1 The Meaning of the Discrete Symmetries

LefLeft–right symmetry—also called space-inversion or parity symmetry—and time-reversal symmetry are two invariances of classical physics—of classical mechanics and of classical gravitational and electromagnetic interactions—which were recognized long before the advent of quantum mechanics and of quantum field theory. We shall review the meaning of those symmetries in classical physics before implementing them in a quantum-mechanical context.

Parity symmetry, usually called P, consists in the invariance of physics under a discrete transformation which changes the sign of the space coordinates x, y, and z. This corresponds to the inversion of the three coordinate axes through the origin, a transformation which changes the handedness of the system of axes. A right-handed system becomes left-handed upon the parity transformation (see Fig. 1.1).

Parity symmetry is sometimes called mirror symmetry, because the inversion of the coordinate axes may be achieved in two steps, through a mirror reflection on a coordinate plane followed by a rotation by an angle $π$ around the axis perpendicular to that plane (see Fig. 1.2). From the basic assumption of isotropy of space it follows that physics is invariant under a rotation. Therefore, the relevant point is whether physics is invariant under the mirror reflection too. Thus, P symmetry is in practice equivalent to symmetry under a mirror reflection. As the mirror interchanges left and right—for instance, our right arm is the left arm of our mirror image—parity symmetry is also called left–right symmetry.

Applying two parity transformations in succession is equivalent to no transformation at all. The square of the parity transformation is the identity transformation.

The parity transformation changes the sign of the position vector of a particle: $r→→−r→$⁠. As a consequence, the velocity of the particle,

also changes sign under P. The same happens with the momentum

The angular momentum,

is invariant under P, because both $r→$ and $p→$ change sign. According to Newton’s law, the force which acts on a particle is equal to the rate of change of its momentum,

Therefore, under parity $F→→−F→$⁠.

The Lorentz force acting on a particle with electric charge q is given by

where $E→$ is the electric-field strength and $B→$ is the magnetic-field strength. Under parity $F→Lorentz$ and $v→$ change sign. Therefore, P must transform $E→→−E→$ and $B→→B→$⁠.

The scalar potential V and the vector potential $A→$ are defined through

The operator $∇→=∂/∂r→$ changes sign under P. Therefore, parity transforms $A→→−A→$ while V is left invariant.

We shall not demonstrate the invariance of the whole body of classical mechanics and electromagnetism under the parity transformation delineated above. That invariance is established by surveying all the equations of classical physics and checking that they are invariant under parity.

Vectors are generically defined as three-component objects which transform in the same way as $r→$ under a rotation of the system of coordinate axes. This prescription does not tell us how vectors should transform under parity. As the square of the parity transformation is the identity transformation, there are two types of vectors: those which change sign under P and those which do not. Vectors which change sign under parity, like $p→$ and $E→$⁠, are called polar vectors, or simply vectors. Entities like $B→$ and $J→$⁠, which do not change their sign under a mirror reflection, are called axial vectors, or pseudovectors.

Analogously, there are quantities which are invariant under a space rotation but change sign under parity. This is the case in particular of the scalar product of a vector and a pseudovector, e.g. $E→⋅B→$⁠. Those quantities are called pseudoscalars, as opposed to (proper) scalars, which are mirror-invariant.

Let us now consider the time-reversal transformation, usually called T. This consists of changing the sign of the time coordinate t. From eqn (1.1) we see that, when $t→−t$⁠, the velocity $v→→−v→$⁠. The momentum $p→$ also changes sign. The angular momentum $J→→−J→$⁠. On the other hand, from eqn (1.4), as both $p→$ and t change sign under time reversal, the force $F→$ remains invariant.

As $F→Lorentz$ in eqn (1.5) must be invariant while $v→$ changes sign, $E→→E→$ but $B→→−B→$ under time reversal. From eqns (1.6) we see that $V→V$ but $A→→−A→$ under T.

The mathematical transformation delineated above, under which t,  $p→$⁠, and other entities change sign, may be called $T̂$⁠. The genuine time-reversal transformation T goes beyond $T̂$⁠, since it also interchanges final states and initial states. Time reversal is related to the following fundamental question that one may ask about the laws of Nature: let us consider the final state of some process, invert the velocities of all particles in that state, and let it evolve; shall we obtain the former initial state with all velocities reversed?

Spin is a concept extraneous to classical physics. However, there is no problem in integrating it as an ad hoc quantity. Some particles are postulated to have associated with them an intrinsic angular momentum $s→$⁠, which is in everything identical to a classical angular momentum $J→$⁠. As such, $s→→s→$ under parity and $s→→−s→$ under time reversal.

The spin $s→$ is observable through interactions which are proportional to it. If a particle with spin $s→$ moves in an electromagnetic field with field strengths $E→$ and $B→$⁠, there may exist in the Hamiltonian terms of the form

and

the numerical coefficients $de$ and $dm$ having the appropriate dimensions. If the interaction in eqn (1.7) exists the particle is said to possess an electric dipole moment $de$⁠. If the interaction in eqn (1.8) is present the particle has a magnetic dipole moment $dm$⁠.

Checking the transformation rules of $E→$ and of $B→$ under parity and under time reversal, we conclude that $dm→dm$ under any of those transformations, while $de→−de$ under any of them. Therefore, electric dipole moments violate both P and T. On the other hand, magnetic dipole moments violate neither P nor T and, indeed, they provide a practical way of measuring the spin of a particle.

An important quantity is the sign of the projection of a particle’s spin $s→$ along the direction of its momentum $p→$⁠. This is called the helicity h of the particle:

Helicity is a pseudoscalar, because it is the dot product of a polar vector $(p→)$ and an axial vector $(s→)$⁠. On the other hand, $h→h$ under $T$⁠.

The preceding results are summarized in Table 1.1, in which we have indicated whether the relevant quantities are invariant (denoted by a ‘+’ sign) or change their sign (denoted by a ‘–’ sign) under the P and $T̂$ transformations.

When one makes the transition to relativistic mechanics, the time and the position vector get united in the position four-vector¹  $xμ=(t,r→)$⁠, while the energy and three-momentum become components of the momentum four-vector $pμ=(E,p→).$ The derivative four-vector is $∂μ=(∂/∂t,−∇→)$⁠. The scalar and vector potentials of electromagnetism are united in the four-vector $Aμ=(V,A→)$⁠. The angular momentum $J→$ becomes part of an antisymmetric tensor $Mμν=xμpν−pμxν$⁠. In the same way, $E→$ and $B→$ are united in an electromagnetic-field tensor $Fμν=∂μAν−∂νAμ$⁠.

All four-vectors behave in the same way under parity: their time component is left unchanged, while their space components change sign. We denote this by $xμ→xμ,∂μ→∂μ,pμ→pμ$⁠, and $Aμ→Aμ$⁠. Tensors also have their indices lowered by P: $Mμν→Mμν$ and $Fμν→Fμν$⁠. Parity invariance may thus be extended to relativistic mechanics.

The transformation properties under time reversal are more complex, since not all four-vectors behave in the same way. The four-vectors $xμ→−xμ$ and $∂μ→−∂μ$ behave in a different way² from the four-vectors $pμ→pμ$ and $Aμ→Aμ$⁠. The electromagnetic-field tensor $Fμν→−Fμν$⁠, and the angularmomentum tensor $Mμν→−Mμν$ under $T̂.$ All equations of relativistic mechanics are invariant under the time-reversal transformation delineated above.

In Table 1.2 we have indicated how relativistic tensors of interest transform under P and under $T̂$⁠.

Parity and time reversal are closely related to some of the most basic questions that one may ask about the laws of Nature.

Suppose that one watches some physical event in a mirror. Does the event that one sees there look real? Does the event seen in the mirror correspond to something allowed by the laws of Nature? This is the basic question that P symmetry addresses.

Now suppose instead that one has the physical event filmed and then watches the film running backwards. Will the events seen in the backward-running film look possible and realistic, or will they be at odds with the laws of Nature? This is the issue raised by T symmetry.

In a certain sense, it is an obvious fact that left and right are distinct in Nature. They represent more than mere conventions. Most people display greater skilfulness with their right than with their left hand. Our liver is located in the right side, our heart in the left side of our body. Therefore, no one would confuse the mirror image of a human being with a real person! At a more fundamental level, the aminoacids in life’s chemistry are not identical with their mirror images. Many organic molecules have a right-handed and a left-handed version, and one of the versions occurs much more often in the biosphere than the other one. However, the above asymmetries are in general considered to be accidents of life’s evolution on Earth, and not the consequence of a fundamental left–right asymmetry in the laws of Nature.

As for T, a ‘time arrow’ seems to exist in observed events, not only in biology, but also in the more fundamental realms of physics. A piece of wood burns down to ashes and smoke, but ashes and smoke have never been seen to absorb heat from their surroundings and generate a piece of wood. Naively one might think that this asymmetry in the time evolution of physical systems is in contradiction with the laws of classical physics, since Newton’s law is invariant under time reversal, in the sense that, if $r→(t)$ is a possible trajectory, then $r→(−t)$ is also an allowed trajectory. The asymmetry in the time evolution is one of the postulates of classical thermodynamics, which states that, if a system at a certain instant is in a non-equilibrium macroscopic state, it will evolve into another state with higher entropy. This is the second law of thermodynamics, for which an explanation is given within the framework of statistical mechanics. Systems evolve in the time direction that they do because the final macroscopic configuration is microscopically more probable than the initial one. This in no way implies a time-reversal asymmetry in the fundamental laws of microscopic physics.

The fundamental laws of classical physics are T invariant but, because of the large number of individual particles and collisions involved, macroscopic systems display T-asymmetry. In order to understand how this comes about, a simple thought experiment (Lee 1990) may help.

Suppose that, in a large country with lots of intersecting roads, one thousand drivers start from the same place and drive one thousand kilometres each. Suppose that no directions are marked on the roads, and each driver meets multiple road crossings, each time choosing at will, with no outside help, which new road he shall take.

Now consider the time-reversed situation. Each of the thousand drivers starts from the final point that he has reached in the previous journey, and drives once again one thousand kilometres, once again choosing, more or less at random, which new road he will take at each crossing. One asks oneself, will all drivers, at the end of their second journey, meet at the starting point of the first one? Clearly, the probability that this happens is extremely small.

Although the individual motion of each driver has obeyed time-reversalinvariant laws—they have driven one thousand kilometers in each journey, and at each crossing they have chosen, according to the same chance rules, which new road to take—the observed motion of the total system was time-reversal-asymmetric. This was due to two reasons: first, the numerous road crossings at which each driver had to make a choice; second, the large number of drivers involved.

In the same way, in classical mechanics, the large number of particles and the large number of collisions among them render the time-reversed motion of a macroscopic system extremely improbable.

One may wonder, where does the observed difference between right and left (see § 1.2.1) come from? Is it built into the fundamental equations of physics? Or is it just a chance consequence of the particular development that life took on Earth?

One may translate this question into a thought experiment. Suppose that we were able to build a live being, for instance a dog or a fly, completely made up of organic molecules of the wrong handedness. The question then is, would this artificial being be able to live and function properly? Would it be competitive in a Darwinian sense with the existing forms of life, or would it suffer from some intrinsic disadvantage because of the opposite handedness of its biochemistry?

For a mechanical analogue of this question (Lee 1990), consider two mirrorsymmetric cars. They are of the same model, but each of them is the mirror image of the other one.³ One asks oneself, will these cars run in the same way? If the two cars are accelerated with the gas pedal tilted at the same angle, will they move forward at the same speed? Or might one of them, for instance, stay stuck or even move backwards?

In spite of the observed reality that life is mirror-asymmetric, the equations of classical physics are left–right symmetric. In spite of the obvious arrow of time in real physical events, classical mechanics and classical gravitational and electromagnetic interactions do not have a preferred time direction. The question then is whether left–right symmetry and time-reversal symmetry carry over to the microscopic world. Is there somewhere a fundamental P asymmetry which might explain the observed left–right asymmetry of the biosphere? Or should that asymmetry be assigned to fortuitous initial conditions? And is there, somewhere in the fundamental interactions beyond classical physics’ realm, a T asymmetry? Could such an asymmetry help explain the observed time arrow of events?

Contrary to P and T, charge-conjugation symmetry C does not have an analogue in classical physics. This symmetry is related to the existence of an antiparticle for every particle. This is a prediction of relativistic quantum theory which has been brilliantly confirmed by experiment, in particular through the discovery of the positron (Anderson 1933) and of the antiproton (Chamberlain et al. 1955). It should be emphasized that the notion of antiparticle exists neither in classical physics nor in non-relativistic quantum mechanics.

In relativistic quantum field theory, one can associate both positively and negatively charged particles with each (complex) field $ϕ$⁠. Moreover, there is a C transformation which transforms $ϕ$ into a related field, e.g. $ϕ†$⁠, which has opposite U(1) charges—electric charge, baryon and lepton number, and flavour quantum numbers such as strangeness, the third component of isospin, and so on. The transformed field obeys the same relativistic equation of motion as the original one. It has the same mass, but its interaction with an electromagnetic potential is characterized by opposite electric charge.

C symmetry asserts that antiparticles behave in exactly the same way as the corresponding particles, and that it is a mere matter of convention which of them we call ‘particles’ and which we call ‘antiparticles’.

Why should C symmetry be important? Experimentally it is known that, when a particle and its antiparticle collide, they have a high probability of annihilating. Let us then consider the following thought experiment (Lee 1990).

Suppose that our civilization came into contact with another civilization on a distant planet. The contact might take place via exchange of electromagnetic messages, without any charged particle ever being exchanged. After years of friendly correspondence, the two civilizations might want to physically meet, for instance through the sending of a space vessel. The problem would then be to know whether the other civilization is made out of matter or of antimatter. Indeed, if it were made out of antimatter, physical contact would be impossible, lest annihilation destroys both meeting parties. How could civilizations communicate to each other whether each one’s ‘matter’ is the same as the other one’s, or whether they are made out of the antimatter of each other?

In order to communicate this, one needs some absolute way of distinguishing matter from antimatter. If C symmetry holds, matter and antimatter are distinguishable only by practical example, i.e., they are a convention. C symmetry must be violated, some physical event must occur differently with matter and antimatter, in order that an explanation to our distant partners of how that event happens in our world lets them know what ‘matter’ means to us.

It turns out that the whole body of weak interactions works differently for matter and antimatter. Moreover, weak interactions also are left–right asymmetric. On the other hand, after simultaneous C and P transformations, (most) weak interactions remain identical to themselves—cross sections and decay rates remain unchanged.

The composite transformation CP, made out of simultaneous C and P transformations, then acquires relevance. Namely, the conceptual problem of distinguishing matter from antimatter (see § 1.3.1) can only acquire a solution if we are able to eliminate the convention of what is ‘left’ and what is ‘right’ from the game. It is not enough that C be violated, CP must be violated too in order that matter may be distinguished from antimatter.

Historically, the possibility that weak interactions violate parity was first suggested by Lee and Yang (1956). They examined the experimental evidence then available and concluded that parity invariance of the weak interactions was ‘only an extrapolated hypothesis unsupported by experimental evidence’, i.e,, it had not yet been probed. They went on to suggest experiments which might test whether parity is conserved. They observed that, in order to test parity violation, one should try and measure some pseudoscalar quantity, like for instance the helicity of some particle, or more generally the scalar product of the spin of a particle and the momentum of some other particle. If the expectation value of a pseudoscalar observable is found to be non-zero, then there is parity violation.

Parity violation in the $β$ decay of $60Co$ was discovered soon afterwards (Wu et al. 1957). The nuclide $60Co$ decays through $β−$emission to an excited state of $60Ni$⁠, which then decays to its fundamental state through the emission of two successive gammas. In the experiment of Wu et al., a $60Co$ source was incorporated into a crystal of cerium magnesium. When a small magnetic field $(∼0.05T)$ is applied to the crystal, there is an alignment of the electronic spins, generating inside the crystal a strong magnetic field $(∼10−100T)$⁠. This in turn polarizes the $60Co$ nuclei through the hyperfine coupling, provided the whole system is at a sufficiently low temperature $(∼0.01K)$⁠. The low temperature was achieved through a process of adiabatic demagnetization.

A scintillation counter was used to measure the intensity of the $β−$ emission relative to the orientation of the polarizing field and as a function of the temperature. It was found that the electrons are emitted preferentially in the direction opposite to the one of the applied magnetic field, and that that preference disappears when the crystal warms up. This means that the scalar product of the spin of the $60Co$ nuclei and the momentum of the emitted electron has a non-zero expectation value. As that scalar product changes sign under P, parity violation in $β$ decay was established.

After the experiment of Wu et al. (1957), another experiment of great importance in testing the nature of the weak interactions was the one of Goldhaber et al. (1958). There, the helicity of the neutrino emitted in the electron capture by $152Eu$ was measured. The experiment was particularly ingenious because, as the neutrino hardly interacts with matter, some way of indirectly measuring its helicity had to be devised.

The nuclide $152Eu$ has zero spin. It captures an electron from the K shell—with zero orbital angular momentum—of the atom, giving rise to an excited state with spin 1 of $152Sm$⁠, and emitting a neutrino $νe$⁠. The excited state then decays to the fundamental state of $152Sm$⁠, which is spinless, through emission of a photon.

Conservation of angular momentum along the direction of flight of the neutrino and of the $152Sm∗$ nucleus, in the rest frame of $152Eu$⁠, implies that the handedness⁴ of the neutrino is the same as the handedness of $152Sm∗$—see Fig. 1.3. (This is because the angular momentum of the initial state, constituted by the $152Eu$ nucleus and by the K-shell electron, is just the spin of the electron.) Again, it can easily be shown that, if the photon from the decay of that excited state is emitted in the same direction as the neutrino, that photon has the same handedness as the excited state and, therefore, as the neutrino. Thus, the measurement of the helicity of the neutrino is reduced to a selection of the events in which the gamma is emitted in the same direction as the neutrino, and to a measurement of the helicity of the gamma in those events. It was found that the gammas have helicity –1. Thus, the $νe$ emitted in the electron capture must have helicity –1 , i.e., be left-handed.

The fact that the neutrino emitted in electron capture always has helicity –1 constitutes a violation of parity, as can be seen in Fig. 1.3.

After the original experiment of Goldhaber et al. (1958) on the electron neutrino, many other experiments have attempted to measure the helicity of the neutrinos. In particular, the helicity of the muonic neutrino has been directly measured in a nice experiment by Roesch et al. (1982). It has always been found that neutrinos have helicity –1 , while antineutrinos have helicity +1.

In general, one finds that C is violated together with P in the weak interactions. Let us give an example of this. The charged pion $π+$ decays predominantly to $μ+νμ$⁠. The muon neutrino from the decay is left-handed (Garwin et al. 1957; Friedman and Telegdi 1957), see Fig. 1.4 (a). The P-conjugate process, in which the $νμ$ would be right-handed, never occurs—see Fig. 1.4 (b). The C-conjugate process is the decay of $π−$ to $μ−ν¯μ$⁠, the $ν¯μ$ being left-handed—Fig. 1.4 (c). It turns out that this process never occurs either. Instead, $π−→μ−ν¯μ$ occurs at the same rate as $π+→μ+νμ$⁠, only the $ν¯μ$ is right-handed, as in Fig. 1.4 (d).

This fact represents a simultaneous violation of C and of P. P is violated because the neutrinos have a definite handedness. If P were not violated the neutrinos should, with equal probability, be either left-handed or right-handed, both in $π+→μ+νμ$ and in $π−→μ−ν¯μ$⁠. C is violated because the neutrinos in the two observed decays have opposite handedness, and therefore the decays are not C-conjugates of each other.

On the other hand, the combined symmetry CP is preserved. Indeed, when we simultaneously interchange $π+↔π−,μ+↔μ−$⁠, and $νμ↔ν¯μ$ (C transformation), and also interchange left- and right-handedness of the neutrinos and of the muons (P transformation), the decay rates are equal. C and P are violated, but the combined symmetry CP is not.

Clearly, $π±$ decays cannot provide a solution to the communication problem of § 1.3.1. The distinction of matter from antimatter using these decays requires a previous convention about the handedness of the coordinate system. We are unable to explain to our far-away partners what we mean by ‘a positively charged pion’ because, if we try and tell them that ‘it decays into a neutral particle with negative helicity’ they will ask us how do we define the sign of the helicity, and such a definition is equivalent to a convention for the handedness of the coordinate system.

The fact that CP symmetry is preserved even while C and P symmetries are violated was first pointed out by Landau (1957), who suggested that neutrinos are always left-handed and antineutrinos are always right-handed.⁵ Only much later was CP discovered to be violated too (Christenson et al. 1964).

The clearest evidence for CP violation is the charge asymmetry in $Kl3$ decays. The kaon $KL$ is a neutral particle with well-defined mass and decay width. There is no other particle with equal mass. Therefore, $KL$ must be its own antiparticle. It decays both to $π+e−ν¯e$ and to the C-conjugate mode $π−e+νe$⁠. However, it decays slightly less often to the first than to the second mode. This fact unequivocally establishes both C violation and CP violation.

Indeed, when we consider the total decay rates, we have already performed an integration over the momenta of all the particles resulting from the decay, as well as a sum over their spin states. These integrations and sums eliminate parity from consideration. Then, if the decay rates to $π+e−ν¯e$ and to $π−e+νe$ are different, there is violation of CP, not only of C.

We now have the solution to the thought experiment of § 1.3.1. We should tell our partners on a distant planet to observe $Kl3$ decays: the decay which occurs less often gives rise to a pion with the same electric charge as the proton that we are made of. They would thus learn what we mean by ‘matter’. No reference to right-handed and left-handed coordinate systems would be needed. Indeed, we might afterwards use pion decays to explain to our distant friends what we mean by ‘left’ and ‘right’.

Besides being a fascinating effect because of its elusiveness at both the experimental and theoretical levels, CP violation might also play an important role in our understanding of cosmology. This is because the observed baryon asymmetry of the Universe, i.e., the fact that there is much more matter than antimatter in the observed Universe, could only be generated from an initial situation in which the amounts of matter and antimatter would be equal if there is CP violation. This fact was first shown by Sakharov (1967), who pointed out that baryon-number violation, C and CP violation, and a departure from thermal equilibrium, are all necessary in order for it to be possible to generate a net baryon asymmetry in the early Universe.

Another interesting consequence of CP violation would be the possibility that elementary particles have electric dipole moments. We have seen in § 1.1.3 that electric dipole moments violate both P and T. Composite particles like nuclei or atoms may display a net electric dipole moment because of the existence of degenerate states with different properties under parity and time reversal; on the contrary, any electric dipole moment of an elementary particle would in a downright manner violate both P and T. Now, T violation is connected through the CPT theorem to CP violation—this means that it would be very difficult to conceive of a theory which would violate T without simultaneously violating CP (see § 2.5 below). Thus, although not strictly necessary for the existence of electric dipole moments of elementary particles, theoreticians certainly do not expect them to exist unless CP is violated.