Mirror, Mirror, On the Wall

One of the most useful, general tools in physics is symmetry.  To a scientist, a symmetry is any change we can make to reality that is undetectable.  For example, if I moved the entire universe five metres to my right, how would anyone know?  All we can measure are the relative positions of objects, e.g. the Earth relative to the Sun.  If I move the everything the same amount, then relative locations are unchanged.  This is then a symmetry of nature.  Hopefully, it is clear that there is nothing special about the distance and direction I chose; all such shifts are unobservable.

Why is this helpful?  The key lies in Noether's Theorem, named after the German mathematician Emmy Noether.  This states that whenever we have a symmetry in nature, there is a conserved quantity associated with it.  Conserved quantities make our lives easier, since they simplify calculations.  Amount in equals amount out is a pretty easy expression, after all.  (Not all physics demands advanced mathematics!)  For example, the symmetry I mentioned in the previous paragraph leads to conservation of momentum, while the fact that physics is constant in time means energy is conserved.

What are the complete list of possible symmetries?  This is a broad question that seems difficult to answer.  However, in 1967 Sidney Coleman and Jeffrey Mandula tried to do so.  In a seminal paper published in Physical Review they proved that, assuming special relativity and quantum mechanics, the only possibilities are:

1. The symmetries of special relativity, and the associated conservation laws: momentum, energy and angular momentum;
2. The symmetries of electromagnetism, and the associated conservation of electric charge;
3. Certain well-defined generalisations of electromagnetism, called gauge theories.

In particular, since all of these symmetries were known already, it suggested that we had reached an important milestone in our understanding of nature.  This theorem is known, unsurprisingly, as the Coleman-Mandula theorem.

I am not going to explain the proof.  Indeed, it's been nearly a decade since I read the proof and I don't remember exactly how it went.  And indeed, the most important point about this result was the loophole people found in it.  You see, Coleman and Mandula had made one additional assumption beyond those I have mentioned, an assumption about the mathematical structure of the symmetry group.

To understand the assumption they made, let us consider rotational symmetry.  Here's a simple experiment you can do.  Get a six-sided die, and note which faces are visible and where.  Then, rotate the die through 90 degrees twice, around different axes.

For example, you could rotate the die as shown by the red lines in the top row above.  Note where the faces are now.  Then, returning to the original configuration, do the same thing but with the two rotations in different order, as in the bottom row with the green lines.  Note that the outcome is different.

This example is related to symmetries (in that rotations are a symmetry of the universe), but mainly serves to illustrate that some transformations depend on the order they are applied.  When this is the case, it shows up in the relevant mathematical structure.  Let us use the letters f and g to describe two rotations.  Then the above shows that, in general,

$f \times g \neq g \times f$

Or

$f \times g - g \times f \neq 0.$

The object on the left hand side of the latter equation is called the commutator of f and g.  Coleman and Mandula assumed that any symmetry could be characterised by the commutator of its elements.  However, some mathematical structures are instead defined by the anti-commutator:

$f \times g + g \times f.$

If we allow symmetry groups to be in this type, we get the one exception to the Coleman-Mandula theorem: Supersymmetry.  Which will be the subject of my next physics blog post.