Symmetry Principles and Conservation Laws in Atomic and ...

Symmetry Principles and ConservationLaws inAtomic and Subatomic Physics – Part I

GENERAL ARTICLESymmetry Principles and Conservation Laws inAtomic and Subatomic Physics – 1P C Deshmukh and J Libby(left) P C Deshmukh is aProfessor of Physics at IITMadras. He leadsan active research group inthe field of atomic andmolecular physics andis involved in extensiveworldwide researchcollaborations in boththeoretical and experimentalinvestigations in thisfield. He enjoysteaching both undergraduateand advanced graduatelevel courses.(right) Jim Libby is anAssociate Professor in theDepartment of Physics atIIT, Madras. He is anexperimental particlephysicist specialising in CPviolating phenomena.KeywordsSymmetry, conservation laws,Noether’s theorem.The whole theoretical framework of physics restsonly on a few but profound principles. Wignerenlightened us by elucidating that \It is now naturalfor us to try to derive the laws of natureand to test their validity by means of the laws ofinvariance, rather than to derive the laws of invariancefrom what we believe to be the laws ofnature." Issues pertaining to symmetry, invarianceprinciples and fundamental laws challengethe most gifted minds today. These topics requirea deep and extensive understanding of both`quantum mechanics' and the `theory of relativity'.We attempt in this pedagogical article topresent a heuristic understanding of these fascinatingrelationships based only on rather elementaryconsiderations in classical and quantummechanics. An introduction to some fundamentalconsiderations regarding continuous symmetries,dynamical symmetries (Part 1), and discretesymmetries (Part 2) (parity, charge conjugationand time-reversal), and their applicationsin atomic, nuclear and particle physics, will bepresented.1. IntroductionThe principal inquiry in classical mechanics is to seeka relationship between position, velocity, and acceleration.This relationship is rigorously expressed in whatwe call the `equation of motion'. The equation of motionis not self-evident, but rests on some fundamentalprinciple that must be discovered. A prerequisite forthis discovery is the principle of inertia, discovered by832 RESONANCE September 2010

GENERAL ARTICLEGalileo, contrary to common experience, that the velocityof an object is self-sustaining and remains invariantin the absence of its interaction with an external agency.This principle identi¯es an inertial frame of referencein which physical laws apply. This great discovery byGalileo was soon incorporated in Newton's scheme asthe First law of mechanics, the law of inertia. Newtonrecognised, following his invention of calculus, thatit is the change in velocity that seeks a cause. Newton'scalculus expressed the rate of change of velocityas acceleration which is interpreted as the `e®ect' of thephysical interaction that generated it. Newton's secondlaw expresses this `cause-e®ect' relationship as a linearresponse of the system to the physical interaction it experienced:¡! F = m ¡! a . The mass m of the object is theconstant of proportionality between the e®ect ( ¡! a ) andthe cause ( ¡! F ).1 This article is partly based onthe talk given by PCD at theKarnataka Science and TechnologyAcademy’s special lecturesat the Bangalore Universityon 23rd March, 2009.In the following section we will begin by considering howNewton's third law introduces a simple illustration ofthe relation between a symmetry and a conservation law.In the remainder of the article we will explore similar relationshipsthat impact much of the frontiers of physics,which are being investigated today; these studies usepowerful theoretical frameworks and sophisticated technology.2. Translational Invariance and Conservation ofMomentumWe consider a closed system of N point particles in homogeneousisotropic space. The force on the kth particleis the sum of forces on it due to all the other particles¡! F k =NX ¡! f kj : (1)j=1j6=kWe now consider `virtual' translational displacement ofthe entire N-particle system in the homogeneous space.Newton’s third lawintroduces asimple illustrationof the relationbetween asymmetry and aconservation law.RESONANCE September 2010833

GENERAL ARTICLEIn such a process, the internal forces can do no work,since the process amounts to merely displacing the entiresystem to an adjacent region, displaced from theoriginal by an amount ¡! ±s. As this displacement is beingconsidered in a homogeneous medium, it is referred to asbeing `virtual' as no work is done by the internal forces.This phenomenon is then expressed by the relationNX ¡! F k ¢ ¡! ±s =k=1NXk=1d ¡! P kdt¢ ¡! ±s = 0 ; (2)where ¡! P k is the momentum of the kth particle. In expressingthis quantitative result, we have made use ofNewton's ¯rst two laws (the ¯rst law implicitly and thesecond law explicitly) and also the notion of translationalinvariance in homogeneous space. Now, for an arbitrarydisplacement ¡! ±s, this relationship requires thatConservation of linearmomentum isgoverned by thesymmetry principle oftranslationalinvariance inhomogeneous space.Likewise, one can seethat the conservationof angular momentumemerges fromrotationaldisplacements inisotropic space.NXk=1d ¡! P kdt= 0 : (3)If we write this result for a two-body closed system, wediscover Newton's third law, that action and reactionare equal and opposite:d ¡! P 1dt= ¡ d¡! P 2dt: (4)In other words, we discover that conservation of linearmomentum is governed by the symmetry principle oftranslational invariance in homogeneous space. Likewise,one can see that the conservation of angular momentumemerges from rotational displacements in isotropicspace.It is interesting to observe that Newton actually inventedcalculus to explain departure from equilibrium ofan object which manifests as its acceleration, and proposeda linear relationship between the physical interaction(force) which he interpreted as the `cause' of the834 RESONANCE September 2010

GENERAL ARTICLEacceleration. Newton's second law contains the heartof this stimulus{response relation, expressed as a di®erentialequation. It is interesting that laws of classicalmechanics can be built alternatively on the basis of an`integral principle', namely the `principle of variation',discussed in the next section.3. Principle of VariationThe connection between symmetry and conservation lawsbecomes even more transparent in the alternative formalismof classical mechanics, namely the Lagrangian/Hamiltonian formulation. It is instructive to ¯rst seethat this alternative formalism is based not on the linearresponse relationship embodied in the Newtonian principleof causality, but in a completely di®erent approach,namely the `principle of variation'.It is interesting thatlaws of classicalmechanics can bebuilt alternativelyon the basis of an‘integral principle’,namely the‘principle ofvariation’,Newtonian mechanics o®ers an accurate description ofclassical motion by accounting for the same by the `causeand e®ect' relationship. An alternative and equivalentdescription makes it redundant to invoke such a causaldescription. This alternative description dispenses theNewtonian notion of the 'cause-e®ect' relationship, andinstead of it invokes a variational principle, namely, thatthe `action integral' is an extremum. Those who are usedto thinking in terms of the Newtonian formulation alonewould ¯nd it strange that one gets equivalent descriptionof classical mechanics without invoking the notion offorce at all!Let us ¯rst state the principle of extremum action. Webegin on common ground with the Newtonian formulation,namely that the position q and velocity : q specifythe mechanical state of a system. A well-de¯ned functionof q and : q would also then specify the mechanicalstate of the system. What is known as the Lagrangian ofa system L(q; : q) is just that; it is named after its originatorLagrange (1736{1813). Furthermore, in a homoge-RESONANCE September 2010835

GENERAL ARTICLESimply stated, theprinciple of leastaction is that themechanicalstate of a systemevolves along aworld-line such thatthe ‘action’,Zt 2S = L(q; q; : t)dt ;t 1is an extremum.2A ‘world-line’ is a trajectory inthe phase space, or the mathematicalspace, along which themechanical state of a systemevolves over a period of time.neous isotropic system, L(q; : q) can depend only quadraticallyon the velocity, so that it could be independent ofits direction. The simplest form the Lagrangian wouldthen have is L(q; : q) = f 1 (q) + f 2 ( : q 2 ), wherein the functionsf 1 and f 2 must be suitably chosen. It turns outthat the choice f 1 (q) = ¡V (q), i.e., the negative of theparticle's potential energy, and f 2 ( : q 2 ) = (m=2) : q 2 , i.e.,the kinetic energy T of the particle, renders this new formalismcompletely equivalent to Newtonian mechanics.This relationship o®ers us with a simple interpretationof the Lagrangian as L(q; : q) = T ¡ V .Simply stated, the principle of least action is that themechanical state of a system evolves along a world-line 2such that the `action',S =Zt 2t 1L(q; : q; t)dt ; (5)is an extremum. This principle was formulated by Hamilton(1805{1865). It has an interesting development beginningwith Fermat's principle about how light travelsbetween two points, and subsequent contributions byMaupertius (1698{1759), Euler (1707{1783), and Lagrangehimself. The principle that `action' is an extremumis equivalent to stating that the mechanical systemevolves over the period t 1 to t 2 along a world-linetraced by the points (q; q) : such that if the `action integral'S is evaluated along any other alternative pathdisplaced in¯nitesimally from the one it actually evolvesover, (q + ±q; q : + ± q), : then:±S =Zt 2t 1L(q + ±q; : q + ± : q; t)dt ¡Zt 2t 1L(q; : q; t)dt = 0: (6)The above equation is a mathematical expression of thestatement of the `principle of extremum action'. Thenecessary and su±cient condition that this principle836 RESONANCE September 2010

GENERAL ARTICLEmust hold good provides us the well-known Lagrange'sequation of motion:@L@q ¡ d @Ldt @ q : = 0: (7)The quantity (@L)=(@ q) : in the above equation is knownas the generalised momentum (written as p) conjugate tothe generalised coordinate q. The power of Lagrangianmechanics comes from the fact that there are very manypairs of variables (q; p) which can be considered conjugateto each other in the Lagrangian sense { q and p neednot have the physical dimensions of [L] and [MLT ¡1 ] respectively.The dimension of the product of q and p,however, must always be ML 2 T ¡1 , that of the angularmomentum. From Lagrange's equation, it follows immediatelythat if the Lagrangian is independent of q,(i.e., if (@L)=(@q) = 0) then the generalized momentump = (@L)=(@ q) : conjugate to this coordinate is constant.The independence of the Lagrangian with respect to q isan expression of `symmetry', since the Lagrangian wouldthen be the same no matter what the value of q is. Thisresults in a conservation principle since the generalisedmomentum conjugate to this q becomes independent oftime, remains constant. One may pair (time, energy)as (q; p), and see from this that (@L=(@t) = 0 wouldresult in energy being constant. This result immediatelyfollows from the following expression for the timederivativeof the Lagrangian:0 = dLdt= @L :q + @L ::@q @ q: q + @L@t½ ¾ d @L :=dt @ q: q + @L ::@ q: q + @L@t ; (8)where Lagrange's equation is used to re-express the ¯rstterm.The independence ofthe Lagrangian withrespect to q is anexpression of‘symmetry’, since theLagrangian wouldthen be the same nomatter what the valueof q is. This resultsin a conservationprinciple since thegeneralisedmomentumconjugate to this qbecomesindependent of time,remains constant.It thus follows that:ddt¸@L :@ q: q ¡ L = ¡ @L@t : (9)RESONANCE September 2010837

GENERAL ARTICLEConservation ofenergy followsfrom the symmetryprinciple that theLagrangian isinvariant withrespect to time.From the above, it immediately follows that when theLagrangian depends on time only implicitly through itsdependence on q and q, : then:¸d @L :q ¡ L = 0 ; (10)h³which implies@L@ : qdt@ q:´ i :q ¡ L is a conserved quantity. Thisquantity is called the Hamiltonian, or Hamilton's principalfunction, of the system, which for a conservativesystem is essentially the same as the total energy of thesystem. This can be seen easily by identifying the generalizedmomentum and substituting T ¡ V for the Lagrangian.We thus see that conservation of energy followsfrom the symmetry principle that the Lagrangianis invariant with respect to time.These results illustrate an extremely powerful theoremin physics, known as the Noether's theorem, which canbe stated informally as:If a system has a continuous symmetry property, thenthere are corresponding quantities whose values are conservedin time [1].This theorem is named after Noether (1882{1935), ofwhom Einstein said:If a system has acontinuoussymmetryproperty,then there arecorrespondingquantities whosevalues areconserved in time.In the judgement of the most competent living mathematicians,FrÄaulein Noether was the most signi¯cantcreative mathematical genius thus far produced since thehigher education of women began [2].4. Symmetry Principles and Physical LawsWe have now seen that both the equation of motionand the conservation principles result from the singleprinciple of least action. Moreover, the same principleprovides for the connection between symmetry and conservationlaws, exalted by Noether to one of the mostprofound principles in contemporary physics. We now838 RESONANCE September 2010

GENERAL ARTICLEask if the conservation principles are consequences of thelaws of Nature, or, rather the laws of Nature are consequencesof the symmetry principles that govern them?Until Einstein's special theory of relativity, it was believedthat conservation principles are the result of thelaws of Nature. Since Einstein's work, however, physicistsbegan to analyze the conservation principles asconsequences of certain underlying symmetry considerationsfrom which they could be deduced, enablingthe laws of Nature to be revealed from this analysis.Wigner's profound impact on physics is that his explanationsof symmetry considerations using `group theory'resulted in a change in the very perception of just whatis most fundamental, and physicists began to regard`symmetry' as the most fundamental entity whose formwould govern the physical laws. Wigner was awardedthe 1963 Nobel Prize in Physics for these insights [3].The conservation of linear and angular momentum weillustrated above are consequences of invariance undercontinuous displacements and rotations respectively inhomogenous and isotropic space. Likewise, the conservationof energy is a consequence of invariance undercontinuous temporal displacement.A detailed exposition of the governing symmetry principlesrequires group theoretical methods, and is clearlybeyond the scope of this article, but we continue to dwellon some other kinds of symmetries now and examinetheir connections with conservation principles.Figure 1. Masters of symmetry.5. Dynamical Symmetry: Laplace{Runge{LenzVectorIt is well known that in the classical two-body Keplerproblem (gravitational Sun{Earth system, or the Coulombicproton{electron planetary model of the old-quantumtheoryof the hydrogen atom), both energy and angularmomentum are conserved. We have already discussedRefer to Resonance issues on:Einstein, Vol.5, March and April2000.Noether, Vol.3,September 1998.Wigner, Vol.14, October 2009.RESONANCE September 2010839

GENERAL ARTICLEthe associated symmetries. What is interesting is thatthe elliptic orbit of the Kepler system for bound statesis ¯xed, i.e., the ellipse does not precess (Figure 2).Figure 2. If the eclipse wereto precess it would generatewhat is called a ‘rosette’motion since the trajectoryof the planet would seem togo over the petals of a rose,if seen from a distance.Can we then ¯nd a symmetry that would explain theconstancy of the orbit? It turns out that the orbit itselfremains ¯xed if and only if the potential in which motionoccurs is strictly of the form ¡1=r and the associatedforce is of the form ¡1=r 2 . This is true for both thegravitational and the Coulomb potential, and hence theKepler elliptic orbits remain ¯xed. This is rigorouslyexpressed as the constancy of the Laplace{Runge{Lenz(LRL) vector. The LRL vector is de¯ned as:³~A = ~v £ H ~ ´¡ ^e ½ (11)and is shown in Figure 3 [4]. In the above equation ~vis the `speci¯c' linear momentum and H ~ is the `speci¯c'angular momentum. The term `speci¯c' denotes the factthat the physical quantities linear momentum and angularmomentum, which are being referred to, are de¯nedper unit mass. Likewise in the second term of the LRLvector, is the proportionality in the inverse distancegravitational potential per unit mass of the planet. Itcan be easily veri¯ed that the time derivative of the LRLvector vanishes, and the A ~ is therefore a conserved quantity.Its direction is from the focus of the ellipse to theperihelion (Figure 3) [4], which has a direction along themajor axis of the ellipse, thus holding the ellipse ¯xed.Figure 3. Schematic diagramshowingtheLaplace–Runge–Lenz vector, ~A .840 RESONANCE September 2010

GENERAL ARTICLEThe constancy of the LRL vector is a conservation principle,and since the governing criterion involves dynamics(namely that the force must have a strict inversesquare form), the associated symmetry is called `dynamicalsymmetry'. Sometimes, it is also called an `accidental'symmetry. This symmetry breaks down when thereis even a minor departure from the inverse square lawforce, as would happen in a many-electron atom, suchas the hydrogen-like sodium atom. The potential experiencedby the `outer-most' electron goes as 1=r only inthe asymptotic (r ! 1) region. Close to the center,the potential goes rather as ¡Z=r, due to the reducedscreening of the nuclear charge by the orbital electrons,and thus departs from 1=r. This di®erence in the hydrogenatom potential and that in the sodium atom is dueto the quantum analogue of the breakdown of the LRLvector constancy in the sodium atom. Using group theoreticalmethods, Vladmir Fock (1898{1974) explainedthe dynamical symmetry of the hydrogen atom [5].Using the language of group theory, the Fock symmetryaccounts for the (2l + 1)-fold degeneracy of the hydrogenatom eigenstates. This degeneracy is lifted for thehydrogen-like sodium atom due to the breakdown of theassociated symmetry. In atomic physics, this is oftenexpressed in terms of what is called as `quantum defect'¹ n;l which makes the hydrogenic energy eigenvalues dependnot merely on the principal quantum number n butalso on the orbital angular momentum quantum numberl. This enables the use of the hydrogenic formulafor energy with n replaced by n e®ective = n ¡ ¹ n;l . The`quantum-defect theory' has very many applications inthe analysis of the atomic spectrum, including the `autoionizationresonances' [6,7]. As pointed out above,the conservation of angular momentum is due to the rotationalsymmetry, referred to as the symmetry underthe group SO(3). All central ¯elds have this symmetry.However, the inverse-square-law force (such as gravity orRESONANCE September 2010841

GENERAL ARTICLEThe conservationof the generalizedmomentum whichis conjugate to acyclic coordinate isa genericexpression of adeeper relationshipbetween symmetryand conservationlaws.Coulomb) has symmetry under a bigger group, SO(4)or SO(3; 1), where SO(4) is the rotational group in 4dimensions, and SO(3; 1) is the Lorentz group. The dimensionalityof the SO(N) group is N(N¡1)/2, so theSO(4) group is 6-dimensional and corresponds to the 6conserved quantities, namely the 3 components of theangular momentum vector and the three componentsof Pauli{Runge{Lenz vector which is the quantum analogueof the LRL vector [8].6. ConclusionThe conservation of the generalized momentum which isconjugate to a cyclic coordinate is a generic expression ofa deeper relationship between symmetry and conservationlaws. In the next part of this article we shall discussdiscrete symmetries, the CPT symmetry and commenton spontaneous symmetry breaking and the search forthe Higgs boson.Suggested ReadingAddress for CorrespondenceP C Deshmukh and J LibbyDepartment of PhysicsIndian Institute of TechnologyMadrasChennai 600036.Email: pcd@physics.iitm.ac.inlibby@physics.iitm.ac.in[1] W J Thompson, Angular Momentum, Wiley, p.5, 2004.[2] From a letter to the New York Times on May 5th, 1935 from AlbertEinstein shortly after Emmy Noether’s death.[3] Details of the 1963 Nobel Prize in physics can be found at http://nobelprize.org/nobel\_prizes/physics/laureates/1963/index.html[4] For a detailed discussion of the Laplace–Runge–Lenz vector see H Goldstein,Classic Mechanics, Second Edition, Addison-Wesley, p102ff,1980.[5] W Fock, Z. Phys., Vol.98, p.145, 1935.[6] M J Seaton, Rep. Prog. Phys., Vol.46, p.167, 1983.[7] S B Whitfield, R Wehlitz, H R Varma, T Banerjee, P C Deshmukh andS T Manson, J. Phys. B: At. Mol. Opt. Phys., Vol.39, p.L335, 2006.[8] V Bargmann, Z. Physik Vol.99, pp.576–582, 1936.842 RESONANCE September 2010

Symmetry Principles and ConservationLaws inAtomic and Subatomic Physics – Part II

GENERAL ARTICLESymmetry Principles and Conservation Laws inAtomic and Subatomic Physics – 2P C Deshmukh and J Libby(left) P C Deshmukh is aProfessor of Physics at IITMadras. He leadsan active research group inthe field of atomic andmolecular physics andis involved in extensiveworldwide researchcollaborations in boththeoretical and experimentalinvestigations in thisfield. He enjoysteaching both undergraduateand advanced graduatelevel courses.(right) Jim Libby is anAssociate Professor in theDepartment of Physics atIIT, Madras. He is anexperimental particlephysicist specialising in CPviolating phenomena.Part 1: Resonance, Vol.15, No.9,p.832.KeywordsDiscrete symmetries, violationof parity and CP, Higgs mechanism,LHC.This article is the second part of our review of theimportant role that symmetry plays in atomicand subatomic physics. We will concentrate onthe discrete symmetries { parity, charge conjugation,and time reversal { that have played a signi¯cantpart in the development of the `standardmodel' of particle physics during the latter partof the 20th century. The importance of experimentaltests of these symmetries, in both atomicand particle physics, and their sensitivity to newphenomena is also discussed. To conclude, wedescribe how `symmetry breaking' in the standardmodel leads to the generation of mass viathe Higgs mechanism and how the search forevidence of this symmetry violation is one ofthe principal goals of the Large Hadron Collider,which began operating at CERN, Switzerland in2009.1. Discrete SymmetriesApart from continuous and dynamical symmetries, thereare other kinds of symmetries that are of importance inphysics. In particular, we have three discrete symmetriesof central importance in what is known as the `standardmodel' of particle physics. These discrete symmetriesare: (i) P (Parity), (ii) C (Charge conjugation, i.e.,matter/antimatter) and (iii) T (Time-reversal), oftenknown together as PCT symmetry. In physical reactionsof particle physics, these symmetries lead to conservationprinciples operating either separately or in combination.We shall now discuss these discrete symmetries.926 RESONANCE October 2010

GENERAL ARTICLE1.1 ParityParity is the symmetry we see between an object andits mirror image. It is interesting that in a mirror, weusually see the left go to right, and the right go to left,but we do not see top go to bottom and the bottomto the top. This feature typi¯es the di®erence betweenre°ection and rotation. If we represent the transformationof a vector ~r to its image in a mirror placed in theCartesian yz-plane, then we can express the transformation~r = (x; y; z) to its image ~r I = (x I ; y I ; z I ) by a matrixequation:~r I =

GENERAL ARTICLEThe physicalphenomena for whichparity is violatedresult from aninteraction known asthe weak interaction;its most widelyknownmanifestationis nuclear decay.from rotation and one may ask, as Alice would (in Throughthe Looking Glass), if the physical laws are thesame in the world of images in a mirror. In other words,this question amounts to asking, given the fact thatthere is a certain degree of invariance when one comparesan object with its image in a mirror, whether parityis conserved in nature.The parity operator ¦ is a unitary operator which anticommuteswith the position operator and also with theoperator for linear momentum, since both position andmomentum are polar vectors. However the parity operatorcommutes with the operator for angular momentumwhich is a pseudovector.While most of the everyday physical phenomena couldtake place just as well in essentially the same mannerin the image world as in the real world, certain physicalphenomena occur di®erently. The physical phenomenafor which parity is violated result from an interactionknown as the weak interaction; its most widely-knownmanifestation is nuclear ¯ decay. The search for parityviolation in weak interactions was advocated strongly byLee and Yang [1], after a careful review of the subject indicatedthat parity conservation, though often assumed,had not been veri¯ed in weak interactions. Acting onthe proposals of Lee and Yang, Wu and collaboratorsclearly observed parity violation in the ¯ decay of polarisednuclei via asymmetries in the distribution of the¯-decay electron with respect to the spin of the nucleus(Figure 2).The violation ofparity wasunexpected. Itallowed the firstunambiguousdefinition of leftand right in nature.These and subsequent measurements showed that theweak interaction was maximally parity violation, whichmeant that it only couples to left-handed chiral statesof matter and right-handed chiral states of antimatter;i.e., for a massless fermion this would correspond to thestate where the spin is in the opposite direction to itsmomentum.928 RESONANCE October 2010

GENERAL ARTICLEa) b)Parity violation is observed in nuclear and subatomic interactions,and through the uni¯cation of the weak andelectromagnetic interactions, parity is violated in certainatomic processes as well. Atomic transitions are normallygoverned by the parity selection rule, which thenbreaks down for those transitions in which parity is notconserved. The electroweak uni¯cation achieved in theGlashow{Weinberg{Salam model triggered the search inthe 1970s for parity nonconservation (PNC) in atomicprocesses [2].The gauge bosons W § have a charge of +1 and ¡1 unit,but the Z 0 boson of the standard model is neutral. Thelatter can mediate an interaction between atomic electronsand the nucleus. The nuclear weak charge QWof the standard model plays the same role with regardto Z 0 that the `usual' electric charge plays with regardto the Coulomb interaction. PNC e®ect in atomic cesiumyields the value of QW ( 133 Cs) ¼ ¡72:90, not farfrom the value of QW ( 133 Cs) ¼ ¡73:09 obtained fromhigh-energy experiments extrapolated to atomic scale[3]. The Z-boson has a very large mass and the weakinteractionis `contact' type. It includes a parity-evenpart and a parity-odd (PNC) part. While the parityevenpart leads to a correction to isotope shift and tohyper¯ne structure, the PNC part leads to the `pseudoscalar'correlations in atomic processes.Figure 2. Schematic (a) isof the direction of the decayelectron, characterizedby momentum p e,with respectto the spin of the 60 Conucleus, J . Schematic60 Co(b) is the same processtransformed by the parityoperation. Unequal probabilitiesfor these two processesto occur were observedby Wu and collaborators;this was the first experimentalevidence forparity violation in nature.Atomic transitionsare normallygoverned by theparity selectionrule, which thenbreaks down forthose transitions inwhich parity is notconserved.RESONANCE October 2010929

GENERAL ARTICLEThe anapole momentis a newelectromagneticmoment that can bepossessed by anelementary particle (aswell as compositesystems like thenucleon or nucleus)and this wouldcorrespond to a PNCcoupling to a virtualphoton.A significantly largevalue of the anapolemoment of thenucleon has beenestimated in the caseof cesium, augmentedby collective nucleareffects.The usual radiative transitions in atomic processes aregoverned by parity-conserving selection rules imposedby the electromagnetic Hamiltonian. However, once theHamiltonian is modi¯ed to include the electroweak interaction,it does not commute with the parity operatorand provides for non-zero probability for parityviolatingatomic transitions. The two sources of paritynonconservation (PNC) in atoms are: (1) the electronnucleusweak interaction and (2) the interaction (sometimescalled as PNC hyper¯ne interaction) of electronswith the nuclear anapole moment. The anapole momentwas predicted by Vaks and Zeldovich [4] soon after Leeand Yang's proposal that weak interactions violate parity.The anapole moment is a new electromagnetic momentthat can be possessed by an elementary particle(as well as composite systems like the nucleon or nucleus)and this would correspond to a PNC coupling toa virtual photon. The anapole moment can be seen toresult from a careful consideration of the magnetic vectorpotential at a ¯eld point after taking into accountthe constraints of current conservation and the boundednessof the current density.A signi¯cantly large value of the anapole moment of thenucleon has been estimated in the case of cesium, augmentedby collective nuclear e®ects. Recently, Dunfordand Holt [5] recommended parity experiments on atomichydrogen and deuterium using UV radiation from freeelectron laser (FEL) to probe new physics beyond thestandard model. The Dunford{Holt proposal is basedon the consideration that if an isolated hydrogen atom2p 12existed in an excited state that is a mix of states 2s 1 and2which have opposite parity, then parity would be violatedif the electromagnetic interactions alone were toexist. These two energy states are very nearly degenerateand thus very sensitive to the electroweak interactionwhich would mix them. More recently, atomic parity violationhas been observed in the 6s 2 1 S 0 ! 5d6s 3 D 1930 RESONANCE October 2010

GENERAL ARTICLE408 nm forbidden transition of ytterbium [6]. In thiswork, the transition that violates parity was found tobe two orders of magnitude stronger than that found inatomic cesium. Atomic physics experiments provide alow-energy test of the standard model and also providerelatively low-cost tools to explore physics beyond it.1.2 Charge Conjugation and CP SymmetriesThe discrete symmetry of charge conjugation (C) convertsall particles into their corresponding antiparticles.For example, C operation transforms an electron into apositron. The chirality of the state is preserved undercharge conjugation. For example, a left-handed neutrinobecomes a left-handed antineutrino; the latter does notinteract weakly and shows that C, as well as P, are maximallyviolated in weak interactions. However, the combinedoperation CP, on a process mediated by the weakinteraction was anticipated to be invariant because, forexample, a left-handed neutrino is transformed into aright-handed antineutrino. However, violation of CP isessential to describe the observed state of the universe asbeing matter dominated. Only di®erences in behaviourbetween matter and antimatter, in other words CP violation,can produce such an asymmetry. The presence ofCP-violation is one of the three conditions for producingbaryons (baryogenesis) in the early universe put forwardby the Soviet physicist and dissident Sakharov (1921{1989). He had been inspired to propose CP-violationas an essential ingredient of baryogenesis by the experimentsof Cronin, Fitch and collaborators in 1964 thathad clearly shown that CP-violation occurs in the weakdecays of hadrons containing a strange quark [7].The origin of CP-violation in weak hadronic decays tooksome time to describe. It required the bold hypothesisof Kobayashi and Maskawa in 1973 that there was athird generation of quarks to complement the alreadydiscovered up (u), down (d), and strange (s) quarks,Atomic physicsexperimentsprovide a lowenergytest of thestandard modeland also providerelatively low-costtools to explorephysics beyond it.Violation of CP isessential to describethe observed state ofthe universe as beingmatter dominated.RESONANCE October 2010931

GENERAL ARTICLEIt was Kobayashiand Maskawa’sgreat insight that a3 3 matrixallowed a complexphase to beintroduced, whichcan describe CPviolationin weakhadronic decays.and that time, postulated charm (c) quark. The additionof a third generation of bottom (b) and top (t)quarks leads to a 3£3 matrix being required to describethe weak couplings between the di®erent quarks, whichallow for the change of quark type unlike the strongor electromagnetic interactions. It was Kobayashi andMaskawa's great insight that a 3 £ 3 matrix allowed acomplex phase to be introduced, which can describe CPviolationin weak hadronic decays. The postulated thirdgeneration was not discovered until Lederman and collaboratorsobserved evidence of the b quark in 1977.The CP-violating parameters of Kobayashi and Maskawamatrix have now been measured accurately principallyin experiments at the Stanford Linear Accelerator Center,US, the High Energy Accelerator Research Organisation(KEK), Japan, and the Fermilab National AcceleratorLaboratory, US [8]. This con¯rmation of thethree generation model to describe CP-violation led tothe award of the Nobel Prize for Physics to Kobayashiand Maskawa in 2008 [9].This confirmationof the threegeneration modelto describe CPviolationled to theaward of the NobelPrize for Physicsto Kobayashi andMaskawa in 2008.Despite the success of this model of CP-violation in thestandard model of particle physics, the rate at which itis observed in weak hadronic decays is insu±cient to describethe large matter-antimatter asymmetry observedin universe. Therefore, theories that go beyond thestandard model must accommodate new sources of CPviolationto explain the rate of baryogenesis. This meansthat the further study of CP-violation is extremely important.Therefore, °avour experiments are planned atthe Large Hadron Collider (see Section 2) and elsewhere.CP-violation may also occur in the lepton sector nowthat the non-zero mass of the neutrino has been established[10]; however, an exposition of this exciting topicis beyond the scope of this article.932 RESONANCE October 2010

GENERAL ARTICLE1.3 CPT SymmetryThe `Time Reversal Symmetry' (T) is another discretesymmetry. This has a characteristically di®erent formin quantum mechanics that has no classical analogue.The name time-reversal is perhaps inappropriate, becauseit would make a layman suspect that it is merelythe inverse of the `time evolution', which is not thecase. In quantum theory, the operator for `time evolution'is unitary, but that for time-reversal is antiunitary.The quantum mechanical operator ¦ for parity anticommuteswith the position and the momentum operator,but commutes with the operator for angular momentum.On the other hand, the operator for time-reversal,£ commutes with the position operator, but anticommuteswith both the linear and the angular momentumoperators.In quantum theory,the operator for‘time evolution’ isunitary, but that fortime-reversal isantiunitary.An important consequence of these properties is thefact that the response of a wavefunction to time-reversalwould include not merely t going to ¡t in the argumentof the wavefunction, but also simultaneous complex conjugationof the wavefunction. This property connectsthe quantum mechanical solutions of an electron{ioncollision problem with those of electron{atom scatteringthrough time-reversal symmetry. The physical contentof this connection is depicted in Figure 3 which representsthe fact that in a photoionization experiment it isthe escape channel for the photoelectron which is uniquewhereas in an electron{ion scattering experiment it isthe entrance channel of the projectile electron which is(a)(b)Figure 3. Schematic diagramshowing the time-reversalrelation betweenphotoionization and scatteringprocesses in atomicphysics.RESONANCE October 2010933

GENERAL ARTICLEThe Lorentz symmetryof the standard modelof physics conservesPCT. Violation of Tsymmetry wouldrequire an elementaryparticle, atom ormolecule to possess apermanent electricdipole moment(EDM).The standard modelof particle physicspredicts that thesedipole moments wouldbe too small to beobservable. EDMmeasurementstherefore provide anexciting probeto explore newphysics beyond thestandard model.unique. Despite the fact that the ingredients of theelectron{ion collision experiment and that of photoionizationare completely di®erent, both the processes resultin the same ¯nal state consisting of an electronand an ion. The initial state, being obviously di®erent,implies that the quantum mechanical solutions ofelectron{ion scattering and photoionization are relatedto each other via the time-reversal symmetry [11]. Theboundary condition for electron{ion collision and foratomic photoionization are therefore appropriately referredto as `outgoing wave boundary condition' and `ingoingwave boundary condition'. The employment of thesolutions corresponding to the ingoing wave boundaryconditions in atomic photoionization gives appropriateexpressions for not just the photoionization transitionintensities, but also for the angular distribution and thespin polarization parameters of the photoelectrons.The Lorentz symmetry of the standard model of physicsconserves PCT. The discovery of CP violation in the decayof K mesons [7] therefore made it pertinent to lookfor the violation of the time-reversal symmetry. Violationof T symmetry would require an elementary particle,atom or molecule to possess a permanent electricdipole moment (EDM), since the only direction withwhich an electric dipole moment d ~ =j d j ^e s could bede¯ned will have to be along the unit vector ^e s , thedirection of the particle's spin. Crudely, this can beschematically shown in Figure 4 which shows an angulardirection to represent a rotation, and a charge distributionto depict a dipole moment. As t goes to ¡t, thespin reverses, but not the electric dipole moment.We thus expect from the above equations that the electricdipole moment (EDM) of an elementary particlemust be zero, unless both P and T are violated. Thestandard model of particle physics predicts that thesedipole moments would be too small to be observable.EDM measurements therefore provide an exciting probe934 RESONANCE October 2010

GENERAL ARTICLEa) b)Figure 4. Schematic diagramexplaining that the dipolemoment of an elementaryparticle must be zerounless T symmetry is broken.The existence of anEDM also requires that Psymmetry is violated.to explore new physics beyond the standard model. Highprecisionmeasurements in agreement with predictions ofa robust theoretical formulation would therefore providea valuable test of the standard model, since limits onEDMs would put conditions on supersymmetric gaugetheories [12,13].2. Spontaneous Symmetry Breaking and theSearch for the Higgs BosonHere we will discuss how symmetry plays an importantpart in attempts to address another outstanding issuein the standard model of particle physics: How doesan elementary particle, such as an electron, attain itsmass? The standard model answers this question byassuming that there exists a scalar (spin-less) particlethat was predicted in 1964 by Higgs, which is believedto impart a mass to other particles that interact with it.The particle predicted by Higgs is called a Higgs boson,so named after Higgs and Bose (1894{1974).The standard model of particle physics is a relativisticquantum ¯eld theory, which can be expressed in termsof a Lagrangian. The Lagrangian that describes the interactionsof a scalar ¯eld Á is:L = 1 2 (@ ¹Á) 2 ¡ 1 2 ¹2 Á 2 ¡ 1 4¸Á4 ; (2)RESONANCE October 2010935

GENERAL ARTICLEFigure 5. Potential V for aone-dimensional scalarfield for two cases , 0, as defined in the text.where @ ¹ is the covariant derivative and ¹ is the particlemass and ¸ is the strength of the coupling of Á to itself.The ¯rst term on the right-hand side is consideredthe kinetic energy whereas the other two terms are thepotential.Figure 5 shows the potential as a function of the scalar¯eld Á for two cases: ¹ 2 > 0 and ¹ 2 < 0. For the caseof an imaginary mass (¹ 2 < 0) there are two minima atr¡¹2Á min = §À = §¸ : (3)In considering weak interactions we are interested insmall perturbations about the minimum energy so weexpand the ¯eld about one of the minima, À or ¡ÀÁ = À + ¾(x) ; (4)The breaking ofsymmetryprovidesa hypothesis forthe generation ofall particle masses– the Higgsmechanism.where ¾(x) is the variable value of the ¯eld above theconstant uniform value of À. Substituting this expressionfor Á into (2) one gets:L = 1 2 (@ ¹¾) 2 ¡ ¸À 2 ¾ 2 ¡µ¸À¾ 3 + 1 4¸¾4 + constant ;(5)where the constant term depends on À 2 and À 4 and thethird term (in parenthesis) on the right-hand side describesself interactions. The second term correspondsto a mass term with real massm = p 2¸À 2 = p ¡2¹ 2 : (6)936 RESONANCE October 2010

GENERAL ARTICLEThe perturbative expansion about one of the two minimahas led to a real mass appearing. Since the expansionis made about one or other of the minima, chosenat random, the symmetry of Figure 5 is broken. This isthe process of spontaneous symmetry breaking.Nambu and Jona-Lasinio ¯rst applied spontaneous symmetrybreaking as mechanism of mass generation in 1961.In recognition of this work Nambu was awarded a shareof the 2008 Nobel Prize [9]. There are many examplesof spontaneous symmetry breaking in other areas ofphysics. For example a bar magnet heated above theCurie temperature has its elementary magnetic domainsorientated randomly, leading to zero net ¯eld. The Lagrangiandescribing the ¯eld of the magnet would beinvariant under rotations. However, on cooling, the domainsset in a particular direction, causing an overall¯eld and breaking the rotational symmetry. Thereare further examples of spontaneous symmetry breakingin the description of superconductivity; these inspiredNambu and Jona-Lasinio's work in particle physics.The introduction of such a scalar ¯eld interaction anda spontaneous symmetry breaking within the standardmodel allows the weak force carrying bosons, W § andZ 0 , to obtain mass as well as all quarks and leptons. Inaddition, this leads to the physical Higgs boson. TheHiggs boson is the only part of the standard model ofparticle physics that has not been experimentally veri-¯ed. However, the precise measurements of the propertiesof the Z 0 and the W § by experiments at theLarge Electron Positron (LEP) collider, which operatedat the European Centre for High Energy Particle Physics(CERN) in Geneva, Switzerland, and of the W § and theheaviest quark (the top) at Fermilab, have led to an upperlimit on the mass of the Higgs boson of 157 GeV=c 2with a 95% con¯dence level. In addition, unsuccessfulsearches for the production of a standard model Higgsboson at LEP placed a lower limit on the mass of theNambu and Jona-Lasinio first appliedspontaneoussymmetry breakingas mechanism ofmass generation in1961. In recognitionof this work Nambuwas awarded a shareof the 2008 NobelPrize.The Higgs boson isthe only part of thestandard model ofparticle physics thathas not beenexperimentallyverified.RESONANCE October 2010937

GENERAL ARTICLEFigure 6. Computer-generatedimage shows the locationof the 27-km LHC tunnel(in blue) on the Swiss–France border. The fourmain experiments (ALICE,ATLAS,CMS,andLHCb) arelocated in undergroundcaverns connected to thesurface by 50 m to 150 mpits. Part of the pre-accelerationchain is shown ingrey.The centre-of-masscollision energy is14~TeV which is eighttimes greater than theprevious highestenergy collider. Suchenergies have notbeen produced sinceapproximately 10 –25 safter the big bang.Higgs boson of 114 GeV=c 2 with a 95% con¯dence level.The search for the Higgs boson is one of the principalgoals of the largest and the biggest experiment donein the world at the LHC (Large Hadron Collider), a 27km-long particle accelerator built at CERN near Geneva(Figure 6). The LHC stores and collides two beamsof protons which are circulating clockwise and counterclockwiseabout the accelarator [14]. Superconductingdipole magnets generate 8.3 Tesla ¯elds to keep thebeams in orbit. The magnets are cooled to 1.9 K, colderthan outer space, to achieve these ¯elds. The centreof-masscollision energy is 14 TeV which is eight timesgreater than the previous highest energy collider. Suchenergies have not been produced since approximately10 ¡25 s after the big bang.There are three experiments around the LHC which willrecord the particles generated in the proton{proton collisions.Two, ATLAS and CMS, are the largest colliderparticle physics experiments ever built with dimensionsof 46 m £ 25 m £ 25 m and 21 m £ 15 m £ 15 m, respectively.ATLAS and CMS will search for collisions thatcontain Higgs bosons or other exotic phenomena. Thethird experiment for proton{proton collisions is LHCb,which is dedicated to studying beauty quarks that exhibitCP violation in their decay as discussed in Section1.2. There is a fourth experiment, ALICE, which willstudy the strong interaction via events produced whenthe LHC collides gold nuclei together.938 RESONANCE October 2010

GENERAL ARTICLEBeams of protons were successfully circulated in bothdirections about the LHC in September 2008. Unfortunatelyshortly afterward a fault in one of the 1232superconducting dipole magnets led to signi¯cant damagein one part of the accelerator. Repairs and implementationof additional safeguards has taken just over ayear, leading to colliding beams restarting successfullyin December 2009. In March 2010 a new world recordcollision energy of 7 TeV was achieved. The LHC willrun at this energy until late 2011, before upgrades tothe accelerator will allow collisions at 14 TeV.Within the next fiveyears the LHC willeither confirm theHiggs mechanism orshed light on analternative model ofmass generation.3. ConclusionsThis article (Parts 1 and 2) presents a pedagogical summaryof the importance of symmetry principles in describingmany aspects of physical theories, in particularthose related to atomic, particle and nuclear physics.The continuous symmetries in classical mechanics thatlead to conservation of momentum, angular momentumand other quantities such as the Laplace{Runge{Lenzvector, were the starting point. Then discrete symmetriesP, C and T were discussed, along with how their violationis embedded within the standard model of particlephysics. The particular importance of the combinedoperation of C and P was emphasised as it maps matterinto antimatter. P and T violating phenomena in atomicphysics were discussed as the study of these are at theheart of some of the most exciting current atomic physicsresearch. Finally, spontaneous symmetry breaking andthe search for this phenomenon in particle physics atthe Large Hadron Collider was discussed. We hope thereader is left with a sense of the importance of symmetryand the many areas in which it is signi¯cant.Suggested Reading[1] Details of Lee and Yang’s 1957 Nobel Prize can be found athttp:nobelprize.org/nobel\_prizes/physics/laureates/1957/index.htmlRESONANCE October 2010939

GENERAL ARTICLEAddress for CorrespondenceP C Deshmukh and J LibbyDepartment of PhysicsIndian Institute of TechnologyMadrasChennai 600036.Email: pcd@physics.iitm.ac.inlibby@physics.iitm.ac.in[2] D Budker, D F Kimball and D P DeMille, Atomic Physics: An explorationthrough problems and solutions, Oxford Press, 2004.[3] I B Khriplovich, Physica Scripta, Vol.T112, p.52, 2004.[4] Ya B Zeldovich, Sov. Phys. JETP, Vol.6, p.1184, 1958.[5] R WDunford and R J Holt, J.Phys.G:Nucl.Part.Phys., Vol.34,pp.2099–2118, 2007.[6] K Tsigutkin, D Dounas-Frazer, A Family, J E Stalnaker, V V Yashchukand D Budker, Observation of a Large Atomic Parity Violation Effectin Ytterbium, http://arxiv.org/abs/0906.3039v3 2009.[7] Details of Cronin and Fitch’s 1980 Nobel Prize can be found athttp://nobelprize.org/nobel\_ prizes/physics/laureates/1980/index.html.[8] For a popular review of experimental results related to the CKMmatrix see T Gershon, A Triangle that Matters, Physics World, April2007.[9] Details of the 2008 Nobel Prize in physics can be found athttp://nobelprize.org/nobel\_prizes/physics/laureates/2008/index.html[10] For a popular review of neutrino oscillations and evidence for theirmass see D Wark, Neutrinos: ghosts of matter, Physics World, June2005.[11] U Fano and A R P Rau, Atomic collision and spectra, Academic Press,INC, 1986.[12] R Hasty et al, Science, Vol.290, p.15, 2000.[13] J J Hudson, B E Sauer, M R Tarbutt and E A Hinds, Measurement ofthe electron electric dipole moment using YbF molecules, 2002.http://arxiv.org/abs/hepex/0202014v2.[14] More details and the latest news about the LHC can be found athttp://public.web.cern.ch/public/en/LHC/LHC-en.html .940 RESONANCE October 2010