Have you ever wondered why gauge symmetry is synonymous with electromagnetism equations? Anytime the topic comes up, it reminds us of Maxwell’s electrodynamics. Gauge field theories are the characteristic of gauge symmetries. Interestingly, you were most probably taught about these theories if you went through a physics class, albeit unknowingly. In this guide, we will discuss the meaning of this local symmetry. We will also discover the symmetry’s relation to electromagnetism. Since this is a guide to anyone new to the topic, we will use a simple analogy to drive the point home.
Meaning of Symmetry
Perhaps the best way to understand this term is to look at the idea of symmetry in common language. In symmetry, an object is transformed without any change in its nature or shape. A circle is very symmetric; you can rotate it at any angle and not change it. On the contrary, a rectangle does not remain the same when you rotate it at say, 45 degrees.
Background of Gauge Symmetry
Gauge symmetry is one of the fundamental concepts of theoretical physics. Together with Lorentz symmetry, this symmetry went unrecognized until the twentieth century after the emergence of quantum mechanics.
Gauge symmetry is different from global symmetry as seen in conventional symmetries. Observations do not change. In global symmetry, the situation is different. Its simulation is solely reliant on laws of nature.
Economic Analogy for Gauge Symmetry
The following economic analogy describes this symmetry clearly.
In this comparison, we assume we have several countries placed on a regular grid. A bridge connects each country to its neighbours. A bank is placed at the bridge. As you pass through the bridge, you are required to change currency from the immediate country to the country you want to visit.
No central entity controls the transactions. Therefore, each bank has the autonomy to use any exchange rate. In addition, there are no commissions for using bank services.
Another condition is that you need to move to the immediate country before moving to the next. In other words, there is no flying over countries. There is also an assumption that you can only carry money-no silver or gold.
Symmetry comes in when one country decides to slash zeros when they become too many. We have seen real life case scenarios of this kind of action when inflation hits, for example in Zimbabwe. This is symmetry because despite the slashing, there is no change in real sense. No economic opportunities from the manipulation: poverty and wealth levels remain the same.
This is a “local” symmetry because each country is at liberty to make any changes, without regard for what other countries are doing. Some countries may never perform the changes while others do it countless times.
There is another interesting aspect of this analogy- the opportunity to make money. A speculator can move from one country to another and make money depending on the exchange rates.
Back to physics, each of these components has a meaning. The exchange rates relate to magnetic potentials within space. Magnetic fields could be equated to the situation of earning money through speculation. Speculators represent electrons/particles that operate in a magnetic field.
To fit in physics, you need to assume that everything in this story happens in extremely short distances. The discrete structure as described here is just an assumption in physics.
Electromagnetism borrows from the gauge symmetry analogized above. Magnetic and electric fields follow what is called Maxwell equations. The first gauge theory was Maxwell theory. Going back to the economic analogy, consider a certain bank on a certain bridge. The number of speculators moving in either side determines what the bank does to the exchange rate. The bank may run out of a particular currency if there is an imbalance in the two sets of speculators. Maxwell equations are equal to the condition for the banks not to run out of any of the two currencies in question.
The role of symmetry in unravelling the secrets of nature has really come out this century. Gauge symmetry has been at the center of things because of its advanced form. Using the economic analogy, we have managed to relate travelers to particles, total monetary benefit to magnetic field, and exchange rate to magnetic potentials. The point is that despite all the activities, there are spacetime points that make any change in currency irrelevant. Doesn’t gauge symmetry define all this?