Symmetry transformations in physical processes


In our discussion concerning particle decays we have used terms like 'reactions' and 'processes' to describe how matter changes from one form to another. We need to be a bit more precise about what we mean by these terms. A particle (or nuclear) reaction is a change in the identity of a particle (or nucleus) leading to the production of other particles (or nuclei). A particle reaction is an example of a specific process -one that is governed by a physical law but how we observe it depends on the way we look at it. Consider viewing a clock with moving hands. Now look at the image of the clock in a mirror. You now see a different process. The hands of the clock move in the opposite direction. In the real world clock hands move clockwise but in the 'mirror world' they move anticlockwise . Now it is important to understand that a clock which has hands that move anticlockwise is a perfectly possible process. Any competent clockmaker could make you a clock whose hands move backwards and such a clock could be constructed without violating any known law of physics. This leads us to ask the following questions:


Is there any physical process whose mirror image is impossible?

If the answer to this question is 'yes' then it means the nature makes a physical distinction between left and right.


One symmetry that is encountered in nuclear and particle physics is called parity and is denoted by the symbol P. Parity is akin to looking at a physical process in a mirror. A measurement of a particle reaction is usually observed from a co-ordinate reference frame which may be chosen in an arbitrary way and we refer to its space inversion as when the particle system is viewed from a 'mirror' perspective. For example if we started looking at the system using a right-handed co-ordinate set then under P we now view it from a left handed set. If the particle system behaves in the same way under a space inversion then it is invariant under P and we say that the parity is preserved.

In particle decays, each particle is assigned a parity odd or even (-1 or +1). The total parity of the decay is found by multiplying the individual parties of the particles together to give either a positive or negative value of one. Note that unlike lepton number or strangeness which is additive, parity is a multiplicative quantum number and we must multiply the various parties together to find the net parity. The main point to understand about parity is that it is the conserved quantum number that expresses the conservation of symmetry between left and right in a process when the physical law involved is invariant under space inversion. Processes in which parity is not conserved would look different in a mirror image world.

However, some puzzling features of the decay of the K+ meson hinted that all was not well with parity conservation. The K+ sometimes decayed into states of positive parity and sometimes into states of negative parity. This anomaly was investigated by two American physicists Tsung Dao Lee (1926-) and C.N.Yang who concluded that while there was strong evidence that the electromagnetic and strong interactions always conserved parity, the weak interaction might not. They proposed an experiment to test this by checking for parity conservation in the beta decay of a nucleus. The result of this experiment showed that parity conservation is not conserved in the weak interaction, and its non conservation implies that nature makes a distinction between left and right. The mirror-image process of the K+ decay is not the same as its non-mirror counterpart!

Charge conjugation

Dirac's prediction of antimatter and the subsequent discovery of anti-particles shows that there is a physical difference between matter and antimatter. But can we distinguish matter from antimatter? What we want to know is that when particle physicists study a process that involves particles of matter would the same process be possible with particles of antimatter. If the answer is no, then we have found a physical distinction between matter and antimatter. We don't have large lumps of antimatter available to us to experiment on, but particle physicists have a ready collection of antiparticles to use in an 'anti-process'. for example the anti-process of a neutron decaying into a proton

neutron decay

would have the anti-process of an anti-neutron decaying into an anti-proton.

anti neutron decay

Notice how the charges are opposite. The operation of replacing a process with its anti-particle counterparts is called charge conjugation and is given the symbol C. In a charge conjugation version of the a process involving a system of particles, the mass, lifetime, spin and position and velocity co-ordinates do not change. The charge conjugate version is simply obtained by replacing all the particles by their corresponding antiparticles. Particle Physicists have found that the elctromagnetic, and strong interactions do not distinguish from antimatter and there is no evidence that the gravitational interaction does so either. However when the weak interaction was found to distinguish left from right by violating P, they wondered whether the weak interaction might be able to tell the difference between matter and antimatter as well. The conversion of the neutron into a proton by the emission of


particle is a weak decay process and this provided a method by which to see if the weak interaction does in fact distinguish matter from antimatter. Amazingly, it turned out that the spin of the neutrino in weak decays showed that there was a loss of invariance under charge conjugation! To see why, quantum theory tells us that in every weak decay in which neutrino is emitted, the spin of the neutrino is related by the left hand rule. In other words the neutrino from the decay of an anti-neutron is spinning in a sense that appears clockwise as it travels towards you but the antineutrino from the decay of a neutron is spinning in a sense that is anticlockwise as it travels towards you. Particle physicists call this 'handeness' of a quantum particle which is related to its spin, helicity.

e.g. the decay of the neutron and its charge counterpart

neutron decay

(antineutrino right-handed helicity)

anti neutron decay

(neutrino left-handed helicity)

While other particles can exist in either state, particle physicists always see right-handed spinning neutrinos and left handed spinning antineutrinos. We can now make a physical distinction between matter and antimatter. If you begin with a beam of neutrons and observe them changing into protons you will always find right-handed antineutrinos. But the charge conjugate version of this process will not reduce right-handed neutrinos but left-handed ones (remember that spin is unchanged under C). The symmetry between processes starting as charge conjugate versions is not invariant. So just like loss of P, weak processes do distinguish matter from antimatter.

CP Invariance

Particle physicists have found that symmetry can be restored in weak interaction processes where both C and P transformations are applied together. Such a process is said to be CP invariant. In the late 1950's and early 1960's it was hoped that the invariance in C and P that had been found in weak processes would cancel each other out so that the combination of CP invariance would always be preserved. However they found to their surprise that even this symmetry was violated! The process involving the decay of neutral K meson (K0) provides a very sensitive test of CP invariance. Recall that Lee and Yang explained the

theta tua

puzzle by theorising that the K+ meson cannot decay both into two pions


and three pions


unless the law of conservation of parity is violated. The K0 or neutral K-meson as a single particle was not allowed under the CP theory, to decay by two pion and three pion processes. This was confirmed when further experiments revealed that there were in fact two versions of the natural K meson.

The short lived tau

meson which decayed via


and the long lived


meson which decayed via


The long lived neutral K meson has a half-life about 600 times greater than the


meson. However in 1964, a team at the Brookhaven National Accelerator Laboratory in the USA found a small number of


meson events in which the long-lived neutral K meson decayed into two pions by


. The number of



was small, about 0.3% of all


decays, but has been confirmed in many experiments involving neutral K-mesons. This was a totally unexpected result, and established that the weak interaction does not conserve CP invariance although the violation happens rarely.


Time reversal symmetry

Despite its exotic sounding title, this simply means that the behaviour of a quantum system is invariant if we reverse the direction of motion of the particles in the system rather like running a film backwards. Time reversal is denoted by T and transforms the time co-ordinate of the wave function to its negative value. Under T, particle reactions are equally capable of proceeding forward in time as well as backward. A general theorem of quantum field theory (the proof of which is beyond the scope of this book) called the CPT theorem, asserts that any violation of CP invariance must be compensated by a violation of time reversal invariance. This is because the CPT theorem states that

Every law of physics is invariant under the symmetry transformations of charge conjugation, space inversion and time reversal.

It is because of this invariance across all the fundamental interactions, that particle physicists say that CPT invariance is an exact symmetry. The violation of CP invariance by weak processes thus has important consequences for T as it implies that some microscopic processes are not reversible. In large scale systems we are used to seeing asymmetries in time in terms of thermodynamic processes- for example a hot cup of coffee will eventually get cold as time flows into the future. But an asymmetry in time in sub-microscopic processes was totally unexpected by particle physicists. Its existence is only inferred by the CPT theorem, its occurrence is rare, and only in conjunction with the decay of the long lived neutral K meson. This is the only evidence we have to date, that nature distinguishes the past from the future on the sub-microscopic level and why it should do so is completely unknown.

So far, no other effects of this kind have been detected with any of the other interactions, but this is not ruled out.

The Theory of Parity
The Fall of Parity
Tsung Dao Lee - Biography
Tsung Dao Lee and C.N. Yang