For centuries, symmetries in the natural world, such as in a snowflake, have both fascinated scientists and inspired artists. In the world of elementary particle physics, symmetries are not just interesting patterns, but a fundamental part of our explanation of how particles behave.

What is a Symmetry?

A symmetry is a transformation, which, when applied to a system, may or may not change some of its properties. The most well known symmetries are reflecting a system in a mirror and rotating a system, but in particle physics, there are a few more. A symmetrical system is invariant under a symmetry, ie all its properties are the same after the transformation. For example, a snowflake looks the same after it has been rotated through sixty degrees. Confusingly, the word 'symmetry' also refers to this invariance; the 'symmetry' of a snowflake refers to its invariance under the symmetry operation of rotation.

Space-time Symmetries

Rotational symmetry is an example of a space-time symmetry; another is a translation through space, such as moving a coordinate system two metres to the left, or a translation in time, such as putting the clock forward one hour.

The laws of physics are invariant under these operations. We are free to choose whatever coordinate system we like when doing calculations, so moving it forwards and backwards or rotating it has no effect on physical laws. The results of an experiment remain the same if we move it to the lab next door; the speed at which an apple falls is the same if measured at midday or midnight, in GMT (Greenwich Mean Time) or BST (British Summer Time). This freedom is very useful, as some physical equations can only be solved if we select a specific coordinate system.

This may seem obvious, but these symmetries are of fundamental importance as they allow us to derive the conservation laws. Doing so requires a bit of maths, but the results are straightforward:

• The invariance of physical laws under translations in space implies the conservation of momentum.

• The invariance of physical laws under translations in time implies the conservation of energy.

• The invariance of physical laws under rotations in space implies the conservation of angular momentum.

Physical laws are also invariant under translations of velocity. Experiments performed inside a railway carriage moving at a perfectly steady speed produce exactly the same results as an experiment in a stationary laboratory. This means that there is no absolute velocity; we can only measure the speed of one object relative to another. This fact is fundamental to Einstein's laws of special relativity.

Discrete Symmetries

The symmetries outlined above are all continuous, as they have a magnitude - how far you move in space or time, or what angle you rotate a system. However, discrete symmetries simply flip a system from one state to another, such as reflecting a coordinate system in a mirror. These symmetries do not have corresponding conservation laws, however they are important in particle physics to determine which particle interactions are possible.

P for Parity

P, or Parity, is the operation of space inversion, or reflecting a coordinate system in a mirror, except it reverses all three dimensions, so you need three mirrors. If physical laws are invariant under parity, then particle interactions should be perfectly symmetric, however as explained below, this is not always the case.

C for Charge Conjugation

C is the operation of reversing the Charges of particles in a physical system. This means exchanging all the particles present for their corresponding antiparticles. Invariance under this operation means the interaction between a proton and neutron is the same as that between an antiproton and an antineutron. However again, this is not always so.

T for Time Reversal

Finally, T corresponds to the operation of reversing the direction of Time. If invariant under the T operation, a particle reaction which changes an initial state to a final state can also change the final state back to the initial state.

There are also combinations of these operations, for example, a CP operation changes a particle system to the space-inverted version of its antiparticle system.

Broken Symmetries

Unlike the continuous space-time symmetries, there is no fundamental reason why physical laws should be invariant under discrete symmetries. Indeed, there are physical systems in which these symmetries are broken or violated. For example, it was discovered in the 1950s that some radioactive atoms are asymmetric and emit more radioactivity in the direction of their spin axis; this asymmetry is known as parity violation. It was later discovered that charge conjugation is not always conserved either.

These discoveries upset many physicists. Although there was never any evidence to suggest that C and P should be conserved, the symmetry was beautiful; it made the maths easier, and so many physical systems did conserve C and P that it was a great disappointment to learn that it was not a universal law.

As belief in symmetry was strong, particle physicists hoped that the combination of charge conjugation and parity (CP) would be an invariant symmetry - that the antimatter world would look like the 3D mirror image of the matter world. At first this did appear to be the case. However, more precise experiments performed in the 1960s showed that CP violation does in fact take place, and in a tiny fraction of cases the CP image of a system is not an exact copy.

CP Violation

Tests for CP violation have been carried out at particle physics laboratories around the world. By studying the decay of kaon particles physicists have been able to determine the difference between the lifetimes of the kaon and antikaon to great precision. The first tests in 1964 showed that indirect CP violation did exist, and recent experiments at CERN - the European Laboratory for Particle Physics - and at Fermilab in the USA have confirmed the existence of direct CP violation.

So it appears that antimatter is not precisely the three-coordinate mirror image of matter; there is in fact a small flaw in the symmetry. While this effect may seem to be unimportant, it could explain one of the largest questions in antimatter physics: 'Why is the universe made almost entirely of matter?'

The big bang is believed to have produced equal amounts of matter and antimatter. However, whenever matter and antimatter meet, they annihilate in a burst of gamma rays, so this theory would suggest that both matter and antimatter should all have annihilated long ago, and the universe should today simply consist of radiation.

However, since CP violation exists, we know there is a small difference between matter and antimatter. This may explain why, after all the early annihilating, there was a tiny surplus of matter left over to form galaxies.

Finally, although it is now clear that the laws of physics are not invariant under C, P, or CP, scientists are still optimistic that the ultimate discrete symmetry CPT is not broken. The mirror image of the antimatter world with time running backwards should look exactly the same as ours.