Physics from Finance: A gentle introduction to gauge theories

Book Preface

The main idea of this book is that we can understand many of the most important insights in modern physics as dis-coveries about the arena modern physics takes place in. In particular, physicists discovered that there is not just space-time but also internal spaces. Spacetime, together with these internal spaces, yields the arena that we use to make sense of modern physics.
Arguably the most important insight was that this arena is not some kind of static background structure but a dynam-ical part of nature itself. Describing the interplay between elementary particles and this arena is what modern physics is all about.

So to understand modern physics we need to answer a few key questions:
– What’s an internal space?
– How can the structure of an internal space be non-trivial? Where does such a non-trivial structure come from?
– How does the non-trivial structure affect objects in the Admittedly all of this sounds horribly abstract.
But, as mentioned already in the preface, we can answer all these questions by thinking about the financial market. This is a useful analogy because, while financial transactions take place at locations in the real world (spacetime), we need abstract internal spaces to keep track of them.
Maybe you don’t care about financial markets at all. But let me assure you that it makes sense to think about the follow-ing simplified version of the financial market. As soon as we understand how we can describe what is going on here con-sistently, we automatically understand how modern physics works, too.1
We will start by talking about symmetries. Symmetries
are something which we can grasp intuitively and there
is a close connection between the internal spaces and the symmetries of a system.
1.1 Symmetry
First of all, what is a symmetry?

Imagine a friend stands in front of you and holds a per-fectly round ball in her hand. Then you close your eyes, your friend performs a transformation of the ball and after-ward you open your eyes again. If she rotates the ball while your eyes are closed, it is impossible for you to determine whether or not she did anything at all. Hence, rotations are symmetries of the ball. But if she moves the ball to another location, you will immediately notice it. Therefore, transla-tions are not symmetries of the ball.

Moreover, if she holds a cube, only very special rotations can be done without you noticing it. In general, all trans-formations which, in principle, change something but lead to an indistinguishable result are symmetries. Formulated differently, a symmetry takes us from one state to a different one, which happens to have the same properties.2
With this in mind, let’s talk about an important symmetry of the financial market. Talking about this symmetry will lead us directly to the internal space of our financial toy model.