Symmetry Principles and ... - Department of Physicsphysics.iitm.ac.in/~labs/amp/Symmetry_Principles_Conservation_Laws.pdfAtomic and Subatomic Physics – 1 P C Deshmukh and J Libby

Symmetry Principles and Conservation Laws in

Atomic and Subatomic Physics – Part I

832 RESONANCE September 2010

GENERAL ARTICLE

Symmetry Principles and Conservation Laws inAtomic and Subatomic Physics – 1

P C Deshmukh and J Libby

Keywords

Symmetry, 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 nat-ural for 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 in-variance from what we believe to be the laws ofnature." Issues pertaining to symmetry, invari-ance principles and fundamental laws challengethe most gifted minds today. These topics re-quire a deep and extensive understanding of both`quantum mechanics' and the `theory of relativ-ity'. We attempt in this pedagogical article topresent a heuristic understanding of these fas-cinating relationships based only on rather ele-mentary considerations in classical and quantummechanics. An introduction to some fundamen-tal considerations regarding continuous symme-tries, dynamical symmetries (Part 1), and dis-crete symmetries (Part 2) (parity, charge conju-gation and time-reversal), and their applicationsin atomic, nuclear and particle physics, will bepresented.

1. Introduction

The principal inquiry in classical mechanics is to seeka relationship between position, velocity, and accelera-tion. This relationship is rigorously expressed in whatwe call the èquation of motion'. The equation of mo-tion is not self-evident, but rests on some fundamentalprinciple that must be discovered. A prerequisite forthis discovery is the principle of inertia, discovered by

(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 experimen-

tal investigations in this

field. He enjoys

teaching both undergradu-

ate 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.

833RESONANCE September 2010

GENERAL ARTICLE

1This article is partly based on

the talk given by PCD at the

Karnataka Science and Tech-

nology Academy’s special lec-

tures at the Bangalore Univer-

sity on 23rd March, 2009.

Galileo, contrary to common experience, that the veloc-ity of 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. New-ton recognised, following his invention of calculus, thatit is the change in velocity that seeks a cause. New-ton's calculus expressed the rate of change of velocityas acceleration which is interpreted as the è®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 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 ).

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 re-lationships that impact much of the frontiers of physics,which are being investigated today; these studies usepowerful theoretical frameworks and sophisticated tech-nology.

2. Translational Invariance and Conservation ofMomentum

We consider a closed system of N point particles in ho-mogeneous isotropic space. The force on the kth particleis the sum of forces on it due to all the other particles

¡!F k =

NX

j=1j 6=k

¡!f kj : (1)

We now consider `virtual' translational displacement ofthe entire N -particle system in the homogeneous space.

Newton’s third law

introduces a

simple illustration

of the relation

between a

symmetry and a

conservation law.


GENERAL ARTICLE

In such a process, the internal forces can do no work,since the process amounts to merely displacing the en-tire system to an adjacent region, displaced from theoriginal by an amount

¡!±s. As this displacement is being

considered 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 relation

NX

k=1

¡!F k ¢

¡!±s =

NX

k=1

d¡!Pkdt

¢¡!±s = 0 ; (2)

where¡!Pk is the momentum of the kth particle. In ex-

pressing this quantitative result, we have made use ofNewton's ¯rst two laws (the ¯rst law implicitly and thesecond law explicitly) and also the notion of transla-tional invariance in homogeneous space. Now, for an ar-bitrary displacement

¡!±s , this relationship requires that

NX

k=1

d¡!Pkdt

= 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¡!P1

dt= ¡

d¡!P2

dt: (4)

In other words, we discover that conservation of linearmomentum is governed by the symmetry principle oftranslational invariance in homogeneous space. Like-wise, one can see that the conservation of angular mo-mentum emerges from rotational displacements in isotro-pic space.

It is interesting to observe that Newton actually in-vented calculus to explain departure from equilibrium ofan object which manifests as its acceleration, and pro-posed a linear relationship between the physical inter-action (force) which he interpreted as the `cause' of the

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.


GENERAL ARTICLE

It is interesting that

laws of classical

mechanics can be

builtalternatively

on the basis of an

‘integral principle’,

namely the

‘principle of

variation’,

acceleration. Newton's second law contains the heartof this stimulus{response relation, expressed as a di®er-ential equation. It is interesting that laws of classicalmechanics can be built alternatively on the basis of anìntegral principle', namely the `principle of variation',discussed in the next section.

3. Principle of Variation

The connection between symmetry and conservation lawsbecomes even more transparent in the alternative for-malism of 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 prin-ciple of causality, but in a completely di®erent approach,namely the `principle of variation'.

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 àction 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 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 ofa system L(q;

:q) is just that; it is named after its origina-

tor Lagrange (1736{1813). Furthermore, in a homoge-


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 =

t2Z

t1

L(q;:q; t)dt ;

is an extremum.

neous isotropic system, L(q;:q) can depend only quadrat-

ically on the velocity, so that it could be independent ofits direction. The simplest form the Lagrangian wouldthen have is L(q;

:q) = f1(q) + f2(

:q2), wherein the func-

tions f1 and f2 must be suitably chosen. It turns outthat the choice f1(q) = ¡V (q), i.e., the negative of the

particle's potential energy, and f2(:q

2) = (m=2)

:q2, i.e.,

the kinetic energy T of the particle, renders this new for-malism completely 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-line2

such that the àction',

S =

t2Z

t1

L(q;:q; t)dt ; (5)

is an extremum. This principle was formulated by Hamil-ton (1805{1865). It has an interesting development be-ginning with Fermat's principle about how light travelsbetween two points, and subsequent contributions byMaupertius (1698{1759), Euler (1707{1783), and La-grange himself. The principle that àction' is an ex-tremum is equivalent to stating that the mechanical sys-tem evolves over the period t1 to t2 along a world-linetraced by the points (q;

:q) such that if the àction in-

tegral' S is evaluated along any other alternative pathdisplaced in¯nitesimally from the one it actually evolvesover, (q + ±q;

:q + ±

:q), then:

±S =

t2Z

t1

L(q + ±q;:q + ±

:q; t)dt¡

t2Z

t1

L(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 principle

2 A ‘world-line’ is a trajectory in

the phase space, or the math-

ematical space, along which the

mechanical state of a system

evolves over a period of time.


GENERAL ARTICLE

must hold good provides us the well-known Lagrange'sequation of motion:

@L

@q¡

d

dt

@L

@:q

= 0: (7)

The quantity (@L)=(@:q) in the above equation is known

as 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 conju-gate to each other in the Lagrangian sense { q and p neednot have the physical dimensions of [L] and [MLT¡1] re-spectively. The dimension of the product of q and p,however, must always be ML2T¡1, that of the angularmomentum. From Lagrange's equation, it follows im-mediately that 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 immedi-ately follows from the following expression for the time-derivative of the Lagrangian:

0 =dL

dt=

@L

@q

:q +

@L

@:q

::q +

@L

@t

=

½d

dt

@L

@:q

¾:q +

@L

@:q

::q +

@L

@t; (8)

where Lagrange's equation is used to re-express the ¯rstterm.

It thus follows that:

d

dt

@L

@:q

:q ¡ L = ¡

@L

@t: (9)

¸

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.


GENERAL ARTICLE

If a system has a

continuous

symmetryproperty,

then there are

corresponding

quantities whose

values are

conserved in time.

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 theLagrangian depends on time only implicitly through itsdependence on q and

:q, then:

d

dt

@L

@:q

:q ¡ L = 0 ; (10)

which impliesh³

@L@:q

´:q ¡ L

iis a conserved quantity. This

quantity is called the Hamiltonian, or Hamilton's prin-cipal function, 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 gen-eralized momentum and substituting T ¡ V for the La-grangian. We thus see that conservation of energy fol-lows from 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 con-served in time [1].

This theorem is named after Noether (1882{1935), ofwhom Einstein said:

In the judgement of the most competent living math-ematicians, FrÄaulein Noether was the most signi¯cantcreative mathematical genius thus far produced since thehigher education of women began [2].

4. Symmetry Principles and Physical Laws

We 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 con-servation laws, exalted by Noether to one of the mostprofound principles in contemporary physics. We now

¸


GENERAL ARTICLE

Refer to Resonance issues on:

Einstein, Vol.5, March and April

2000.

Noether, Vol.3,September 1998.

Wigner, Vol.14, October 2009.

Figure 1. Masters of sym-

metry.

ask if the conservation principles are consequences of thelaws of Nature, or, rather the laws of Nature are conse-quences of the symmetry principles that govern them?

Until Einstein's special theory of relativity, it was be-lieved that conservation principles are the result of thelaws of Nature. Since Einstein's work, however, physi-cists began to analyze the conservation principles asconsequences of certain underlying symmetry consid-erations from which they could be deduced, enablingthe laws of Nature to be revealed from this analysis.Wigner's profound impact on physics is that his expla-nations of 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 conser-vation of energy is a consequence of invariance undercontinuous temporal displacement.

A detailed exposition of the governing symmetry prin-ciples requires 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.

5. Dynamical Symmetry: Laplace{Runge{LenzVector

It is well known that in the classical two-body Keplerproblem (gravitational Sun{Earth system, or the Coulom-bic proton{electron planetary model of the old-quantum-theory of the hydrogen atom), both energy and angularmomentum are conserved. We have already discussed


GENERAL ARTICLE

the 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).

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=r2. 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:

~A =³~v £ ~H

´¡ e½ (11)

and is shown in Figure 3 [4]. In the above equation ~v

is 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 angu-lar momentum, 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 quan-tity. 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 2. If the eclipse were

to precess it would gener-

ate 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.

Figure 3. Schematic dia-

gramshowingthe Laplace–

Runge–Lenz vector, ~A .


GENERAL ARTICLE

The constancy of the LRL vector is a conservation prin-ciple, and since the governing criterion involves dynam-ics (namely that the force must have a strict inversesquare form), the associated symmetry is called `dynam-ical symmetry'. Sometimes, it is also called an àcciden-tal' 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 expe-rienced by the òuter-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 hydro-gen atom 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 the-oretical methods, 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 hydro-gen atom 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'¹n;l which makes the hydrogenic energy eigenvalues de-pend not merely on the principal quantum number n butalso on the orbital angular momentum quantum num-ber l. This enables the use of the hydrogenic formulafor energy with n replaced by ne®ective = n ¡ ¹n;l. The`quantum-defect theory' has very many applications inthe analysis of the atomic spectrum, including the àu-toionization resonances' [6,7]. As pointed out above,the conservation of angular momentum is due to the ro-tational symmetry, referred to as the symmetry underthe group SO(3). All central ¯elds have this symmetry.However, the inverse-square-law force (such as gravity or


GENERAL ARTICLE

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 di-mensionality of 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 ana-logue of the LRL vector [8].

6. Conclusion

The conservation of the generalized momentum which isconjugate to a cyclic coordinate is a generic expression ofa deeper relationship between symmetry and conserva-tion laws. 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.

Address for Correspondence


Department of Physics

Indian Institute of Technology

Madras

Chennai 600036.

Email: [email protected]

[email protected]

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://

nobelprize.org/nobel\_prizes/physics/laureates/1963/index.html

[4] For a detailed discussion of the Laplace–Runge–Lenz vector see H Gold-

stein, 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 and

S 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.

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.

Symmetry Principles and Conservation Laws in

Atomic and Subatomic Physics – Part II

926 RESONANCE October 2010

GENERAL ARTICLE

Symmetry Principles and Conservation Laws inAtomic and Subatomic Physics – 2


Keywords

Discrete symmetries, violation

of parity and CP, Higgs mecha-

nism, LHC.

(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 experimen-

tal investigations in this

field. He enjoys

teaching both undergradu-

ate 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.

Part 1: Resonance, Vol.15, No.9,

p.832.

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 conjuga-tion, and time reversal { that have played a sig-ni¯cant part in the development of the `standardmodel' of particle physics during the latter partof the 20th century. The importance of experi-mental tests of these symmetries, in both atomicand particle physics, and their sensitivity to newphenomena is also discussed. To conclude, wedescribe how `symmetry breaking' in the stan-dard model 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 Symmetries

Apart from continuous and dynamical symmetries, thereare other kinds of symmetries that are of importance inphysics. In particular, we have three discrete symme-tries of central importance in what is known as the `stan-dard model' of particle physics. These discrete symme-tries are: (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 conserva-tion principles operating either separately or in combi-nation. We shall now discuss these discrete symmetries.

927RESONANCE October 2010

GENERAL ARTICLE

Figure 1. Depending on the

plane of reflection, right

goes to left and top to bot-

tom; the primary feature

discussed in the text is

that ‘parity’ is an operation

that is essentially different

from ‘reflection’.

1.1 Parity

Parity 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 transforma-tion of a vector ~r to its image in a mirror placed in theCartesian yz-plane, then we can express the transforma-tion ~r = (x; y; z) to its image ~rI = (xI; yI; zI) by a matrixequation:

~rI = <~r : (1)

Now, in the case of re°ection, the determinant of thematrix < in the above relation is ¡1, whereas if onewrites a similar relation for the rotation of the vector~r to a new orientation ~rR, the corresponding matrix oftransformation would have for its determinant the value+1. The reason left goes to right and right to left, butnot the top to the bottom and bottom to the top, in amirror is that we usually tend to imagine the image tohave gone to the opposite side of the mirror through arotation about the vertical axis. If we imagine the rota-tion to be about the horizontal axis, we would certainlysee the top go to the bottom, the bottom to the top,but not left to right or right to left. Figure 1 illustratesthis. Of course, the fundamental reason is the intrin-sic di®erence between rotation and re°ection, exhibitedby the di®erent signs of the determinants of their ma-trices. The parity transformation is thus very di®erent

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.


GENERAL ARTICLE

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.

from rotation and one may ask, as Alice would (in Th-rough the 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 com-pares an object with its image in a mirror, whether par-ity is conserved in nature.

The parity operator ¦ is a unitary operator which anti-commutes with the position operator and also with theoperator for linear momentum, since both position andmomentum are polar vectors. However the parity opera-tor commutes 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 in-dicated that 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 po-larised nuclei via asymmetries in the distribution of the¯-decay electron with respect to the spin of the nucleus(Figure 2).

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.

The violation of

parity was

unexpected. It

allowed the first

unambiguous

definition of left

and right in nature.


GENERAL ARTICLE

Figure 2. Schematic (a) is

of the direction of the de-

cay electron, characterized

by momentum ,ep

with re-

spect to the spin of the 60Co

nucleus, .60CoJ

Schematic

(b) is the same process

transformed by the parity

operation. Unequal prob-

abilities for these two pro-

cesses to occur were ob-

served by Wu and collabo-

rators; this was the first ex-

perimental evidence for

parity violation in nature.

a) b)

Atomic transitions

are normally

governed by the

parity selection

rule, which then

breaks down for

those transitions in

which parity is not

conserved.

Parity violation is observed in nuclear and subatomic in-teractions, and through the uni¯cation of the weak andelectromagnetic interactions, parity is violated in certainatomic processes as well. Atomic transitions are nor-mally governed 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 Z0 boson of the standard model is neutral. Thelatter can mediate an interaction between atomic elec-trons and the nucleus. The nuclear weak charge QWof the standard model plays the same role with regardto Z0 that the ùsual' electric charge plays with regardto the Coulomb interaction. PNC e®ect in atomic ce-sium yields the value of QW (133Cs) ¼ ¡72:90, not farfrom the value of QW (133Cs) ¼ ¡73:09 obtained fromhigh-energy experiments extrapolated to atomic scale[3]. The Z-boson has a very large mass and the weak-interaction is `contact' type. It includes a parity-evenpart and a parity-odd (PNC) part. While the parity-even part leads to a correction to isotope shift and tohyper¯ne structure, the PNC part leads to the `pseudo-scalar' correlations in atomic processes.


GENERAL ARTICLE

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 in-teraction, it does not commute with the parity oper-ator and provides for non-zero probability for parity-violating atomic transitions. The two sources of paritynonconservation (PNC) in atoms are: (1) the electron-nucleus weak interaction and (2) the interaction (some-times called 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 par-ity. The anapole moment is a new electromagnetic mo-ment that can be possessed by an elementary particle(as well as composite systems like the nucleon or nu-cleus) 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 vec-tor potential at a ¯eld point after taking into accountthe constraints of current conservation and the bound-edness of the current density.

A signi¯cantly large value of the anapole moment of thenucleon has been estimated in the case of cesium, aug-mented by 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 atomexisted in an excited state that is a mix of states 2s 1

2and

2p 12

which have opposite parity, then parity would be vi-olated if the electromagnetic interactions alone were toexist. These two energy states are very nearly degener-ate and thus very sensitive to the electroweak interactionwhich would mix them. More recently, atomic parity vi-olation has been observed in the 6s2 1S0 ! 5d6s 3D1

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.


GENERAL ARTICLE

408 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 Symmetries

The discrete symmetry of charge conjugation (C) con-verts all 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 max-imally violated in weak interactions. However, the com-bined operation 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 vio-lation, 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 exper-iments of 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 physics

experiments

provide a low-

energy test of the

standard model

and also provide

relatively low-cost

tools to explore

physics beyond it.

Violation of CP is

essential to describe

the observed state of

the universe as being

matter dominated.


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 CP-

violation in weak

hadronic decays.

This confirmation

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.

and that time, postulated charm (c) quark. The ad-dition of 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 CP-violation in weak hadronic decays. The postulated thirdgeneration was not discovered until Lederman and col-laborators observed 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 Cen-ter, US, the High Energy Accelerator Research Organ-isation (KEK), Japan, and the Fermilab National Ac-celerator Laboratory, 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].

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 de-scribe the large matter-antimatter asymmetry observedin universe. Therefore, theories that go beyond thestandard model must accommodate new sources of CP-violation to explain the rate of baryogenesis. This meansthat the further study of CP-violation is extremely im-portant. 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 estab-lished [10]; however, an exposition of this exciting topicis beyond the scope of this article.


GENERAL ARTICLE

(a) (b)


gram showing the time-re-

versal relation between

photoionization and scat-

tering processes in atomic

physics.

1.3 CPT Symmetry

The `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, be-cause it 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 evolu-tion' is unitary, but that for time-reversal is antiunitary.The quantum mechanical operator ¦ for parity anticom-mutes with the position and the momentum operator,but commutes with the operator for angular momen-tum. On the other hand, the operator for time-reversal,£ commutes with the position operator, but anticom-mutes with both the linear and the angular momentumoperators.

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 con-jugation of 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 repre-sents the 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

In quantum theory,

the operator for

‘time evolution’ is

unitary, but that for

time-reversal is

antiunitary.


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.

unique. Despite the fact that the ingredients of theelectron{ion collision experiment and that of photoion-ization are completely di®erent, both the processes re-sult in the same ¯nal state consisting of an electronand an ion. The initial state, being obviously di®er-ent, 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 re-ferred to as òutgoing wave boundary condition' and ìn-going wave 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 de-cay of K mesons [7] therefore made it pertinent to lookfor the violation of the time-reversal symmetry. Viola-tion of T symmetry would require an elementary par-ticle, atom or molecule to possess a permanent electricdipole moment (EDM), since the only direction with

which an electric dipole moment ~d =j d j es could bede¯ned will have to be along the unit vector es, 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 distribu-tion to 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 elec-tric dipole 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 probe


GENERAL ARTICLE


gram explaining that the di-

pole moment of an elemen-

tary particle must be zero

unless T symmetry is bro-

ken. The existence of an

EDM also requires that P

symmetry is violated.

a) b)

to explore new physics beyond the standard model. High-precision measurements 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 Boson

Here 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 in-teractions of a scalar ¯eld Á is:

L =1

2(@¹Á)

2 ¡1

2¹2Á2 ¡

1

4¸Á4 ; (2)


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 particlemass and ¸ is the strength of the coupling of Á to it-self. 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: ¹2 > 0 and ¹2 < 0. For the caseof an imaginary mass (¹2 < 0) there are two minima at

Ámin = §À = §

r¡¹2

¸: (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 ¡À

Á = À + ¾(x) ; (4)

where ¾(x) is the variable value of the ¯eld above theconstant uniform value of À. Substituting this expres-sion for Á into (2) one gets:

L =1

2(@¹¾)2 ¡ ¸À2¾2 ¡

µ

¸À¾3 +1

4¸¾4

¶

+ constant ;

(5)where the constant term depends on À2 and À4 and thethird term (in parenthesis) on the right-hand side de-scribes self interactions. The second term correspondsto a mass term with real mass

m =p

2¸À2 =p¡2¹2 : (6)

The breaking of

symmetryprovides

a hypothesis for

the generation of

all particle masses

– the Higgs

mechanism.


GENERAL ARTICLE

The perturbative expansion about one of the two min-ima has led to a real mass appearing. Since the expan-sion is 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 sym-metry breaking 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 La-grangian describing the ¯eld of the magnet would beinvariant under rotations. However, on cooling, the do-mains set in a particular direction, causing an over-all ¯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§ andZ0, 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 prop-erties of the Z0 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 up-per limit on the mass of the Higgs boson of 157 GeV=c2

with 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 the

Nambu and Jona-

Lasinio 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.


GENERAL ARTICLE

Figure 6. Computer-gener-

ated image shows the loca-

tion of the 27-km LHC tun-

nel (in blue) on the Swiss–

France border. The four

main experiments (ALICE,

ATLAS,CMS,andLHCb)are

located in underground

caverns connected to the

surface by 50 m to 150 m

pits. Part of the pre-accel-

eration chain is shown in

grey.

Higgs boson of 114 GeV=c2 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 coun-terclockwise about the accelarator [14]. Superconduct-ing dipole 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 centre-of-mass collision 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 col-lisions. 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, respec-tively. 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 ex-hibit CP 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.

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.


GENERAL ARTICLE

Beams of protons were successfully circulated in bothdirections about the LHC in September 2008. Unfor-tunately shortly afterward a fault in one of the 1232superconducting dipole magnets led to signi¯cant dam-age in one part of the accelerator. Repairs and imple-mentation of 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.

3. Conclusions

This article (Parts 1 and 2) presents a pedagogical sum-mary of the importance of symmetry principles in de-scribing many 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 symme-tries P, C and T were discussed, along with how their vi-olation is embedded within the standard model of parti-cle physics. 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 at

http:nobelprize.org/nobel\_prizes/physics/laureates/1957/index.html

Within the next five

years the LHC will

either confirm the

Higgs mechanism or

shed light on an

alternative model of

mass generation.


GENERAL ARTICLE

Address for Correspondence


Department of Physics

Indian Institute of Technology

Madras

Chennai 600036.

Email: [email protected]

[email protected]

[2] D Budker, D F Kimball and D P DeMille, Atomic Physics: An explora-

tion 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 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 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://nobelprize .org/nobel\_ prizes/physics/laureates/1980/

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.

[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 of

the 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 at

http://public.web.cern.ch/public/en/LHC/LHC-en.html .

Documents

Symmetry Principles and ... - Department of Physicsphysics.iitm.ac.in/~labs/amp/Symmetry_Principles_Conservation_Laws.pdfAtomic and Subatomic Physics – 1 P C Deshmukh and J Libby