheless useful, as there are many cases where neither is the case. The best known example is the movement of the planets around the sun. This equation is, however, not yet complete. It states everything that is to known about the particle, but nothing about the force. It is what is called a kinematic equation.

We need to supplement it by an equation for the force. In case of the movement of a planet around the sun, it is Newton's law of gravitation: The force is given by a constant, which has to be measured, times the mass of the sun times the mass of the planet, divided by the square of the distance between the sun and the planet. With this, we know enough to solve the equation, and find after some tedious calculations (to be done by every first semester student of physics) that the planet moves on an elliptical orbit around the sun. With the force given, the equation therefore describes the motion of the planet. It is thus called an equation of motion. Generically, if we can formulate the equations of motion for a theory, we have everything at our disposal to describe the solutions of the theory. However, in general we have to supplement the equations with the situation we want to actually describe with the theory. In the case of the planet, we have to add where the planet was and where it moved to at a certain instance of time. Otherwise the equation of motion would give us the solutions for all possible initial positions and velocities of the planet, and thus an infinite number of possible solutions to the theory. Such additional information are called boundary conditions. They select out of any possible kind of behavior described by a theory the particular one which is compatible with the state a system is in.

This concept now sounds at first like something which is very much tied to Newton's law. In fact, it is not. Already before 1900 we have known how to write down the equations of motion for all kinds of non-quantum physics happening at a speed much less than the one of light. Unfortunately, knowing how to write down the equations is not the same as being able to solve them. For example, we know very well the equations of motions describing how a river flows. But as soon as it flows quickly over rough grounds, such that it becomes turbulent, we are no longer able to solve the equations. In such cases we are often forced to revert to the simulation methods discussed previously.

Now, what happens, when things get fast or quantum? Well, when things get fast, not much changes. The equations just look a bit different, and are much nastier, but that is more or less all. When things go quantum, it becomes more weird. Since in a quantum world we have this problem with being either wave or particle, it is no longer really possible to talk about moving objects anymore. Nonetheless, people have been able to formulate something which is in spirit close to the equations of motions, the so-called quantum equations of motions (or some times called Dyson-Schwinger or Schwinger-Dyson equations, honoring those people who have developed them). These equations describe, in a way, the average behavior of particles in a quantum theory. Nonetheless, supplemented once more by boundary conditions, they describe the contents of a theory completely. Thus, they are powerful indeed. But as with anything powerful, it gets complicated. Thus, only for very, very simple theories it is possible to solve these equations exactly. For theories like the standard model, one has to introduce severe approximations (often called truncations) to be able to solve them. If these approximations are made wisely and with insight, these approximations are such that still questions we have to the theory can be answered correctly. But it takes often very long to understand how to do approximations right.

The way these quantum equations of motions (or also the ones for non-quantum physics) look is by no means unique. We are free to do mathematical reformulations of them. These leave us always with the same physics of course, but the equations look rather different. This is often very helpful, as the different formulations have very different properties, and very different advantages and disadvantages when it comes to doing calculations. Thus, by exploiting the different reformulations in a wise way, one can go a long way in solving the equations.

In case of the quantum equations of motions a particular useful reformulation is given by the so-called functional renormalization group equations. That sounds like a awful big thing, but the idea behind it is rather straightforward. The idea behind this reformulation is not to swallow the whole theory as one big thing, but chop it off in simpler bites, taking one after the other. Technically, it is realized by slicing the energy which particles are allowed to have, and only include particles with a particular range of energy values in each single step. Building up the whole reality is then done by adding up the particles with different energies one after the other. Though also this cannot be done exactly for most theories, it a very useful complementary way of solving the equations, with great successes.

Both approaches together are often collected under the common name of functional methods. Functional here stands for the fact that on a mathematical level both are strictly speaking not dealing with ordinary functions. Rather, they deal with functions of functions, so-called functionals. This sounds awful and is in fact as awful as it sounds. But it is the price one has to pay when one wants to venture into quantum physics mathematically. Nonetheless, this name is nowadays attached to a collection of different formulations of the quantum equations of motions. These are a great help in describing and understanding physics in every detail. In contrast to the lattice methods, it is easy to disassembled the equations, and to understand what each every part is doing. Though, while very complicated to solve, they are a vital part of the physicists tool box in one way or the other, and thus remain a thing I am working with on a daily basis.