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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
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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.
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Newton’s third law
introduces a
simple illustration
of the relation
between a
symmetry and a
conservation law.
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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.
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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
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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-
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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
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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
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(9)
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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
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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
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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
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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.
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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.
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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.
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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’.
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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.
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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
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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