Monthly Archives: February 2018

Why does light exist?

Light is something that we probably take for granted, but have you ever thought about why it should exist at all? From the viewpoint of quantum mechanics, it turns out that it must be there to satisfy a fundamental symmetry of the universe.

Electric and magnetic fields are created by electric charges, and electric charges also move in response to those fields. The electromagnetism that you probably learned at school is one description of this. The electric potential and the electric field are related to each other because the field is given by the gradient (or slope) of the potential. It’s not possible to directly measure a potential, so in some sense it is only the field that has a physical reality. Charged objects moving in the field experience a force which changes their speed or direction of travel.

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Electric fields created by small charged objects. Picture credit: Wikipedia.

This description works very nicely for many everyday situations and, on the face of it, there is no obvious role for light here. But, at the scales of individual atoms and electrons, electromagnetism has to be described in the framework of quantum mechanics. It turns out that light is the thing that carries the forces that charged objects feel in electromagnetic fields.

The point of this post is to try and explain why that is.

As I’ve said before, looking at symmetries can help to simplify physics problems. Symmetries are transformations which leave the object that is transformed in an identical state from how it started. For example, a square can be rotated by 90 degrees about its centre point and the result will look the same as the unrotated version. This is an example of a discrete symmetry – there are only four rotations of the square that are symmetry transformations (90, 180, 270, and 360 degrees). In contrast, a circle has a continuous symmetry – you can rotate a circle about its centre point by any angle and end up with a circle that looks just the same as the unrotated one.

This is the moment that things start to get a bit less easy to visualise, because talking about a different kind of symmetry is unavoidable: We have to delve into gauge transformations and gauge symmetries.

To try and explain the concept of a gauge transformation, look at the left-hand picture below. The lines represent potentials at each position. Start with the lower, blue line. It has a field associated with it, which is the slope of the line at each point. To get the red line from the blue line, you have to add on a fixed amount of potential at every position. This is represented by the three dashed black arrows, which are all the same length.

But here is the crucial point: The field associated with the red line is exactly the same as the field associated with the blue line, because the slopes of the two lines are the same at every position. Adding on the extra potential hasn’t changed this.

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Global and local gauge transformations.

So, adding a fixed amount to the potential at every position does not change the field at all. And remember, the field is the only thing that is physically observable – we can’t measure the potential. Therefore, this is a continuous symmetry.

The fact that many different potentials give the same field is called “gauge symmetry”. The type of gauge symmetry illustrated here is a “global” symmetry, because the amount of potential that you add or subtract at each point in space is the same (i.e. it’s a global amount).

However, the crucial part for the existence of light comes from a slightly different type of gauge symmetry, called a “local” one. For a local gauge transformation, instead of adding the same amount of potential at every point, you have the freedom to add different amounts of potential in different places. This is shown in the right-hand graph above. The red line is obtained from the blue line by adding different amounts of potential at each position. Notice that the three dashed black arrows are now different lengths.

Electromagnetism in quantum mechanics works under the assumption that this local gauge transformation is also a symmetry. For this to be true, the physical observables, including the field, must stay the same after the local transformation.

But, looking at the graph, it immediately becomes apparent that adding a different amount of potential at different positions to the blue line means that the red line has a different slope. This would change the field, and so this local transformation should not be a symmetry at all!

The only way that this can work is if our description of the quantum field is incomplete: In addition to the potential, there must be another part which also feels the effect of the local gauge transformation. When the combined transformation of the potential and this additional object are added together, the fields remain unchanged so that the local gauge symmetry is intact.

For the electromagnetic field in quantum mechanics, it turns out that this secondary part is a photon. Looking deeper into the mathematics, we find that their existence explains how charged objects feel a force from the field: they emit and absorb photons.

But photons are also the particles which carry light! So, one answer to the question “why do we have light?” is simply that photons must exist to preserve local gauge symmetry.

I appreciate that this has the whiff of a magic trick to it: Why should local gauge symmetry be something that we insist must exist? Perhaps there is some deep answer to this question that I don’t know, but the best response might simply be that this theory works.

To add more to the “it just works” line of reasoning, local gauge symmetry is also the reason that gluons (which carry the strong interaction) and the W and Z bosons (which carry the weak interaction) exist. In those cases the symmetry operations are more complicated than adding a potential, but the fundamental assumptions and logic are the same. So, this is a powerful concept which seems to be important in describing quantum physics, and gives one explanation for light comes from.

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