Introduction
to
Groups, Invariants
and
Particles
Frank W. K. Firk, Emeritus Professor of Physics, Yale University
2000
CONTENTS 2
1. INTRODUCTION 3
2. GALOIS GROUPS 8
3. SOME ALGEBRAIC INVARIANTS 20
4. SOME INVARIANTS IN PHYSICS 29
5. GROUPS CONCRETE AND ABSTRACT 43
6. LIE’S DIFFERENTIAL EQUATION, INFINITESIMAL
ROTATIONS, AND ANGULAR MOMENTUM 57
7. LIE’S CONTINUOUS TRANSFORMATION GROUPS 68
8. PROPERTIES OF nVARIABLE rPARAMETER
LIE GROUPS
9. MATRIX REPRESENTATIONS OF GROUPS
10. SOME LIE GROUPS OF TRANSFORMATIONS
11. THE GROUP STRUCTURE OF LORENTZ
TRANSFORMATIONS
12. ISOSPIN
78
83
94
107
115
13. LIE GROUPS AND THE STRUCTURE OF MATTER 128
14. LIE GROUPS AND THE CONSERVATION LAWS
OF THE PHYSICAL UNIVERSE
159
15. BIBLIOGRAPHY 163
1 3
INTRODUCTION
The notion of geom etrical symmetr y in Art and in Natu re is a
familiar one. In Modern Physics, this notion ha s evolved to include
symmetries of an abstract kind. These new symmetries play an essential
par t in the theories of the microstructure of matter. The basic symmetries
found in Natu re seem to originate in t he ma thematical stru cture of t he laws
themselves . laws that go vern the motions of the g alaxies on the one
hand and the motions of quarks in nucleons on the oth er.
In the N ewtonian era, t he laws of Natu re we re de duced from a sm all
number of imperfect obs ervations by a small number of renowned
scientists and mathematicians. It was not until the Einsteinian era ,
however, that the significance of the symmetries associated with the laws
was fully app reciated. The discovery of spacetime symmetries has led to
the widelyheld belief that the laws of Nature c an be derived from
symmetry , or invariance, principles. Our incomplete knowledge of the
fundamental interactions means th at we are not yet in a position to confirm
this belief. We therefore use arguments based on emp irically established
laws and rest ricted sym metry principles to guide us in our search for the
fundamental symmetries. Fre quently, it is important to understand w hy
the symm etry of a system is observed t o be broken.
In Geometry, an object with a definite shape, size, location, and 4
orientation constitutes a st ate whose symmetry proper ties, or i nvariants,
are to be studied. Any transformation that leaves the state un changed in
form is called a s ymmetry transformation. The greater the number of
symmetry transformations that a s tate can underg o, the higher its
symmetry . If the number of conditions that define th e state is reduced
then the symm etry of the state is increased. For exa mple, an object
characterized by oblateness alone is symmet ric under all transformat ions
except a change of shape.
In describing the symmetry of a state of the most general kind (not
simply geometric), the algebraic structure of the set of symmetry operators
must be given; it is not sufficient to give the number of operations, and
not hing else. The law of combination of the operators must be stated. It
is the algebraic group that fully characterizes the symmetry of t he general
state.
The Theory of Groups came ab out unexpectedly. G alois showed
that an equation of degree n , where n is an integer greater than or equal to
five cannot, in general, be solved by algebraic means. In the course of this
gre at wo rk, h e developed the ideas of Lagra nge, Ruffini, and Ab el and
introduced th e concept of a group. Galois discussed the functional
relationships amon g the roots of an equation, and showed t hat the
relationships have symm etries associated with them un der permut ations of
the roots.
The operators that transform one functional relationship into 5
another are elements of a se t that is characteristic of the equation ; the set
of operators is called the G alois group of the equati on.
In the 1 850’s, Cayley showed that ever y finite group is isomorp hic
to a cer tain permu tation group. The geometrical symm etries of crystals
are described in t erms of finite group s. These symmetries are discussed in
many standard works (see bibliography) and therefore, they will not be
discussed in this book.
In the b rief period bet ween 1924 and 1 928, Quantum Mechanics
was developed. Almost immediately, it was recognized by Weyl, and by
Wigner, that certain parts o f Group Theory could be used as a p owerful
analytical tool in Quan tum Physics. Their ideas have been developed over
the decades in man y are as that range from the Theory of Solids to Particle
Phy sics.
The essential role played by groups th at are characterized by
par ameters th at va ry continuously in a given range was first emphasized
by Wigner. T hese group s are know n as Lie Groups. They have become
increasingly important in many branches of contempora ry physics,
par ticularly Nuclear an d Particle Phys ics. Fifty years after Galois had
introduced th e concept of a group in t he Theory of Eq uations, Lie
introduced th e concept of a continuous transformation group in the Theory
of Differential Equations. Lie’s theo ry unified many of t he disconnected
methods of solving differential equations t hat h ad evolved over a period of
two hundred years. Infinitesimal unitary transformat ions play a key role in 6
discussions of the fundamental conservation laws of Physics.
In Classical Dynamics, the invariance of the equations of motion of a
particle, or system of particles, unde r the Galilean transformation is a basic
par t of everyday relativity. The search for the transformation that leaves
Maxwell’s equations of Electromag netism unc hanged in form (invariant)
under a linear transformation of the spacetime coordinates, led to the
discovery of the Lorent z transformation. The fundamental importance of
this tra nsformation, and its related invariants, cannot be overstated.
This introduction to Group Theory , with its emph asis on Lie Groups
and their application to the stud y of symmetries of t he fundamental
constituents of matter, has its origin in a onesemester course that I ta ught
at Yale University for more than ten y ears. The course was developed for
Seniors, and advanced Juniors, majoring in the Physical Sciences. The
students had generally completed the core c ourses for their majors, and
had taken intermediate level courses in Linear A lgebra, Real and Com plex
Analysis, Ord inary Linear Differential Equations, and some of t he Sp ecial
Fun ctions of Physics. Group Theory was not a mathematical requirement
for a degree in the Phy sical Sciences. The majority of existing
undergra duate textbooks on Group Theory and its applications in Phys ics
tend to be either highly qua litative or highly mathematical. The pu rpose of
this introduction is to steer a middle course that provides the stud ent w ith
a sound mathematical basis for st udying the symm etry prope rties of t he
fundamental particles. It is not generally appreciated by Physicists that 7
continuous tr ansformation groups originated in t he Theory of Differential
Equ ations. The infinitesimal generators of Lie Group s therefore have
forms that involve differential operators and their commutators, and these
operators and their algebraic properties have found, and continue to find, a
natural place in the development of Quantum Physics.
Guilford, CT.
June, 2000
2 8
GALOIS GROUPS
In the early 19th  century, Abel proved that it is not possible to solve the
general polynomial equation of degree greater than four by algebraic means.
He attempted to characterize all equations that can be solved by radicals.
Abel did not solve this fundamental problem. The problem was taken up and
solved by one of the greatest innovators in Mathematics, namely, Galois.
2.1. Solving cubic equations
The main ideas of the Galois procedure in the Theory of Equations,
and their relationship to later developments in Mathematics and Physics, can
be introduced in a plausible way by considering the standard problem of
solving a cubic equation.
Consider solutions of the general cubic equation
Ax3 + 3Bx2 + 3Cx + D = 0, where A  D are rational constants.
If the substitution y = Ax + B is made, the equation becomes
y3 + 3Hy + G = 0
where
H = AC  B2
and
G = A2D  3ABC + 2B3.
The cubic has three real roots if G2 + 4H3 < 0 and two imaginary roots if G2 9
+ 4H3 > 0. (See any standard work on the Theory of Equations).
If all the roots are real, a trigonometrical method can be used to obtain
the solutions, as follows:
the Fourier series of cos3u is
cos3u = (3/4)cosu + (1/4)cos3u.
Putting
y = scosu in the equation y3 + 3Hy + G = 0
(s > 0),
gives
cos3u + (3H/s2)cosu + G/s3 = 0.
Comparing the Fourier series with this equation leads to
s = 2 v(H)
and
cos3u = 4G/s3.
If v is any value of u satisfying cos3u = 4G/s3, the general solution is
3u = 2np ± 3v, where n is an integer.
Three different values of cosu are given by
u = v, and 2p/3 ± v.
The three solutions of the given cubic equation are then
scosv, and scos(2p/3 ± v).
Consider solutions of the equation
x3 3x + 1 = 0. 10
In this case,
H = 1 and G2 + 4H3 = 3.
All the roots are therefore real, and they are given by solving
cos3u = 4G/s3 = 4(1/8) = 1/2
or,
3u = cos1(1/2).
The values of u are therefore 2p/9, 4p/9, and 8p/9, and the roots are
x1 = 2cos(2p/9), x2 = 2cos(4p/9), and x3 = 2cos(8p/9).
2.2. Symmetries of the roots
The roots x1, x2, and x3 exhibit a simple pattern. Relationships among
them can be readily found by writing them in the complex form 2cos. = ei. +
ei. where . = 2p/9 so that
x1 = ei. + ei. ,
x2 = e2i. + e2i. ,
and
x3 = e4i. + e4i. .
Squaring these values gives
x12 = x2 + 2,
x22 = x3 + 2,
and
x32 = x1+ 2. 11
The relationships among the roots have the functional form f(x1,x2,x3) = 0.
Other relationships exist; for example, by considering the sum of the roots we
find
x1 + x22 + x2  2 = 0
x2 + x32 + x3  2 = 0,
and
x3 + x12 + x1  2 = 0.
Transformations from one root to another can be made by doublingthe
angle, .
The functional relationships among the roots have an algebraic
symmetry associated with them under interchanges (substitutions) of the
roots. If is the operator that changes f(x1,x2,x3) into f(x2,x3,x1) then
f(x1,x2,x3) . f(x2,x3,x1),
2f(x1,x2,x3) . f(x3,x1,x2),
and
3f(x1,x2,x3) . f(x1,x2,x3).
3
The operator = I, is the identity.
In the present case,
 2) = (x22  x3  2) = 0,
(x12  x2
and
2(x12  x2  2) = (x32  x1  2) = 0. 12
2.3. The Galois group of an equation.
The set of operators {I, , 2} introduced above, is called the Galois
group of the equation x3  3x + 1 = 0. (It will be shown later that it is
isomorphic to the cyclic group, C3).
The elements of a Galois group are operators that interchange the
roots of an equation in such a way that the transformed functional
relationships are true relationships. For example, if the equation
x1 + x22 + x2  2 = 0
is valid, then so is
(x1 + x22 + x2  2 ) = x2 + x32 + x3  2 = 0.
True functional relationships are polynomials with rational coefficients.
2.4. Algebraic fields
We now consider the Galois procedure in a more general way. An
algebraic solution of the general nth  degree polynomial
aoxn + a1xn1 + ... an = 0
is given in terms of the coefficients ai using a finite number of operations (+,
,×,÷,v). The term "solution by radicals" is sometimes used because the
operation of extracting a square root is included in the process. If an infinite
number of operations is allowed, solutions of the general polynomial can be
obtained using transcendental functions. The coefficients ai necessarily belong 13
to a field which is closed under the rational operations. If the field is the set
of rational numbers, Q, we need to know whether or not the solutions of a
given equation belong to Q. For example, if
x2  3 = 0
we see that the coefficient 3 belongs to Q, whereas the roots of the equation,
xi = ± v3, do not. It is therefore necessary to extend Q to Q', (say) by
adjoining numbers of the form av3 to Q, where a is in Q.
In discussing the cubic equation x3  3x + 1 = 0 in 2.2, we found
certain functions of the roots f(x1,x2,x3) = 0 that are symmetric under
permutations of the roots. The symmetry operators formed the Galois group
of the equation.
For a general polynomial:
xn + a1xn1 + a2xn2 + .. an = 0,
functional relations of the roots are given in terms of the coefficients in the
standard way
x1 + x2 + x3 .. .. + xn = a1
x1x2 + x1x3 + .. x2x3 + x2x4 + ..+ xn1xn = a2
x1x2x3 + x2x3x4 + .. .. + xn2xn1xn = a3
.
.
x1x2x3 .. .. xn1xn = ±an.
Other symmetric functions of the roots can be written in terms of these 14
basic symmetric polynomials and, therefore, in terms of the coefficients.
Rational symmetric functions also can be constructed that involve the roots
and the coefficients of a given equation. For example, consider the quartic
x4 + a2x2 + a4 = 0.
The roots of this equation satisfy the equations
x1 + x2 + x3 + x4 = 0
x1x2 + x1x3 + x1x4 + x2x3 + x2x4 + x3x4 = a2
x1x2x3 + x1x2x4 + x1x3x4 + x2x3x4 = 0
x1x2x3x4 = a4.
We can form any rational symmetric expression from these basic
equations (for example, (3a43  2a2)/2a42 = f(x1,x2,x3,x4)). In general, every
rational symmetric function that belongs to the field F of the coefficients, ai, of
a general polynomial equation can be written rationally in terms of the
coefficients.
The Galois group, Ga, of an equation associated with a field F therefore
has the property that if a rational function of the roots of the equation is
invariant under all permutations of Ga, then it is equal to a quantity in F.
Whether or not an algebraic equation can be broken down into simpler
equations is important in the theory of equations. Consider, for example, the
equation
x6 = 3.
It can be solved by writing x3 = y, y2 = 3 or 15
x = (v3)1/3.
To solve the equation, it is necessary to calculate square and cube roots
. not sixth roots. The equation x6 = 3 is said to be compound (it can be
broken down into simpler equations), whereas x2 = 3 is said to be atomic.
The atomic properties of the Galois group of an equation reveal
the atomic nature of the equation, itself. (In Chapter 5, it will be seen that a
group is atomic ("simple") if it contains no proper invariant subgroups).
The determination of the Galois groups associated with an arbitrary
polynomial with unknown roots is far from straightforward. We can gain
some insight into the Galois method, however, by studying the group
structure of the quartic
x4 + a2x2 + a4 = 0
with known roots
x1 = ((a2 + µ)/2)1/2 , x2 = x1,
x3 = ((a2  µ)/2)1/2 , x4 = x3,
where
µ = (a 22  4a4)1/2.
The field F of the quartic equation contains the rationals Q, and the
rational expressions formed from the coefficients a2 and a4.
The relations
x1 + x2 = x3 + x4 = 0
are in the field F.
16
Only eight of the 4! possible permutations of the roots leave these
relations invariant in F; they are
x1 x2 x3 x4 x1 x2 x3 x4 x1 x2 x3 x4
{ P1 = , P2 = , P3 = ,
x1 x2 x3 x4 x1 x2 x4 x3 x2 x1 x3 x4
x1 x2 x3 x4 x1 x2 x3 x4 x1 x2 x3 x4
P4 = , P5 = , P6 =
x2 x1 x4 x3 x3 x4 x1 x2 x3 x4 x2 x1
x1 x2 x3 x4 x1 x2 x3 x4
P7 = , P8 = }.
x4 x3 x1 x2 x4 x3 x2 x1
The set {P1,...P8} is the Galois group of the quartic in F. It is a subgroup of
the full set of twentyfour permutations. We can form an infinite number of
true relations among the roots in F. If we extend the field F by adjoining
irrational expressions of the coefficients, other true relations among the roots
can be formed in the extended field, F'. Consider, for example, the extended
field formed by adjoining µ (= (a22  4a4)) to F so that the relation
x12  x32 = µ is in F'.
We have met the relations
x1 = x2 and x3 = x4
so that
x12 = x22 and x32 = x42.
Another relation in F' is therefore
x22  x42 = µ. 17
The permutations that leave these relations true in F' are then
{P1, P2, P3, P4}.
This set is the Galois group of the quartic in F'. It is a subgroup of the set
{P1,...P8}.
If we extend the field F' by adjoining the irrational expression
((a2  µ)/2)1/2 to form the field F'' then the relation
x3  x4 = 2((a2  µ)/2)1/2 is in F''.
This relation is invariant under the two permutations
{P1, P3}.
This set is the Galois group of the quartic in F''. It is a subgroup of the set
{P1, P2, P3, P4}.
If, finally, we extend the field F'' by adjoining the irrational
((a2 + µ)/2)1/2 to form the field F''' then the relation
x1  x2 = 2((a2  µ)/2)1/2 is in F'''.
This relation is invariant under the identity transformation, P1 , alone; it is
the Galois group of the quartic in F''.
The full group, and the subgroups, associated with the quartic equation
are of order 24, 8, 4, 2, and 1. (The order of a group is the number of
distinct elements that it contains). In 5.4, we shall prove that the order of a
subgroup is always an integral divisor of the order of the full group. The
order of the full group divided by the order of a subgroup is called the index 18
of the subgroup.
Galois introduced the idea of a normal or invariant subgroup: if H is a
normal subgroup of G then
HG  GH = [H,G] = 0.
(H commutes with every element of G, see 5.5).
Normal subgroups are also called either invariant or selfconjugate subgroups.
A normal subgroup H is maximal if no other subgroup of G contains H.
2.5. Solvability of polynomial equations
Galois defined the group of a given polynomial equation to be either
n, (see 5.6). The Galois method
therefore involves the following steps:
the symmetric group, Sn, or a subgroup of S
1. The determination of the Galois group, Ga, of the equation.
2. The choice of a maximal subgroup of H
max(1). In the above case, {P
4.
1, ...P8}
is a maximal subgroup of Ga = S
3. The choice of a maximal subgroup of Hmax(1) from step 2.
In the above case, {P1,..P4} = Hmax(2) is a maximal subgroup of Hmax(1).
The process is continued until Hmax = {P1} = {I}.
The groups Ga, Hmax(1), ..,Hmax(k) = I, form a composition series. The
composition indices are given by the ratios of the successive orders of the
groups:
gn/h(1), h(1)/h(2), ...h(k1)/1. 19
The composition indices of the symmetric groups Sn for n = 2 to 7 are found
to be:
n Composition Indices
2 2
3 2, 3
4 2, 3, 2, 2
5 2, 60
6 2, 360
7 2, 2520
We shall state, without proof, Galois' theorem:
A polynomial equation can be solved algebraically if and only if its
group is solvable.
Galois defined a solvable group as one in which the composition indices are
all prime numbers. Furthermore, he showed that if n > 4, the sequence of
maximal normal subgroups is Sn, An, I, where An is the Alternating Group
with (n!)/2 elements. The composition indices are then 2 and (n!)/2. For n >
4, however, (n!)/2 is not prime, therefore the groups Sn are not solvable for n
> 4. Using Galois' Theorem, we see that it is therefore not possible to solve,
algebraically, a general polynomial equation of degree n > 4.
3 20
SOME ALGEBRAIC INVARIANTS
Although algebraic invariants first appeared in the works of Lagrange and
Gauss in connection with the Theory of Numbers, the study of algebraic
invariants as an independent branch of Mathematics did not begin until the
work of Boole in 1841. Before discussing this work, it will be convenient to
introduce matrix versions of real bilinear forms, B, defined by
mn
B = .. aijxiyj
I=1 j=1
where
x = [x1,x2,...xm], an mvector,
y = [y1,y2,...yn], an nvector,
and aij are real coefficients. The square brackets denote a
column vector.
In matrix notation, the bilinear form is
B = xTAy
where
a11 . . . a1n A = am1. . . amn
The scalar product of two nvectors is seen to be a special case of a
bilinear form in which A = I.
If x = y, the bilinear form becomes a quadratic form, Q:
21
Q = xTAx.
3.1. Invariants of binary quadratic forms
Boole began by considering the properties of the binary
quadratic form
Q(x,y) = ax2 + 2hxy + by2
under a linear transformation of the coordinates
x' = Mx
where
x = [x,y],
x' = [x',y'],
and
 i j
M = .
 k l
The matrix M transforms an orthogonal coordinate system into an
oblique coordinate system in which the new x' axis has a slope (k/i), and the
new y' axis has a slope (l/j), as shown:
y 22
y'
[i+j,k+l]
[j,l]
x'
[0,1]
x'
[i,k]
[1,1]
[0,0] [1,0] x
Fig. 1. The transformation of a unit square under M.
The transformation is linear, therefore the new function Q'(x',y') is a
binary quadratic:
Q'(x',y') = a'x'2 + 2h'x'y' + b'y'2.
The original function can be written
Q(x,y) = xTDx
where
 a h
D = ,
 h b
and the determinant of D is
detD = ab  h2, called the discriminant of Q.
The transformed function can be written
Q'(x',y') = x'TD'x' 23
where
a' h'
D' = ,
h' b'
and
detD' = a'b'  h'2, the discriminant of Q'.
Now,
Q'(x',y') = (Mx)TD'Mx
= xTMTD'Mx
and this is equal to Q(x,y) if
MTD'M = D.
The invariance of the form Q(x,y) under the coordinate transformation M
therefore leads to the relation
(detM)2detD' = detD
because
detMT = detM.
The explicit form of this equation involving determinants is
(il  jk)2(a'b'  h'2) = (ab  h2).
The discriminant (ab  h2) of Q is said to be an invariant
of the transformation because it is equal to the discriminant (a'b'  h'2) of Q',
apart from a factor (il  jk)2 that depends on the transformation itself, and not
on the arguments a,b,h of the function Q.
3.2. General algebraic invariants 24
The study of general algebraic invariants is an important branch of
Mathematics.
A binary form in two variables is
n
f(x1,x2) = aox1n + a1x1n1x2 + ...anx2
= . aix1nix2i
If there are three or four variables, we speak of ternary forms or quaternary
forms.
A binary form is transformed under the linear transformation M as
follows
f(x1,x2) => f'(x1',x2') = ao'x1'n + a1'x1'n1x2' + ..
The coefficients
ao, a1, a2,... ao', a1', a2' ..
and the roots of the equation
f(x1,x2) = 0
differ from the roots of the equation
f'(x1',x2') = 0.
Any function I(ao,a1,...an) of the coefficients of f that satisfies
rwI(ao',a1',...an') = I(ao,a1,...an)
is said to be an invariant of f if the quantity r depends only on the
transformation matrix M, and not on the coefficients ai of the function being
transformed. The degree of the invariant is the degree of the coefficients, and
the exponent w is called the weight. In the example discussed above, the 25
degree is two, and the weight is two.
Any function, C, of the coefficients and the variables of a form f that is
invariant under the transformation M, except for a multiplicative factor that is
a power of the discriminant of M, is said to be a covariant of f. For binary
forms, C therefore satisfies
rwC(ao',a1',...an'; x1',x2') = C(ao,a1,...an; x1,x2).
It is found that the Jacobian of two binary quadratic forms, f(x1,x2) and
g(x1,x2), namely the determinant
.f/.x1 .f/.x2
.g/.x1 .g/.x2
where .f/.x1
is the partial derivative of f with respect to xsimultaneous covariant of weight one of the two forms.
The determinant
1 etc., is a
.2f/.x12 .2f/.x1.x2
,
.2g/.x2.x1 .2g/.x22
called the Hessian of the binary form f, is found to be a covariant of weight
two. A full discussion of the general problem of algebraic invariants is outside
the scope of this book. The following example will, however, illustrate the 26
method of finding an invariant in a particular case.
Example:
To show that
(aoa2  a12)(a1a3  a22)  (aoa3  a1a2)2/4
is an invariant of the binary cubic
f(x,y) = aox3 + 3a1x2y + 3a2xy2 + a3y3
under a linear transformation of the coordinates.
The cubic may be written
f(x,y) = (aox2+2a1xy+a2y2)x + (a1x2+2a2xy+a3y2)y
= xTDx
where
x = [x,y],
and
aox + a1y a1x + a2y
D = .
a1x + a2y a2x + a3y
Let x be transformed to x': x' = Mx, where
i j
M =
k l
then 27
f(x,y) = f'(x',y')
if
D = MTD'M.
Taking determinants, we obtain
detD = (detM)2detD',
an invariant of f(x,y) under the transformation M.
In this case, D is a function of x and y. To emphasize this point, put
detD = f(x,y)
and
detD'= f'(x',y')
so that
f(x,y) = (detM)2f'(x',y'
= (aox + a
1y)(a2x + a3y)  (a1x + a2y)2
= (aoa2  a12)x2 + (aoa3  a1a2)xy + (a1a3  a22)y2
= xTEx,
where
E =
Also, we have
(aoa2  a12 ) (aoa3  a1a2)/2
(aoa3  a1a2)/2 (a1a3  a22 )
.
f'(x',y') = x'TE'x'
= xTMTE'Mx 28
therefore
xTEx = (detM)2xTMTE'Mx
so that
E = (detM)2MTE'M.
Taking determinants, we obtain
detE = (detM)4detE'
= (aoa2  a12)(a1a3  a22)  (aoa3 a1a2)2/4
= invariant of the binary cubic f(x,y) under the transformation
x' = Mx.
4 29
SOME INVARIANTS IN PHYSICS
4.1. Gal ilean invariance.
Eve nts o f finite extension and du ration are part of t he ph ysical
wor ld. It will be conv enient to introduce the n otion of ideal ev ents that
have neither extension nor duration. Ideal events may be repre sented as
mathematical points in a spacetime ge ometr y. A particular eve nt, E, is
described by the four components [t,x,y,z] where t is the time of the event,
and x,y,z, are its three spatial coordinates. The time and space coordinates
are referred to arbitrarily chosen origins. The spatial mesh need not be
Cartesian.
Let an event E[t,x], recorded by an observer O at the origin of an x
axis, be reco rded as the event E'[t ',x'] by a second ob server O', moving at
constant speed V along the x axis. We suppose that their clocks are
synchron ized at t = t' = 0 when t hey coincide at a common origin, x = x' =
0.
At time t, we write the plausible equations
t' = t
and
x' = x  Vt,
where Vt is the distance travelled by O' in a time t. The se equations can
be written
E' = GE
where 30
1 0
G = .
V 1
G is the operator of the Galilean transformation.
The inverse equations are
t =t'
and
x = x ' + V t'
or
E = G1E'
where G1 is the inverse Galilean ope rator. (It undoes the effect of G).
If we mu ltiply t and t' by the constants k and k ', respectively, where
k a nd k' have dimensions of velocity then all terms h ave dimensions of
length.
In spacespace, we have the Pythagorean form x2 + y2 = r2, an
invariant under rotations. We are the refore led to ask th e question: is
(kt)2 + x2 invariant under the op erator G in spacetime? C alculation gives
(kt)2 + x2 = (k't ')2 + x'2 + 2Vx't' + V 2t'2
= (k't' )2 + x'2 o nly if V = 0.
We see, therefore, that Galilean spacetime is not Pythagorean in its
algebraic form. We not e, however, the key role played by acceleration in
GalileanNewtonian physics:
The velocities of the events according to O and O' are obtained by 31
differentiating the equation x' = Vt + x with respect to time, giving
v' = V + v,
a p lausible result, based up on ou r experience.
Differentiating v' with resp ect t o time gives
d v'/dt' = a ' = d v/dt = a
where a and a' are the accelerations in the two frames of reference. The
classical acceleration is invariant un der the Galilean transformation. If the
relationship v' = v  V is used to describe the motion of a pulse of light,
moving in emp ty space at v = c . 3 x 108 m/s, it does not fit the facts. All
studies of very high speed particles that e mit electromagnetic radiation
show tha t v' = c f or all va lues of the relative speed, V.
4.2. Lorentz invariance and Einstein's spacetime
symmetry .
It was E instein, above all others, who adv anced our understanding of
the true natu re of spacetime and relative motion. We shall see tha t he
made use of a symm etry argum ent to find the changes that must b e made
to the G alilean transformation if it is to account for the relative motion of
rapidly moving objects and o f beams of light. H e rec ognized an
inconsistency in t he GalileanNewtonian equ ations, based as they are , on
everyday experience. H ere, we shall restrict the discussion to non
accelerating, or so called inertial, frames
We have seen that the classical equations relating the events E an d 32
E' a re E' = GE, and the inverse E = G1E'
where
1 0 1 0
G = and G1 = .
V 1 V 1
These equations ar e connected by the substitution V .V; this is an
algebraic statement of the N ewtonian principle of relativity. Einstein
incorporated this principle in his theory. He a lso retained th e lin earity of
the classical equations in t he absence of a ny ev idence to the c ontra ry.
(Equispaced interv als of time and distance in one inertial frame remain
equispaced in any other inertial frame). He the refore symmetrized the
spacetime eq uations as follows:
t' 1 V t
= .
x' V 1 x
Note, however, the inconsistency in the dimensions of the timeequation
that has now been introduced:
t' = t  Vx.
The term Vx has dimensions of [L]2/[T], and not [T]. This can be
cor rected by introducing the invariant speed of light, c . a postulate in
Einstein's theory that is consistent w ith experiment:
ct' = ct  Vx/c
so that all terms now h ave dimensions of length.
Einstein went further, and introd uced a dimensionless quantity . 33
instead of the scaling factor of unity that appears in the Galilean equations
of spacetime. This factor must be consistent w ith all observations. The
equations then become
ct ' = .ct  ß.x
x ' = ß.ct + .x, where ß=V/c.
These can be written
E' = LE,
where
. ß.
L = , and E = [ct,x]
ß. .
L is the operator of the Lorentz transformat ion.
The inverse equation is
E = L1E'
where
. ß.
L1 = .
ß. .
This is the inverse Lorentz transformation, obta ined from L by changing
ß . ß (or ,V .V); it h as the effect o f undoing the transformat ion L.
We can t herefore write
LL1 = I
or
. ß. .ß. 1 0 34
= .
ß.. ß.. 0 1
Equ ating elements gives
.2 ß2.2 = 1
therefore,
. = 1/v(1 ß2) (taking the positive root).
4.3. The invariant interval.
Pre viously, it was shown tha t the spacetime of Galileo and Newton
is not Pythagorean in form. We now ask the question: is Einsteinian space
time Pythagor ean in for m? Direct calculation leads to
(ct)2 + (x)2 = .2(1 + ß2)(ct')2 + 4ß.2x'c t'
+.2(1 + ß2)x'2
. (ct')2 + (x')2 if ß > 0.
Note, however, that the dif ference of squares is an
invariant under L:
(ct)2  (x)2 = (ct')2  (x')2
because
.2(1 ß2) = 1.
Spacetime is said to be pseudoEuclidean.
The negative sign that characterizes Lorent z invariance can be
included in t he th eory in a general way as follows.
We introduce two k inds of 4vectors
xµ = [x0, x1, x2, x3], a contravariant vector, 35
and
xµ = [x0, x1, x2, x3], a covariant ve ctor, where
xµ = [x0,x1,x2,x3].
The scalar product of t he vectors is defined as
x µTxµ = (x0, x1, x2, x3)[x0,x1,x2,x3]
= (x0)2  ((x1)2 + (x2)2 + (x3)2)
The event 4vector is
Eµ = [ct, x, y, z] and the covariant for m is
Eµ = [ct,x,y,z]
so that the Lorent z invariant scalar produc t is
EµTEµ = (ct)2  (x2 + y2 + z2).
The 4vector xµ tr ansforms a s follows:
x'µ = Lxµ
where
. ß. 0 0
ß. . 0 0
L = .
0 0 1 0
0 0 0 1
This is the o perator of the Lorentz transformation if the motion of O' is
along the xaxis of O's frame of reference.
Importan t con sequences of the Lorentz transformation are that
intervals of time measured in two different inertial frames are not the same
but are related by the equation
.t' = ..t 36
where .t is an interval measured on a clock at res t in O's frame, and
distances are given by
.l' = .l/.
where .l is a length measured on a ruler at rest in O's frame.
4.4. The energymomentum invariant.
A d ifferential time interval, dt, cannot be used in a Lorentzinvariant
way in kinematics. We must use the proper time differential interval, dt,
defined by
( cdt)2 dx2 = (cdt')2 dx'2 = (cdt)2.
The Newt onian 3velocity is
vN = [dx/dt, dy/dt, dz/dt],
and this must be replaced by the 4velocity
Vµ = [d(ct)/dt, dx/dt, dy/dt, dz/dt]
= [ d(ct)/dt, dx/dt, dy/dt, d z/dt]dt/dt
= [.c,.vN] .
The scalar product is then
VµVµ = (.c)2 (.vN)2
= (.c)2(1  (vN/c)2)
= c2.
(In form ing the scalar produ ct, the transpose is unde rstood).
The magn itude of t he 4velocity is .Vµ . = c, the invariant speed of light.
In Classical Mechanics, the concept of momentum is importa nt because 37
of its role as an invariant in an isolated system. We the refore introduce the
concept of 4momentum in Relativistic Mechanics in order to find
possible Lorentz invariants involving this new q uantity. The contra variant
4momentum is defined as:
Pµ = mVµ
where m is the mass of the p article. (It is a Lorentz scalar, t he ma ss
measured in t he frame in which th e par ticle is at res t).
The scalar product is
P µPµ =
Now,
Pµ =
therefore,
P µPµ =
Writing
(mc)2.
[m.c, m.vN]
(m.c)2  (m.vN)2.
M = .m, the relativistic mass, we obta in
PµPµ = (Mc)2  (MvN)2 = (mc)2.
Multiplying throug hout by c2 gives
M2c4  M2vN2c2 = m2c4.
The quan tity Mc2 ha s dimensions of ener gy; we the refore write
E = Mc2
the total energy of a freely moving pa rticle.
This leads to the fundamental invariant of dynamics 38
c2PµPµ = E2  (pc)2 = Eo2
where
Eo = mc2 is the rest energy of the p article, and
p is its relativistic 3momentum.
The total energy can be written:
E = .Eo = Eo + T,
where
T = Eo(. 1),
the relativistic kinetic energy.
The magn itude of t he 4momentum is a Lorent z invariant
.Pµ . = mc.
The 4 m omentum transforms a s follows:
P'µ = LPµ.
For relative motion along the xaxis, this equation is equivalent to the
equations
E' = .E  ß.cpx
and
cpx = ß.E + .cpx .
Using the PlanckEinstein equations E = h. an d
E = pxc for photons, the ener gy eq uation becomes
.' = ..  ß..
= ..(1 ß) 39
= .(1 ß)/(1 ß2)1/2
= .[(1 ß)/(1 + ß)]1/2 .
This is the relativistic Dop pler shift for the frequency .', measured in an
inertial frame (primed) in t erms of the frequency . me asured in another
inertial frame (unprimed).
4.5. The frequencywavenumber invarian t
Par ticleWave duality, one o f the most prof ound
discoveries in Physics, has its origins in Lorentz invariance. It w as
pro posed by deBroglie in the early 192 0's. He used the following
arg ument.
The displacement o f a wave can be written
y(t ,r) = Acos(.t k•r)
where . =2p. (the angular frequency), .k. =2p/. (the wavenumber),
and r = [x, y , z] (the position vector). The ph ase (.t  k•r) can be
written ((./c)ct  k•r), and t his has the form of a Lorentz invariant
obtained from the 4vectors
Eµ[ct , r], and Kµ[./c, k]
where Eµ is the event 4vector, and Kµ is the "frequencywavenumber" 4
vector.
deB roglie not ed th at the 4momentum Pµ is conn ected to the event 4 
vector E µ through the 4velocity Vµ, and the frequencywavenumber 4
vector Kµ is conn ected to the event 4 vector Eµ through the Lorentz
invariant pha se of a wa ve ((./c)ct  kr). He therefore proposed that a 40
direct connectionmustexist betw een P µ an d Kµ; it is illustrated
in the following diagram:
Eµ[ct ,r]
( Einstein) Pµ Pµ =inv.
P µ [E/c,p]
Eµ Kµ =inv. (deBroglie)
Kµ[./c,k]
(deBroglie)
Fig. 2. The coup ling between Pµ an d Kµ vi a Eµ.
deB roglie proposed that the conn ection is the
simplest possible, name ly, Pµ an d Kµ are proportional to ea ch ot her. He
realized that there could be only one value for the c onstant of
pro portionality if the PlanckEinstein resu lt for photons E = h ./2p is but a
special case of a general result . it must be h/2p, where h is Planck’s
constant. Therefore, d eBroglie proposed th at
Pµ . Kµ
or
Pµ = (h/2p)Kµ.
Equ ating the elements of the 4vectors gives
E = (h/2p).
and
p = (h/2p)k .
In these remarkable equations, our notions of particles and waves are 41
forever merged. The smallness of the value of Planck's co nstant prevents
us from observing the d uality directly; however, it is clearly observed at
the molecular, atomic, nuclear, and pa rticle level.
4.6. deBroglie's invariant.
The invariant form ed from the frequencywavenumber 4vector is
KµKµ = (./c, k)[./c,k]
=(./c)2  k2 = (.o/c)2, where .o is the proper
angular frequency.
This invariant is the w ave version of Einstein's
energymomentum invariant; it gives the dispersion relation
.o2 = .2  (kc)2.
The ratio ./k is the phase velocity of the w ave, vf.
For a wa vepacket, the group velocity vG is d./dk; it can be obt ained by
differentiating the dispersion eq uation as follows:
.d. kc 2dk = 0
therefore,
vG = d./dk = kc2/..
The deBr oglie invariant involving the produ ct of the phase and group
velocity is therefore
vfvG = (./k).(kc2/.) = c2.
This is the w aveequivalent of Einstein's famous
E = Mc2. 42
We see that
v fvG = c2 = E/M
or,
v G = E/Mvf = Ek/M. = p/M = vN, the particle
velocity.
This result played an important p art in the development of Wave
Mechanics.
We shall find in later chapters, that Lorentz transformations form a
gro up, and that invariance principles are related directly to symmetry
tra nsformations and the ir associated g roups.
5 43
GROUPS — CONCRETE AND ABSTRACT
5.1 Some concrete examples
The elements of the set {±1, ±i}, where i = v1, are the roots of the
equation x4 = 1, the “fourth roots of unity”. They have the following special
properties:
1. The product of any two elements of the set (including the same two
elements) is always an element of the set. (The elements obey closure).
2. The order of combining pairs in the triple product of any elements
of the set does not matter. (The elements obey associativity).
3. A unique element of the set exists such that the product of any
element of the set and the unique element (called the identity) is equal to the
element itself. (An identity element exists).
4. For each element of the set, a corresponding element exists such
that the product of the element and its corresponding element (called the
inverse) is equal to the identity. (An inverse element exists).
The set of elements {±1, ±i} with these four properties is said to form
a GROUP.
In this example, the law of composition of the group is multiplication; this
need not be the case. For example, the set of integers Z = {.., 2, 1, 0, 1, 2,
...} forms a group if the law of composition is addition. In this group, the
identity element is zero, and the inverse of each integer is the integer with the
same magnitude but with opposite sign.
In a different vein, we consider the set of 4×4 matrices: 44
1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0
{M} = 0 1 0 0 , 1 0 0 0 , 0 0 0 1 , 0 0 1 0 .
0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1
0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0
If the law of composition is matrix multiplication , then {M} is found to obey:
1 closure
and
2 associativity,
and to contain:
3 an identity, diag(1, 1, 1, 1),
and
4 inverses.
The set {M} forms a group under matrix multilication.
5.2. Abstract groups
The examples given above illustrate the generality of the group
concept. In the first example, the group elements are real and imaginary
numbers, in the second, they are positive and negative integers, and in the
third, they are matrices that represent linear operators (see later discussion).
Cayley, in the mid19th century, first emphasized this generality, and he
introduced the concept of an abstract group, Gn which is a collection of n
distinct elements (...gi...) for which a law of composition is given. If n is finite,
the group is said to be a group of order n. The collection of elements must
obey the four rules:
1. If gi, gj . G then gn = gj •gi . G . gi, gj . G (closure) 45
2. gk(gjgi) = (gkgj)gi [leaving out the composition symbol•] (associativity)
3. . e . G such that gie = egi = gi . gi . G (an identity exists)
4. If gi . G then . gi1 . G such that gi1gi = gigi1 = e (an inverse exists).
For finite groups, the group structure is given by listing all
compositions of pairs of elements in a group table, as follows:
e . gi gj . .(1st symbol, or operation, in pair)
e
gi . . gigi gigj .
gj . gjgi gjgj .
gk . gkgi gkgj .
.
.
If gjgi = gigj . gi, gj . G, then G is said to be a commutative or abelian
group. The group table of an abelian group is symmetric under reflection in
the diagonal.
A group of elements that has the same structure as an abstract group is
a realization of the group.
5.3 The dihedral group, D3
The set of operations that leaves an equilateral triangle invariant under
rotations in the plane about its center, and under reflections in the three
planes through the vertices, perpendicular to the opposite sides, forms a
group of six elements. A study of the structure of this group (called the 46
dihedral group, D3) illustrates the typical grouptheoretical approach.
The geometric operations that leave the triangle invariant are:
Rotations about the zaxis (anticlockwise rotations are positive)
Rz(0) (123) . (123) = e, the identity
Rz(2p/3)(123) . (312) = a
Rz(4p/3)(123) . (231) = a2
and reflections in the planes I, II, and III:
RI (123) . (123) = b
RII (123) . (321) = c
RIII (123) . (213) = d
This set of operators is D3 = {e, a, a2, b, c, d}.
Positive rotations are in an anticlockwise sense and the inverse rotations are in
a clockwise sense., so that the inverse of e, a, a2 are
e1 = e, a1 = a2, and (a2)1 = a.
The inverses of the reflection operators are the operators themselves:
b1 = b, c1 = c, and d1 = d.
We therefore see that the set D3 forms a group. The group
multiplication table is:
e 2 b
e 2 b
a 2 e
a2 a2 e
b 2
c 2 e
d 2 e
aa d c
aa e d c
aa c b d
b d c a
aa e d c b
ab d c a
aa c b d
In reading the table, we follow the rule that the first operation is written on 47
the right: for example, ca2 = b. A feature of the group D3 is that it can be
subdivided into sets of either rotations involving {e, a, a2} or reflections
involving {b, c, d}. The set {e, a, a2} forms a group called the cyclic group
of order three, C3. A group is cyclic if all the elements of the group are
powers of a single element. The cyclic group of order n, Cn, is
Cn = {e, a, a2, a3, .....,an1},
where n is the smallest integer such that an = e, the identity. Since
k
nk = a
n = e,
a
a
an inverse ank exists. All cyclic groups are abelian.
The group D3 can be broken down into a part that is a group C3, and a
part that is the product of one of the remaining elements and the elements of
C3. For example, we can write
D3 = C3 + bC3 , b . C3
= {e, a, a2} + {b, ba, ba2}
= {e, a, a2} + {b, c, d}
= cC3 = dC3.
This decomposition is a special case of an important theorem known as
Lagrange’s theorem. (Lagrange had considered permutations of roots of
equations before Cauchy and Galois).
5.4 Lagrange’s theorem
The order m of a subgroup H
m of a finite group Gfactor (an integral divisor) of n.
n of order n is a 48
Let
Gn = {g1=e, g2, g3, ...gn} be a group of order n, and let
Hm = {h1=e, h2, h3, ...hm} be a subgroup of Gn of order m.
If we take any element gk of Gn which is not in Hm, we can form the set of
elements
{gkh1, gkh2, gkh3, ...gkhm} = gkHm.
This is called the left coset of Hm with respect to gk. We note the important
facts that all the elements of gkhj, j=1 to m are distinct, and that none of the
elements gkhj belongs to Hm.
Every element gk that belongs to Gn but does not belong to Hm
belongs to some coset gkHm so that Gn forms the union of Hm and a number
of distinct (nonoverlapping) cosets. (There are (n  m) such distinct cosets).
Each coset has m different elements and therefore the order n of Gn is
divisible by m, hence n = Km, where the integer K is called the index of the
subgroup Hm under the group Gn. We therefore write
Gn = g1Hm + gj2Hm + gk3Hm + ....g.KHm
where
gj2 . Gn . Hm, 49
gk3 . Gn . Hm, gj2Hm
.
gnK . Gn . Hm, gj2Hm, gk3Hm, ...gn1, K1Hm.
The subscripts 2, 3, 4, ..K are the indices of the group.
As an example, consider the permutations of three objects 1, 2, 3 ( the
group S3
) and let Hm = C3 = {123, 312, 231}, the cyclic group of order
three. The elements of S3 that are not in H3 are {132, 213, 321}. Choosing
gk = 132, we obtain
gkH3 = {132, 321, 213},
and therefore
S3 = C3 + gk2C3 ,K = 2.
This is the result obtained in the decomposition of the group D3 , if we make
the substitutions e = 123, a = 312, a2 = 231, b = 132, c = 321, and d = 213.
The groups D3 and S3 are said to be isomorphic. Isomorphic groups have
the same group multiplication table. Isomorphism is a special case of
homomorphism that involves a manytoone correspondence.
5.5 Conjugate classes and invariant subgroups
If there exists an element v . Gn such that two elements a, b . Gn are 50
related by vav1 = b, then b is said to be conjugate to a. A finite group can
be separated into sets that are conjugate to each other.
The class of Gn is defined as the set of conjugates of an element a .
Gn. The element itself belongs to this set. If a is conjugate to b, the class
conjugate to a and the class conjugate to b are the same. If a is not conjugate
to b, these classes have no common elements. Gn can be decomposed into
classes because each element of Gn belongs to a class.
An element of Gn that commutes with all elements of Gn forms a class
by itself.
The elements of an abelian group are such that
bab1 = a for all a, b . Gn,
so that
ba = ab.
If Hm is a subgroup of Gn, we can form the set
{aea1, ah2a1, ....ahma1} = aHma1 where a . Gn .
Now, aHma1 is another subgroup of Hm in Gn. Different subgroups may be
found by choosing different elements a of G
n. If, for all values of a . Gn
aHma1 = Hm
(all conjugate subgroups of Hm in Gn are identical to Hm), 51
then Hm is said to be an invariant subgroup in Gn.
Alternatively, Hm is an invariant in Gn if the left and rightcosets
formed with any a . Gn are equal, i. e. ahi = hka.
An invariant subgroup Hm of Gn commutes with all elements of Gn.
Furthermore, if hi . Hm then all elements ahia1 . Hm so that Hm is an
invariant subgroup of Gn if it contains elements of Gn in complete classes.
Every group Gn contains two trivial invariant subgroups, Hm = Gn and
Hm = e. A group with no proper (nontrivail) invariant subgroups is said to
be simple (atomic). If none of the proper invariant subgroups of a group is
abelian, the group is said to be semisimple.
An invariant subgroup Hm and its cosets form a group under
multiplication called the factor group (written Gn/Hm) of Hm in Gn.
These formal aspects of Group Theory can be illustrated by considering
the following example:
The group D3 = {e, a, a2, b, c, d} ~ S3 = {123, 312, 231, 132, 321, 213}.
C3 is a subgroup of S3: C3 = H3 = {e, a, a2} = {123, 312, 231}.
Now,
bH3 = {132, 321, 213} = H3b
cH3 = {321, 213, 132} = H3c 52
and
dH3 = {213,132, 321} = H3d.
Since H3 is a proper invariant subgroup of S3, we see that S3 is not simple.
H3 is abelian therefore S3 is not semisimple.
The decomposition of S3 is
S3 = H3 + bH3 = H3 + H3b.
and, in this case we have
H3b = H3c = H3d.
(Since the index of H3 is 2, H3 must be invariant).
The conjugate classes are
e = e
eae1 = ea = a
aaa1 = ae = a
a2a(a2)1 = a2a2 = a
bab1 = bab = a2
cac1 = cac = a2
dad1 = dad = a2
The class conjugate to a is therefore {a, a2}.
The class conjugate to b is found to be {b, c, d}. The group S3 can be
decomposed by classes:
S3 = {e} + {a, a2} + {b, c, d}.
S3 contains three conjugate classes.
If we now consider Hm = {e, b} an abelian subgroup, we find
53
aHm = {a,d}, Hma = {a.c},
a2Hm = {a2,c}, Hma2 = {a2, d}, etc.
All left and right cosets are not equal: Hm = {e, b} is therefore not an
invariant subgroup of S3. We can therefore write
S3 = {e, b} + {a, d} + {a2, c}
= Hm + aHm + a2Hm.
Applying Lagrange’s theorem to S3 gives the orders of the possible
subgroups: they are
order 1: {e}
order 2: {e, d}; {e, c}: {e, d}
order 3: {e, a, a2} (abelian and invariant)
order 6: S3.
5.6 Permutations
A permutation of the set {1, 2, 3, ....,n} of n distinct elements is an
ordered arrangement of the n elements. If the order is changed then the
permutation is changed. the number of permutations of n distinct elements is
n!
We begin with a familiar example: the permutations of three distinct
objects labelled 1, 2, 3. There are six possible arrangements; they are
123, 312, 231, 132, 321, 213.
These arrangements can be written conveniently in matrix form: 54
1 2 3 1 2 3 1 2 3
p1 = , p2 = , p3 = ,
1 2 3 3 1 2 2 3 1
1 2 3 1 2 3 1 2 3
p4 = , p5 = , p6 = .
1 3 2 3 2 1 2 1 3
The product of two permutations is the result of performing one arrangement
after another. We then find
p2p3 = p1
and
p3p2 = p1
whereas
p4p5 = p3
and
p5p4 = p2.
The permutations p1, p2, p3 commute in pairs (they correspond to the
rotations of the dihedral group) whereas the permutations do not commute
(they correspond to the reflections).
A general product of permutations can be written
s1 s2 . . .sn 1 2 . . n 1 2 . . n
= .
t1 t2 . . .tn s1 s2 . . sn t1 t2 . . tn
The permutations are found to have the following properties:
1. The product of two permutations of the set {1, 2, 3, ...} is itself a
permutation of the set. (Closure)
2. The product obeys associativity: 55
(pkpj)pi = pk(pjpi), (not generally commutative).
3. An identity permutation exists.
4. An inverse permutation exists:
s1 s2 . . . sn
p1 =  
1 2 . . . n
such that pp1 = p1p = identity permutation.
The set of permutations therefore forms a group
5.7 Cayley’s theorem:
Every finite group is isomorphic to a certain permutation group.
Let Gn ={g1, g2, g3, . . .gn} be a finite group of order n. We choose any
element gi in Gn, and we form the products that belong to Gn:
gig1, gig2, gig3, . . . gign.
These products are the nelements of Gn rearranged. The permutation pi,
associated with gi is therefore
g1 g2 . . gn pi = .
gig1 gig2 . . gign
If the permutation pj associated with gj is
g1 g2 . . gn pj = ,
gjg1 gjg2 . . gjgn
where gi . gj, then
g1 g2 . . gn pjpi = .
(gjgi)gi (gjgi)g2 . . (gjgi)gn
This is the permutation that corresponds to the element gjgi of Gn. 56
There is a direct correspondence between the elements of Gn and the n
permutations {p1, p2, . . .pn}. The group of permutations is a subgroup of
the full symmetric group of order n! that contains all the permutations of the
elements g1, g2, . . gn.
Cayley’s theorem is important not only in the theory of finite groups
but also in those quantum systems in which the indistinguishability of the
fundamental particles means that certain quantities must be invariant under
the exchange or permutation of the particles.
6 57
LIE’S DIFFERENTIAL EQUATION, INFINITESIMAL ROTATIONS
AND ANGULAR MOMENTUM OPERATORS
Although the field of continuous transformation groups (Lie groups)
has its origin in the theory of differential equations, we shall introduce the
subject using geometrical ideas.
6.1 Coordinate and vector rotations
A 3vector v = [vx, vy, vz] transforms into v´ = [vx´, vy´, vzgeneral coordinate rotation R about the origin of an orthogonal coordinate
´] under a
system as follows:
v´ = R v,
where
i.i´ j.i´ k.i´
R =.i.j´ j.j´ k.j´.
i.k´ j.k´ k.k´
cos.ii´ . .
= .cos.ij´ . . .
cos.ik´ . cos.kk´
where i, j, k, i´, j´, k´ are orthogonal unit vectors, along the axes, before and
after the transformation, and the cos.ii´’s are direction cosines.
The simplest case involves rotations in the xy plane:
. vx´ . = . cos.ii´ cos.ji´ ..vx .
vy´ cos.ij´ cos.jj´ vy
= . cosf sinf. = Rc(f)v 58
sinf cosf
where Rc(f) is the coordinate rotation operator. If the vector is rotated in a
fixed coordinate system, we have f . f so that
v´ = Rv(f)v,
where
Rv(f) = . cosf sinf. .
sinf cosf
6.2 Lie’s differential equation
The main features of Lie’s Theory of Continuous Transformation
Groups can best be introduced by discussing the properties of the rotation
operator Rv(f) when the angle of rotation is an infinitesimal. In general,
Rv(f) transforms a point P[x, y] in the plane into a “new” point P´[x´, y´]:
P´ = Rv(f)P. Let the angle of rotation be sufficiently small for us to put
cos(f) . 1 and sin(f) . df, in which case, we have
Rv(df) = . 1 df.
df 1
and
x´ = x.1  ydf = x  ydf
y´ = xdf + y.1 = xdf + y
Let the corresponding changes x . x´ and y . y´ be written
59
so that
We note that
where
x´ = x + dx and y´ = y +dy
dx = ydf and dy = xdf.
Rv(df) = .1 0. + .0 1. df
0 1 1 0
= I + idf
i = .0 1. = Rv(p/2).
1 0
Lie introduced another important way to interpret the operator
i = Rv(p/2), that involves the derivative of Rv(f) evaluated at the identity
value of the parameter, f = 0:
dRv(f)/df. = .sinfcosf.. = .0 1. = i
f =0 cosfsinf 1 0
f = 0
so that
Rv(df) = I + dRv(f)/df. .df,
f = 0
a quantity that differs from the identity I by a term that involves the
infinitesimal, df: this is an infinitesimal transformation.
Lie was concerned with Differential Equations and not Geometry. He
was therefore motivated to discover the key equation
dRv(f)/df =
. 0 1..cosf sinf. 60
1 0 sinf cosf
= iRv(f) .
This is Lie’s differential equation.
Integrating between f = 0 and f = f, we obtain
Rv(f) f
.dRv(f)/Rv(f) = i .df
I 0
so that
ln(Rv(f)/I) = if,
or
Rv(f) = Ie if , the solution of Lie’s equation.
Previously, we obtained
Rv(f) = Icosf + isinf.
We have, therefore
Ie if = Icosf + isinf .
This is an independent proof of the famous CotesEuler equation.
We introduce an operator of the form
O = g(x, y, ./.x, ./.y),
and ask the question: does
dx = Of(x, y; df) ?
Lie answered the question in the affirmative; he found
dx = O(xdf) = (x./.y  y./.x)xdf = ydf
and
dy = O(ydf) = (x./.y  y./.x)y.f = xdf . 61
Putting x = x1 and y = x2, we obtain
dxi = Xxidf , i = 1, 2
where
X = O = (x1./.x2  x2./.x1), the “generator of rotations” in the plane.
6.3 Exponentiation of infinitesimal rotations
We have seen that
Rv(f) = eif,
and therefore
Rv(df) = I + idf, for an infinitesimal rotation, df
Performing two infinitesimal rotations in succession, we have
Rv2(df) = (I + idf)2
= I + 2idf to first order,
= Rv(2df).
Applying Rv(df) ntimes gives
Rvn(df) = Rv(ndf) = eindf = e if
= Rv(f) (as n .8 and df . 0, the
product ndf . f).
This result agrees, as it should, with the exact solution of Lie’s differential
equation.
A finite rotation can be built up by exponentiation of infinitesimal 62
rotations, each one being close to the identity. In general, this approach has
the advantage that the infinitesimal form of a transformation can often be
found in a straightforward way, whereas the finite form is often intractable.
6.4 Infinitesimal rotations and angular momentum operators
In Classical Mechanics, the angular momentum of a mass m, moving in
the plane about the origin of a cartesian reference frame with a momentum p
is
Lz = r × p = rpsinfnz
where nz is a unit vector normal to the plane, and f is the angle between r
and p.
In component form, we have
Lzcl = xpy  ypx, where px and py are the cartesian
components of p.
The transition between Classical and Quantum Mechanics is made by
replacing
px by i(h/2p)./.x (a differential operator)
and py by i(h/2p)./.y (a differential operator),where h
is Planck’s constant.
We can therefore write the quantum operator as
LzQ = i(h/2p)(x./.y  y./.x) = i(h/2p)X
and therefore
X = iLzQ/(h/2p), 63
and
dxi = Xxi df = (2piLzQ/h)xi df, i = 1,2.
Let an arbitrary, continuous, differentiable function f(x, y) be
transformed under the infinitesimal changes
x´ = x  ydf
and
y´ = y + xdf .
Using Taylor’s theorem, we can write
f(x´, y´) = f(x + dx, y + dy)
= f(x  ydf, y + xdf)
= f(x, y) + ((.f/.x)dx + ((.f/.y)dy)
= f(x, y) + df(y(./.x) + x(./.y))f(x, y)
= I + 2pidfLz/h)f(x, y)
= e2pidfLz/h f(x, y)
= Rv(2pLzdf/h) f(x, y).
The invatriance of length under rotations follows at once from this result:
If f(x, y) = x2 + y2 then
.f/.x = 2x and .f/.y = 2y, and therefore
f(x´, y´) = f(x, y) + 2xdx + 2ydy
= f(x, y)  2x(ydf) + 2y(xdf)
= f(x, y) = x2 + y2 = invariant.
This is the only form that leads to the invariance of length under rotations.
64
6.5 3dimensional rotations
Consider three successive counterclockwise rotations about the x, y´,
and z´´ axes through angles µ, ., and f, respectively:
z
z'
y
y'
y
µ about x
x x, x'
z' y' z'' y', y''
. about y´
x' x'' x'
y''
y'''
z''
z'''
f about z´´
'' '' '''xxxThe total transformation is
Rc(µ, ., f) = Rc(f)Rc(.)Rc(µ)
cosfcos. cosfsin.sinµ + sinfcosµcosfsin.cosµ + sinfsinµ
= .sinfcos.sinfsin.sinµ + cosfcosµ sinfsin.cosµ + sinfsinµ.
sin.cos.sinµ cos.cosµ
For infinitesimal rotations, the total rotation matrix is, to 1storder in the d’s:
1 df d.
Rc(dµ, d., df) = .df 1 dµ. .
d. dµ 1
The infinitesimal form can be written as follows: 65
1 df 0 1 0 d. 1 0 0
Rc(dµ, d., df) = .df 1 0.. 0 1 0 ..0 1 dµ.
0 0 1 d. 0 1 0 dµ 1
= . I + Y3df .. I + Y2d. .. I + Y1dµ.
where
0 0 0 0 0 1 0 1 0
Y1 = . 0 0 1. , Y2 = . 0 0 0. , Y3 = . 1 0 0. .
0 1 0 1 0 0 0 0 0
To 1storder in the d’s, we have
Rc(dµ, d., df) = I + Y1dµ + Y2d. + Y3df .
6.6 Algebra of the angular momentum operators
The algebraic properties of the Y’s are important. For example, we find
that their commutators are:
0 0 0 0 0 1 0 0 1 0 0 0
[Y1, Y2] =  0 0 1 0 0 0   0 0 0 0 0 1
0 1 0 1 0 0 1 0 0 0 1 0
= Y3 ,
[Y1, Y3] = Y2 ,
and
[Y2, Y3] = Y1 .
These relations define the algebra of the Y’s. In general, we have
[Yj, Yk] = ± Yl = ejkl Yl ,
where ejkl is the antisymmetric LeviCivita symbol. It is equal to +1 if jkl is 66
an even permutation, 1 if jkl is an odd permutation, and it is equal to zero if
two indices are the same.
Motivated by the relationship between Lz and X in 2dimensions, we
introduce the operators
Jk = i(2p/h)Yk , k = 1, 2, 3.
Their commutators are obtained from those of the Y’s, for example
[Y1, Y2] = Y3 . [2piJ1/h, 2piJ2/h] = 2piJ3/h
or
[J1, J2](2p/h)2 = 2piJ3/h
and therefore
[J1, J2] = ihJ3/2p .
These operators obey the general commutation relation
[Jj, Jk] = ihejkl Jl /2p .
The angular momentum operators form a “Lie Algebra”.
The basic algebraic properties of the angular momentum operators in
Quantum Mechanics stem directly from this relation.
Another approach involves the use of the differential operators in 3
dimensions. A point P[x, y, z] transforms under an infinitesimal rotation of
the coordinates as follows
P´[x´, y´, z´] = Rc(dµ, d., df]P[x, y, z]
Substituting the infinitesimal form of Rc
in this equation gives
dx = x´  x = ydf zd. 67
dy = y´  y = xdf + zdµ
dz = z´  z = xd.  ydµ .
Introducing the classical angular momentum operators: Licl, we find that
these small changes can be written
3 dxi = . dak Xkxi
k = 1
For example, if i = 1
dx1 = dx = dµ(z./.y y./.z)x
+ d.(z./.x + x./.z)x
+ df(y./.x x./.y)x = zd. + ydf .
Extending Lie’s method to three dimensions, the infinitesimal form
of the rotation operator is readily shown to be
3
Rc(dµ, d., df) = I + . (.Rc/.ai) · dai .
i=1 All ai’s = 0
7 68
LIE’S CONTINUOUS T RANSF ORMATION G ROUPS
In the p revious chapter, we discussed the p roper ties of infinitesimal
rot ations in 2 and 3dimensions, and we found that they are related
directly to the angular momentum operators of Quantum Mechanics.
Importan t algebraic propert ies of the matrix representations of the
operators also were introduc ed. In this chapter, we shall consider the
subject in general terms.
Let xi, i = 1 to n be a set of n v ariables. They may be considered to
be the coordinates of a point in an ndimensional vector s pace, Vn. A set
of equations involving the x i’s is obtained by the transformat ions
xi´ = fi(x1, x2, ...xn: a1, a2, ....ar), i = 1 to n
in which the set a1, a2, ...ar co ntains rindependent para meters. The set Ta,
of transformations maps x . x´. We shall write
x´ = f(x;a) or x´ =
for the set of functions.
Tax
It is assumed that the functions fi are differentiable with res pect to
the x’s and t he a’ s to any required order. These functions necessarily
depend o n the essential para meters, a. This means th at no two
tra nsformations with different numbers of p arameters are the same. r is
the smallest number required to characterize the transformation,
completely.
The set of functions fi forms a finite co ntinu ous group if:
1. The result of two successive transformations x . x´ . x´´ is equivalent 69
to a single transformat ion x . x´´:
x´ = f(x´; b) = f(f(x; a); b)
= f(x; c)
= f(x; .(a; b))
where c is the set of p arameters
c. = .. (a; b) , . = 1 to r,
and
2. To e very transformation there corresponds a unique inverse that
belongs to the set:
. a such that x = f (x´; a) = f(x´; a)
We have
1
TaTa = Ta1Ta = I, the identity.
We shall see that 1) is a highly restrictive req uirement.
The transformation x = f(x; a0) is the identity. Without loss of
generality, we can take a0 = 0. The essential point of Lie’s th eory of
continuous tr ansformation groups is to consider that part of the group that
is close to the identity, and not to consider the group as a whole.
Successive infinitesimal changes can be used to build up the finite change.
7.1 Oneparam eter group s
Consider the transformation x . x´ unde r a f inite change in a single
par ameter a, and t hen a change x´ + dx´. There are two pa ths from x .
x´ + dx´; they are as shown:
x´ 70
an “infinitesimal”
da
a , a finite para meter change
x´ + dx´
a + da
a “differential”
x (a = 0)
We have
x ´ + d x´ = f(x; a + da)
= f(f(x; a); da) = f(x´; da)
The 1storder Taylor expansion is
dx´ = .f(x´; a)/.a
The lie group cond itions then dem and
a + da = .(a; da).
But
.(a; 0) = a, (b =0)
therefore
da = u(x´) da
a = 0
a + da = a + ..(a; b)/. b da
b = 0
so that
d a = ..(a; b)/. b da
b = 0
or
da = A(a)da.
Therefore
dx ´ = u (x´)A(a)da,
leading to
dx´/u(x´)
so that
x´
.dx´/u(x´)
x
We therefore obtain
=
=
A(a)da 71
a
.
A(a)da = s, (s = 0 . the identity).
0
U (x´)  U(x) = s.
A t ransformat ion of coordinates (new v ariables) therefore transfers all
elements of t he group b y the same transformation: a o nepa rameter group
is equivalent to a group of translatio ns.
Two cont inuous tra nsformation groups are said to be s imilar when
they can be obtained from one another by a change of variable. For
example, consider the g roup defined by
x1´ a 0 x1
x2´ = 0 a2 x2
The identity coprr esponds to a = 1. The infinitesimal transformation is
therefore
x1´ ( 1 + da) 0 x1
x2´ = 0 (1+ da)2 x2 .
To 1storder in da we have
x 1´ = x1 +x1da
and
x 2´ = x2 + 2x2da
or
dx1 = x1da
and
dx2 = 2x2da. 72
In the limit, these equations give
dx1/x1 = dx2/2x2 = da.
These are the differential equations t hat c orrespond to the infinitesimal
equations above.
Integrating, we have
x1´ a x2´ a
.dx 1/x1 = .da and .dx 2/2x2 = da ,
x10x2 0
so that
lnx1´ lnx1 = a = ln (x1´/x1)
and
ln(x2´/x2) = 2a = 2ln(x1´/x1)
or
U´ = (x2´/x1´2) = U = (x2/x12) .
Put ting V = lnx1, we obt ain
V´ = V + a and U ´ = U, the translation group.
7.2 Determin ation of t he finite equations from the infinitesim al
for ms
Let the finite equations of a oneparameter group G(1) be
x1´ = f(x1, x2 ; a)
and
x2´ = .(x1, x2 ; a),
and let the identity correspond to a = 0. 73
We consider the transformation of f(x1, x2) to f(x1´, x2´). We expand
f(x1´, x2´) in a Maclaurin series in the p arameter a (at definite values of x 1
and x2):
f(x1´, x2´) = f(0) + f´(0)a + f´´(0)a2/2! + ...
where
f(0) = f(x1´, x2´) a=0 = f(x1, x2),
and
f´(0) = (df(x1´, x2´)/da a=0
= {(.f/.x1´)(dx1´/da) + (.f/.x2´)(dx2´/da)} a= 0
= {(.f/.x1´)u(x1´, x2´) + (.f/.x2´)v(x1´, x2´)}a=0
therefore
f ´(0) = {(u(./. x1) + v( ./.x2))f}a=0
= Xf(x1, x2).
Continuing in this way, we have
f´´(0) = {d2f(x1´, x2´)/da2}a=0 = X2f(x1, x2), etc....
The function f(x1´, x2´) can be expanded in t he series
f(x1´, x2´) = f(0) + af ´(0) + (a2/2!)f´´(0) + ...
= f(x1, x2) + aXf + (a 2/2!)X2f + ...
Xnf is the symb ol for operating ntimes in succession of f with X.
The finite eq uations of the group are therefore
x1´ = x1 + aXx1 + (a2/2!)X2x1 + ...
and
x2
´
= x2 + aXx2 + (a2/2!)X2x2 + = ...
If x1 and x2 are definite values to which x1´and x2´ reduce for the identity 74
a=0, then the se equations ar e the series solutions of the differential
equations
dx1´/u(x1´, x2´) = d x2´/v(x1´, x2´) = d a.
The group is referred to as the g roup Xf.
For exam ple, let
Xf = (x1./.x1 + x2./.x2)f
then
x1´ = x1 + aXx1 + (a2/2!)X2f ...
= x1 + a(x1./.x1 + x2./.x2)x1 + ...
= x1 + ax1 + (a2/2!)(x1./.x1 + x2./.x2)x1 +
= x1 + ax1 + (a2/2!)x1 + ...
= x1(1 + a + a2/2! + ...)
= x1ea.
Also, we find
´
x2 = x2ea.
Put ting b = ea, we have
x1
´
= bx1, and x2´ = bx2.
The finite group is the group of magnifications.
If X = (x./.y y./.x) we find, for example, that the finite group is the
gro up of 2dimensional rotat ions.
7.3 Invariant function s of a group 75
Let
Xf = (u./.x1 + v./.x2)f define a oneparameter
gro up, and let a=0 give the identity. A fu nction F(x1, x2) is ter med a n
invariant un der the transformation group G (1) if
F(x1´, x2´) = F (x1, x2)
for all values of the p arameter, a.
The function F(x1´, x2´) can be expanded as a series in a:
F(x1´, x2´) = F(x 1, x2) + aXF + (a 2/2!)X(XF) + ...
If
F(x1´,x2´) = F (x1, x2) = in variant for all values of a,
it is necessary fo r
XF = 0,
and this mean s that
{u(x1, x2)./.x1 +
Consequently,
v(x 1, x2)./.x2}F = 0 .
F(x1, x2) = constant
is a solution of
dx1/u(x1, x2) = dx2/v(x1, x2) .
This equation has one solution that depends on one ar bitrary constant, and
therefore G(1) ha s only one basic invariant, and all othe r pos sible invariants
can be given in terms o f the basic invariant.
For exam ple, we now rec onsider the the invariants of rotat ions:
The infinitesimal transformations are given by 76
Xf = (x 1./.x2 x2./.x1),
and the differential equation tha t gives the invariant function F of the
gro up is obta ined by solving the characteristic differential equations
dx1/x2 =
so that
dx1/x2 + dx2/x1 =
The solution of this equation is
22
x1 + x2 =
df, and dx 2/x1 = df,
0.
constant,
and therefore the invariant function is
22
F (x1, x2) = x1 + x2.
2
All functions of x 1 + x22 are the refore invariants of the 2dimensional
rot ation group.
This method c an be generalized. A group G(1) in nvariables defined
by the e quation
xi´ = f(x1, x2, x3, ...xn; a), i = 1 to n,
is equivalent to a unique infinitesimal transformation
Xf = u1(x1, x2, x3, ...xn).f/.x1 + ...un(x1, x2, x3, ...xn).f/.xn .
If a is the g roup parameter then the infinitesimal transformation is
x i´ = xi + ui(x1, x2, ...xn)da (i = 1 t o n),
then, if E(x1, x2, ...xn) is a function that can be differentiatedntimes with
respect to its arg uments, we have
E (x1´, x2´, ...xn´) = E(x1, x2, ...xn) + aXE + (a 2/2!)X2E + . 77
Let (x1, x2, ...xn) be the coordinates of a point in nspace and let a be a
par ameter, independent of the xi’s. As a var ies, the p oint (x1, x2, ...xn) will
describe a trajectory, starting from the initial point (x1, x2, ...xn). A
necessary and sufficient con dition tha t F(x 1, x2, ...xn) be an invariant
function is that XF = 0. A curve F = 0 i s a t rajectory and t herefore a n
invariant curve if
XF(x1, x2, x3, ...xn) = 0.
8 78
PROPERTIES OF nVARIABLE, rPARAMETER LIE GROUPS
The change of an nvariable function F(x) produced by the
infinitesimal transformations associated with ressential parameters is:
n
dF = . (.F/.xi)dxi
i = 1
where
r
dxi = . ui.(x)da. , the Lie form.
. = 1
The parameters are independent of the xi’s therefore we can write
rn
dF = .da.{. ui.(x)(./.xi)F}
. = 1 i = 1
r
= .da. X. F
. = 1
where the infinitesimal generators of the group are
n
X.= . ui.(x)(./.xi) , .= 1 to r.
i = 1
The operator
r
I + . X.da.. = 1
differs infinitesimally from the identity.
The generators X. have algebraic properties of basic importance in the
Theory of Lie Groups. The X.’s are differential operators. The problem is
therefore one of obtaining the algebraic structure of differential operators.
This problem has its origin in the work of Poisson (1807); he
introduced the following ideas:
The two expressions
X1f = (u11./.x1 + u12./.x2)f
and
X2f = (u21./.x1 + u22./.x2)f 79
where the coefficients ui. are functions of the variables x1, x2, and f(x1, x2)
is an arbitrary differentiable function of the two variables, are termed
linear differential operators.
The “product” in the order X2 followed by X1 is defined as
X1X2f = (u11./.x1 + u12./.x2)(u21.f/.x1 + u22.f/.x2)
The product in the reverse order is defined as
X2X1f = (u21./.x1 + u22./.x2)(u11.f/.x1 + u12.f/.x2).
The difference is
X1X2f X2X1f = X1u21.f/.x1 + X1u22.f/.x2
X2u11.f/.x1 X2u12.f/.x2.
= (X1u21 X2u11).f/.x1 + (X1u22 X2u12).f/.x2
= [X1, X2]f.
This quantity is called the Poisson operator or the commutator of the
operators X1f and X2f.
The method can be generalized to include . = 1 to r essential parameters
and i = 1 to n variables. The athlinear operator is then
Xa = uia.f/.xi
n
= . uia.f/.xi , ( a sum over repeated indices).
i = 1
Lie’s differential equations have the form
.xi/.a. = uik(x)Ak.(a) , i = 1 to n, . = 1 to r.
Lie showed that
(.ckts/.a.)uik = 0
in which 80
ujs.uit/.xj ujt.uis/.xj = ckts (a)uik(x),
so that the c
kts’s are constants. Furthermore, the commutators can be
written
[X., Xs] = ( ck.sujk)./.xj
= ck.sXk.
The commutators are linear combinations of the Xk’s. (Recall the earlier
discussion of the angular momentum operators and their commutators).
The ck.s’s are called the structure constants of the group. They have the
properties
ck.s = cks. ,
cµ.sc.µt + cµstc.µ. + cµt.c.µs = 0.
Lie made the remarkable discovery that, given these structure constants,
the functions that satisfy
.xi/.a. = uikAk.(a) can be found.
(Proofs of all the above important statements, together with proofs of
Lie’s three fundamental theorems, are given in Eisenhart’s
standard work Continuous Groups of Transformations, Dover Publications,
1961).
8.1 The rank of a group
Let A be an operator that is a linear combination of the generators
of a group, Xi:
A = aiXi (sum over i),
and let 81
X = xjXj .
The rank of the group is defined as the minimum number of commuting,
linearly independent operators of the form A.
We therefore require all solutions of
[A, X] = 0.
For example, consider the orthogonal group, O+(3); here
A = aiXi i = 1 to 3,
and
X = xjXj j = 1 to 3
so that
[A, X] = aixj[Xi, Xj] i, j = 1 to 3
= aixjeijkXk .
The elements of the sets of generators are linearly independent, therefore
aixjeijk = 0 (sum over i, j,, k = 1, 2, 3)
This equation represents the equations
a2 a1 0 x1 0
.a3 0 a2..x2. = .0. .
0 a3 a2 x3 0
The determinant of is zero, therefore a nontrivial solution of the xj’s
exists. The solution is given by
xj = aj (j = 1, 2, 3)
so that
A = X .
O+(3) is a group of rank one.
8.2 The Casimir operator of O+(3)
The generators of the rotation group O+(3) are the operators. Yk’s, 82
discussed previously. They are directly related to the angular momentum
operators, Jk:
Jk = i(h/2p)Yk (k = 1, 2, 3).
The matrix representations of the Yk’s are
0 0 0 0 0 1 0 1 0
Y1 = . 0 0 1., Y2 = . 0 0 0. , Y3 = . 1 0 0. .
0 1 0 1 0 0 0 0 0
The square of the total angular momentum, J is
3
J2 = . Ji2
1
= (h/2p)2 (Y12 + Y22 + Y32)
= (h/2p)2(2I).
Schur’s lemma states that an operator that is a constant multiple of I
commutes with all matrix irreps of a group, so that
[Jk, J2] = 0 , k = 1,2 ,3.
The operator J2 with this property is called the Casimir operator of the
group O+(3).
In general, the set of operators {Ci} in which the elements commute
with the elements of the set of irreps of a given group, forms the set of
Casimir operators of the group. All Casimir operators are constant multiples
of the unit matrix:
Ci = aiI.
The constants ai are characteristic of a particular representation of a group.
9 83
MATRIX REPRESENTATIONS OF GROUPS
Matrix r epresentations of linear operators are important in Linear
Algebra; we s hall see that they are eq ually important in Group Theory.
Ifa group of m × m matrices
Dn(m) = {D1(m) (g1),...Dk(m) (gk), ...Dn(m) (gn)}
can be f ound in which each element is associated with the corre sponding
element gk of a group o f ord er n
Gn = {g1,...gk,....gn},
and the matrices obey
Dj(m) (gj)Di(m) (gi) = Dji(m) (gjgi),
and
D1(m) (g1) = I, the identity,
then the matrices Dk(m) (gk) are said to form an mdimensional
rep resentation of Gn. If the association is onetoone we have an
isomorph ism and the rep resentation is said to be fai thful .
The subject o f Group Representations forms a ver y large bra nch o f
Gro up Theory. The re are man y standard works on this topic (see the
bibliography), each one cont aining numerous definitions, lemmas and
theorems. He re, a rather brief account is given of some o f the more
importan t res ults. The read er sh ould delve into the deeper asp ects of the
subject as the need arises. The subject will be introduced by considering
representations of the rotat ion groups, and their corresponding cyclic 84
gro ups.
9.1 The 3di mensional representation of rotations in the plane
The rotation of a vector through an angle f in the plane is
characterized by the 2 x 2 matrix
c osf sinf
Rv(f) = .
sinf cos f
The group of symmetry transformat ions that leaves an equilateral
triangle invariant unde r rotations in the p lane is of order three, and each
element of the group is of d imension two
Gn ~ R3(2) = {R(0), R(2p /3), R(4p /3)}
= 1 0 , 1/2 v3/2 , 1/2 v3/2 .
0 1 v3/2 1/2 , v3/2 1/2
˜ { 123, 312, 231} = C3.
These matrices form a 2dimensional representation of C3 .
A 3 dimensional representation of C3 can be obtained as follows:
Consider an equilateral triangle located in the plane and let the
coordinates of the three ver tices P1[x, y], P2[x´, y´], and P3[x´´, y´´] be
written as a 3vector P13 = [P1, P2, P3], in normal order . We introduce
3 × 3 matrix ope rators Di(3) that ch ange the o rder of the elements of P13,
cyclically. The identity is
P13 = D1(3) P13, where D1(3) = diag(1, 1, 1).
The rearrangement 85
P13 . P23[P3, P1, P2] is given by
P23 = D2(3) P13,
where
0 0 1
D2(3) = 1 0 0 ,
0 1 0
and the rearrangement
P13 . P33[P2, P3, P1] is given by
P33 = D3(3) P13
where
0 1 0
D3(3) = 0 0 1 .
1 0 0
The set of matrices {Di(3) } = {D1(3) , D2(3) , D3(3) } is said to form a 3
dimensional r epresentation of the original 2dimensional representation
{R3(2) }. The elements Di(3) ha ve th e same group multiplication ta ble as
that associated with C3.
9.2 The mdi mensional representation of sy mmetry
tra nsformations in ddi mensions
Consider the case in which a group of order n
Gn = {g1, g2, ...gk, ...gn}
is repre sented by
Rn(m) = {R1(m) , R2(m) , .....Rn(m) 86
where
Rn(m) ~ Gn,
and Rk(m) is an m × m matrix representation of gk. Let P1d be a vector in
ddimensional space, written in normal order:
P1d = [P1, P2, ...Pd],
and let
P1m = [P1d, P2d, ....Pmd]
be an mvector, written in normal order, in which the comp onents are each
dvectors. Introd uce t he m × m matrix ope rator Dk(m) (gk) such t hat
P1m = D1(m) (g1)P1m
P2m = D2(m) (g2)P1m
.
.
Pkm = Dk(m) (gk)P1m , k = 1 to m, the number of
symmetry operations,
where Pkm is the kth (cyclic) permuta tion of P1m , and Dk(m) (gk) is called
the “md imensional representation of g k”.
Infinitely many representations of a given repre sentation can be 87
found, for, if S is a matrix repre sentation, and M is any definite matrix
with an inverse, we can form T(x) = MS(x)M1, . x . G. Sin ce
T(xy) = MS(xy)M1 = MS(x)S(y)M1 = MS(x)M1MS(y)M1
= T(x)T(y),
T is a representation of G. The new re presentation simply involves a
change o f variable in t he corresponding substitutions. Representations
related in the man ner of S an d T are equivalent , and are not regarded as
different representations. All representations that are e quivalent to S are
equivalent to each othe r, and the y for m an infinite class. Two equivalent
rep resentations will be written S ~ T.
9.3 Direct sums
If S is a representation of dimension s, and T is a representation of
dimension t of a group G, the mat rix
S(g) 0
P = , (g . G)
0T(g)
of dimension s + t is called the direct sum of the matrices S(g) and T(g),
written P = S . T. Therefore, given two representations (they can be the
same), we can obta in a third by adding them directly. Alternatively, let P
be a rep resentation of dimension s + t; we suppose that, for all x . G, the
matrix P(x) is of the form
A(x) 0
0B(x)
where A(x) and B(x) are s × s and t × t matrices, respectively. (The 0’s 88
are s × t and t × s zero matrices). Define the matrices S an d T as follows:
S(x) = A(x) and T(x) = B(x), . x . G.
Since, by the group property , P(xy) = P(x)P(y),
A(xy) 0A(x) 0A(y) 0
=
0B(xy) 0B(x) 0B(y)
A(x)A(y) 0
= .
0B(x)B(y)
Therefore, S(xy) = S(x)S(y) and T(xy) = T(x)T(y), so that S an d T are
rep resentations. The representation P is said to be decomposable, with
components S an d T. A representation is indecomposable if it cannot be
decomposed.
If a com ponent of a decomposable repre sentation is itself
decomposable, we c an continue in this manner to decompose any
rep resentation into a finite number of indecomposable comp onents. (It
should be not ed th at the property of indecomposablity depends on the field
of the representation; the real field must sometimes be extended to the
complex field to check for indecomposability).
A w eaker form of d ecomposability arises when we consider a
matrix of the form
A(x) 0
P(x) =
E(x) B(x)
where A(x), and B(x) are matrices of dimensions s × s and t × t 89
respectively and E(x) is a matrix that de pends on x, and 0 is the s × t zero
matrix. The matrix P, and an y equ ivalent form, is said to be reducible.
An irreducible representation is one t hat cannot be reduced. Every
decomposable matrix is reducible (E(x) = 0), whereas a reducible
rep resentation need not be decomposable.
If S an d T are red ucible, we can continue in this way to obtain a se t
of irreducible com ponents. The components are d eterm ined uniquely, up
to an equivalence. The set of distinct irr educible representations of a finite
gro up is (in a given field) an invariant of the group . The com ponents form
the building bloc ks of a representation of a group.
In Physics, decomposable representations are ge nerally referred to as
reducible representatio ns (reps).
9.4 Sim ilarity and unitary transforma tions and matrix
dia gonal ization
Before discussing the q uestion of the possibility of reducing the
dimension of a given re presentation, it will be useful to consider s ome
importan t res ults in theTheory of Matr ices. The proofs of these statements
are given in the standard works o n Matrix Theory. (See bibliography).
If there exists a matrix Q such th at
Q1AQ = B ,
then the matrices A an d B are related by a similarity transformation.
If Q is unitary (QQ† = I: Q† =(Q*)T , the hermitian conjugate) 90
then A an d B are related by a unitary transformation.
If A´ = Q1AQ; B´ = Q1BQ; C´ = Q1CQ..then any algebraic
relation amon g A, B, C...is also satisfied by A´, B´, C´ ...
If a similarity transformation produces a d iagonal matrix then the
pro cess is called dia gonal ization.
If A an d B can be diagonalized by the same matrix the n A an d B
commute.
If V is form ed from the eigenvectors of A then th e similarity
tra nsformation V1AV will produce a diagonal matrix whose elements are
the eigenvalues of A.
If A is hermitian then V will be unitary a nd therefore an hermitian
matrix can always be diagonalized by a unitary transformat ion. A real
symmetric mat rix can always be diagonalized by an orthogonal
tra nsformation.
9.5 The SchurAuerbach theorem
This theorem states
Every matrix representation of a finit e group is equi valent to a
uni tary matrix representatio n
Let Gn = {D1, D2, ....Dn} b e a matrix group, and let D be the matrix
formed b y tak ing the sum of pairs of elements
n
D = . DiDi†
i = 1
where Di† is the hermitian conjugate of Di.
Since Di is nonsingular, each term in the sum is positive definite. 91
Therefore D itself is positive definite. Let Ld be a diagonal matrix that is
equivalent to D, and let Ld1/2 be the positive definite ma trix formed by
replacing the elements of Ld by their positive square roots. Let U be a
unitary matrix with the property that
Ld = UDU1.
Introdu ce th e mat rix
S = Ld1/ 2U,
then SDiS1 is unitary. (This property can be dem onstrated by co nsidering
(SDiS1)(SDiS1)†, and showing that it is equal to the identity.). S will
tra nsform the original matrix representation Gn into diagonal form. Every
unitary matrix is diagonalizable, and therefore every matrix in ever y finite
matrix r epresentation can be diagonalized.
9.6 Schur’s lemmas
A m atrix representation is reducible if every element of t he
representation can be put in blockdiagonal form by a single similarity
tra nsformation. Invoking the result of the previous section, we need only
discuss unitary representations.
IfGn = {D(.)(R)} is an irreducible rep resentation of dimension . of
a g roup Gn, and {D(µ)(R)} is an irreducible representation of dimension µ
of the same g roup, Gn, and if there exists a matrix A such th at
D(.)(R)A = AD(µ)(R) . R . Gn
then either
i) A = 0 92
or
ii) A is a sq uare nonsingular matrix (so that . = µ)
Let the µ co lumns of A be written c1, c2, ...cµ, then, for any ma trices
D(.) an d D(µ) we have
D(.)A =
an
AD(µ) =
therefore
D(.)cj =
(D(.)c1, D(.)c2, ...D(.)cn)
µµ µ
( . D(µ)
k1ck, . D(µ)
k2ck, ....D(µ)
kµck).
k = 1 k = 1 k = 1
µ
. D(µ)
kjck
k = 1
and therefore the µ cvectors span a sp ace that is invariant under the
irreducible set of .dimensional matrices {D(.)}. The cvectors are
therefore the nullvector or they span a .dimensional vector spa ce. The
first case corresponds to A = 0, and the second to µ= . an d A . 0.
In the second case, the hermitian conjugates D(.)
1†, ...D(.)
n† an d D(µ)
1†,
...D(µ)
n† also ar e irreducible . Furthermo re, since D(.)
i(R)A = AD(µ)
i(R)
†
D(µ)
i†A† = A†D(.)
i ,
and therefore, following the method ab ove, we find that .= µ. We must
therefore have . = µ, so tha t A is square.. Since the .columns of A span
a .dimensional space, the matrix A is necessarily nonsingular.
As a cor ollary, a matrix D that commutes with an irr educible set of
matrices must be a scalar ma trix.
9.7 Characters 93
If D(.)(R) and D(µ)(R) are related by a similarity transformat ion then
D(.)(R) gives a representation of G t hat is equivalent to D(µ)(R). These two
sets of matrices are ge nerally different, whereas their structu re is the same.
We wish, therefore, to answer the question: what intrinsic properties of the
matrix r epresentations are invariant u nder coord inate transformations?
Consider
. [ CD(R)C1]ii = . CikDkl(R)Cli1
i ikl
= .dklDkl(R)
kl
= . Dkk(R) , the trace of D(R).
k
We see that the trace, or character, is an invariant under a change of
coordinate ax es. We write the character as
.(R) = . Dii(R)
i
Equ ivalent representations have t he same set of characters. The
ch aracter of R in the repre sentation µ is written
.(µ)(R) or [ µ; R].
Now, the conjugate elements of G have the form S = URU1, and then
D(R) = D(U)D(R)[D(R)]1
therefore
.(S) = .(R).
We can describe G by giving its characters in a particular representation;
all elements in a class have the same ..
10 94
SOME LIE GROU PS OF TRANSFORM ATIONS
We shall consider those Lie group s that can be described b y a finite
set of continuously varying essential parameters a1,...ar:
xi´ = fi(x1,...xn; a1,...ar) = f(x; a) .
A set of para meters a e xists that is associated with the inverse
tra nsformations:
x = f(x´; a).
These equations mu st be solvable to give th e xi’s in terms o f the xi´’s.
10.1 Linear group s
The general linear group GL (n) in ndimensions is given b y the set
of equations
n
xi´ = . aijxj , i = 1 to n,
j = 1
in which det aij . 0.
The group con tains n2 pa rameters t hat h ave values covering an infinite
ran ge. The group GL(n) is said to be not closed.
All linear groups with n > 1 are nonabelian. The group GL(n ) is
isomorph ic to the group of n × n matrices; the law of comp osition is
therefore mat rix multiplication.
The special linear group of transformations SL(n) in ndimensions is
obt ained from GL(n ) by imposing the condition de taij = 1. A functional
relation therefore exists among the n2 parameters so th at the num ber of
required para meter s is reduced to (n2 1).
10.2 Orthogonal groups 95
If the transformat ions of the general linear group GL(n) are such
that
n . xi2 . invariant ,
i = 1
then the rest ricted group is called th e orthogonal gr oup, O(n), in n
dimensions. There are [n + n(n 1)/2] conditions imposed on the n2
par ameters of GL(n ), and the refore the re are n(n  1)/2 essential
par ameters of O(n).
For exam ple, in three d imensions
x´ = Ox ; O = { O3×3: OOT = I, detO = 1, aij . R}
where
a11 a 12 a 13
O = a21 a 22 a 23 .
a 31 a32 a 33
We have
x1´2 +x2´2 + x3´2 = x12 +x22 +x32 . invariant under O(3).
This invariance imposes six conditions on the original nine par ameters, and
therefore O(3) is a threeparameter group.
10.3 Unitary grou ps
If the x i’s and t he aij’s of the general linear group GL(n) are
complex, and the transformat ions are required to leave xx† invariant in the
complex space, then we obtain the unitary group U(n) in ndimensions:
U(n) = { Un×n: UU† = I, detU . 0, uij . C}.
There are 2n2 independent real parameters ( th e real and imaginary parts of
the aij’s), and the unitary condition imposes n + n(n 1) c onditions on
them so the group has n2 real paramet ers. The unitary condition means 96
that
.j a ij2 = 1,
and therefore
aij2 = 1 fo r all i, j.
The para meters are limited to a f inite range of values, and the refore the
gro up U(n) is said to be closed.
10.4 Sp ecial unit ary g roups
If we impose the restriction detU = +1 on the unitary group U (n)
then we obtain the special unita ry group SU(n) in ndimensions. We have
SU(n) = { Un×n: UU† = I, detU = +1, uij . C}.
The dete rminantal condition reduces the num ber of required real
par ameters to (n2 1). We shall see th at the groups S U(2) and S U(3) play
an important part in Modern Physics.
10.5 The group SU(2), the infinitesimal form of SU(2), and the
Pauli sp in matrices
The special unitary group in 2dimensions, SU(2), is defined as
SU (2) = { U2×2: UU† = I, detU = +1, uij . C}.
It is a threeparameter group.
The defining conditions can be used to obta in the mat rix
rep resentation in its simplest form; let
a b
U =
c d
where a, b, c, d . C. 97
The hermitian conjugate is
a* c*
U† = ,
b* d*
and therefore
 a2 + b2 ac* + bd*
UU† = .
a*c + b*d c 2 + d2
The unitary c ondition gives
a2 + b2 = c2 + d2 = 1,
and the determinantal condition gives
ad bc = 1 .
Solving these equations , we obta in
c = b*, and d = a*.
The general form of SU(2) is therefore
a b
U = .
b* a*
We now study the infinitesimal form of SU(2); it must have the
structur e
1 0 da db 1 + da db
Uinf = + = .
0 1 db* da* db* 1 + da*
The dete rminantal condition therefore gives
d etUinf = (1+ da)(1 + da*) +dbdb* = 1 .
To first order in the d’s, we obtain
1 + da* + da = 1,
or 98
da = da*.
so that
1+ da db
Uinf = .
db* 1da
The matrix elements can be written in their comp lex forms:
da = ida/2 , db = dß/2 + id./2.
(The factor of two has been introduced for later conv enience).
1 + ida/2 dß/2 + id./2
Uinf = .
dß/2 + id./2 1  ida/2
Now, any 2×2 matrix can be written as a linear co mbination of the
matrices
1 0 0 1 0 i 1 0
, , , .
0 1 1 0 i 0 0 1
as follows
a b 1 0 0 1 0 i 1 0
= A + B + C + D ,
c d 0 1 1 0 i 0 0 1
where
a = A + D, b= B iC, c = B + iC, and d = A  D.
We then have
a b (a + d) 1 0 (b+ c) 0 1 i(b c) 0 i (a d) 1 0
= + + + .
c d 2 0 1 2 1 0 2 i 0 2 0 1
The infinitesimal form of SU(2) can th erefore be written
Uinf = I + (id./2) 1 + (idß/2) 2 + (ida/2) 3 , 99
or
Uinf = I + (i/2). dtjThis is the Lie form.
j . j = 1 to 3.
The
j’s are the Pau li sp inma trices:; they are the generators o f the group
SU(2):
0 1 0 i 1 0
=
=
=
,2 ,3
.
1
1 0 i 0 0 1
They play a f undamental role in t he description of spin1/2 particles in
Quantum mechanics. (See later discussions).
10.6 Commutators of th e spi n mat rices and structure constants
We have previously introduced the commutators of the infinitesimal
generators of a Li e group in conn ection with the ir Lie Algebra. In this
section, we c onsider the com mutat ors o f the generators of SU(2); they are
found to have the symmetric forms
[ 1, 2] = i 3, 2 [ 2, 1] = 2i
[ 1, 3] = i 2, [ 2 3, 1] =
[ 2, 3] = i 1, 2 [3, 2] =
3,
i 2, 2
i 1. 2
We see that the commutator of any pair of t he th ree matrices gives a
constant multiplied by the v alue of the remaining matrix, thus
[ j, k] = ejkl2i l .
where the quantity ejkl = ±1, depending on the permutations of the indices.
(e(xy )z = +1, e(yx )z = 1 ..etc...).
The quan tities 2iejkl are the structure constants associated with the group. 100
Other propert ies of the spin matrices are found to be
2
=
2
=
2
= I;
1
2
= i
3,
2
3
= i
1,
3
1
= i
2.
1
2 3
10.7 Homomor phism of S U(2) and O+(3)
We can form the ma trix
P = xT = xj
j, j = 1, 2, 3
from the matrices
x = [x1, x2, x3] and = [
1,
2,
3] :
therefore
x 3 x1 ix2
P = .
x1 + ix2 x3
We see that
x3 x1 ix2
P† = (P*)T = = P,
x1 + ix2 x3
so that P is hermitian.
Furtherm ore,
TrP = 0,
and
detP = (x12 + x22 + x32).
Another matrix, P´, can be formed b y car rying out a similarity
tra nsformation, thus
P´ = UPU †, (U . SU(2)).
A similarity transformation leaves bot h the trace and the determinant
unchanged, therefore 101
T rP = TrP´,
and
d etP = detP´.
However, the condition d etP = detP´ means that
xxT = x´x´T,
or
x 12 + x22 + x32 = x1´2 + x2´2 + x3´2 .
The transformation P´ = UPU † is therefore equivalent to a three
dimensional or thogonal transformat ion that leaves xxT invariant.
10.8 Ir reducible representations of S U(2)
We have seen that the b asic form of the 2×2 matrix representation
of
the group SU(2) is
a b
U = , a,b . C; a2 + b2 =1.
b* a*
Le t the basis vectors of this space be
1 0
x1 = and x2 = .
0 1
We then have
a
x1´ = Ux1 = = ax1 b*x2 ,
b*
and
b
x2´ = Ux2 = = bx1 + a*x2 ,
a*
and therefore
x´ = Utx.
If we wr ite a 2dimensional vector in this complex space as c = [u, v ]
then the components transform under SU(2) as 102
u ´ = au + bv
and
v ´ = b*u + a*v ,
and therefore
c´ = Uc .
We see that the components o f the vect or c tr ansform dif ferently
from those of the basis vect or x — the transformat ion matrices are the
tra nsposes of each othe r. T he vector c = [u, v ] in this complex space is
called a spinor (Cartan, 1913).
To find an irreducible repre sentation of SU(2) in a 3dimensional
space, we need a s et of three linearly independent ba sis functions.
Following Wigner (see bibliograph y), we can choo se the polynomials
u2, uv, and v2,
and introduce the polynomials defined by
1 + m 1 m
j = 1 u v
f =
m
v { (1 + m)! (1 + m)!}
where
j = n/2 (the dimension of the space is n + 1) .
and
m = j, j 1,...j .
Inthe p resent case, n= 2,j = 1 , and m = 0, ±1.
(The factor 1/v{(1 + m)! (1  m)!} is chosen to make the representative
matrix unitary). 103
We have, therefore
f11 = u2/v2 , f01 = uv, and f11 = v2/v2.
A 3 ×3 r epresentation of an element U . SU(2) in this space can be found
by defining the transformation
Ufm1(u, v) = fm1(u´, v´).
We then obtain
m
Ufm1(u, v) = (au + bv)1 + m(b*u +a*v)1 ,m = 0, ±1,
v{(1 + m)!(1 m)!}
so that
Uf11(u, v) = (au + bv)2/v 2
= (a2u2 + 2abuv + b2v2)/v 2 ,
Uf01(u, v) = (au + bv)(b*u + a*v)
= ab*u2 + (a2 b2)uv + a*bv2 ,
and
Uf11(u, v) = (b*u + a*v)2/v 2
= (b*2u2 2a*b*uv + a*2v2)/v 2 .
We then have
1
a2 v 2ab b2 f1 f11´
v 2ab* a2 b2 v2a*b f01 = f01´
1
b*2 v2a*b* a*2 f1 f11´
or
UF = F´.
We find that UU † = I an d the refore U is, indeed, unitary.
This pro cedure can be generalized to an (n + 1)dimensional space as
follows
Let
fmj(u, v) = uj + mvj m , m = j, j  1, ...j. 104
v{(j + m)!(j m)!}
(Note that j = n/2 = 1/ 2, 1/1, 3/2, 2/1, ..).
For a given v alue of j, there are 2j + 1 linearly independent p olynomials,
and therefore we c an form a (2j + 1) × (2j + 1 ) representative mat rix of an
element U of SU(2):
Ufmj(u, v) = fmj(u´, v´).
The deta ils of this general case are g iven in Wigner’ s classic text. He
demonstrates the irredu cibility of the (2j + 1)dimensional representation
by showing that an y matrix M wh ich commut es with Uj for all a, b such
that a2 + b2 = 1 mus t nec essarily be a constant ma trix, and therefore, by
Schur’s lemma, Uj is an irredu cible representation.
10.9 Representations o f rotations and the concept of tens ors
We have discussed 2 a nd 3dimensional representations of the
ort hogonal group O (3) and their connection to an gular momentum
operators. Higherdimensional representations of the orthogonal group can
be obtained b y con sidering a 2index q uantity , Tij — a tensor — that
consists of a set of 9 elements t hat transform under a rotation of t he
coordinates as follows:
Tij . Tij´ = RilRjmTlm (sum over repeated indices 1, 2, 3).
If Tij = Tji (Tij is symm etric), then th is symmetr y is an invariant un der
rot ations; we have
Tji´ = RjlRimTlm = RjmRilTml = RilRjmTlm = Tij´ . 105
If TrTij = 0, then so is TrTij´, for
Tii´ = RilRimTlm = (RTR)lmTlm = dlmTlm = Tll = 0.
The comp onents of a symmetric traceless 2index tensor contains 5
members so th at the transformation Tij . Tij´ = RilRjmTlm de fines a new
rep resentation of them of dimension 5.
Any tensor Tij can be written
Tij = (Tij + Tji)/2 + (Tij Tji)/2 ,
and we have
Tij = (Tij + Tji)/2 = (Tij (dijTll)/3) + (dijTll)/3 .
The decomposition of the ten sor Tij gives any 2index tensor in terms of a
sum of a single component, proportional to the identity, a set of 3
independent q uantities combined in an antisymmetric tensor (Tij Tji)/2,
and a se t of 5 independent c ompon ents of a symmetric traceless tensor.
We write the dimensional eq uation
9 = 1 . 3 . 5.
This is as far as it is possible to go in t he process of d ecomposition: no
oth er su bsets of 2 index ten sors can be found that preserve the ir identities
under the defining transformation of t he coordinates. Rep resentatio ns with
no subsets of tensors that preserve th eir identities under the defining
rotation s of tensors are irreducible representations.
We shall see that the decomposition of tensor products into 106
symmetric and antisymmetric parts is important in the Quark Mo del of
elementary particles.
The representations of the o rthog onal group O(3) are found to be
importan t in defining the int rinsic spin of a particle. The dy namics of a
par ticle of finite mass can always be descibed in its rest frame (all inertial
frames are eq uivalent!), and therefore the particle can be char acterized by
rot ations. All known p articles have dynamical states that can be described
in terms of t he te nsors of some irredu cible representation of O (3). If the
dimension of the irrep is (2j + 1 ) then the particle spin is found to be
pro portional to j. In Particle Physics, irreps with values of j = 0 , 1, 2,... and
with j = 1/2, 3/2, ... are found that corre spond to the fundamental bosons
and fermions, resp ectively.
The three dimensional ortho gonal group SO(3) (det = +1) and the
two dimensional group SU(2) have the same Lie algebra. In the case of
the group SU(2), the (2j + 1 )dimensional representations are a llowed for
bot h integer and h alf integer va lues of j, whereas, the representations of
the group SO(3) are limited to integer values of j. Since all the
rep resentations ar e allowed in SU(2), it is called th e covering gr oup. We
not e tha t rotations thr ough f a nd f +2p ha ve different effects on the 1/2
integer repre sentations, and therefore they are (spinor) t ransfomations
associated with SU(2).
11 107
THE GROUP STRUCTURE OF LORENTZ TRANSFORMATIONS
The square of the invariant interval s, between the o rigin [0, 0, 0, 0]
of a spacetime coordinate system and an arbitrary event xµ = [x0, x1, x2,
x3] is, in index notation
s2 = xµxµ = x´µx´µ , (sum over µ = 0, 1, 2, 3 ).
The lower indices can be raised using the metric tensor
.µ. = diag(1, –1, –1, –1),
so that
s2 = .µ.xµx. = .µ.x´µx´v , (sum over µ an d .).
The vect ors n ow ha ve contrav ariant for ms.
In matrix notation, the invariant is
s2 = xT x = x´T x´ .
(The transpose must be written explicitly).
The primed and unp rimed column matrices (contrav ariant vec tors) are
related by the Lorentz matrix ope rator, L
x´ = Lx .
We therefore have
xT x =(Lx)T (Lx)
= xTLT Lx .
The x’s are arbitrary, therefore
LT L =.
This is the defining propert y of the Lorent z transformations. 108
The set of all Lor entz transformations is the set L of all 4 × 4
matrices that satisfies the defining proper ty
L ={L: LT L =; L: all 4 × 4 real matrices;
= diag(1, –1, –1, –1}.
(Note that each L ha s 16 (independent) r eal matrix elements, and therefore
belongs to the 16dimensional space, R16).
11.1 The group structure of L
Consider the result of two successive Lorentz transformations L1
and L2 that transform a 4vector x as follows
x . x´ . x´´
where
x´ = L1x ,
and
x´´ = L2x´.
The resu ltant vector x´´ is given b y
x´´ = L2(L1x)
= L2L1x
= Lcx
where
Lc = L2L1 (L1 followed by L2).
If the c ombined op eration Lc is always a Lorentz transformation then it
must satisfy
T
Lc Lc = . 109
We must therefore have
(L2L1)T (L2L1) =
or
L1T(L2T L2)L1 =
so that
T
L1 L1 =, (L1, L2 . L)
therefore
Lc = L2L1 . L .
Any number of successive Lorentz transformations may be carried out to
give a resultant that is itself a Lorentz transformat ion.
If we take th e det erminant o f the defining equation of L,
d et(LT L) = det
we obtain
(detL)2 = 1 (d etL = detLT)
so that
detL = ±1.
Since t he determinant of L is not zero, an inverse transformation L–1
exists, and t he equation L–1L = I, the identity, is always valid.
Consider the inverse of the defining equation
(LT L)–1 = –1 ,
or
L–1 –1(LT)–1 = –1 . 110
Using = –1, and rearran ging, gives
L–1 (L–1)T = .
This result shows that the inverse L–1 is always a member of the set L.
We therefore see that
1. If L1 an d L2 . L , then L2 L1 . L
2. If L . L , then L–1 . L
3. The identity I = diag(1, 1, 1, 1 ) . L
and
4. The matrix operators L ob ey associativity.
The set of all Lor entz transformations therefore form s a group.
11.2 The rotation grou p, revisited
Spatial rotat ions in two and three dimensions are Lorentz
tra nsformations in which the timecomponent remains unchanged.
Let R be a real 3×3 matrix that is part of a Lorent z transformation
with a constant timecomponent. In this case, the defining property of t he
Lorentz transformations leads to
RTR = I , the identity matrix, diag(1,1,1).
This is the d efining propert y of a threedimensional ortho gonal matrix
If x = [x1, x2, x3] is a t hreevector that is transformed und er R to
give x´ then
x´Tx´ = xTRTRx 111
= xTx = x12 + x22 + x32
= invariant unde r R.
The action of R on any threevector preserves length. The set of all 3×3
ort hogonal matrices is denoted by O(3),
O(3) = {R: RTR = I, r ij . R}.
The elements of this set satisfy the four group axioms.
The group O(3) can be split into two parts that are said to be
disconnected:: one w ith detR = +1 an d the othe r with det R = 1. The
two parts are written
O+(3) = {R: detR = +1}
and
O(3) = {R: detR = 1} .
If we define the parity operator , P, t o be the o perator that reflects
all points in a 3dimensional cartesian system throug h the origin then
1 0 0
P = 0 1 0 .
0 0 1
The two parts of O(3) are related by the operator P:
if R . O+(3) then PR . O(3),
and
if R´ . O(3) then PR´ . O+(3). 112
We can t herefore c onsider only that pa rt of O(3) that is a group, namely
O+(3), together with the operator P.
11.3 Connected and disconnected parts of the Lorentz grou p
We have shown, previously, that e very Lorent z transformation, L,
has a determinant equal to ±1. The matrix elements o f L ch ange
continuously as the relative velocity changes continuously. It is not
possible, however, to move continuously in such a way that we c an go
from the set of tr ansformations with detL = +1 to thos e with det L = 1; we
say that the set { L: detL = +1} is disconnected fr om the set {L: detL =
1}.
If we wr ite the Lorentz transformation in its component form
L . Lµ
.
where µ = 0,1,2,3 labels the rows, and . = 0,1,2,3 labels the c olumns then
the time comp onent L00 ha s the values
L00 = +1 or L00 = 1.
The set of tr ansformations can th erefore be split into four
disconnected parts, labelled as follows:
{ L.
+} = {L: detL = +1, L00 = +1}
{L.
} = {L: detL = 1, L00 = +1}
{L.
+} = {L: detL = +1, L00 = 1},
and
{L.
} = {L: detL = 1, L00 = 1}. 113
The identity is in {L.
+}.
11.4 Parity, timereversal and orthochronous transformations
Two discrete Lorentz transformations are
i) the p arity transformation
P ={P: r . r, t . t}
= di ag(1, 1, 1, 1),
and
ii) the timerever sal transfprmat ion
T ={T: r . r, t . t}
= di ag(1, 1, 1, 1} .
The disconnected p arts of {L} are related by the transformations
that involve P, T, and PT, as shown:
PT
L.
+ L.

P T
L.
L.

Fig. 3 Connections b etween the disconnected p arts of the Lorentz
transformations
The proper orthochronou s tra nsformations ar e in the g roup L.
+. We 114
see that it is not necessary to consider the complete set {L} o f Lorentz
tra nsformations — we need consider only tha t sub set { L.
+} t hat forms a
gro up by itself, and either P, T, or PT co mbined. Experiments have
shown clear v iolations under the parity transformation, P an d violations
under T ha ve been inferred fro m exp eriment an d the ory, combined.
However, not a single experiment has been carried out that shows a
violation of the p roper orthochro nous transformations, {L.
+}.
12 115
ISOSPIN
Par ticles can be distinguished from one another by their intrinsic
pro perties: mass, charge, spin, parity, and their electric and magnetic
mom ents. In o ur ongoing que st for an under standing of the true natu re of
the fundamental particles, and their interactions, other intrinsic proper ties,
with nam es such as “isospin” and “stra ngeness”, have been discovered.
The intrinsic properties are defined b y qua ntum numbers; for example, the
quantum number a is defined b y the eigenvalue equation
Af = a.f
where A is a linear operator, f is the wavefunction of the system in the
zeromomentum frame, and a is an eigenvalue of A.
In this chapter, we shall discuss the first of t hese new p roper ties to
be introduced, namely, isospin.
The building blocks of n uclei are proto ns (positively charged) and
neutrons (neutral). Nu merou s experiments on the scattering of proto ns by
pro tons, and proto ns by neut rons, have shown tha t the nuclear forces
between pairs have the same strength, provided t he angular momentum
and spin states are the same. These observations form the basis of an
importan t con cept — the chargeindependence of.the n ucleonnuc leon.
force. (Corrections for the coulomb e ffects in proto nproton s cattering
must be made). The origin of this concept is found in a new symmetry
principle. In 1932, Chadwick no t only identified th e neutron in studying
the interaction of alphaparticles on beryllium nuclei but also showed that 116
its mass is almost equal to the mass of the proton. (Recent measurements
give
m ass of proton = 938·27231(28) MeV/c2
and
ma ss of neut ron = 939·56563(28) MeV/c2)
Within a few month s of Chadwick’s discovery , Heisenberg introdu ced a
theory of nuclear forces in which he considered the n eutro n and the proto n
to be tw o “st ates” of t he same ob ject — the nucleon. He introd uced an
intrinsic var iable, later ca lled isospin, that p ermits the char ge states (+, 0) of
the nucleons to be distinguished. This new variable is needed (in addition
to the traditional spacespin variables) in the description of nucleon
nucleon scattering.
In nuclei, protons and neutr ons behave in a remarkably sym metrical
way: the binding energy of a nucleus is closely propo rtional to the number
of neutr ons and protons, and in light nuclei (mass number <40), the
number of neutrons can be equal to the number of protons.
Before discussing the isospin of particles and n uclei, it is necessary to
introduce an extended Pa uli Exclusion Principle. In its or iginal form, th e
Pauli Exclusion Principle was introduc ed to account for features in the
observed spectra of atoms th at co uld not be unde rstood using the the n
cur rent models of atomic str uctur e:
no.two electrons in an.atom.can exist.in the same quantum.state defined.117
by.the q uantum num bers.n, , m , m s wh ere n is the principal quantum
number, is the.orbital angular momentum.quantum nu mber, m is the.
magnetic quan tum n umber, and ms is the.spin.quantum nu mber.
For a sy stem of N particles, the complete wavefunction is written as
a p roduct of singleparticle wavefunctions
.(1, 2, ...N) = .(1).(2)....(N).
Consider this form in t he simplest case — for two identical particles. L et
one be in a s tate labelled .a an d the othe r in a state .b. For i dentical
particles, it makes no difference to the probability density .2 of the 2
particle system if the particles are e xchanged:
.(1, 2)2 = .(2, 1)2 , (the .’s are n ot measurable)
so that, either
.(2, 1) = .(1, 2) (symmetric)
or
.(2, 1) = .(1, 2) (antisymmetric).
Let
.I = .a(1).b(2) (1an a, 2 in b)
and
.II = .a(2).(1) (2in a, 1 in b).
The two particles are indistinguishable, therefore we have no way of 118
knowing whether .I or .II de scribes the system; we po stulate that the
system spends 50% of its time in .I an d 50% of its time in .II. The two
particle system is considered to be a lin ear combination of .I an d .II:
We have, therefore, either
.sym m = (1 /v2){.a(1).b(2) + .a(2).b(1)} (BOS ONS)
or
.ant isymm = (1/v2){.a(1).b(2) .a(2).b(1)} (FER MIONS) .
(The coefficient (1/v2) normalizes the sum of the squares to be 1).
Exc hanging 1. 2 leaves .sym m un changed, whereas exchanging pa rticles
1. 2 rever ses the sign of .ant isymm .
If two p articles are in .S, both particles can exist in the same state with
a = b. If two particles are in .AS , and a = b, we have .AS = 0 — they
cannot e xist in the same quantum state. Electrons (fermions, spin = (1/2)h)
are described by antisymmetric wavefunctions.
We can now introdu ce a more general Pauli Exclusion Principle.
Write the nucleon wavefunction as a product:
.(., q) = .(.)fN(q) ,
where
. = .(r, s)
in which r is the space vect or, s is the spin, and q is a charge or isospin
label.
For two nucleons, we write 119
.(.1, q1; .2, q2),
for two proto ns:
.2p = .1(.1, .2)fN(p1)fN(p2),
for two neutr ons:
.2n = .2(.1, .2)fN(n1)fN(n2),
and for an np pair:
.np = .3(.1, .2)fN(p1)fN(n2)
or
= .4(.1, .2)fN(n1)fN(p2).
If we re gard the p roton and neutr on as different states of the same object,
labelled by the “c harge or i sospin coo rdinate”, q, we must extend the Pau li
principle to cover the new coordinate: the total wavefunction is then
.(.1, q1; .2, q2) = .(.2, q2; .1, q1) .
It.must.be an tisymmetric un der the fu ll ex change..
For a 2p ora 2npair, the exchange q 1. q2 is symm etrical, and therefore
the spacespin part mus t be antisymmetrical.
For an np pair, the symmetr ic (S) and antisymmetric (AS)
“is ospin” wav efunctions are
I) FS = (1/v2){fN(p1)fN(n2) + fN(n1)fN(p2)}
(symmetr ic under q1 . q2),
and therefore the spacespin part is antisymmetrical,
II) FAS = (1/v2){fN(p1)fN(n2) fN(n1)fN(p2)} 120
(antisymmetr ic under q1 . q2),
and therefore the spacespin part is symmet rical.
We shall need these results in later discussions of t he symmetr ic and ant i
symmetric properties of quark systems.
12.1 Nuclear decay
Nuclei are bound states of n eutro ns and protons. If the n umber of
pro tons in a nucleus is Z and the number of neut rons is N then the mass
number of the nucleus is A = N + Z. Some n uclei are natur ally unstable.
A p ossible mode of decay is by the emission of a n electron ( th is is
ßdecay — a process that typifies the fundamental “wea k interaction”) .
We write the decay as
A
ZXN . AZ+1 XN1 + e–1 + .e ( ß–decay)
or, we c an have
AA
ZXN . Z1 XN1 + e+ + .e ( ß+ decay).
A r elated process is that of electron capture of an orbital electron that is
sufficiently close to the po sitively charged nuc leus:
e
–A
+ AZXN . Z+1 XN+1 + .e.
Other related processes are
.e + AZXN . AZ1 XN1 + e+
and
.e + AZXN . AZ+1 XN1 + e
–
.
The decay of the free.pr oton has not been observed at the present time. 121
The experimental limit on the halflife of the p roton is > 1031 ye ars! Many
cur rent theor ies of the microstructure of m atter predict that the proton
decays. If, however, t he lifetime is > 1032  1033 ye ars t hen t here is no
realistic possibility of observing the decay directly (The limit is set by
Avogadro ’s nu mber and t he finite number of proto ns th at can be
assembled in a suitable experimental apparatus).
The fundamental ßdecay is that of the free n eutro n, first o bserved in
1946. The process is
–
n0 . p + +e + .e0 ,t1/2 = 10·37 + 0·19 minutes.
This measured lifetime is of fundamental importance in Particle Phy sics
and in Cosmology.
Let us set up an algebraic description of t he ßdecay process, recognizing
that we have a 2state system in which the transformation p . n occurs:
In the ß–decay of a free n eutro n
–
n . p + +e + .e,
and in t he ß+decay of a p roton , bound in a nucleus,
p . n + e+ + .e .
12.2 Isospin of t he nucleon
The spontaneous tr ansformations p. n observed in ßdecay lead us
to introduce the o perators
= fp , +fp = 0, (eliminates a p roton )
± that transform p . n:
+fn
and 122
fp = fn , fn = 0, (eliminates a n eutro n).
Since we are dealing with a twostate system, we choo se the “isospin”
par ts of the proto n and neut ron wavefunctions to be
1 0
f(p) = a nd f(n) = ,
0 1
in which case the operators must have the forms:
0 1 0 0
= and

=
.
+
0 0 1 0
They are singular and n onhermitian.
We have, for example
0 1 0 1
+fn = = , fn .fp,
0 0 1 0
and
0 1 1 0
= =
0 0 0 0
+fp
( + removes a proton).
To make the p resent algebraic description analogous to the twostate
system of the intrinsic spin of the e lectron, we introduce linear
combinations of the
± :
0 1
= =
1 0
1, a Pauli matrix,
+ +

=
1
and
0 i 123
+) = =
2
= i( 
2, a Pauli matrix.
i 0
A t hird operator that is diagonal is, as expected
1 0
= =
3
3, a Pauli matrix.
0 1
The three operators { 1,
3} t herefore obey t he commutation
2,
relations
[ j/2,
k/2] = iejkl
l/2 ,
where the factor of(1/2) is introduced because of the 2:1 homom orphism
between SU(2) and O+(3): the vect or operator
t =
is called the isospin operator o f the nucl eon.
/2
To classify the isospin states of the nucleon we may use the
pro jection of t on the 3rd axis, t3. The eigenvalues, t3, of t3 co rrespond to
the proton (t3 = +1/2) and neutr on (t3 = 1/2) states. The nucleon is said to
be an isospin doublet with isospin quan tum number t = 1/2. (The number
of states in the multiplet is 2t + 1 = 2 fo r t = 1/2).
The char ge, QN of the nucleon can be written in terms of the isospin
quantum numbers:
QN = q(t 3 +(1/2)) = q or 0,
where q is the proton c harge. (It is one o f the grea t unsolved problems of
Par ticle Phys ics to und erstand wh y the char ge on the proto n is equal to the
charge on the electron).
12.3 Isospin in nuclei. 124
The concept o f isospin, and of ro tations in isospin space, associated
with individual nucleons can be applied to nuclei — systems of many
nucleons in a boun d state.
Let the isospin of the ithnucleon be ti, and let ti = i /2. The
operator of a system of A nucleons is defined as
T = .Ai=1 ti = .Ai=1 i/2 .
The eigenvalue of T3 of the isospin operator T3 is the sum o f the individual
components
T3 = .Ai=1 t3i = .Ai=1 t3i/2
= (Z – N)/2.
The char ge, QN of a nucleus can be written
QN = q.Ai=1 (t3i + 1)/2
= q( T3 + A/2) .
For a given eigenvalue T of the o perator T, the state is (2T + 1) fold
degenerate. The eigenvalues T3 of T3 are
T3 = T, T + 1,...0,...T + 1, T .
If the H amiltonian H of the nucleus is char geindependent then
[H, T] = 0.
and T is said to be a good q uantum number. In light nuclei, where the
isospinviolating coulomb interaction between pairs of pro tons is a small
effect, the c oncept of isospin is particularly useful. The study of isospin
effects in nuclei was first applied to the observed p roper ties of the lowest
lying states in the three nuclei with mass number A = 14: 14C, 14N, and 14O. 125
The relative energies of the states are shown in the following diagram:
Ene rgy ( MeV)
6
0+ T = 1, T3 =1
4
0+ T = 1 , T3 =0
2
0+ T = 1 , T3 =1 1+ T = 0 , T3 =0
0
Fig 4. An isospin si nglet (T = 0) a nd an isospin triplet (T = 1) in
the A = 14 system. In the absence of the coulomb in teraction, the three
T = 1 states woul d be degenerate.
The spin and parity of the g round state of 14C, the first excited state o f 14N
and the groun d state of 14O a re measured to be 0+; these three states are
characterized by T = 1. The ground st ate o f 14N h as spin and parity 1+; it
is an isospin singlet (T = 0).
12.4 Isospin and mesons
We have seen that it is possible to classify the char ge states of
nucleons and nuclear isobars using the concept o f isospin, and the a lgebra
of SU(2). It will be useful to classify other particles, including field
par ticles (quanta) in t erms of their isospin.
Yukawa (1935), first pr oposed tha t the stro ng nu clear force bet ween
a p air of nucleons is carried by massive field particles called mesons.
Yukawa’s method was a masterful development of the theory of the 126
electrom agnetic field to include the c ase of a m assive field particle. If .p is
the “mes on wavefunction” the n the Yukawa differential equation for t he
meson is
.µ.µ .p + (E0/hc)2.p = 0.
where
.µ.µ = (1/c2).2/.t2 .2 .
The rdependent (spatial) form of .2 is
.2 . (1/r2)d/dr(r2d/dr)
The static (timeindependent) solution of t his equation is read ily checked to
be
.(r) = (g2/r)exp(r/rN)
where
rN = h/mpc = hc/mpc2 = hc/Ep
0,
so that
2
1 /rN = (Ep
0/hc)2
The “ran ge of the nuclear force” is defined by the condition
r = rN = h/mpc ˜ 2 ×1013 cm .
This gives the mass of the meson to be close to the measured value. It is
importan t to note that the “ range of t he force” . 1/(mass of t he field
quantum). In the case of the electrom agnetic field, the mass of the field
quantum (the photon) is zero , and therefore the force has an infinite ran ge.
The mesons come in three charge states: +, , and 0. The mesons 127
have intrinsic spins equal to zer o (they ar e field particles and the refore they
are bosons), and t heir rest energies are measured to be
Ep±
0 = 139·5 MeV, and Ep00 = 135·6 MeV.
They are therefore considered to be me mbers of an isospin triplet:
t = 1,t3 = ±1,0.
In Particle Physics, it is the custom to de signate the isospin quantum
number by I, we shall follow this conv ention fro m now on.
The third com ponent of the isospin is an additive quantum number.
The comb ined values of the isospin projections of the two particles, one
with isospin projection I3(1) , and t he ot her with I3(2) , is
I3(1+ 2) = I3(1) + I3(2) .
Their isospins combine to give states with different numbers in each
multiplet. For example, in pion (meson)nucleon scattering
p + N . st ates with I3(1 + 2) = (3/2) or (1/2).
These values are o btained by noting that
Ip
(1) = 1, and IN(2) = 1/2, so that
I3p
(1) + I3N(2) = (±1, 0) + (±1/2)
= (3/2), an isospin quartet, or (1/2), an
isospin doublet.
Sym bolically , we write
3 . 2 = 4 . 2.
(This is the rule for forming the product (2I3(1) + 1).(2I3(2) + 1).
13 128
GROUPS AND THE STRUCTURE OF MATTER
13.1 St rangeness
In the e arly 1950’ s, our understanding of t he ultimate str uctur e of
matter s eemed to be com plete. We required neutr ons, proto ns, electrons
and neut rinos, and mesons and pho tons. Our optimism was shortlived.
By 1953, excited states of t he nu cleons, and more massive mesons, had
been discovered. Some of the new particles had completely unex pected
pro perties; for example, in the interaction betw een proton s and pmesons
(pions) the following decay mode was observed:
Pr oton (p+)
.
Pion (p +)
Sig ma (.+) Pion
(p0 )
Kaon (K+) .
Pion
(p +)
..
Initial interaction Final decay
lasts ~1023 seconds ta kes ~ 1010 seconds
( Strong force a cting) (Wea k force a cting)
GellMann, and independently Nishijima, pro posed that the kaons (heavy
mesons) were endowed with a new intrinsic proper ty not affected by the
strong force. GellMann called this proper ty “s trang eness”. S trang eness
is conserved in t he strong interactions but cha nges in the weak
interactions. The GellMann  Nishijima interpretation of the strangeness129
changing involved in the protonpion interaction is
p + (S = 0) .+ (S = –1)
p0 (S = 0)
p+ (S = 0)
. K + (S = +1) .
p+ (S = 0)
..
. S = 0 . S = 1
In the strong part of t he interaction, there is no ch ange in the num ber
defining the strangeness, whereas in t he weak part, the strangeness changes
by one u nit. Having de fined the values of S for the particles in this
interaction, they are d efined for ever. All subsequent exp eriments involving
these objects have been consistent with the original assignments.
13.2 Pa rticl e patterns
In 1961, GellMann, and independently Ne’eman, introd uced a
scheme t hat c lassified the strong ly interacting particles into family groups
(Mendeleev again?). They were not, at this point in the d evelopment of t he
theory, concerned with the n ature of p ossible building blocks of the
nucleons and mesons. They w ere c oncerned, however, with the inclusion
of “stra ngeness” i n the theo ry, and therefore they were led to study the
arr angements of particles in an abstract space de fined by their electric
charge a nd st rangeness. The common feature of each family was chosen
to be their intrinsic spin; the family of spin1/2 baryons (strongly 130
interacting particles) has eight members: n0, p+ ,.± ,.0 ,.– ,.0 , and .0 .
Their strangeness quantum numbers are:
S = 0: n0, p+ ; S = –1: .± ,.0 , and .0 ;and S = –2: .0,– . (No negatively
charged proto n exists, and n o positivelycharged . ex ists). If the positions
of these eight particles are given in chargestrangeness space, a remarkable
pattern emerges:
Str angen ess
n0 p+ .
0
.–
0
.0
.– .+
.0
+1
.0
Charge
Fig. 5.
chargestrangeness space
–1
The family po rtrait of spin1/2 baryons in
There are two particles at the center, each with zero char ge and zer o
strangeness; they are the .0 an d the .0. ( They have different rest masses).
–1
–2
They studied the struct ure of other families. A particularly 131
importan t set of p articles consists of all baryo ns with sp in 3/2. A t the time,
there we re nine kn own p articles in this category: .0, . ±1, . +2, .*0, .*±1,
.0, and .1 . They have the following pattern in char gestrangeness space:
Charge: –1 0 +1 +2 Str angen ess
.
0
.
.0 .+ . ++
.*0 .+
.*– .*0
–1
.*–
–2
O–
The symm etry pattern of the family of spin3/2 baryon s, shown b y the
known nine ob jects was sufficiently compelling for GellMann, in 196 2, to
suggest that a ten th member of the family should exist. Furthe rmore, if
the symm etry has a physical basis, the tenth member s hould have spin3/2,
charge –1, strangeness –3, and its mass should be about 150MeV greater
than the mass of t he .0 pa rticle. Two ye ars after this suggestion, the tenth
member of the family was identified in high ener gy pa rticle collisions; it
decayed via weak interactions, and possessed the predicted properties.
This could not hav e been by chance. The discovery of the O– pa rticle was
–3
crucial in helping to establish the concept of the GellMann – Ne’eman 132
symmetry mode l.
In addition to the symm etries of baryo ns, groupe d by their spins, the
model was used to obtain sym metries of mesons, also grouped by their
spins.
13.3 The special unitary group SU(3) and particle st ructu re
Several years before the work of GellMann and Ne’eman, Sakata
had atte mpted to buildup the kno wn pa rticles from {neutro n proton
lambda0} t riplets. The lambda particle was required to “car ry the
strangeness”. Alt hough the model was shown not to be valid, Ikeda et al.
(1959) introd uced an importa nt mathematical analysis of the threestate
system that involved th e group SU(3). The notion tha t an under lying
gro up structu re of elementary particles might ex ist was popular in t he
early 1960’s. (Special Unitary Groups were used by J. P. Elliott in the
late1950’s to describe symmetry proper ties of light nuclei).
The problem facing Particle Physicists, at the time, was to find the
appropriate g roup and its fundamental repre sentation, and to co nstruct
higherdimensional representations that wou ld account for the w ide variety
of symmetries observed in chargestrangeness space. We have seen th at
the char ge of a particle can be written in terms of its isospin, a concept that
has its origin in the c hargeindependence of the nucleonnucleon for ce.
When app ropriate, we shall discuss the symm etry properties of p articles in
isospinstrangeness space.
Pre viously, we discussed the properties of the Lie group SU(2). It is 133
a g roup characterized by its three generators, the Pa uli spin matrices.
Twostate systems, such as t he electron with its quan tized spinup and spin
down, and the isospin states of n ucleons and nuc lei, can be treated
quantitatively using this group. The symmetries of n ucleon and meson
families discovered by GellMann and N e’eman, implied an underlying
structur e of nucleons and me sons. It could not be a structure simply
associated with a twostate system bec ause the o bserved particles were
endowed not only with positive, negative, and zero charge but a lso with
strangeness. A threestate system was therefore considered necessary, at
the very least; the most promising candidate was the group SU(3). We
shall discuss the infinitesimal form of this gro up, and we shall find a
suitable set of generators.
13.3.1 The a lgebra of SU(3)
The group of special unitary transformations in a 3dimensional
complex space is defined as
SU( 3) = {U3×3 : UU† = I, detU = +1, uij .C}.
The infinitesimal form of SU(3) is
SU(3)inf = I +idaj j/2, j = 1 to 8.
(There a re n2 1 = 8 generators).
The quan tities daj are rea l and infinitesimal, and the 3 ×3 matrices j are
the linearly independent gen erators of the group . The rep eated index, j,
means that a sum over j is taken.
The defining properties of the group restrict the form of the 134
generators. For example, the unitary condition is
UU† = (I +idaj j/2)(I –idaj
†
j/2)
= I –idaj j†/2 + idaj j/2 to 1s torder,
= I if j = j†.
The generators must be hermitian.
The dete rminantal condition is
det = +1; and the refore Tr j
= 0.
The generators must be traceless.
The finite form of U is obta ined by ex ponentiation:
U = exp{iaj j/2}.
We can find a suitable set of 8 g enerators by ex tending the met hod
used in our discussion of isospin, thus:
Let three fundamental states of t he system be chosen in the simplest
way, namely:
1 0 0
u = 0 , v = 1 , an d w = 0 .
0 0 1
If we wish to transform v . u, we can do so by de fining the operator A+:
0 1 0 0 1
A+ v = u, 0 0 0 1 = 0 .
0 0 0 0 0
We can introd uce other operators that transform the states in pairs, thus 135
0 0 0
A– =1 0 0 ,
0 0 0
0 0 0 0 0 0
B+ = 0 0 1 , B– = 0 0 0 ,
0 0 0 0 1 0
0 0 0 0 0 1
C+ = 0 0 0 , C– = 0 0 0 .
1 0 0 0 0 0
These matrices are singular and n onhermitian. In the discussion of isospin
and the group SU(2), the non singular, traceless, hermitian matrices
1, and
2 are formed from the ra ising and lowering operators
± ma trices by
introducing the complex linear co mbinations
= + + = 1 an d = i( 1 – 2) =
1– 2
The generators of SU(3) are formed fro m the operators A±, B±, C± by
constructing complex linear combinations. For example:
the isospin operator 1 = 1 = + +
0
0 = A+ + A– = 1, a generator of SU(3).
2.
–, a generator of SU(2) becomes
1
0 0 0
Continuing in this way, we obtain
A+ = 1/2 + i 2/2, 136
where
0
2
2
= 0 ,
0 0 0
and
C+ + C– = 4, C+ – C– = –i 5,
B+ + B– =
6 a nd B + – B– = i 7 .
The remaining generators, 3 an d 8 are traceless, diagonal, 3×3 matrices:
0 1 0 0
3 0 , 8
= 0 1 0 .
=
3
0 0 0 0 0 2
The set of matrices { 1, ..... 8} are ca lled the G ellMann m atrices,
introduced in 1961 . They ar e normalized so that
Tr (
j k) = 2djk.
The normalized for m of 8 is therefore
1 0 0
8
= (1/v3) 0 1 0 .
0 0 –2
If we pu t Fi = i/2. we find
A± = F1 ± iF2 ,
B± = F6 ± iF7,
and 137
C±= F4 + iF5 .
Let A3 = F3, B3 = –F3/2 + (v3/4)F8 , and C3 = (–1/2)F3 (v3/4)F8., so th at
A3 + B3 + C3 = 0.
The last cond ition mean s that only eight of the nine operators are
independent.
The generators of the g roup are readily shown to obey the Lie
commutation relations
[Fi, Fj] = ifijkFk , i,j,k = 1 to 8.
where the quantities fijk are the nonzero structure constants of the group;
they are found to obey
fijk = –fjik,
an d the Jacobi identity.
The commutation relations [Fi, Fj] can be written in terms of the operators
A±, ...Some typ ical results ar e
[ A+, A] = 2A3, [ A+, A3] = A+, [ A, A3] = +A,
[A3, B3] = 0, [A3, C3] = 0, [B3, C3] = 0
[B+, B] = 2B3, [ B+, B3] = B, [B, B3] = +B, etc.
The two diagonal operators c ommut e:
[F3, F8] = 0 .
Now, F1, F2, and F3 co ntain the 2×2 isospin ope rators (Pauli matrices),
each with zer os in the third row and column; they obe y the commutation
relations of isospin. We therefore make th e identifications
F1 = I1, F2 = I2, and F3 = I3 138
where the Ij’s are the components o f the isospin.
Par ticles that exp erience th e str ong n uclear interaction are ca lled
hadrons; they are separated into two sets: the baryons, with halfinteger
spins, and the mesons with zero or integer spins. Pa rticles that do not
experience th e str ong interaction are called leptons. In or der to qua ntify
the difference bet ween baryo ns and leptons, it h as been found n ecessary to
introduce the baryon nu mber B = +1 to denote a baryon, B = –1 to
denote a n ant ibaryon a nd B = 0 f or all other pa rticles. Lepto ns ar e
characterized by the lepton number L = +1, antileptons ar e assigned L =
–1, and all other particles are a ssigned L = 0. It is a p resentday fact,
based up on nu merou s observations, that the total baryon an d lepton
number in any interaction is conserved. For exa mple, in t he decay of the
free neutron we find
–
n0 =p+ +e + .e0
B = +1 = +1 + 0 + 0
L = 0 = 0 + 1 +(–1).
The fundamental symmetr ies in Nat ure responsible for these conservation
laws are not known at this time. These conservation laws may, in all
likelihood, be broken.
In discussing the patterns o f baryon families in char gestrangeness
space, we wish to incorporat e the fact that we a re de aling with baryons
that interact via the strong nuclear force in which isospin an d strangeness
are conserved. We therefore choose to describe their patterns in isospin139
hypercharge s pace, where the hype rchar ge Y is defined to i nclude bot h the
strangeness and the bar yon a ttribute o f the particle in an additive way:
Y = B + S.
The diagonal operator F8 is therefore assumed to be directly associated
with the hype rchar ge op erator,
F8 = (v3/2)Y.
Because I3 an d Y co mmute, states can be chosen th at are
simultaneous eigenstates of the o perators F3 an d F8. Since no other SU(3)
operators commute with I3 an d Y, no other ad ditive quantum numbers are
associated with the SU(3) symmetr y. The op erators F4,...F8 are considered
to be new con stantsofthemotion of t he strong interaction ham iltonian.
13.4 Irreducible r epresentations of SU(3)
In an earlier discussion of the irredu cible representations of SU(2),
we found that the commutation relations of the g enerators of the group
were satisfied not only by the fundamental 2×2 matrices but also by
matrices of h igher dimension [(2J + 1) . (2J + 1)], where J can have the
values 1/2, 1, 3/2, 2, ....The Jvalues cor respond to the spin of the par ticle
whose state is given by a sp inor (a column vector with special
tra nsformation properties). In t he 2×2 r epresentation, both covariant an d
contravariant spinors are allowed:
i) covariant spinors (with lower indices) are written as 2component 140
columns that transform under U . SU(2) as
i´ = Uij
j ,
where
a1
= ,
a2
and
ii) contravariant spinors (with uppe r indices) are written as
2compon ent rows that transform as:
j´ = i Uij † ,
where
= (b1, b2).
The coand contra variant spinors are transformed with the aid of t he anti
symmetric ten sors ij an d ij. For example,
j
=
i ij
tra nsforms as a covariant sp inor with the form
b2
i = .
b1
The higherdimensional repre sentations are built up f rom the fundamental
form by taking ten sor p roduc ts of the fundamental spinors i , j , or i
and by symmetrizing and antisymmetrizing the re sult. We state, without
pro of, the th eorem that is used in this method:
when a tensor product o f spi nors has been broken down into its symmetric
and antisymmetric parts, it has been decomposed into irreducible 141
representatio ns of the SU(n). (See Wigner’ s standard work for the
original discussion of the method, and de Swart in Rev. Mod. Ph ys. 35,
(1963) for a detailed discussion of tensor analysis in the stud y of the irreps
of SU(n))
As an example, we wr ite the te nsor produ ct of two covariant spinors
i an d j in the following way
i. j = i j =( i j + j i)/2 + ( i j j i)/2
There are four elements associated with the product (i,j can have values 1
and 2).
The symm etric part of t he product has three independent elements,
and transforms as an ob ject that has spin J=1. (There are 2J + 1 members
of the symmetric set). The antisymmetric part has one element, and
therefore transforms as an object with spin J = 0. This result is familiar in
the theo ry of angu lar momentum in Quantum M echanics. The explicit
forms of the four elements are:
J3 =+1: 1 1
J = 1 J3 =0 :(1/v2)(12 + 2 1)
J 3 =–1 : 2 1
and
J = 0 J3 = 0 : (1/v2)(12 – 21) .
Higherdimensional representations are built up from the tensor products 142
of covariant and contra variant 3spinors, an d respectively. The
pro ducts are then written in terms of their symm etric and antisymmetric
par ts in order to obtain the irreducible representations. For example, the
pro duct i
j, i,j = 1,2,3, can be written
i
j = ( i
j (1/3)dij
k
k) + (1/3)dij
k
k
,
in which the trace has been separated out. The trace is a zero rank tensor
with a single co mponent. The other tensor is a traceless, symmetr ic tensor
with eight independent components. The decomposition is written
symbolically as:
3 . 3 = 8 . 1.
We can form the te nsor produ ct of two covariant 3spinors, i j as
follows:
= (1/2)( i j + j i) + (1/2)( i j
–
j
i), i,j = 1,2,3.
ij
Sym bolically, we have
3 . 3 = 6 . 3 ,
in which the symmetric tensor has six components and the a ntisymmetric
tensor h as three c omponents.
Other tensor produ cts that w ill be of interest are
3 . 3 . 3 = 10 . 8 . 8 . 1 ,
and
8 . 8 = 27 . 10 . 10 . 8 . 8´ . 1 .
The appearance of the octet “8” in the 3 . 3 decomposition (recall 143
the observed octet of spin1/2 baryons), and the decuplet “10” in the triple
pro duct 3 . 3 . 3 de composition (recall the o bserved decuplet of spin3/2
bar yons), was of p rime importance in t he development of the group theory
of “elementary” pa rticles.
13.4.1 Weight diagrams
Two of t he GellMann ma trices, 3 an d 8, are diagonal. We can
write the eigenvalue equations:
3u = auu, 3v = avv, and 3w = aww,
an d
8u = ßuu, 8v = ßvv, and 8w = ßww ,
wh ere ai an d ßi are the eigenvalues.
Let a and b be normalization factors associated with the operators 3
and 8, r epectively, so that
a 0 0 b 0 0
N
= 0 – a 0 , an d 8
N= 0 b 0 .
3
0 0 0 0 0 –2b
If
u = [1, 0 , 0], v = [0, 1 , 0], and w = [0, 0 , 1] (columns), we find
3Nu = au , 8Nu =bu,
3Nv = –av , 8Nv =bv ,
and
3Nw =0w , 8
Nw = –2bw.
The weight v ectors are formed from the pairs of eigenvalues:
[au, ßu] = [a, b], 144
[av, ßv] = [a, b],
and
[aw, ßw] = [0, 2b].
A w eight diagram is obt ained by plotting these vectors in the a–ß
space, thus:
ß
2b
b
–2a –a a 2a
b
–2 b
a
This weight diagram for the fundamental “3” representation of SU(3) was
wellknown to Mathematicians at the time of the first use of SU(3)
symmetry in P article Physics. It was to play a key role in the development
of the q uark model.
13.5 The 3quark model of m atter
Although the octet and decuplet patterns of hadrons of a given spin
and parity emerge as irreducible repre sentations of t he group SU(3),
major problems rem ained that resu lted in a great deal of scepticism
concerning the validity of t he SU (3) model of fundamental particles. The
most pressing problem was: why are there no known particles associated 145
with the fundamental triplets 3, 3 of SU(3) that exh ibit the symmetry of
the weight diagram discussed in t he last section? In 1964 , GellMann, and
independently, Zweig, proposed th at three fundamental entities do ex ist
that correspond to the base states of SU(3), and that they form boun d
states of the hadrons. That such entities have not been observed in the
free state is simply related to their enorm ous binding energy. The three
entities were called quarks by GellMann, and aces by Zweig. The Gell
Mann term has surv ived. The antiquarks ar e associated with the
conjugate 3 representation. The three quarks, denoted by u, d, and s (u
and d fo r the upand d ownisospin states, and s for strangeness) have
highly unusual properties; they are
Lab el B Y I I3 Q=I3 +Y/2 S = Y  B
u 1/3 1/3 1/2 +1/2 +2 /3 0
d 1/3 1/3 1/2 –1/2 –1 /3 0
s 1/3 –2/3 0 0 –1/3 –1
s –1 /3 2/3 0 0 + 1/3 + 1
d –1 /3 –1/3 1 /2 +1 /2 + 1/3 0
u –1 /3 –1/3 1/2 –1/2 – 2/3 0
The quarks occupy the following positions in I3  Y space
Y Y 146
d u
I3
I3
s
u d
s
These diagrams have the same relative forms as t he 3 an d 3 we ight
diagrams of SU(3).
The baryons are made up of q uark triplets, and t he me sons are made
up of the simplest possible structures, namely quark–antiquark pairs. The
covariant and cont ravar iant 3spinors introduced in t he previous section
are now given physical significance:
= [u, d , s], a covariant co lumn 3spinor,
and
= (u, d, s), a contrav ariant row 3spinor.
where u = [1, 0, 0 ], d = [0, 1, 0 ], an d s = [0, 0, 1] represent the unitary
symmetry part of t he to tal wavefunctions of the three quarks.
The form al operators A±, B±, and C±, introd uced in section 13.3.1,
are now viewed as operators that transform one fla vor (type)of quark into
another flavor (th ey ar e shift op erators):
A± = I±(I3) . I3 ± 1 ,
B± = U±(U3) . U3 ± 1, called the U spin operator,
and
C± = V±(V3) . V3 ± 1, called the Vspin operator. 147
Exp licitly, we have
I+(–1/2) . 1 /2 : d . u
I–(+1/2) . –1 /2 : u . d
U+(–1/2) . 1/2 : s . d
U–(+1/2) . –1 /2 : d . s
V+(–1/2) . 1/2 : u . s
and
V–(+1/2) . 1/2 : s . u.
The quarks can be characterized by the three quantum numbers I3, U3, V3.
Their positions in the I3U3V3 space again show the unde rlying
symmetry :
d (1/2, 1/2, 0)
I3
U3 V3
+1/2
u(1/2, 0, 1/2)
1/2 + 1/2 I3
+1/ 2
s( 0, 1/2, 1/2)
V3 Y U3
The members of the octet of mesons with JP = 0– are formed from qqpairs 148
that belong to the fundamental 3, 3 representation of the quark s. The p0
and .0 mesons are linear combinations of the q q(bar) states, thus
K0 d s Y K+ us
s
d u
p– d u p 0 p+ ud
1 .0 +1
u d
s
K– su K0 sd
I3
The none t for med from the tensor produ ct 3 . 3 is split into an octe t
that is even under the label exchange of two particles, and a s inglet that is
odd unde r label exchange :
3 . 3 = 8 . 1
where the “1” is
.0´ = (1/v 3)(uu + dd + ss ),
an d the two members of the octet at the center are:
p0 = (1/v2)(uu – dd) and .0 = (1/v6)(uu + dd  2ss).
The action of I– on p+ is to transform it into a p 0. This operation has the
following mea ning in terms o f I– acting on the ten sor p roduc t, u . d:
I–(u . d) = (I–u) . d + u . (I–d) (c.f. derivative rule) 149
.. .
I ( p+) = d . d+ u u
–
.p 0
Omitting the tensor product sign, normalizing the amplitudes, and ch oosing
the phases in the generally accepted w ay, we have:
p0 = (1/v2)(uu – dd).
The singlet .0´ is said to be orthogonal to p0 an d .0 at the origin.
If the symmetry of the octet were exact, th e eight members of t he
octet wo uld have t he same ma ss. This is not quite the case; the symmetry
is broken by the d ifference in effective ma ss between the u and dquark
(essentially the same effective masses: ~ 300 MeV/c2) and the squark
(effective ma ss ~ 500 MeV/c2). (It should be noted that the effective
masses of the quarks, derived from the mass differences of hadronpairs, is
not the same as the “cu rrentquark” ma sses that appear in the
fundamental theory . The discrepancy b etween the effective masses and the
fundamental masses is not fully understood at this time).
The decomposition of 3 . 3 . 3 is
3 . 3 . 3 =(6 . 3) . 3
= 10 . 8 . 8´ . 1
in which the states of the 10 are symmetric, the 1 is antisymmetric, and the
8, 8´ states are of mixed symmetry. The decuplet that appears in t his
decomposition is associated with the o bserved decuplet of spin3/2 baryon s.
In terms of t he th ree fundamental quarks — u, d, and s, the mak e up of
the individual members of the decuplet is shown schematically in the 150
following diagram:
d dd ~ du d ~ u ud uuu
~ dds ~ dus ~ uus
~ sds ~ sus
sss
The prec ise makeup of each state, labelled by ( Y, I, I3,) is given in the
following tab le:
(1, 3/2, +3/2)
(1, 3/2, +1/2)
(1, 3/2, –1/2)
(1, 3/2, –3/2)
(0, 1, +1)
(0, 1, 0)
(0, 1, – 1)
(–1, 1/2, +1/2)
(–1, 1/2, –1/2)
(2, 0, 0)
=
=
=
=
=
=
=
=
=
=
uuu(++ )
(1/v3)(udu + duu + uud)
(1/v3)(ddu + udd + dud)
ddd(–)
(1/v3)(usu + suu + uus)
(1/v6)(uds + dsu + sud + dus + sdu + usd)
(1/v3)(dsd + sdd + dds )
(1/v3)(ssu + uss + s us)
(1/v3)(ssd + dss + s ds)
sss(–)
The general theory of t he permuta tion group of n entities, and its
rep resentations, is out side the scope of this introdu ction. The use of t he
Young ta bleaux in obtaining the mixed symmetry s tates is treated in
Hamermesh (1962).
The char ges of the .++
of the q uarks, namely
, .–, and O–
pa rticles fix the fractional values
quark flavor charge (in units of the electron charge) 151
u +2/3
d –1/3
s –1/3
The char ges of the antiquarks ar e opp osite in sign to the se values.
Extensive rev iews of the 3quark model and its application to the
physics of the lowenergy pa rt of the hadron spectrum can be found in
Gasiorowicz (1966) and Gibson and Pollard (1976).
13.6 The need for a new quantum number: hidden color
Immediately after the introduction of the 3quark model by
GellMann and Zweig, it was recognized that the model was not c onsistent
with the exte nded Pauli principle when applied to bou nd states of three
quarks. For example, the structu re of the spin3/2 .+ st ate is such tha t, if
each qua rk is assigned a spin sq = 1/2, the three spins must be aligned ...
to give a net spin of 3 /2. (It is assumed that the relative orbital angu lar
mom entum of t he qu arks in the .+ is zero (a symmetric sstate) — a
reasonable assumption to mak e, as it corresponds to minimum kinetic
energy, and t herefore to a s tate of lowest total ener gy). The quark s are
fermions, and therefore they must obey the generalized Pau li Principle;
they cannot e xist in a completely aligned spin state when they are in an s
state that is symm etric unde r particle (quark) exchange. The unitary spin
component of the total wavefunction must be antisymmetric. Greenberg
(1964) proposed th at a new d egree of freedom mus t be assigned t o the
quarks if the Pauli Principle is not to be violated. The new proper ty was 152
later ca lled “color”, a property with profound c onsequences. A quark
with a certain flavor possesses color (red, blue, gre en, say) that
cor responds to the triplet representation of another form of SU(3) —
namely SU(3)C, where the subscript C differentiates the group from that
introduced by GellMann and Zweig — the flavor group SU(3)F. The anti
quarks (that possess anticolor) have a triplet repre sentation in SU(3)C that
is the c onjugate representation (the 3). Although the SU(3)F symmetr y is
known no t to be exact, we have evidence tha t the SU( 3)C symmetry is an
exact symmetry of Nature. Baryons and mesons are found to be colorless;
the color singlet of a baryo n occurs in the decomposition
S U(3)C = 3 . 3 . 3 = 10 + 8 + 8´ + 1 .
The meson singlets consist of linear c ombinations of the form
1 = (RR + BB + GG)/v3 .
Although the hadro ns ar e colorless, certain observable quantities are
directly related to the number of colors in the model. For exa mple, the
pur ely electromagnetic decay of t he neutral pion, p0, into two ph otons
p0 = . + .,
has a lifetime tha t is found to be closely propo rtionl to the square of the
number of colors. (Adler (1970) gives G = h/t = 1(eV) (no.of colors)2 .
The measurements o f the lifetime give a value of G ~8 eV, consistent with
Ncol s = 3. Since these early mea surements, refined experiments have
demonstrated that there are three, and only three, colors associated with 153
the quarks.
In studies of electronpositron interactions in the G eVregion, the
rat io of cros s sections:
––
R= s(e+e . ha drons)/s(e+e .µ+µ –)
is found to depend linearly on the num ber of colors. Good agre ement
between the theoretical mode l and the measured value of R, over a wide
ran ge of ener gy, is obt ained for three co lors.
The color attribute of the q uarks has been responsible for the
development o f a t heory of t he strongly interacting particles, called
quantum chrom odynamics. It is a field theo ry in which the quarks ar e
generators of a new typ e of field — the color field. The mediators of the
field are called gluons; they p ossess color, the at tribute of the source of the
field. Consequently, they can interact with each other through the color
field. This is a field quite unlike the electrodynamic field of classical
electrom agnetism, in which the field quanta do not ca rry the at tribute of
the sour ce of the field, namely electric charge. The photons, therefore, do
not interact with each other .
The gluons tr ansform a quark of a particular color into a quark of a
different color. For example, in the interaction bet ween a red quark and a
blue quark, t he colors are e xchanged. This requires that the e xchanged
gluon carry color and anticolor, as shown:
qb qr 154
the color lines are continuo us.
q r
gluon, g rb(bar)carries red and antiblue:
qb
Three different co lors permit nine different way s of coupling quarks
and gluons. Three of t hese are redred, blueblue, and greengreen that do
not change th e colors. A linear combination ~(R.R + B.B + G.G) is
symmetric in the c olor labels, and this combination is the singlet state of
the group SU(3)C. Eight gluons, each with two co lor indices, are the refore
req uired in t he 3color theo ry of quarks.
13.7 More ma ssive quarks
In 1974, the results of two independent ex periments, one a study of
–
the reac tion p + Be . e+ + e .. (Ting et al.) and t he ot her a stud y of
–
e+ + e . ha drons ..(Richter et al) — showed th e pre sence of a shar p
resonance at a centerofmass energy of 3.1 GeV. The lifetime of the
resonant state was found to be ~10–20 seconds — more than 10 3 seconds
longer than expected for a state formed in the strong interaction. The
resonant state is called the J/.. It was quickly realized t hat the state
cor responds t o the ground st ate o f a n ew qu ark–antiquark system, a
bound st ate cc, where c is a four th, m assive, quark e ndowed with one unit
of a new quan tum number c, called “cha rm”. The quantum numbers
assigned to the cquark are
JP=1/2+, c = 1, Q/e= +2/3, and B = 1/3. 155
Sou nd theoretical argum ents for a four th quark, carry ing a new
quantum number, had been put forward s everal years be fore the
experimental observation of the J/. st ate. Since 197 4, a complex set of
states of the “cha rmonium” s ystem has been observed, and t heir decay
pro perties studied. De tailed com parisons have been m ade with
sophisticated theo retical models of the system.
The inclusion of a char med q uark in the set of q uarks mean s that the
gro up SU(4)F must be used in place of the original GellMannZweig group
SU(3)F. Although the SU (4)F symmetr y is badly broken because the
effective mass of the c harmed qua rk is ~ 1.8 GeV/c2, some u seful
applications have been made using the model. The fundamental
rep resentations ar e
[ u, d, s, c], a covariant column spinor,
an d
( u, d, s, c), a contr avariant row spinor.
The irreps ar e constructed in a way th at is analogous to that u sed in
SU(3)F, namely, by finding the symmetric and antisymmetric
decompositions of the v arious tensor p roducts. The most u seful are:
4 . 4 = 15 . 1,
4 . 4 = 10 . 6,
4 . 4 . 4 = 20sym . 20mix . 20´mix . 4ant i,
an d
15 . 15 = 1 . 15sym . 15ant i . 20sym . 45 . 45 . 84. 156
The “15” includesthe n oncharmed (JP = 0– ) mesons and the following
charmed mesons:
D0 = cu, D0 = cu, mass = 1863 MeV/c2 ,
D+ = cd, D– = cd, mass = 1868 MeV/c2,
F + = cs, F– = cs, mass = 2.04 MeV/c2.
In order to discuss the baryons, it is necessary to i nclude the quark spin,
and therefore the group must be extended to SU(8)F. Relatively few
bar yons have been studied in deta il in this extended framework.
In 1977, welldefined resonant st ates were observed at ene rgies of
9.4, 10.01, and 10 .4 GeV, and were interpre ted a s bound st ates of another
quark, t he “b ottom ” qua rk, b , and its antipartn er, t he b. Mesons can be
formed that include the bquark, thus
00
Bu = bu, Bd = bd, Bs = bs, and Bc = bc .
The stud y of the w eak decay modes of t hese states is curre ntly fashionable.
In 1994, definitive evidence was obtained for the existence of a sixth
quark, c alled the “top” quark, t. It is a massive entity with a mass almost
200 times the mass of t he proton!
We have seen that the q uarks interact strongly via gluon e xchange.
They also tak e par t in the w eak interaction. In an earlier discussion of
isospin, the group generators were introduc ed by considering the ßdecay
of the free n eutro n:
–
n0 . p+ +e + .0 .
We now know that, at the microscopic level, this process involves the 157
tra nsformation of a dquark into a uquark, and the p roduc tion of the
––
carrier of the weak force, the ma ssive W pa rticle. The W bo son (spin 1)
decays instantly into a n electron –antineutrino pair, as shown:
.0
–
W–1 e
neutron, n0 d(– 1/3) . u(+2/3)
d u
proton, p+
u u
d d
The carriers of the Weak Force, W± , Z0, were first identified in pp
collisions at high cent erofmass ener gy. The processes involve
quark–antiquark interactions, and the dete ction of t he decay electrons and
positrons.
e+ e
–
Z0
u(+2/3) u (–2/3)
.0
W+ e+
u(+2/3) d (+1/3)
.0
W– e
–
d(1/3) u (2/3)
The char ge is conserved at each vertex.
The carriers have very large measured masses: 158
mass W± ~ 8 1 GeV/c2, and ma ss Z0 ~ 93 Ge V/c2.
(Recall that the range of a force . 1/(mass of carrier); the W and Z masses
cor respond to a very sh ort range,~1018 m, for the Weak F orce).
Any qua ntitative discussion of curren t work using Group Theory to
tackle Grand Unified Theories, requires a k nowledge o f Quantum Field
Theory that is not expected of readers of t his introd uctor y boo k.
14 159
LIE GROU PS AND THE CONSERVATION LAWS OF THE
PHY SICAL UNIVERSE
14.1 Po isson and Dirac Brackets
The Poisson Bracket of two differentiable functions
A(p1, p2, ...pn, q1, q2, ...qn)
and
B(p1, p2, ...pn, q1, q2, ...qn)
of two sets of variables (p1, p2, ...pn) and (q1, q2, ...qn) is defined as
{A, B} =.1n (.A/.qi)(. B/.pi) – (.A/.pi)(. B/.qi) .
If A = (pi, qi), a dyn amical variable, and
B = H(pi, qi), the h amiltonian of a dyna mical system,
where pi is the (canonical) momentum and qi is a (g eneralized) coordinate,
then
{ , H} = .1n (. /.qi)(. H/.pi) – (. /.pi)(.H/.qi) .
(n is the”num ber of degrees of freedom” of the system).
Hamilton’s eq uations ar e
.H/.pi = dqi/dt and .H/.qi = – dpi/dt ,
and therefore
{, H} = .1n (. /.qi)(dqi/dt) + (. /.qi)(dpi/dt) .
The total differential of (pi, qi) is
d = .1n (. /.qi)dqi + (. /.pi)dpi. 160
and its time derivative is
( d /dt) = .1n (. /.qi)(dqi/dt) + (. /.pi)(dpi/dt)
•
= { , H} = .
If the Poisson Bra cket is zero, the ph ysical quantity is a constant
of the m otion .
In Quantum Mechanics, the re lation
is replaced by
(d
(d
/dt) = { , H}
/dt) = (i/h))[ , H],
Heisenberg’s equation of motion. It is the custom to refer to the
commutator [, H] as the Dirac Bracket.
If the D irac Bracket is zero, the quan tum m echanical quantity is
a c onstant of the motion..
(Dirac p roved that the classical Poisson Br acket { , H} can be
identified with the Heisenberg commuta tor –(i/h)[ , H] b y mak ing a
suitable choice of the order of the q’s a nd p’s in the Poisson Bra cket).
14.2 Infinitesimal unitary transforma tions in Q uantum Mec hanic s
The Lie form of an infinitesimal unitary transformation is
U = I +idaX/h ,
where da ia real infinitesimal parameter, and X is an hermitian operator.
(It is straightforward to sh ow th at this form of U is, indeed, unitary).
Let a dynamical operator change under an infinitesimal unitary 161
tra nsformation:
. ´ = UU–1
= (I +idaX/h) (I– idaX/h)
= –ida X/h +idaX /h to 1storder
= +i(daX – daX)/h
= +i(F – F)/h.
where
F = daX.
The infinitesimal change in is therefore
d = ´ –
= i[F, ]/h
If we identify F with –Hdt (the c lassical form f or a purely temporal change
in the system) then
d = i[Hdt, ]/h,
or
–d = i[H, ]dt/h ,
so that
–d /dt = i[H, ]/h.
For a tempora l change in the system, d /dt = – d /dt.
The fundamental Heisenberg equati on of motion 162
d /dt = i[ , ]/h
is therefore deduced from th e uni tary infin itesimal transformation of the
operator .
This approach was taken by Schwinger in his form ulation of Quan tum
Mechanics.
F = Hdt is directly related to the generator, X, of a Quantum
Mechanical infinitesimal transformation, and the refore we can associate
with every sy mmetr y transformation of the system an hermitian operator
F that is a constant of the motion  its eigenvalues do not change with
time. This is an example of Noether’s Theorem:
A c onservation law is associated with every symmetry of the
equation s of motio n. I f the equations of m otion are unchanged by the
tra nsformations of a Group then a property of the system w ill remain
constant as t he system evolves with time. As a wellknown exam ple, if the
equations of motion of an ob ject are invariant u nder translations is space,
the linear momentum of the system is conserved.
15 163
BIBLIOGRAPHY
The following book s are typical of those that a re suitable for
Undergra duates:
Arm strong, M. A., Groups and Sy mmetry, S pringerVerlag, New York,
1988.
Burns, Gerald, Int roduction to Group Theory, Academic Press, New York,
1977.
Fritzsch, Harald, Quarks: the S tuff of Matter, Basic Books, New York,
1983.
Jones, H. F., Groups, Representations and Physics, Adam H ilger, Bristol,
1990.
The following book s are of a specialized nature; they are typical of
what lies beyond the present introduction.
Carter, Roger; Segal, Graeme; and Macdonald, Ian, Lectures on Lie
Groups and Lie Algebras, Cambridge University Press, Cambridge, 1995.
Commins, E. D., and Bucksbaum, P. H., Weak Interactions of Leptons and
Quarks, Cambridge U niversity Press, Cambridge, 1983
Dickson, L. H., Lin ear Groups, Dover, New York, 1960 .
Eisenhart, L. P., Continuo us Groups of Tr ansformations, Dover, New
York, 1961.
Elliott, J. P ., and Dawber, P. G., Symmetry in P hysics, Vol. 1,
Oxford University Press, New York, 1979.
GellMann, Murray, and Ne’eman, Yuval, The Eightfold Way, 164
Benjamin, New York, 1964.
Gibson, W. M., and Pollard, B. R., Symmetry Principles in Elementary
Particle Phys ics, Cambridge U niversity Press, Cambridge, 1976.
Hamermesh, Morton, Group Theory and its Applications to P hysical
Problems, Dover, New York, 1989 .
Lichtenberg, D. B., Unitary Symmetry and Elementary P articles,
Academic Pres s, New York, 1978.
Lipkin, Harry J., Lie Groups fo r Ped estrians, NorthHolland, Amsterdam,
1966.
Lomont, J. S., Applications of Finite Groups, Dover, New York, 1993.
Racah, G., Group Theory and S pectroscopy, Reprinted in CER N(6168),
1961.
Wigner, E. P. , Group Theory and its Applications to the Qu antum
Mechanics of Atomic Spectra, Academic Press, New York, 1959.