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Symmetry Principles and Conservation Laws in Atomic and Subatomic Physics – Part I GENERAL ARTICLE Symmetry Principles and Conservation Laws in Atomic and Subatomic Physics – 1 P C Deshmukh and J Libby (left) P C Deshmukh is a Professor of Physics at IIT Madras. He leads an active research group in the field of atomic and molecular physics and is involved in extensive worldwide research collaborations in both theoretical and experimental investigations in this field. He enjoys teaching both undergraduate and advanced graduate level courses. (right) Jim Libby is an Associate Professor in the Department of Physics at IIT, Madras. He is an experimental particle physicist specialising in CP violating phenomena. The whole theoretical framework of physics rests only on a few but profound principles. Wigner enlightened us by elucidating that \It is now natural for us to try to derive the laws of nature and to test their validity by means of the laws of invariance, rather than to derive the laws of invariance from what we believe to be the laws of nature." Issues pertaining to symmetry, invariance principles and fundamental laws challenge the most gifted minds today. These topics require a deep and extensive understanding of both `quantum mechanics' and the `theory of relativity'. We attempt in this pedagogical article to present a heuristic understanding of these fascinating relationships based only on rather elementary considerations in classical and quantum mechanics. An introduction to some fundamental considerations regarding continuous symmetries, dynamical symmetries (Part 1), and discrete symmetries (Part 2) (parity, charge conjugation and time-reversal), and their applications in atomic, nuclear and particle physics, will be presented. 1. Introduction Keywords Symmetry, conservation laws, Noether’s theorem. 832 The principal inquiry in classical mechanics is to seek a relationship between position, velocity, and acceleration. This relationship is rigorously expressed in what we call the `equation of motion'. The equation of motion is not self-evident, but rests on some fundamental principle that must be discovered. A prerequisite for this discovery is the principle of inertia, discovered by RESONANCE September 2010 GENERAL ARTICLE Galileo, contrary to common experience, that the velocity of an object is self-sustaining and remains invariant in the absence of its interaction with an external agency. This principle identi¯es an inertial frame of reference in which physical laws apply. This great discovery by Galileo was soon incorporated in Newton's scheme as the First law of mechanics, the law of inertia. Newton recognised, following his invention of calculus, that it is the change in velocity that seeks a cause. Newton's calculus expressed the rate of change of velocity as acceleration which is interpreted as the `e®ect' of the physical interaction that generated it. Newton's second law expresses this `cause-e®ect' relationship as a linear response of the system to the physical interaction it ex! ¡ ! perienced: F = m¡ a . The mass m of the object is the ! constant of proportionality between the e®ect (¡ a ) and ! ¡ the cause ( F ). 1This article is partly based on the talk given by PCD at the Karnataka Science and Technology Academy’s special lectures at the Bangalore University on 23rd March, 2009. In the following section we will begin by considering how Newton's third law introduces a simple illustration of the relation between a symmetry and a conservation law. In the remainder of the article we will explore similar relationships that impact much of the frontiers of physics, which are being investigated today; these studies use powerful theoretical frameworks and sophisticated technology. 2. Translational Invariance and Conservation of Momentum We consider a closed system of N point particles in homogeneous isotropic space. The force on the kth particle is the sum of forces on it due to all the other particles N X¡ ! ¡ ! Fk= f kj : (1) j=1 j6=k We now consider `virtual' translational displacement of the entire N -particle system in the homogeneous space. RESONANCE September 2010 Newton’s third law introduces a simple illustration of the relation between a symmetry and a conservation law. 833 GENERAL ARTICLE In such a process, the internal forces can do no work, since the process amounts to merely displacing the entire system to an adjacent region, displaced from the ¡ ! original by an amount ±s. As this displacement is being considered in a homogeneous medium, it is referred to as being `virtual' as no work is done by the internal forces. This phenomenon is then expressed by the relation ¡ ! N N X ¡ ! ¡ ! X d Pk ¡ ! F k ¢ ±s = ¢ ±s = 0 ; dt k=1 k=1 (2) ¡ ! where Pk is the momentum of the kth particle. In expressing this quantitative result, we have made use of Newton's ¯rst two laws (the ¯rst law implicitly and the second law explicitly) and also the notion of translational invariance in homogeneous space. Now, for an ar¡ ! bitrary displacement ±s, this relationship requires that ¡ ! N X dPk k=1 Conservation of linear momentum is governed by the symmetry principle of translational invariance in homogeneous space. Likewise, one can see that the conservation of angular momentum emerges from rotational displacements in isotropic space. 834 dt =0: (3) If we write this result for a two-body closed system, we discover Newton's third law, that action and reaction are equal and opposite: ¡ ! ¡ ! d P1 d P2 =¡ : (4) dt dt In other words, we discover that conservation of linear momentum is governed by the symmetry principle of translational invariance in homogeneous space. Likewise, one can see that the conservation of angular momentum emerges from rotational displacements in isotropic space. It is interesting to observe that Newton actually invented calculus to explain departure from equilibrium of an object which manifests as its acceleration, and proposed a linear relationship between the physical interaction (force) which he interpreted as the `cause' of the RESONANCE September 2010 GENERAL ARTICLE acceleration. Newton's second law contains the heart of this stimulus{response relation, expressed as a di®erential equation. It is interesting that laws of classical mechanics can be built alternatively on the basis of an `integral principle', namely the `principle of variation', discussed in the next section. 3. Principle of Variation The connection between symmetry and conservation laws becomes even more transparent in the alternative formalism of classical mechanics, namely the Lagrangian/ Hamiltonian formulation. It is instructive to ¯rst see that this alternative formalism is based not on the linear response relationship embodied in the Newtonian principle of causality, but in a completely di®erent approach, namely the `principle of variation'. It is interesting that laws of classical mechanics can be built alternatively on the basis of an ‘integral principle’, namely the ‘principle of variation’, Newtonian mechanics o®ers an accurate description of classical motion by accounting for the same by the `cause and e®ect' relationship. An alternative and equivalent description makes it redundant to invoke such a causal description. This alternative description dispenses the Newtonian notion of the 'cause-e®ect' relationship, and instead of it invokes a variational principle, namely, that the `action integral' is an extremum. Those who are used to thinking in terms of the Newtonian formulation alone would ¯nd it strange that one gets equivalent description of classical mechanics without invoking the notion of force at all! Let us ¯rst state the principle of extremum action. We begin on common ground with the Newtonian formula: tion, namely that the position q and velocity q specify the mechanical state of a system. A well-de¯ned func: tion of q and q would also then specify the mechanical state of the system. What is known as the Lagrangian of : a system L(q; q) is just that; it is named after its originator Lagrange (1736{1813). Furthermore, in a homoge- RESONANCE September 2010 835 GENERAL ARTICLE Simply stated, the principle of least action is that the mechanical state of a system evolves along a world-line such that the ‘action’, S= Zt2 : L(q; q; t)dt ; t1 is an extremum. 2 : neous isotropic system, L(q; q) can depend only quadratically on the velocity, so that it could be independent of its direction. The simplest form the Lagrangian would : :2 then have is L(q; q) = f1(q) + f2 (q ), wherein the functions f1 and f2 must be suitably chosen. It turns out that the choice f1 (q) = ¡V (q), i.e., the negative of the :2 :2 particle's potential energy, and f2 (q ) = (m=2)q , i.e., the kinetic energy T of the particle, renders this new formalism completely equivalent to Newtonian mechanics. This relationship o®ers us with a simple interpretation : of the Lagrangian as L(q; q) = T ¡ V . Simply stated, the principle of least action is that the mechanical state of a system evolves along a world-line2 such that the `action', A ‘world-line’ is a trajectory in S= the phase space, or the math- evolves over a period of time. : L(q; q; t)dt ; (5) t1 ematical space, along which the mechanical state of a system Zt2 is an extremum. This principle was formulated by Hamilton (1805{1865). It has an interesting development beginning with Fermat's principle about how light travels between two points, and subsequent contributions by Maupertius (1698{1759), Euler (1707{1783), and Lagrange himself. The principle that `action' is an extremum is equivalent to stating that the mechanical system evolves over the period t1 to t2 along a world-line : traced by the points (q; q) such that if the `action integral' S is evaluated along any other alternative path displaced in¯nitesimally from the one it actually evolves : : over, (q + ±q; q + ± q), then: ±S = Zt2 t1 : : L(q + ±q; q + ±q; t)dt ¡ Zt2 : L(q; q; t)dt = 0: (6) t1 The above equation is a mathematical expression of the statement of the `principle of extremum action'. The necessary and su±cient condition that this principle 836 RESONANCE September 2010 GENERAL ARTICLE must hold good provides us the well-known Lagrange's equation of motion: @L d @L ¡ : = 0: @q dt @ q (7) : The quantity (@L)=(@ q) in the above equation is known as the generalised momentum (written as p) conjugate to the generalised coordinate q. The power of Lagrangian mechanics comes from the fact that there are very many pairs of variables (q; p) which can be considered conjugate to each other in the Lagrangian sense { q and p need not have the physical dimensions of [L] and [MLT¡1 ] respectively. The dimension of the product of q and p, however, must always be ML2 T¡1 , that of the angular momentum. From Lagrange's equation, it follows immediately that if the Lagrangian is independent of q, (i.e., if (@L)=(@q) = 0) then the generalized momentum : p = (@L)=(@ q) conjugate to this coordinate is constant. The independence of the Lagrangian with respect to q is an expression of `symmetry', since the Lagrangian would then be the same no matter what the value of q is. This results in a conservation principle since the generalised momentum conjugate to this q becomes independent of time, remains constant. One may pair (time, energy) as (q; p), and see from this that (@L=(@t) = 0 would result in energy being constant. This result immediately follows from the following expression for the timederivative of the Lagrangian: 0= dL @L : @L :: @L = q+ : q+ dt @q @q @t ½ ¾ d @L : @L :: @L = q+ : q+ ; : dt @ q @q @t The independence of the Lagrangian with respect to q is an expression of ‘symmetry’, since the Lagrangian would then be the same no matter what the value of q is. This results in a conservation principle since the generalised momentum conjugate to this q becomes independent of time, remains constant. (8) where Lagrange's equation is used to re-express the ¯rst term. It thus follows that: ¸ d @L : @L : : q¡L = ¡ dt @ q @t RESONANCE September 2010 (9) 837 GENERAL ARTICLE Conservation of energy follows from the symmetry principle that the Lagrangian is invariant with respect to time. From the above, it immediately follows that when the Lagrangian depends on time only implicitly through its : dependence on q and q, then: ¸ d @L : (10) : q¡L = 0 ; dt @ q h³ ´ : i @L which implies @ q: q ¡ L is a conserved quantity. This quantity is called the Hamiltonian, or Hamilton's principal function, of the system, which for a conservative system is essentially the same as the total energy of the system. This can be seen easily by identifying the generalized momentum and substituting T ¡ V for the Lagrangian. We thus see that conservation of energy follows from the symmetry principle that the Lagrangian is invariant with respect to time. These results illustrate an extremely powerful theorem in physics, known as the Noether's theorem, which can be stated informally as: If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time [1]. This theorem is named after Noether (1882{1935), of whom Einstein said: If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time. 838 In the judgement of the most competent living mathematicians, FrÄaulein Noether was the most signi¯cant creative mathematical genius thus far produced since the higher education of women began [2]. 4. Symmetry Principles and Physical Laws We have now seen that both the equation of motion and the conservation principles result from the single principle of least action. Moreover, the same principle provides for the connection between symmetry and conservation laws, exalted by Noether to one of the most profound principles in contemporary physics. We now RESONANCE September 2010 GENERAL ARTICLE ask if the conservation principles are consequences of the laws of Nature, or, rather the laws of Nature are consequences of the symmetry principles that govern them? Until Einstein's special theory of relativity, it was believed that conservation principles are the result of the laws of Nature. Since Einstein's work, however, physicists began to analyze the conservation principles as consequences of certain underlying symmetry considerations from which they could be deduced, enabling the laws of Nature to be revealed from this analysis. Wigner's profound impact on physics is that his explanations of symmetry considerations using `group theory' resulted in a change in the very perception of just what is most fundamental, and physicists began to regard `symmetry' as the most fundamental entity whose form would govern the physical laws. Wigner was awarded the 1963 Nobel Prize in Physics for these insights [3]. The conservation of linear and angular momentum we illustrated above are consequences of invariance under continuous displacements and rotations respectively in homogenous and isotropic space. Likewise, the conservation of energy is a consequence of invariance under continuous temporal displacement. A detailed exposition of the governing symmetry principles requires group theoretical methods, and is clearly beyond the scope of this article, but we continue to dwell on some other kinds of symmetries now and examine their connections with conservation principles. Figure 1. Masters of symmetry. 5. Dynamical Symmetry: Laplace{Runge{Lenz Vector It is well known that in the classical two-body Kepler problem (gravitational Sun{Earth system, or the Coulombic proton{electron planetary model of the old-quantumtheory of the hydrogen atom), both energy and angular momentum are conserved. We have already discussed RESONANCE September 2010 Refer to Resonance issues on: Einstein, Vol.5, March and April 2000. Noether, Vol.3,September 1998. Wigner, Vol.14, October 2009. 839 GENERAL ARTICLE the associated symmetries. What is interesting is that the elliptic orbit of the Kepler system for bound states is ¯xed, i.e., the ellipse does not precess (Figure 2). Figure 2. If the eclipse were to precess it would generate what is called a ‘rosette’ motion since the trajectory of the planet would seem to go over the petals of a rose, if seen from a distance. Can we then ¯nd a symmetry that would explain the constancy of the orbit? It turns out that the orbit itself remains ¯xed if and only if the potential in which motion occurs is strictly of the form ¡1=r and the associated force is of the form ¡1=r2 . This is true for both the gravitational and the Coulomb potential, and hence the Kepler elliptic orbits remain ¯xed. This is rigorously expressed as the constancy of the Laplace{Runge{Lenz (LRL) vector. The LRL vector is de¯ned as: ³ ´ ~ = ~v £ H ~ ¡ e^½ A (11) and is shown in Figure 3 [4]. In the above equation ~v ~ is the `speci¯c' is the `speci¯c' linear momentum and H angular momentum. The term `speci¯c' denotes the fact that the physical quantities linear momentum and angular momentum, which are being referred to, are de¯ned per unit mass. Likewise in the second term of the LRL vector, is the proportionality in the inverse distance gravitational potential per unit mass of the planet. It can be easily veri¯ed that the time derivative of the LRL ~ is therefore a conserved quanvector vanishes, and the A tity. Its direction is from the focus of the ellipse to the perihelion (Figure 3) [4], which has a direction along the major axis of the ellipse, thus holding the ellipse ¯xed. Figure 3. Schematic diagram showing the Laplace– ~. Runge–Lenz vector, A 840 RESONANCE September 2010 GENERAL ARTICLE The constancy of the LRL vector is a conservation principle, and since the governing criterion involves dynamics (namely that the force must have a strict inverse square form), the associated symmetry is called `dynamical symmetry'. Sometimes, it is also called an `accidental' symmetry. This symmetry breaks down when there is even a minor departure from the inverse square law force, as would happen in a many-electron atom, such as the hydrogen-like sodium atom. The potential experienced by the `outer-most' electron goes as 1=r only in the asymptotic (r ! 1) region. Close to the center, the potential goes rather as ¡Z=r, due to the reduced screening of the nuclear charge by the orbital electrons, and thus departs from 1=r. This di®erence in the hydrogen atom potential and that in the sodium atom is due to the quantum analogue of the breakdown of the LRL vector constancy in the sodium atom. Using group theoretical methods, Vladmir Fock (1898{1974) explained the dynamical symmetry of the hydrogen atom [5]. Using the language of group theory, the Fock symmetry accounts for the (2l + 1)-fold degeneracy of the hydrogen atom eigenstates. This degeneracy is lifted for the hydrogen-like sodium atom due to the breakdown of the associated symmetry. In atomic physics, this is often expressed in terms of what is called as `quantum defect' ¹n;l which makes the hydrogenic energy eigenvalues depend not merely on the principal quantum number n but also on the orbital angular momentum quantum number l. This enables the use of the hydrogenic formula for energy with n replaced by ne®ective = n ¡ ¹n;l . The `quantum-defect theory' has very many applications in the analysis of the atomic spectrum, including the `autoionization resonances' [6,7]. As pointed out above, the conservation of angular momentum is due to the rotational symmetry, referred to as the symmetry under the group SO(3). All central ¯elds have this symmetry. However, the inverse-square-law force (such as gravity or RESONANCE September 2010 841 GENERAL ARTICLE The conservation of the generalized momentum which is conjugate to a cyclic coordinate is a generic expression of a deeper relationship between symmetry and conservation laws. Coulomb) has symmetry under a bigger group, SO(4) or SO(3; 1), where SO(4) is the rotational group in 4 dimensions, and SO(3; 1) is the Lorentz group. The dimensionality of the SO(N) group is N (N¡1)/2, so the SO(4) group is 6-dimensional and corresponds to the 6 conserved quantities, namely the 3 components of the angular momentum vector and the three components of Pauli{Runge{Lenz vector which is the quantum analogue of the LRL vector [8]. 6. Conclusion The conservation of the generalized momentum which is conjugate to a cyclic coordinate is a generic expression of a deeper relationship between symmetry and conservation laws. In the next part of this article we shall discuss discrete symmetries, the CPT symmetry and comment on spontaneous symmetry breaking and the search for the Higgs boson. Suggested Reading [1] W J Thompson, Angular Momentum, Wiley, p.5, 2004. [2] From a letter to the New York Times on May 5th, 1935 from Albert Einstein shortly after Emmy Noether’s death. [3] Details of the 1963 Nobel Prize in physics can be found at http:// [4] For a detailed discussion of the Laplace–Runge–Lenz vector see H Gold- nobelprize.org/nobel\_prizes/physics/laureates/1963/index.html Address for Correspondence P C Deshmukh and J Libby Department of Physics stein, Indian Institute of Technology 1980. Classic Mechanics, Second Edition, Addison-Wesley, p102ff, Madras [5] W Fock, Z. Phys., Vol.98, p.145, 1935. Chennai 600036. [6] M J Seaton, Rep. Prog. Phys., Vol.46, p.167, 1983. Email: [email protected] [7] S B Whitfield, R Wehlitz, H R Varma, T Banerjee, P C Deshmukh and S T Manson, J. Phys. B: At. Mol. Opt. Phys., Vol.39, p.L335, 2006. [email protected] [8] 842 V Bargmann, Z. Physik Vol.99, pp.576–582, 1936. RESONANCE September 2010 Symmetry Principles and Conservation Laws in Atomic and Subatomic Physics – Part II GENERAL ARTICLE Symmetry Principles and Conservation Laws in Atomic and Subatomic Physics – 2 P C Deshmukh and J Libby (left) P C Deshmukh is a Professor of Physics at IIT Madras. He leads an active research group in the field of atomic and molecular physics and is involved in extensive worldwide research collaborations in both theoretical and experimental investigations in this field. He enjoys teaching both undergraduate and advanced graduate level courses. (right) Jim Libby is an This article is the second part of our review of the important role that symmetry plays in atomic and subatomic physics. We will concentrate on the discrete symmetries { parity, charge conjugation, and time reversal { that have played a signi¯cant part in the development of the `standard model' of particle physics during the latter part of the 20th century. The importance of experimental tests of these symmetries, in both atomic and particle physics, and their sensitivity to new phenomena is also discussed. To conclude, we describe how `symmetry breaking' in the standard model leads to the generation of mass via the Higgs mechanism and how the search for evidence of this symmetry violation is one of the principal goals of the Large Hadron Collider, which began operating at CERN, Switzerland in 2009. Associate Professor in the Department of Physics at IIT, Madras. He is an experimental particle physicist specialising in CP violating phenomena. Part 1: Resonance, Vol.15, No.9, p.832. Keywords Discrete symmetries, violation of parity and CP, Higgs mechanism, LHC. 926 1. Discrete Symmetries Apart from continuous and dynamical symmetries, there are other kinds of symmetries that are of importance in physics. In particular, we have three discrete symmetries of central importance in what is known as the `standard model' of particle physics. These discrete symmetries are: (i) P (Parity), (ii) C (Charge conjugation, i.e., matter/antimatter) and (iii) T (Time-reversal), often known together as PCT symmetry. In physical reactions of particle physics, these symmetries lead to conservation principles operating either separately or in combination. We shall now discuss these discrete symmetries. RESONANCE October 2010 GENERAL ARTICLE 1.1 Parity Parity is the symmetry we see between an object and its mirror image. It is interesting that in a mirror, we usually see the left go to right, and the right go to left, but we do not see top go to bottom and the bottom to the top. This feature typi¯es the di®erence between re°ection and rotation. If we represent the transformation of a vector ~r to its image in a mirror placed in the Cartesian yz-plane, then we can express the transformation ~r = (x; y; z) to its image ~rI = (xI ; yI ; zI) by a matrix equation: ~rI = <~r : (1) Now, in the case of re°ection, the determinant of the matrix < in the above relation is ¡1, whereas if one writes a similar relation for the rotation of the vector ~r to a new orientation ~rR , the corresponding matrix of transformation would have for its determinant the value +1. The reason left goes to right and right to left, but not the top to the bottom and bottom to the top, in a mirror is that we usually tend to imagine the image to have gone to the opposite side of the mirror through a rotation about the vertical axis. If we imagine the rotation to be about the horizontal axis, we would certainly see the top go to the bottom, the bottom to the top, but not left to right or right to left. Figure 1 illustrates this. Of course, the fundamental reason is the intrinsic di®erence between rotation and re°ection, exhibited by the di®erent signs of the determinants of their matrices. The parity transformation is thus very di®erent RESONANCE October 2010 The reason left goes to right and right to left, but not the top to the bottom and bottom to the top, in a mirror is that we usually tend to imagine the image to have gone to the opposite side of the mirror through a rotation about the vertical axis. Figure 1. Depending on the plane of reflection, right goes to left and top to bottom; the primary feature discussed in the text is that ‘parity’ is an operation that is essentially different from ‘reflection’. 927 GENERAL ARTICLE The physical phenomena for which parity is violated result from an interaction known as the weak interaction; its most widelyknown manifestation is nuclear decay. from rotation and one may ask, as Alice would (in Through the Looking Glass), if the physical laws are the same in the world of images in a mirror. In other words, this question amounts to asking, given the fact that there is a certain degree of invariance when one compares an object with its image in a mirror, whether parity is conserved in nature. The parity operator ¦ is a unitary operator which anticommutes with the position operator and also with the operator for linear momentum, since both position and momentum are polar vectors. However the parity operator commutes with the operator for angular momentum which is a pseudovector. While most of the everyday physical phenomena could take place just as well in essentially the same manner in the image world as in the real world, certain physical phenomena occur di®erently. The physical phenomena for which parity is violated result from an interaction known as the weak interaction; its most widely-known manifestation is nuclear ¯ decay. The search for parity violation in weak interactions was advocated strongly by Lee and Yang [1], after a careful review of the subject indicated that parity conservation, though often assumed, had not been veri¯ed in weak interactions. Acting on the proposals of Lee and Yang, Wu and collaborators clearly observed parity violation in the ¯ decay of polarised nuclei via asymmetries in the distribution of the ¯-decay electron with respect to the spin of the nucleus (Figure 2). The violation of parity was unexpected. It allowed the first unambiguous definition of left and right in nature. 928 These and subsequent measurements showed that the weak interaction was maximally parity violation, which meant that it only couples to left-handed chiral states of matter and right-handed chiral states of antimatter; i.e., for a massless fermion this would correspond to the state where the spin is in the opposite direction to its momentum. RESONANCE October 2010 GENERAL ARTICLE a) b) Figure 2. Schematic (a) is of the direction of the decay electron, characterized by momentum pe , with re- 60 spect to the spin of the Co nucleus, J 60 Co . Schematic (b) is the same process transformed by the parity operation. Unequal probabilities for these two processes to occur were observed by Wu and collabo- Parity violation is observed in nuclear and subatomic interactions, and through the uni¯cation of the weak and electromagnetic interactions, parity is violated in certain atomic processes as well. Atomic transitions are normally governed by the parity selection rule, which then breaks down for those transitions in which parity is not conserved. The electroweak uni¯cation achieved in the Glashow{Weinberg{Salam model triggered the search in the 1970s for parity nonconservation (PNC) in atomic processes [2]. The gauge bosons W § have a charge of +1 and ¡1 unit, but the Z 0 boson of the standard model is neutral. The latter can mediate an interaction between atomic electrons and the nucleus. The nuclear weak charge QW of the standard model plays the same role with regard to Z 0 that the `usual' electric charge plays with regard to the Coulomb interaction. PNC e®ect in atomic cesium yields the value of QW (133 Cs) ¼ ¡72:90, not far from the value of QW (133 Cs) ¼ ¡73:09 obtained from high-energy experiments extrapolated to atomic scale [3]. The Z-boson has a very large mass and the weakinteraction is `contact' type. It includes a parity-even part and a parity-odd (PNC) part. While the parityeven part leads to a correction to isotope shift and to hyper¯ne structure, the PNC part leads to the `pseudoscalar' correlations in atomic processes. RESONANCE October 2010 rators; this was the first experimental evidence for parity violation in nature. Atomic transitions are normally governed by the parity selection rule, which then breaks down for those transitions in which parity is not conserved. 929 GENERAL ARTICLE The anapole moment is a new electromagnetic moment that 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 to a virtual photon. A significantly large value of the anapole moment of the nucleon has been estimated in the case of cesium, augmented by collective nuclear effects. 930 The usual radiative transitions in atomic processes are governed by parity-conserving selection rules imposed by the electromagnetic Hamiltonian. However, once the Hamiltonian is modi¯ed to include the electroweak interaction, it does not commute with the parity operator and provides for non-zero probability for parityviolating atomic transitions. The two sources of parity nonconservation (PNC) in atoms are: (1) the electronnucleus weak interaction and (2) the interaction (sometimes called as PNC hyper¯ne interaction) of electrons with the nuclear anapole moment. The anapole moment was predicted by Vaks and Zeldovich [4] soon after Lee and Yang's proposal that weak interactions violate parity. The anapole moment is a new electromagnetic moment that 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 to a virtual photon. The anapole moment can be seen to result from a careful consideration of the magnetic vector potential at a ¯eld point after taking into account the constraints of current conservation and the boundedness of the current density. A signi¯cantly large value of the anapole moment of the nucleon has been estimated in the case of cesium, augmented by collective nuclear e®ects. Recently, Dunford and Holt [5] recommended parity experiments on atomic hydrogen and deuterium using UV radiation from free electron laser (FEL) to probe new physics beyond the standard model. The Dunford{Holt proposal is based on the consideration that if an isolated hydrogen atom existed in an excited state that is a mix of states 2s 1 and 2 2p 1 which have opposite parity, then parity would be vi2 olated if the electromagnetic interactions alone were to exist. These two energy states are very nearly degenerate and thus very sensitive to the electroweak interaction which would mix them. More recently, atomic parity violation has been observed in the 6s2 1 S0 ! 5d6s 3 D1 RESONANCE October 2010 GENERAL ARTICLE 408 nm forbidden transition of ytterbium [6]. In this work, the transition that violates parity was found to be two orders of magnitude stronger than that found in atomic cesium. Atomic physics experiments provide a low-energy test of the standard model and also provide relatively low-cost tools to explore physics beyond it. 1.2 Charge Conjugation and CP Symmetries Atomic physics experiments provide a lowenergy test of the standard model and also provide relatively low-cost tools to explore The discrete symmetry of charge conjugation (C) converts all particles into their corresponding antiparticles. For example, C operation transforms an electron into a positron. The chirality of the state is preserved under charge conjugation. For example, a left-handed neutrino becomes a left-handed antineutrino; the latter does not interact weakly and shows that C, as well as P, are maximally violated in weak interactions. However, the combined operation CP, on a process mediated by the weak interaction was anticipated to be invariant because, for example, a left-handed neutrino is transformed into a right-handed antineutrino. However, violation of CP is essential to describe the observed state of the universe as being matter dominated. Only di®erences in behaviour between matter and antimatter, in other words CP violation, can produce such an asymmetry. The presence of CP-violation is one of the three conditions for producing baryons (baryogenesis) in the early universe put forward by the Soviet physicist and dissident Sakharov (1921{ 1989). He had been inspired to propose CP-violation as an essential ingredient of baryogenesis by the experiments of Cronin, Fitch and collaborators in 1964 that had clearly shown that CP-violation occurs in the weak decays of hadrons containing a strange quark [7]. physics beyond it. The origin of CP-violation in weak hadronic decays took some time to describe. It required the bold hypothesis of Kobayashi and Maskawa in 1973 that there was a third generation of quarks to complement the already discovered up (u), down (d), and strange (s) quarks, Violation of CP is essential to describe RESONANCE October 2010 the observed state of the universe as being matter dominated. 931 GENERAL ARTICLE It was Kobayashi and Maskawa’s great insight that a 3 3 matrix allowed a complex phase to be introduced, which can describe CPviolation in weak hadronic decays. and that time, postulated charm (c) quark. The addition of a third generation of bottom (b) and top (t) quarks leads to a 3 £ 3 matrix being required to describe the weak couplings between the di®erent quarks, which allow for the change of quark type unlike the strong or electromagnetic interactions. It was Kobayashi and Maskawa's great insight that a 3 £ 3 matrix allowed a complex phase to be introduced, which can describe CPviolation in weak hadronic decays. The postulated third generation was not discovered until Lederman and collaborators observed evidence of the b quark in 1977. The CP-violating parameters of Kobayashi and Maskawa matrix have now been measured accurately principally in experiments at the Stanford Linear Accelerator Center, US, the High Energy Accelerator Research Organisation (KEK), Japan, and the Fermilab National Accelerator Laboratory, US [8]. This con¯rmation of the three generation model to describe CP-violation led to the award of the Nobel Prize for Physics to Kobayashi and Maskawa in 2008 [9]. This confirmation of the three generation model to describe CPviolation led to the award of the Nobel Prize for Physics to Kobayashi and Despite the success of this model of CP-violation in the standard model of particle physics, the rate at which it is observed in weak hadronic decays is insu±cient to describe the large matter-antimatter asymmetry observed in universe. Therefore, theories that go beyond the standard model must accommodate new sources of CPviolation to explain the rate of baryogenesis. This means that the further study of CP-violation is extremely important. Therefore, °avour experiments are planned at the Large Hadron Collider (see Section 2) and elsewhere. CP-violation may also occur in the lepton sector now that the non-zero mass of the neutrino has been established [10]; however, an exposition of this exciting topic is beyond the scope of this article. Maskawa in 2008. 932 RESONANCE October 2010 GENERAL ARTICLE 1.3 CPT Symmetry The `Time Reversal Symmetry' (T) is another discrete symmetry. This has a characteristically di®erent form in quantum mechanics that has no classical analogue. The name time-reversal is perhaps inappropriate, because it would make a layman suspect that it is merely the inverse of the `time evolution', which is not the case. In quantum theory, the operator for `time evolution' is unitary, but that for time-reversal is antiunitary. The quantum mechanical operator ¦ for parity anticommutes with 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 anticommutes with both the linear and the angular momentum operators. In quantum theory, the operator for ‘time evolution’ is unitary, but that for time-reversal is antiunitary. An important consequence of these properties is the fact that the response of a wavefunction to time-reversal would include not merely t going to ¡t in the argument of the wavefunction, but also simultaneous complex conjugation of the wavefunction. This property connects the quantum mechanical solutions of an electron{ion collision problem with those of electron{atom scattering through time-reversal symmetry. The physical content of this connection is depicted in Figure 3 which represents the fact that in a photoionization experiment it is the escape channel for the photoelectron which is unique whereas in an electron{ion scattering experiment it is the entrance channel of the projectile electron which is (a) RESONANCE October 2010 (b) Figure 3. Schematic diagram showing the time-reversal relation between photoionization and scattering processes in atomic physics. 933 GENERAL ARTICLE The Lorentz symmetry of the standard model of physics conserves PCT. Violation of T symmetry would require an elementary particle, atom or molecule to possess a permanent electric dipole moment (EDM). The standard model of particle physics predicts that these dipole moments would be too small to be observable. EDM measurements therefore provide an exciting probe to explore new physics beyond the standard model. 934 unique. Despite the fact that the ingredients of the electron{ion collision experiment and that of photoionization are completely di®erent, both the processes result in the same ¯nal state consisting of an electron and an ion. The initial state, being obviously di®erent, implies that the quantum mechanical solutions of electron{ion scattering and photoionization are related to each other via the time-reversal symmetry [11]. The boundary condition for electron{ion collision and for atomic photoionization are therefore appropriately referred to as `outgoing wave boundary condition' and `ingoing wave boundary condition'. The employment of the solutions corresponding to the ingoing wave boundary conditions in atomic photoionization gives appropriate expressions for not just the photoionization transition intensities, but also for the angular distribution and the spin polarization parameters of the photoelectrons. The Lorentz symmetry of the standard model of physics conserves PCT. The discovery of CP violation in the decay of K mesons [7] therefore made it pertinent to look for the violation of the time-reversal symmetry. Violation of T symmetry would require an elementary particle, atom or molecule to possess a permanent electric dipole moment (EDM), since the only direction with which an electric dipole moment d~ =j d j e^s could be de¯ned will have to be along the unit vector e^s , the direction of the particle's spin. Crudely, this can be schematically shown in Figure 4 which shows an angular direction to represent a rotation, and a charge distribution to depict a dipole moment. As t goes to ¡t, the spin reverses, but not the electric dipole moment. We thus expect from the above equations that the electric dipole moment (EDM) of an elementary particle must be zero, unless both P and T are violated. The standard model of particle physics predicts that these dipole moments would be too small to be observable. EDM measurements therefore provide an exciting probe RESONANCE October 2010 GENERAL ARTICLE a) Figure 4. Schematic diagram explaining that the dipole moment of an elementary particle must be zero unless T symmetry is broken. The existence of an EDM also requires that P symmetry is violated. b) to explore new physics beyond the standard model. Highprecision measurements in agreement with predictions of a robust theoretical formulation would therefore provide a valuable test of the standard model, since limits on EDMs would put conditions on supersymmetric gauge theories [12,13]. 2. Spontaneous Symmetry Breaking and the Search for the Higgs Boson Here we will discuss how symmetry plays an important part in attempts to address another outstanding issue in the standard model of particle physics: How does an elementary particle, such as an electron, attain its mass? The standard model answers this question by assuming that there exists a scalar (spin-less) particle that was predicted in 1964 by Higgs, which is believed to 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 relativistic quantum ¯eld theory, which can be expressed in terms of a Lagrangian. The Lagrangian that describes the interactions of a scalar ¯eld Á is: 1 1 1 L = (@¹ Á)2 ¡ ¹2Á2 ¡ ¸Á4 ; 2 2 4 RESONANCE October 2010 (2) 935 GENERAL ARTICLE Figure 5. Potential V for a one-dimensional scalar field for two cases , 0, as defined in the text. where @¹ is the covariant derivative and ¹ is the particle mass and ¸ is the strength of the coupling of Á to itself. The ¯rst term on the right-hand side is considered the kinetic energy whereas the other two terms are the potential. Figure 5 shows the potential as a function of the scalar ¯eld Á for two cases: ¹2 > 0 and ¹2 < 0. For the case of an imaginary mass (¹2 < 0) there are two minima at r ¡¹2 Ámin = §À = § : (3) ¸ In considering weak interactions we are interested in small perturbations about the minimum energy so we expand the ¯eld about one of the minima, À or ¡À Á = À + ¾(x) ; The breaking of symmetry provides a hypothesis for the generation of all particle masses – the Higgs mechanism. 936 (4) where ¾(x) is the variable value of the ¯eld above the constant uniform value of À. Substituting this expression for Á into (2) one gets: µ ¶ 1 1 4 2 2 2 3 L = (@¹ ¾) ¡ ¸À ¾ ¡ ¸À¾ + ¸¾ + constant ; 2 4 (5) where the constant term depends on À 2 and À 4 and the third term (in parenthesis) on the right-hand side describes self interactions. The second term corresponds to a mass term with real mass p p m = 2¸À 2 = ¡2¹2 : (6) RESONANCE October 2010 GENERAL ARTICLE The perturbative expansion about one of the two minima has led to a real mass appearing. Since the expansion is made about one or other of the minima, chosen at random, the symmetry of Figure 5 is broken. This is the process of spontaneous symmetry breaking. Nambu and Jona-Lasinio ¯rst applied spontaneous symmetry breaking as mechanism of mass generation in 1961. In recognition of this work Nambu was awarded a share of the 2008 Nobel Prize [9]. There are many examples of spontaneous symmetry breaking in other areas of physics. For example a bar magnet heated above the Curie temperature has its elementary magnetic domains orientated randomly, leading to zero net ¯eld. The Lagrangian describing the ¯eld of the magnet would be invariant under rotations. However, on cooling, the domains set in a particular direction, causing an overall ¯eld and breaking the rotational symmetry. There are further examples of spontaneous symmetry breaking in the description of superconductivity; these inspired Nambu and Jona-Lasinio's work in particle physics. The introduction of such a scalar ¯eld interaction and a spontaneous symmetry breaking within the standard model allows the weak force carrying bosons, W § and Z 0 , to obtain mass as well as all quarks and leptons. In addition, this leads to the physical Higgs boson. The Higgs boson is the only part of the standard model of particle physics that has not been experimentally veri¯ed. However, the precise measurements of the properties of the Z 0 and the W § by experiments at the Large Electron Positron (LEP) collider, which operated at the European Centre for High Energy Particle Physics (CERN) in Geneva, Switzerland, and of the W § and the heaviest quark (the top) at Fermilab, have led to an upper limit on the mass of the Higgs boson of 157 GeV=c2 with a 95% con¯dence level. In addition, unsuccessful searches for the production of a standard model Higgs boson at LEP placed a lower limit on the mass of the RESONANCE October 2010 Nambu and JonaLasinio first applied spontaneous symmetry breaking as mechanism of mass generation in 1961. In recognition of this work Nambu was awarded a share of the 2008 Nobel Prize. The Higgs boson is the only part of the standard model of particle physics that has not been experimentally verified. 937 GENERAL ARTICLE Figure 6. Computer-generated image shows the location of the 27-km LHC tunnel (in blue) on the Swiss– France border. The four main experiments (ALICE, ATLAS, CMS, and LHCb) are located in underground caverns connected to the surface by 50 m to 150 m pits. Part of the pre-acceleration chain is shown in grey. The centre-of-mass collision energy is 14~TeV which is eight times greater than the previous highest energy collider. Such energies have not been produced since approximately 10–25s after the big bang. 938 Higgs boson of 114 GeV=c2 with a 95% con¯dence level. The search for the Higgs boson is one of the principal goals of the largest and the biggest experiment done in the world at the LHC (Large Hadron Collider), a 27 km-long particle accelerator built at CERN near Geneva (Figure 6). The LHC stores and collides two beams of protons which are circulating clockwise and counterclockwise about the accelarator [14]. Superconducting dipole magnets generate 8.3 Tesla ¯elds to keep the beams in orbit. The magnets are cooled to 1.9 K, colder than outer space, to achieve these ¯elds. The centreof-mass collision energy is 14 TeV which is eight times greater than the previous highest energy collider. Such energies have not been produced since approximately 10¡25 s after the big bang. There are three experiments around the LHC which will record the particles generated in the proton{proton collisions. Two, ATLAS and CMS, are the largest collider particle physics experiments ever built with dimensions of 46 m £ 25 m £ 25 m and 21 m £ 15 m £ 15 m, respectively. ATLAS and CMS will search for collisions that contain Higgs bosons or other exotic phenomena. The third experiment for proton{proton collisions is LHCb, which is dedicated to studying beauty quarks that exhibit CP violation in their decay as discussed in Section 1.2. There is a fourth experiment, ALICE, which will study the strong interaction via events produced when the LHC collides gold nuclei together. RESONANCE October 2010 GENERAL ARTICLE Beams of protons were successfully circulated in both directions about the LHC in September 2008. Unfortunately shortly afterward a fault in one of the 1232 superconducting dipole magnets led to signi¯cant damage in one part of the accelerator. Repairs and implementation of additional safeguards has taken just over a year, leading to colliding beams restarting successfully in December 2009. In March 2010 a new world record collision energy of 7 TeV was achieved. The LHC will run at this energy until late 2011, before upgrades to the accelerator will allow collisions at 14 TeV. Within the next five years the LHC will either confirm the Higgs mechanism or shed light on an alternative model of mass generation. 3. Conclusions This article (Parts 1 and 2) presents a pedagogical summary of the importance of symmetry principles in describing many aspects of physical theories, in particular those related to atomic, particle and nuclear physics. The continuous symmetries in classical mechanics that lead to conservation of momentum, angular momentum and other quantities such as the Laplace{Runge{Lenz vector, were the starting point. Then discrete symmetries P, C and T were discussed, along with how their violation is embedded within the standard model of particle physics. The particular importance of the combined operation of C and P was emphasised as it maps matter into antimatter. P and T violating phenomena in atomic physics were discussed as the study of these are at the heart of some of the most exciting current atomic physics research. Finally, spontaneous symmetry breaking and the search for this phenomenon in particle physics at the Large Hadron Collider was discussed. We hope the reader is left with a sense of the importance of symmetry and the many areas in which it is signi¯cant. Suggested Reading [1] Details of Lee and Yang’s 1957 Nobel Prize can be found at http:nobelprize.org/nobel\_prizes/physics/laureates/1957/index.html RESONANCE October 2010 939 GENERAL ARTICLE [2] D Budker, D F Kimball and D P DeMille, Atomic Physics: An exploration through 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 W Dunford 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 Yashchuk and D Budker, Observation of a Large Atomic Parity Violation Effect in Ytterbium, http://arxiv.org/abs/0906.3039v3 2009. [7] Details of Cronin and Fitch’s 1980 Nobel Prize can be found at http://no be lprize .o r g /no be l\_ pr ize s/physic s/laur e ate s/1 9 8 0 / index.html. [8] For a popular review of experimental results related to the CKM matrix see T Gershon, A Triangle that Matters, Physics World, April 2007. [9] Details of the 2008 Nobel Prize in physics can be found at http://nobelprize.org/nobel\_prizes/physics/laureates/2008/index.html [10] For a popular review of neutrino oscillations and evidence for their mass see D Wark, Neutrinos: ghosts of matter, Physics World, June 2005. Address for Correspondence P C Deshmukh and J Libby [11] U Fano and A R P Rau, Atomic collision and spectra, Academic Press, INC, 1986. Department of Physics Indian Institute of Technology [12] R Hasty et al, Science, Vol.290, p.15, 2000. Madras [13] J J Hudson, B E Sauer, M R Tarbutt and E A Hinds, Measurement of the electron electric dipole moment using YbF molecules, 2002. Chennai 600036. http://arxiv.org/abs/hepex/0202014v2. Email: [email protected] [email protected] [14] More details and the latest news about the LHC can be found at http://public.web.cern.ch/public/en/LHC/LHC-en.html . 940 RESONANCE October 2010