Chapter 1
Introduction into the group
theory
1.1 Introduction
In these lectures on group theory and its application to the particle physics
there have been considered problems of classication of the particles along
representations of the unitary groups, calculations of various characteristics
of hadrons, have been studied in some detail quark model. First chapter are
dedicated to short introduction into the group theory and theory of group
representations. In some datails there are given unitary groups SU (2) and
SU (3) which play eminent role in modern particle physics. Indeed SU (2)
group is aa group of spin and isospin transforation as well as the base of
group of gauge transformations of the electroweak interactons SU (2) U (1)
of the Salam- Weinberg model. The group SU (3) from the other side is
the base of the model of unitary symmetry with three avours as well as
the group of colour that is on it stays the whole edice of the quantum
chromodynamics.
In order to aknowledge a reader on simple examples with the group theory
formalism mass formulae for elementary particles would be analyzed in detail. Other important examples would be calculations of magnetic moments
and axial-vector weak coupling constants in unitary symmetry and quark
models. Formulae for electromagnetic and weak currents are given for both
models and problem of neutral currents is given in some detail. Electroweak
3
current of the Glashow-Salam-Weinberg model has been constructed. The
notion of colour has been introduced and simple examples with it are given.
Introduction of vector bosons as gauge elds are explained. Author would
try to write lectures in such a way as to enable an eventual reader to perform
calculations of many properties of the elementary particles by oneself.
1.2 Groups and algebras. Basic notions.
Denition of a group
Let be a set of elements G g1 g2 ::: gn :, with the following properties:
1. There is a multiplication law gigj = gl , and if gi gj 2 G, then
gigj = gl 2 G, i j l = 1 2 ::: n.
2. There is an associative law gi(gj gl) = (gigj )gl.
3. There exists a unit element e, egi = gi, i = 1 2 ::: n.
4. There exists an inverse element gi;1, gi;1 gi = e, i = 1 2 ::: n.
Then on the set G exists the group of elements g1 g2 ::: gn :
As a simple example let us consider rotations on the plane. Let us dene
a set of rotations on angles :
Let us check the group properties for the elements of this set.
1. Multiplication law is just a summation of angles: 1 + 2 = 3 2 :
2. Associative law is written as (1 + 2) + 3 = 1 + (2 + 3):
3. The unit element is the rotation on the angle 0(+2n):
4. The inverse element is the rotation on the angle ;(+2n):
Thus rotations on the plane about some axis perpendicular to this plane
form the group.
Let us consider rotations of the coordinate axes x y z, which dene Decartes coordinate system in the 3-dimensional space at the angle 3 on the
plane xy around the axis z:
0 01 0
10 1
0 1
x
cos
sin
x
3
3 0
B@ y0 CA = B@ ;sin3 cos3 0 CA B@ y CA = R3(3) B@ xy CA
z0
0
0 1
z
z
(1.1)
Let be an innitesimal rotation. Let is expand the rotation matrix R3()
4
in the Taylor series and take only terms linear in :
3
2
2
R3() = R3(0) + dR
d =0 + O( ) = 1 + iA3 + O( )
where R3(0) is a unit matrix and it is introduced the matrix
0
1
0
;
i
0
3
B@ i 0 0 CA
A3 = ;i dR
=
d =0
0 0 0
(1.2)
(1.3)
which we name 'generator of the rotation around the 3rd axis' (or z-axis).
Let us choose = 3=n then the rotation on the angle 3 could be obtained
by n-times application of the operator R3(), and in the limit we have
n
n
iA3 3 :
R3(3) = nlim
!1R3 (3 =n)] = nlim
!11 + iA33=n] = e
(1.4)
Let us consider rotations around the axis U :
0 01 0
10 1
0 1
x
cos
0 sin2
x
xC
2
B@ y0 CA = B
C
B
C
B
@ 0 1 0 A @ y A = R2(2) @ y A z0
;sin2 0 cos2 z
z
where a generator of the rotation around the axis U is introduced:
0
10 1
0
0
;
i
xC
dR
C
B
0
0
0
A2 = ;i d2 = B
@
A@ y A:
=0
i 0 0
z
(1.5)
(1.6)
Repeat it for the axis x:
0 01 0
10 1
0 1
x
1
0
0
x
x
B
@ y0 CA = B
@ 0 cos1 sin1 CA B@ y CA = R1(1) B@ y CA z0
0 ;sin1 cos1
z
z
where a generator of the rotation around the axis xU is introduced:
0
1
0
0
0
1
B
C
A1 = ;i dR
d =0 = @ 00 0i ;0i A
5
(1.7)
(1.8)
Now we can write in the 3-dimensional space rotation of the Descartes coordinate system on the arbitrary angles, for example:
R1(1)R2(2)R3(3) = eiA11 eiA2 2 eiA33
However, usually one denes rotation in the 3-space in some other way, namely by using Euler angles:
R(
) = eiA3eiA2 eiA3
functions).
(Usually Cabibbo-Kobayashi-Maskawa matrix is chosen as VCKM =
1
0
c12c13
s12c13
s13e;i13
=B
@ ;s12c23 ; c12s23s13ei13 c12c23 ; s12s23s13ei13 s23c13 CA :
s12s23 ; c12c23s13ei13 ;c12s23 ; s12c23s13ei13 c23c13
Here cij = cosij sij = sinij (i j = 1 2 3) while ij - generalized Cabibbo
angles. How to construct it using Eqs.(1.1, 1.5, 1.7) and setting 13 = 0?)
Generators Al l = 1 2 3 satisfy commutation relations
Ai Aj ; Aj Ai = Ai Aj ] = iijk Ak i j k = 1 2 3
where ijk is absolutely antisymmetric tensor of the 3rd rang. Note that
matrices Al l = 1 2 3 are antisymmetric, while matrices Rl are orthogonal,
that is, RTi Rj = ij , where index T means 'transposition'. Rotations could
be completely dened by generators Al l = 1 2 3. In other words, the
group of 3-dimensional rotations (as well as any continuous Lie group up
to discrete transformations) could be characterized by its algebra , that
is by denition of generators Al l = 1 2 3, its linear combinations and
commutation relations.
Denition of algebra
L is the Lie algebra on the eld of the real numbers if:
(i) L is a linear space over k (for x 2 L the law of multiplication to numbers from the set K is dened), (ii) For x y 2L the
commutator is dened as x y], and x y] has the following properties: x y] = x y] x y] = x y] at 2 K and x1 + x2 y] =
x1 y] + x2 y],
x y1 + y2] = x y1] + x y2] for all x y 2L
x x] = 0 for all x y 2L
x y]z] + y z]x] + z x]y] = 0 (Jacobi identity).
6
1.3 Representations of the Lie groups and
algebras
Before a discussion of representations we should introduce two notions: isomorphism and homomorphism.
Denition of isomorphism and homomorphism
Let be given two groups, G and G0.
Mapping f of the group G into the group G0 is called isomorphism or
homomorphis
If f (g1 g2) = f (g1 )f (g2) for any g1g2 2 G:
This means that if f maps g1 into g10 and g2 W g20 , then f also maps g1g2 into
g10 g20 .
However if f (e) maps e into a unit element in G0, the inverse in general
is not true, namely, e0 from G0 is mapped by the inverse transformation f ;1
into f ;1(e0), named the core (orr nucleus) of the homeomorphism.
If the core of the homeomorphism is e from G such one-to-one homeomorphism is named isomorphism.
Denition of the representation
Let be given the group G and some linear space L. Representation of the group G in L we call mapping T , which to every element
g in the group G put in correspondence linear operator T (g) in the
space L in such a way that the following conditions are fullled:
(1) T (g1g2) = T (g1)T (g2) for all g1 g2 2 G
(2) T (e) = 1 where 1 is a unit operator in L:
The set of the operators T (g) is homeomorphic to the group G.
Linear space L is called the representation space, and operators T (g) are
called representation operators, and they map one-to-one L on L. Because of
that the property (1) means that the representation of the group G into L is
the homeomorphism of the group G into the G ( group of all linear operators
in W L, with one-to-one correspondence mapping of L in L). If the space L
is nite-dimensional its dimension is called dimension of the representation
T and named as nT . In this case choosing in the space L a basis e1 e2 ::: en
7
current of the Glashow-Salam-Weinberg model has been constructed. The
notion of colour has been introduced and simple examples with it are given.
Introduction of vector bosons as gauge elds are explained. Author would
try to write lectures in such a way as to enable an eventual reader to perform
calculations of many properties of the elementary particles by oneself.
1.2 Groups and algebras. Basic notions.
Denition of a group
Let be a set of elements G g1 g2 ::: gn :, with the following properties:
1. There is a multiplication law gigj = gl , and if gi gj 2 G, then
gigj = gl 2 G, i j l = 1 2 ::: n.
2. There is an associative law gi(gj gl) = (gigj )gl.
3. There exists a unit element e, egi = gi, i = 1 2 ::: n.
4. There exists an inverse element gi;1, gi;1 gi = e, i = 1 2 ::: n.
Then on the set G exists the group of elements g1 g2 ::: gn :
As a simple example let us consider rotations on the plane. Let us dene
a set of rotations on angles :
Let us check the group properties for the elements of this set.
1. Multiplication law is just a summation of angles: 1 + 2 = 3 2 :
2. Associative law is written as (1 + 2) + 3 = 1 + (2 + 3):
3. The unit element is the rotation on the angle 0(+2n):
4. The inverse element is the rotation on the angle ;(+2n):
Thus rotations on the plane about some axis perpendicular to this plane
form the group.
Let us consider rotations of the coordinate axes x y z, which dene Decartes coordinate system in the 3-dimensional space at the angle 3 on the
plane xy around the axis z:
0 01 0
10 1
0 1
x
cos
sin
x
3
3 0
B@ y0 CA = B@ ;sin3 cos3 0 CA B@ y CA = R3(3) B@ xy CA
z0
0
0 1
z
z
(1.1)
Let be an innitesimal rotation. Let is expand the rotation matrix R3()
4
in the Taylor series and take only terms linear in :
3
2
2
R3() = R3(0) + dR
d =0 + O( ) = 1 + iA3 + O( )
where R3(0) is a unit matrix and it is introduced the matrix
0
1
0
;
i
0
3
B@ i 0 0 CA
A3 = ;i dR
=
d =0
0 0 0
(1.2)
(1.3)
which we name 'generator of the rotation around the 3rd axis' (or z-axis).
Let us choose = 3=n then the rotation on the angle 3 could be obtained
by n-times application of the operator R3(), and in the limit we have
n
n
iA3 3 :
R3(3) = nlim
!1R3 (3 =n)] = nlim
!11 + iA33=n] = e
(1.4)
Let us consider rotations around the axis U :
0 01 0
10 1
0 1
x
cos
0 sin2
x
xC
2
B@ y0 CA = B
C
B
C
B
@ 0 1 0 A @ y A = R2(2) @ y A z0
;sin2 0 cos2 z
z
where a generator of the rotation around the axis U is introduced:
0
10 1
0
0
;
i
xC
dR
C
B
0
0
0
A2 = ;i d2 = B
@
A@ y A:
=0
i 0 0
z
(1.5)
(1.6)
Repeat it for the axis x:
0 01 0
10 1
0 1
x
1
0
0
x
x
B
@ y0 CA = B
@ 0 cos1 sin1 CA B@ y CA = R1(1) B@ y CA z0
0 ;sin1 cos1
z
z
where a generator of the rotation around the axis xU is introduced:
0
1
0
0
0
1
B
C
A1 = ;i dR
d =0 = @ 00 0i ;0i A
5
(1.7)
(1.8)
Now we can write in the 3-dimensional space rotation of the Descartes coordinate system on the arbitrary angles, for example:
R1(1)R2(2)R3(3) = eiA11 eiA2 2 eiA33
However, usually one denes rotation in the 3-space in some other way, namely by using Euler angles:
R(
) = eiA3eiA2 eiA3
functions).
(Usually Cabibbo-Kobayashi-Maskawa matrix is chosen as VCKM =
1
0
c12c13
s12c13
s13e;i13
=B
@ ;s12c23 ; c12s23s13ei13 c12c23 ; s12s23s13ei13 s23c13 CA :
s12s23 ; c12c23s13ei13 ;c12s23 ; s12c23s13ei13 c23c13
Here cij = cosij sij = sinij (i j = 1 2 3) while ij - generalized Cabibbo
angles. How to construct it using Eqs.(1.1, 1.5, 1.7) and setting 13 = 0?)
Generators Al l = 1 2 3 satisfy commutation relations
Ai Aj ; Aj Ai = Ai Aj ] = iijk Ak i j k = 1 2 3
where ijk is absolutely antisymmetric tensor of the 3rd rang. Note that
matrices Al l = 1 2 3 are antisymmetric, while matrices Rl are orthogonal,
that is, RTi Rj = ij , where index T means 'transposition'. Rotations could
be completely dened by generators Al l = 1 2 3. In other words, the
group of 3-dimensional rotations (as well as any continuous Lie group up
to discrete transformations) could be characterized by its algebra , that
is by denition of generators Al l = 1 2 3, its linear combinations and
commutation relations.
Denition of algebra
L is the Lie algebra on the eld of the real numbers if:
(i) L is a linear space over k (for x 2 L the law of multiplication to numbers from the set K is dened), (ii) For x y 2L the
commutator is dened as x y], and x y] has the following properties: x y] = x y] x y] = x y] at 2 K and x1 + x2 y] =
x1 y] + x2 y],
x y1 + y2] = x y1] + x y2] for all x y 2L
x x] = 0 for all x y 2L
x y]z] + y z]x] + z x]y] = 0 (Jacobi identity).
6
1.3 Representations of the Lie groups and
algebras
Before a discussion of representations we should introduce two notions: isomorphism and homomorphism.
Denition of isomorphism and homomorphism
Let be given two groups, G and G0.
Mapping f of the group G into the group G0 is called isomorphism or
homomorphis
If f (g1 g2) = f (g1 )f (g2) for any g1g2 2 G:
This means that if f maps g1 into g10 and g2 W g20 , then f also maps g1g2 into
g10 g20 .
However if f (e) maps e into a unit element in G0, the inverse in general
is not true, namely, e0 from G0 is mapped by the inverse transformation f ;1
into f ;1(e0), named the core (orr nucleus) of the homeomorphism.
If the core of the homeomorphism is e from G such one-to-one homeomorphism is named isomorphism.
Denition of the representation
Let be given the group G and some linear space L. Representation of the group G in L we call mapping T , which to every element
g in the group G put in correspondence linear operator T (g) in the
space L in such a way that the following conditions are fullled:
(1) T (g1g2) = T (g1)T (g2) for all g1 g2 2 G
(2) T (e) = 1 where 1 is a unit operator in L:
The set of the operators T (g) is homeomorphic to the group G.
Linear space L is called the representation space, and operators T (g) are
called representation operators, and they map one-to-one L on L. Because of
that the property (1) means that the representation of the group G into L is
the homeomorphism of the group G into the G ( group of all linear operators
in W L, with one-to-one correspondence mapping of L in L). If the space L
is nite-dimensional its dimension is called dimension of the representation
T and named as nT . In this case choosing in the space L a basis e1 e2 ::: en
7
it is possible to dene operators T (g) by matrices of the order n:
0 t t ::: t 1
BB t1121 t1222 ::: t12nn CC
t(g) = B
@ ::: ::: ::: ::: CA tn1 tn2 ::: tnn
X
T (g)ek = tij (g)ej t(e) = 1 t(g1g2) = t(g1)t(g2):
The matrix t(g) is called a representation matrix T: If the group G itself
consists from the matrices of the xed order, then one of the simple representations is obtained at T (g) = g ( identical or, better, adjoint representation).
Such adjoint representation has been already considered by us above and
is the set of the orthogonal 33 matrices of the group of rotations O(3) in
the 3-dimensional space. Instead the set of antisymmetrical matrices Ai i =
1 2 3 forms adjoint representation of the corresponding Lie agebra. It is
obvious that upon constructing all the represetations of the given Lie algebra
we indeed construct all the represetations of the corresponding Lie group (up
to discrete transformations).
By the transformations of similarity PODOBIQ T 0(g) = A;1T (g)A it is
possible to obtain from T (g) representation T 0(g) = g which is equivalent to
it but, say, more suitable (for example, representation matrix can be obtained
in almost diagonal form).
Let us dene a sum of representations T (g) = T1(g) + T2(g) and say that
a representation is irreducible if it cannot be written as such a sum ( For the
Lie group representations is denition is suciently correct).
For search and classication of the irreducible representation (IR) Schurr's
lemma plays an important role.
Schurr's lemma: Let be given two IR's, t (g ) and t (g ), of the group G.
Any matrix w , such that w t(g ) = t (g )B for all g 2 G either is equal to 0
(if t(g)) and t (g) are not equivalent) or KRATNA is proportional to the unit
matrix I .
Therefore if B 6= I exists which commutes with all matrices of the given
representation T (g) it means that this T (g) is reducible. Really,, if T (g) is
reducible and has the form
!
T
0
1 (g )
T (g) = T1(g) + T2(g) =
0 T2(g) 8
then
!
1
0
1I
B = 0 I 2 6= I
2
and T (g) B ] = 0:
For the group of rotations O(3) it is seen that if Ai B ] = 0 i = 1 2 3 then
i
R B ] = 0,i.e., for us it is sucient to nd a matrix B commuting with all
the generators of the given representation, while eigenvalues of such matrix
operator B can be used for classications of the irreducible representations
(IR's). This is valid for any Lie group and its algebra.
So, we would like to nd all the irreducible representations of nite dimension of the group of the 3-dimensional rotations, which can be be reduced to
searching of all the sets of hermitian matrices J123 satisfying commutation
relations
Ji Jj ] = iijk Jk :
There is only one bilinear invariant constructed from generators of the algebra
(of the group): J~2 = J12 + J22 + J32 for which J~2 Ji] = 0 i = 1 2 3: So IR's
can be characterized by the index j related to the eigenvalue of the operator
J~2.
In order to go further let us return for a moment to the denition of the
representation. Operators T (g) act in the linear n-dimensional space Ln and
could be realized by n n matrices where n is the dimension of the irreducible
representation. In this linear space n-dimensional vectors ~v are dened and
any vector can be written as a linear combination of n arbitrarily chosen
linear independent vectors ~ei, ~v = Pni=1 vi~ei. In other words, the space Ln
is spanned on the n linear independent vectors ~ei forming basis in Ln . For
example, for the rotation group O(3) any 3-vector can be denned, as we
have already seen by the basic vectors
0 1
0 1
0 1
1
0
0
B
C
B
C
B
ex = @ 0 A ey = @ 1 A ez = @ 0 C
A
0
0
1
0 1
x
as ~x = B
@ y CA or ~x = xex + yey + zez : And the 3-dimensional representation (
z
adjoint in this case) is realized by the matrices Ri i = 1 2 3. We shall write
it now in a dierent way.
9
Our problem is to nd matrices Ji of a dimension n in the basis of n
linear independent vectrs, and we know, rst, commutation reltions Ji Jj ] =
iijk Jk , and, the second, that IR's can be characterized by J~2. Besides, it is
possible to perform similarity transformation of the Eqs.(1.8,1.6,1.3,) in such
a way that one of the matrices, say J3, becomes diagonal. Then its diagonal
elements would be eigenvalues of new basical vectors.
0
1
0
1
1 0 ;i
1 0 i
1
1
U=p B
(1.9)
@ 0 1 + i 0 CA U ;1 = p B@ 0 1 ; i 0 CA
2 ;i 0 1
2 i 0 1
0
1
1 0 0
UA2U ;1 = 2 B
@ 0 0 0 CA
0 0 ;1
0
1
0 ;i 0
U (A1 + A3)U ;1 = 2 B
@ i 0 ;i CA
0 i 0
0
1
0 1 0
U (A1 ; A3)U ;1 = 2 B
@ 1 0 1 CA
0 1 0
In the case of the 3-dimensional representation usually one chooses
0 0 p2 0 1
p
p
J1 = 21 B
(1.10)
@ 2 p0 2 CA
0
2 0
0 0
p
J2 = 12 B
@i 2
0
;ip2 0p 1C
0 ;i 2 A
p
i 2
0
(1.11)
0
1
1 0 0
J3 = B
@ 0 0 0 CA :
0 0
Let us choose basic vectors as
0
0 1
1
j1 + 1 >= B@ 0 CA j1 0 >= B@
0
10
(1.12)
;1
1
0
1C
A
0
0 1
0
j1 ; 1 >= B@ 0 CA 1
and
J3j1 + 1 >= +j1 + 1 > J3j1 0 >= 0j1 + 1 >
J3j1 ; 1 >= ;j1 + 1 > :
In the theory of angular momentum these quantities form basis of the representation with the full angular momentum equal to 1. But they could be
identied with 3-vector in any space, even hypothetical one. For example,
going a little ahead, note that triplet of -mesons in isotopic space could be
placed into these basic vectors:
+ ; 0 ! j >= j1 1 > j0 >= j1 0 > :
Let us also write in some details matrices for J = 2, i.e., for the representation
of the dimension n = 2J + 1 = 5:
1
0
0
1
0
0
0
q
BB
C
3=2 q 0 0 C
BB 1 q 0
CC
B
J1 = B 0 3=2 q 0
(1.13)
3=2 0 C
CC
BB
3=2 0 1 C
A
@0 0
0 0
0
1 0
0
q0
BB 0 ;i
BB i q0 ;i 3=2
J2 = B
BB 0 i 3=2 q0
B@ 0 0
i 3=2
1
0
0 C
0 C
CC
q0
(1.14)
;i 3=2 0 CCC
0
;i CA
0 0
0
i
0
1
0
BB 20 01 00 00 00 CC
C
B
(1.15)
J3 = B
BB 0 0 0 0 0 CCC
@ 0 0 0 ;1 0 A
0 0 0 0 ;2
As it should be these matrices satisfy commutation relations Ji Jj ] = iijk Jk i j k =
1 2 3, i.e., they realize representation of the dimension 5 of the Lie algebra
corresponding to the rotation group O(3).
11
Basic vectors can be chosen as:
0 1
0
1
B
BB
B
CC
0C
B
j1 + 2 >= BBB 0 CCC j1 + 1 >= BBBB
@0A
@
0
0
1
0
0
0
1
0
CC
BB 00
CC
B
CC j1 0 >= BBB 1
A
@0
0 1
0 1
0
BB 0 CC
BB 00 CC
j1 ; 1 >= BBBB 0 CCCC j1 ; 2 >= BBBB 0 CCCC :
@1A
@0A
0
1
CC
CC
CC A
0
1
J3j2 +2 >= +2j2 +2 > J3j2 +1 >= +1j2 +1 > J3j2 0 >= 0j2 0 >
J3j2 ;1 >= ;1j2 ;1 > J3j2 ;2 >= ;2j2 ;2 > :
Now let us formally put J = 1=2 although strictly speaking we could
not do it. The obtained matrices up to a factor 1=2 are well known Pauli
matrices:
!
1
0
1
J1 = 2 1 0 !
1
0
;
i
J2 = 2 i 0
!
1
1
0
J3 = 2 0 ;1
These matrices act in a linear space spanned on two basic 2-dimensional
vectors
! !
1 0 :
0
1
There appears a real possibility to describe states with spin (or isospin)
1/2. But in a correct way it would be possible only in the framework of
another group which contains all the representations of the rotation group
O(3) plus PL@S representations corresponding to states with half-integer spin
(or isospin, for mathematical group it is all the same). This group is SU (2).
12
1.4 Unitary unimodular group SU(2)
Now after learning a little the group of 3-dimensional rotations in which
dimension of the minimal nontrivial representation is 3 let us consider more
complex group where there is a representation of the dimension 2. For this
purpose let us take a set of 2 2 unitary unimodular U ,i.e., U yU = 1
detU = 1. Such matrix U can be written as
U = ei a k
k
k k = 1 2 3 being hermitian matrices, ky = k , chosen in the form of Pauli
matrices
!
0
1
1 = 1 0 !
0
;
i
2 = i 0
!
1
0
3 = 0 ;1 and ak k = 1 2 3 are arbitrary real numbers. The matrices U form a group
with the usual multiplying law for matrices and realize identical (adjoint)
represenation of the dimension 2 with two basic 2-dimensional vectors.
Instead Pauli matrices have the same commutation realtions as the generators of the rotation group O(3). Let us try to relate these matrices with
a usual 3-dimensional vector ~x = (x1 x2 x3). For this purpose to any vector
~x let us attribute SOPOSTAWIM a quantity X = ~~x :
!
x
x
;
ix
3
1
2
Xba = x + ix ;x
a b = 1 2:
(1.16)
1
2
3
Its determinant is detXba = ;~x2 that is it denes square of the vector length.
Taking the set of unitary unimodular matrices U , U yU = 1 detU = 1 in 2dimensional space, let us dene
X 0 = U yXU
and detX 0 = det(U yXU ) = detX = ;~x2: We conclude that transformations
U leave invariant the vector length and therefore corresponds to rotations in
13
the 3-dimensional space, and note that U correspond to the same rotation.
Corresponding algebra SU (2) is given by hermitian matrices k k = 1 2 3
with the commutation relations
i j ] = 2iijk k
where U = ei a :
And in the same way as in the group of 3-rotations O(3) the representation
of lowest dimension 3 is given by three independent basis vectors,for example
, x,y,z in SU (2) 2- dimensional representation is given by two independent
basic spinors q, = 1 2 which could be chosen as
k
k
!
q = 10 1
q
2
!
0 :
1
The direct product of two spinors q and q can be expanded into the sum
of two irreducible representations (IR's) just by symmetrizing and antisymmetrizing the product:
q q = 12 fqq + q qg + 12 qq ; q q] T fg + T ]:
Symmetric tensor of the 2nd rank has dimension dnSS = n(n + 1)=2 and for
n = 2 d2SS = 3 which is seen from its matrix representation:
T fg =
T11 T12
T12 T22
!
and we have taken into account that Tf21g = Tf12g.
Antisymmetric tensor of the 2nd rank has dimension dnAA = n(n ; 1)=2
and for n = 2 d2AA = 1 which is also seen from its matrix representation:
T
]
= ;T0 T012
12
!
and we have taken into account that T21] = ;T12] and T11] = ;T22] = 0.
Or instead in values of IR dimensions:
2 2 = 3 + 1:
14
According to this result absolutely antisymmetric tensor of the 2nd rank (12 = ;21 = 1) transforms also as a singlet of the group SU (2) and we can
use it to contract SU (2) indices if needed. This tensor also serves to uprise
and lower indices of spinors and tensors in the SU (2).
= u u :
0
0
0
0
12 = u11u2212 + u21u1221 = (u11u22 ; u21u12)12 = DetU12 = 12
as DetU = 1. (The same for 21.)
1.5
SU (2) as a spinor group
Associating q with the spin functions of the entities of spin 1/2, q1 j "i
and q2 j #i being basis spinors with +1/2 and -1/2 spin projections,
correspondingly, (baryons of spin 1/2 and quarks as we shall see later) we
can form symmetric tensor T fg with three components
T f11g = q1q1 j ""i
T f12g = p1 (q1q2 + q2q1) p1 j("# + #")i
2
2
f
22g
2 2
T = q q j ##i
p
and we have introduced 1= 2 to normalize this component to unity.
Similarly for antisymmetric tensor associating again q with the spin
functions of the entities of spin 1/2 let us write the only component of a
singlet as
(1.17)
T 12] = p1 (q1q2 ; q2q1) p1 j("# ; #")i
2
2
p
and we have introduced 1= 2 to normalize this component to unity.
Let us for example form the product of the spinor q and its conjugate
spinor q whose basic vectors could be taken as two rows (1 0) and (0 1).
Now expansion into the sum of the IR's could be made by subtraction of a
trace (remind that Pauli matrices are traceless)
q q = (qq ; 12 q q ) + 12 q q T + I
15
where T is a traceless tensor of the dimension dV = (n2 ;1) corresponding to
the vector representation of the group SU (2) having at n = 2 the dimension
3 I being a unit matrix corresponding to the unit (or scalar) IR. Or instead
in values of IR dimensions:
2 2 = 3 + 1:
The group SU (2) is so little that its representations T fg and T corresponds
to the same IR of dimension 3 while T ] corresponds to scalar IR together
with I . For n 6= 2 this is not the case as we shall see later.
One more example of expansion of the product of two IR's is given by
the product
T ij] qk =
= 14 (qiqj qk ; qj qiqk ; qiqk qj + qj qk qi ; qk qj qi + qk qiqj ) +
(1.18)
1 (qiqj qk ; qj qiqk + qiqk qj ; qj qk qi + qk qj qi ; qk qiqj ) =
4
= T ikj] + T ik]j
or in terms of dimensions:
2
2
n(n ; 1)=2 n = n(n ;63n + 2) + n(n 3; 1) :
For n = 2 antisymmetric tensor of the 3rd rank is identically zero. So we
are left with the mixed-symmetry tensor T ik]j of the dimension 2 for n = 2,
that is, which describes spin 1/2 state. It can be contracted with the the
absolutely anisymmetric tensor of the 2nd rank eik to give
eik T ik]j tjA
and tjA is just the IR corresponding to one of two possible constructions of
spin 1/2 state of three 1/2 states (the two of them being antisymmetrized).
The state with the sz = +1=2 is just
t1A = p1 (q1q2 ; q2q1)q1 p1 j "#" ; #""i:
(1.19)
2
2
(Here q1 =", q2 =#.)
16
The last example would be to form a spinor IR from the product of the
symmetric tensor T fikg and a spinor qj .
T fijg qk =
= 41 (qiqj qk + qj qiqk + qiqkqj + qj qk qi + qk qj qi + qk qiqj ) +
(1.20)
1 (qiqj qk + qj qiqk ; qiqk qj ; qj qk qi ; qkqj qi ; qk qiqj ) =
4
= T fikjg + T fikgj
or in terms of dimensions:
2
2
n(n + 1)=2 n = n(n +63n + 2) + n(n 3; 1) :
Symmetric tensor of the 4th rank with the dimension 4 describes the state
of spin S=3/2, (2S+1)=4. Instead tensor of mixed symmetry describes state
of spin 1/2 made of three spins 1/2:
eij T fikgj TSk :
TSj is just the IR corresponding to the 2nd possible construction of spin 1/2
state of three 1/2 states (with two of them being symmetrized). The state
with the sz = +1=2 is just
(1.21)
TS1 = p1 (e122q1q1q2 + e21(q2q1 + q1q2)q1 6
p16 j2 ""# ; "#" ; #""i:
17
1.6 Isospin group SU (2)I
Let us consider one of the important applications of the group theory and
of its representations in physics of elementary particles. We would discuss
classication of the elementary particles with the help of group theory. As
a simple example let us consider proton and neutron. It is known for years
that proton and neutron have quasi equal masses and similar properties as
to strong (or nuclear) interactions. That's why Heisenberg suggested to consider them one state. But for this purpose one should nd the group with
the (lowest) nontrivial representation of the dimension 2. Let us try (with
Heisenberg) to apply here the formalism of the group SU (2) which has as
we have seen 2-dimensional spinor as a basis of representation. Let us introduce now a group of isotopic transformations SU (2)I . Now dene nucleon
as a state with the isotopic spin I = 1=2 with two projections ( proton with
I3 = +1=2 and neutron with I3 = ;1=2 ) in this imagined 'isotopic space'
practically in full analogy with introduction of spin in a usual space. Usually
basis of the 2-dimensional representation of the group SU (2)I is written as
a isotopic spinor (isospinor)
!
N = np what means that proton and neutron are dened as
!
!
p = 10 n = 01 :
Representation of the dimension 2 is realized by Pauli matrices 2 2 k k =
1 2 3 ( instead of symbols i i = 1 2 3 which we reserve for spin 1/2 in usual
space). Note that isotopic operator + = 1=2(1 + i2) transforms neutron
into proton , while ; = 1=2(1 ; i2) instead transforms proton into neutron.
It is known also isodoublet of cascade hyperons of spin 1/2 0; and
masses 1320 MeV. It is also known isodoublet of strange mesons of spin 0
K +0 and masses 490 MeV and antidublet of its antiparticles K 0;.
And in what way to describe particles with the isospin I = 1? Say, triplet
of -mesons + ; 0 of spin zero and negative parity (pseudoscalar mesons)
with masses m() = 139 5675 0 0004 MeV, m(0) = 134 9739 0 0006
MeV and practically similar properties as to strong interactions?
18
In the group of (isotopic) rotations it would be possible to dene isotopic
vector ~ = (1 2 3) as a basis (where real pseudoscalar elds 12 are related to charged pions by formula = 1 i2, and 0 = 3), generators
Al l = 1 2 3 as the algebra representation and matrices Rl l = 1 2 3 as the
group representation with angles k dened in isotopic space. Upon using
results of the previous section we can attribute to isotopic triplet of the real
elds ~ = (1 2 3) in the group SU (2)I the basis of the form
ba =
p
p
3= 2 p (1 ; i2p)= 2
(1 + i2)= 2
;3= 2
! p1
2
0
;
!
+
; p12 0 p
where charged pions are described by complex elds = (1 i2)= 2.
So, pions can be given in isotopic formalism as 2-dimensional matrices:
!
!
= 00 10 ; = 01 00 0 =
+
p1
2
0
0
;p
1
2
!
which form basis of the representation of the dimension 3 whereas the representation itself is given by the unitary unimodular matrices 2 2 U.
In a similar way it is possible to describe particles of any spin with the
isospin I = 1: Among meson one should remember isotriplet of the vector
(spin 1) mesons 0 with masses 760 MeV:
p12 0
;
+
;p !
1 0
2
:
(1.22)
Among particles with half-integer spin note, for example, isotriplet of strange
hyperons found in early 60's with the spin 1/2 and masses 1192 MeV 0
which can be writen in the SU (2) basis as
p1
2
0
;
;p
!
+
1
2
0
:
(1.23)
Representation of dimension 3 is given by the same matrices U .
Let us record once more that experimentally isotopic spin I is dened as
a number of particles N = (2I +1) similar in their properties, that is, having
the same spin, similar masses (equal at the level of percents) and practically
identical along strong interactions. For example, at the mass close to 1115
19
MeV it was found only one particle of spin 1/2 with strangeness S=-1 - it is
hyperon ! with zero electric charge and mass 1115 63 0 05 MeV. Naturally,
isospin zero was ascribed to this particle. In the same way isospin zero was
ascribed to pseudoscalar meson (548).
It is known also triplet of baryon resonances with the spin 3/2, strangness
S=-1 and masses M ( + (1385)) = 1382 8 0 4 m\w, M ( 0(1385)) =
1383 7 1 0 m\w, M ( ; (1385)) = 1387 2 0 5 m\w, (resonances are
elementary particles decaying due to strong interactions and because of that
having very short times of life one upon a time the question whether they
are 'elementary' was discussed intensively) 0(1385) ! !00(88 2%)
or 0(1385) ! (12 2%) (one can nd instead symbol Y1(1385) for
this resonance).
It is known only one state with isotopic spin I = 3=2 (that is on experiment it were found four practically identical states with dierent charges)
: a quartet of nucleon resonances of spin J = 3=2 #++ (1232), #+(1232),
#0(1232), #;(1232), decaying into nucleon and pion (measured mass dierence M+ -M0 = 2,70,3 MeV). ( We can use also another symbol N (1232).)
There are also heavier 'replics' of this isotopic quartet with higher spins.
In the system 0;0 it was found only two resonances with spin 3/2
(not measured yet) 0; with masses 1520 MeV, so they were put into
isodublet with the isospin I = 1=2.
Isotopic formalism allows not only to classify practically the whole set of
strongly interacting particles (hadrons) in economic way in isotopic multiplet
but also to relate various decay and scattering amplitudes for particles inside
the same isotopic multiplet.
We shall not discuss these relations in detail as they are part of the
relations appearing in the framework of higher symmetries which we start to
consider below.
At the end let us remind Gell-Mann{Nishijima relation between the particle charge Q, 3rd component of the isospin I3 and hypercharge Y = S + B ,
S being strangeness, B being baryon number (+1 for baryons, -1 for antibaryons, 0 for mesons):
Q = I3 + 12 Y:
As Q is just the integral over 4th component of electromagnetic current, it
means that the electromagnetic current is just a superposition of the 3rd component of isovector current and of the hypercharge current which is isoscalar.
20
1.7 Unitary symmetry group SU(3)
Let us take now more complex Lie group, namely group of 3-dimensional
unitary unimodular matrices which has played and is playing in modern
particle physics a magnicient role. This group has already 8 parameters.
(An arbitrary complex 3 3 matrix depends on 18 real parameters, unitarity
condition cuts them to 9 and unimodularity cuts one more parameter.)
Transition to 8-parameter group SU (3) could be done straightforwardly
from 3-parameter group SU (2) upon changing 2-dimensional unitary unimodular matrices U to the 3-dimensional ones and to the corresponding
algebra by changing Pauli matrices k k = 1 2 3 to Gell-Mann matrices
= 1 :: 8:
0
1
0
1
0 1 0
0 ;i 0
1 = B
(1.24)
@ 1 0 0 CA 2 = B@ i 0 0 CA
0 0 0
0 0 0
1
0
1
0
0 0 1
1 0 0
(1.25)
3 = B
@ 0 ;1 0 CA 4 = B@ 0 0 0 CA
0 0 0
1 0 0
0
1
0
1
0 0 ;i
0 0 0
5 = B
(1.26)
@ 0 0 0 CA 6 = B@ 0 0 1 CA
i 0 0
0 1 0
0
1
0
1
0 0 0
1
0
0
7 = B
(1.27)
@ 0 0 ;i CA 8 = p1 B@ 0 1 0 CA
3
0 i 0
0 0 ;2
12 i 12 j ] = ifijk 21 k
1
1
1
1
1
1
fp123 = 1 f147
p = 2 f156 = ; 2 f246 = 2 f257 = 2 f346 = 2 f367 = ; 2 f458 =
3
3
2 f678 = 2 :
(In the same way being patient one can construct algebra representation
of the dimension n for any unitary group SU (n) of nite n. ) These matrices
realize 3-dimensional representation of the algebra of the group SU (3) with
the basis spinors
0 1 0 1 0 1
B@ 10 CA B@ 01 CA B@ 00 CA :
1
0
0
21
Representation of the dimension 8 is given by the matrices 8 8 in the
linear space spanned over basis spinors
011
001
001
001
B
BB 1 CC
BB 0 CC
BB 0 CC
0C
B
CC
B
B
C
B
C
BB 0 CC
B
BB 0 CC
BB 1 CC
0C
B
C
BB CC
B
CC
BB 0 CC
BB 0 CC
0
B
x1 = B
CC x2 = BB 0 CC x3 = BB 0 CC :x4 = BBB 10 CCC B
0
B
C
BB CC
BB CC
BB CC
B
0C
0C
0C
B
C
B
B
BB 0 CC
B
C
B
C
B
C
0
0
0
@ A
@ A
@ A
@0A
0
0
0
0
001
001
001
001
B
BB 0 CC
BB 0 CC
BB 0 CC
0C
B
C
B
C
B
C
B
C
BB 0 CC
BB 0 C
BB 0 CC
BB 0 CC
C
BB CC
BB 0 C
BB 0 CC
BB 0 CC
C
x5 = B
x6 = B
CC :x7 = BB 0 CC x8 = BBB 00 CCC BB 1 C
C
B
0
BB CC
BB CC
BB CC
BB 0 C
C
1
0
C
B
C
B
C
BB 0 CC
B@ 0 C
B
C
B
C
0
1
A
@ A
@ A
@0A
0
0
0
1
But similar to the case of SU (2) where any 3-vector can be written as a
traceless matrix 2 2, also here any 8-vector in SU (3) X = (x1 ::: x8) can
be put into the form of the 3 3 matrix:
X
(1.28)
X = p1 8k=1 k xk =
2
1
0
1
x
x
x
1 ; ix2
4 ; ix5
3 + p3 x8
p12 BB@ x1 + ix2 ;x3 + p13 x8 x6 ;2 ix7 CCA :
x4 + ix5 x6 + ix7 ; p3 x8
In the left upper angle we see immediately previous expression (1.16) from
SU (2).
The direct product of two spinors q and q can be expanded exactly
at the same manner as in the case of SU (2) (but now = 1 2 3) into
the sum of two irreducible representations (IR's) just by symmetrizing and
antisymmetrizing the product:
q q = 12 fqq + q qg + 12 qq ; q q] T fg + T ]: (1.29)
22
Symmetric tensor of the 2nd rank has dimension dnS = n(n + 1)=2 and for
n = 3 d3S = 6 which is seen from its matrix representation:
0 11 12 13 1
T T T
B
f
g
T = @ T 12 T 22 T 23 C
A
T 13 T 23 T 33
and we have taken into account that T fikg = T fkig T ik (i 6= k i k = 1 2 3).
Antisymmetric tensor of the 2nd rank has dimension dnA = n(n ; 1)=2
and for n = 3 d3A = 3 which is also seen from its matrix representation:
0
1
0 t12 t13
T ] = B
@ ;t12 0 t23 CA
;t13 ;t23 0
and we have taken into account that T ik] = ;T ki] tik (i 6= k i k = 1 2 3)
and T 11] = T 22] = T 33] = 0.
In terms of dimensions it would be
n n = n(n + 1)=2jSS + n(n ; 1)=2jAA
(1.30)
or for n = 3 3 3 = 6 + 3.
Let us for example form the product of the spinor q and its conjugate
spinor q whose basic vectors could be taken as three rows (1 0 0), (0 1
0)and (0 0 1). Now expansion into the sum of the IR's could be made by
subtraction of a trace (remind that Gell-Mann matrices are traceless)
q q = (qq ; n1 q q ) + n1 q q T + I
where T is a traceless tensor of the dimension dV = (n2 ;1) corresponding to
the vector representation of the group SU (3) having at n = 3 the dimension
8 I being a unit matrix corresponding to the unit (or scalar) IR. In terms
of dimensions it would be n n = (n2 ; 1) + 1n or for n = 3 3 3 = 8 + 1.
At this point we nish for a moment with a group formalism and make a
transition to the problem of classication of particles along the representation
of the group SU (3) and to some consequencies of it.
23