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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 classi cation 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 edi ce 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 sucha way astoenablean eventual reader to perform calculations of many properties of the elementary particles by oneself.

De nition of a group Let be a set of elements Gg1 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 associativelaw 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 :

1.2 Groups and algebras. Basic notions.

As a simple example let us consider rotations on the plane. Let us de ne 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. Associativelaw is written as ( 1 + 2)+ 3 = 1 +( 2 + 3): 3. The unit element is the rotation on the angle 0(+2 n): 4. The inverse element is the rotation on the angle ; (+2 n): 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 de ne Decartes coordinate system in the 3-dimensional space at the angle 3 on the plane xy around the axis z:

0 01 0 cos B x0 C = B ;sin @y A @ z0 0

3 3

sin cos 0

3 3

0 x x 0 C B y C = R3 ( 3 ) B y C A@ A @A 1 z z

10 1

01

(1.1)

Let be an in nitesimal rotation. Let is expand the rotation matrix R3( ) 4


in the Taylor series and take only terms linear in : R3( )= R3(0) + dR3 + O( 2)= 1 + iA3 + O( 2) d =0 where R3(0) is a unit matrix and it is introduced the matrix

(1.2)

A3 = ;i dR d

3 =0

0 1 0 ;i 0 =B i 0 0 C @ A
000

(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 wehave

R3( 3) = nlim R3( 3=n)]n = nl!1 1 + iA3 3=n]n = eiA3 3 : im !1
Let us consider rotations around the axis U :

(1.4)

0 01 0 B x0 C = B cos @y A @ 0 z0 ;sin

2 2

0 sin 2 C B x C = R2( 2) B x C 1 0 A@ y A @yA 0 cos 2 z z

10 1

01
(1.5)

where a generator of the rotation around the axis U is introduced:

A2 = ;i dR d
Repeat it for the axis x:

2 =0

0 10 0 0 ;i x = B 0 0 0 CB y @ A@ i00 z 10 1

1 C: A

(1.6)

0 01 B x0 C = @y A z0

0 0 B 1 cos @0 0 ;sin

1 1

0 x x sin 1 C B y C = R1( 1) B y C A@ A @A cos 1 z z

01
(1.7)

where a generator of the rotation around the axis xU is introduced:

A1 = ;i dR d

1 =0

0 1 00 0 = B 0 0 ;i C @ A 0i 0

(1.8)

5


De nition 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 de ned), (ii) For x y 2L the commutator is de ned 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

Nowwe 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)= eiA1 1 eiA2 2 eiA3 3 However, usually one de nes rotation in the 3-space in some other way, namely by using Euler angles: R( )= eiA3 eiA2 eiA3 functions). (Usually Cabibbo-Kobayashi-Maskawa matrixischosen as VCKM = 1 0 c12c13 s12c13 s13e;i 13 = B ;s12c23 ; c12s23s13ei 13 c12c23 ; s12s23s13ei 13 s23c13 C : A @ s12s23 ; c12c23s13ei 13 ;c12s23 ; s12c23s13ei 13 c23c13 Here cij = cos ij sij = sin ij (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 ]= i ijk 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, RT Rj = ij , where index T means 'transposition'. Rotations could i be completely de ned 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 de nition of generators Al l = 1 2 3 , its linear combinations and commutation relations.


1.3 Representations of the Lie groups and algebras
Before a discussion of representations weshould introduce two notions: 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

De nition of isomorphism and homomorphism

If f (g1 g2)= f (g1 )f (g2) for any g1g2 2 G:

0 0 This means that if f maps g1 into g1 and g2 W g2, then f also maps g1g2 into 00 g1g2. However if f (e) maps e into a unit elementin G0, the inverse in general is not true, namely, e0 from G0 is mapped bythe 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.

De nition 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 , whichto every element g in the group G put in correspondence linear operator T (g) in the space L in suchaway that the following conditions are ful lled: (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 abasis 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 sucha way astoenablean eventual reader to perform calculations of many properties of the elementary particles by oneself.

De nition of a group Let be a set of elements Gg1 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 associativelaw 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 :

1.2 Groups and algebras. Basic notions.

As a simple example let us consider rotations on the plane. Let us de ne 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. Associativelaw is written as ( 1 + 2)+ 3 = 1 +( 2 + 3): 3. The unit element is the rotation on the angle 0(+2 n): 4. The inverse element is the rotation on the angle ; (+2 n): 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 de ne Decartes coordinate system in the 3-dimensional space at the angle 3 on the plane xy around the axis z:

0 01 0 cos B x0 C = B ;sin @y A @ z0 0

3 3

sin cos 0

3 3

0 x x 0 C B y C = R3 ( 3 ) B y C A@ A @A 1 z z

10 1

01

(1.1)

Let be an in nitesimal rotation. Let is expand the rotation matrix R3( ) 4


in the Taylor series and take only terms linear in : R3( )= R3(0) + dR3 + O( 2)= 1 + iA3 + O( 2) d =0 where R3(0) is a unit matrix and it is introduced the matrix

(1.2)

A3 = ;i dR d

3 =0

0 1 0 ;i 0 =B i 0 0 C @ A
000

(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 wehave

R3( 3) = nlim R3( 3=n)]n = nl!1 1 + iA3 3=n]n = eiA3 3 : im !1
Let us consider rotations around the axis U :

(1.4)

0 01 0 B x0 C = B cos @y A @ 0 z0 ;sin

2 2

0 sin 2 C B x C = R2( 2) B x C 1 0 A@ y A @yA 0 cos 2 z z

10 1

01
(1.5)

where a generator of the rotation around the axis U is introduced:

A2 = ;i dR d
Repeat it for the axis x:

2 =0

0 10 0 0 ;i x = B 0 0 0 CB y @ A@ i00 z 10 1

1 C: A

(1.6)

0 01 B x0 C = @y A z0

0 0 B 1 cos @0 0 ;sin

1 1

0 x x sin 1 C B y C = R1( 1) B y C A@ A @A cos 1 z z

01
(1.7)

where a generator of the rotation around the axis xU is introduced:

A1 = ;i dR d

1 =0

0 1 00 0 = B 0 0 ;i C @ A 0i 0

(1.8)

5


De nition 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 de ned), (ii) For x y 2L the commutator is de ned 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

Nowwe 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)= eiA1 1 eiA2 2 eiA3 3 However, usually one de nes rotation in the 3-space in some other way, namely by using Euler angles: R( )= eiA3 eiA2 eiA3 functions). (Usually Cabibbo-Kobayashi-Maskawa matrixischosen as VCKM = 1 0 c12c13 s12c13 s13e;i 13 = B ;s12c23 ; c12s23s13ei 13 c12c23 ; s12s23s13ei 13 s23c13 C : A @ s12s23 ; c12c23s13ei 13 ;c12s23 ; s12c23s13ei 13 c23c13 Here cij = cos ij sij = sin ij (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 ]= i ijk 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, RT Rj = ij , where index T means 'transposition'. Rotations could i be completely de ned 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 de nition of generators Al l = 1 2 3 , its linear combinations and commutation relations.


1.3 Representations of the Lie groups and algebras
Before a discussion of representations weshould introduce two notions: 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

De nition of isomorphism and homomorphism

If f (g1 g2)= f (g1 )f (g2) for any g1g2 2 G:

0 0 This means that if f maps g1 into g1 and g2 W g2, then f also maps g1g2 into 00 g1g2. However if f (e) maps e into a unit elementin G0, the inverse in general is not true, namely, e0 from G0 is mapped bythe 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.

De nition 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 , whichto every element g in the group G put in correspondence linear operator T (g) in the space L in suchaway that the following conditions are ful lled: (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 abasis e1 e2 ::: en

7


it is possible to de ne operators T (g)by matrices of the order n:

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 byus aboveand is the set of the orthogonal 3 3 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 equivalentto it but, say, more suitable (for example, representation matrix can be obtained in almost diagonal form). Let us de ne a sum of representations T (g)= T1(g)+ T2(g) and saythat a representation is irreducible if it cannot be written as such a sum ( For the Lie group representations is de nition is su ciently correct). For search and classi cation 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) arenot 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 ! T1(g) 0 T (g)= T1(g)+ T2(g)= 0 T2(g) 8

0 t t ::: t 1 B t11 t12 ::: t1n C 2n t(g)= B ::: ::: ::: ::: C B 21 22 C @ A 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


then

B=

1

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 B ]= 0 ,i.e., for us it is su cient to nd a matrix B commuting with all R the generators of the given representation, while eigenvalues of such matrix operator B can be used for classi cations of the irreducible representations (IR's). This is valid for any Lie group and its algebra. So, wewould 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 J1 2 3 satisfying commutation relations Ji Jj ]= i ijk 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 de nition 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 ~ are de ned and v any vector can be written as a linear combination of n arbitrarily chosen linear independent vectors ~i, ~ = Pn=1 vi~i. In other words, the space Ln ev e i e is spanned on the n linear independent vectors ~i forming basis in Ln . For example, for the rotation group O(3) any 3-vector can be de nned, as we have already seen by the basic vectors 01 01 01 1 0 B 0 C ey = B 1 C ez = B 0 C ex = @ A @A @0A 0 0 1

I 0

1

0 2I

!
2

6

=I

0 as ~ = B x@

x y z adjoint in t it now in a

1 C or ~ = xex + yey + zez : And the 3-dimensional representation ( Ax

his case) is realized by the matrices Ri i =1 2 3. We shall write di erentway. 9


Our problem is to nd matrices Ji of a dimension n in the basis of n linear independentvectrs, and weknow, rst, commutation reltions Ji Jj ]= ~ i ijk 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 away that one of the matrices, say J3, becomes diagonal. Then its diagonal elements would be eigenvalues of new basical vectors. 0 1 0 1 0 0i 1 B 1 1+ i ;i C 1 B 1 1;i 0 C 0A U=p @ 0 U ;1 = p @ 0 (1.9) A 2 ;i 0 1 2i01

0 10 ;1 =2 B 0 0 UA2U @ 00 0 0 ;1 =2 B i U (A1 + A3)U @ 0 0 U (A1 ; A3)U ;1 =2 B @

In the case of the 3-dimensional

1 J1 = 2 00 p J2 = 1 B i 2 @ 20 0 B1 J3 = @ 0 0 Let us choose basic vectors as 01 1 j1+1 >= B 0 C j1 0 > @A 0

0 0C A ;1 1 ;i 0 C 0 ;i A i0 1 010 1 0 1C A 010 representation usually one chooses 0 0 p2 0 1 B p2 0 p2 C (1.10) @ p2 0 A 0

1

;ip2 0p 0 p2 ;i0 2 i
00 0 0 C: A 0 ;1 0 1C A 0

1 C A

(1.11) (1.12)

1

0 =B @
10

1

0 j1 ; 1 >= B @

0 0C A 1

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 resentation with the full angular momentum equal to 1. But they identi ed with 3-vector in any space, even hypothetical one. For going a little ahead, note that triplet of -mesons in isotopic space placed into these basic vectors:
+

the repcould be example, could be

;

0

!j

>= j1 1 >

j

0

>= j1 0 >:

Let us also write in some details matrices for J = 2, i.e., of the dimension n =2J +1 = 5: 0 00 B 0 1 q0 B1 0 3=2 0 0 Bq B B 0 3=2 0 q3=2 0 J1 = B B B 0 0 q3=2 0 1 @ 00 0 10

for the representation

1 C C C C C C C C A 1

(1.13)

0 ;i 0 0 0C q i q0 ;i 3=2 q 0 0C C C (1.14) 0 i 3=2 q0 ;i 3=2 0 C C C 0 ;i C 00 i 3=2 A 00 0 i 0 1 0 200 0 0C B0 1 0 0 0 C B C B (1.15) J3 = B 0 0 0 0 0 C B C B 0 0 0 ;1 0 C A @ 0 0 0 0 ;2 As it should be these matrices satisfy commutation relations Ji Jj ]= i ijk 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

0 B B B B J2 = B B B B @


Basic vectors can b 0 B1 B0 B j1+2 >= B 0 B B0 @ 0

e chosen as:

1 C C C C C C A

j

0 B B B 1+1 >= B B B @ 1

0 B B B j1 ; 1 >= B B B @

0 1 0 0 0

1 C C C C C C A

j

0 B B B 1 0 >= B B B @

J3j2 +2 >=+2j2 +2 J3j2 ;1 >= ;1j2 ;1 Now let us formally not do it. The obtained matrices:

0C B0C B0C 0C C BC 0 C j1 ; 2 >= B 0 C : C BC B0C 1C A @A 0 1 > J3j2 +1 >=+1j2 +1 > J3j2 0 >=0j2 0 > > J3j2 ;2 >= ;2j2 ;2 >: put J = 1=2 although strictly speaking we could matrices up to a factor 1=2 are well known Pauli

01

0 0 1 0 0

1 C C C C C C A

J1 J J
2

3

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).

These matrices act in a linear vectors

=1 2 1 =2 =1 2 space 1 0

!

01 10 ! 0 ;i i0 ! 10 0 ;1 spanned on two basic 2-dimensional

!

!

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
k

k

a

k

y k =1 2 3 being hermitian matrices, k = k ,chosen in the form of Pauli matrices ! 01 1= 10 ! 0 ;i 2= i0 ! 10 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 ~ =(x1 x2 x3). For this purpose to anyvector x ~ let us attribute SOPOSTAWIM a quantity X = ~ ~ : x x ! x ; Xba = x +3ix x1;xix2 a b =1 2: (1.16) 1 2 3

Its determinantis detXba = ;~ 2 that is it de nes square of the vector length. x Taking the set of unitary unimodular matrices U , U yU =1 detU = 1 in 2dimensional space, let us de ne

and detX 0 = det(U yXU )= detX = ;~ 2: We conclude that transformations x U leaveinvariantthe vector length and therefore corresponds to rotations in 13

X 0 = U yXU


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
ij
k k

]= 2i

ij k 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 bytwo independent basic spinors q , =1 2 which could be chosen as

q= 1 0
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 = 1 fq q + q q g + 1 q q ; q q ] T f g + T ]: 2 2 Symmetric tensor of the 2nd rank has dimension dn = n(n +1)=2 and for SS n =2 d2 =3 which is seen from its matrix representation: SS

T

fg

= T11 T T12 T

12 22

!

and wehavetaken into accountthat Tf21g = Tf12g. Antisymmetric tensor of the 2nd rank has dimension dn = n(n ; 1)=2 AA and for n =2 d2 =1 which is also seen from its matrix representation: AA

T

]

=

;

0 T

12 21]

and we have taken into account that T Or instead in values of IR dimensions: 14

= ;T

T12 0

!

12]

and T

11]

= ;T

22]

= 0.

2 2=3 + 1:


According to this ( 12 = ; 21 = 1) t use it to contract and lower indices

result absolutely antisymmetric tensor of the 2nd rank ransforms also as a singlet of the group SU (2) and wecan SU (2) indices if needed. This tensor also serves to uprise of spinors and tensors in the SU (2).
0 0

= u1u2 12 + u2u1 12 12 as DetU = 1. (The same for
12

21 21

=(u1u2 ; u2u1) 12 12 .)

=u u
0

0

:

12

= DetU

12

=

12

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 pro jections,
correspondingly, (baryons of spin 1/2 and quarks as we shall see later) we can form symmetric tensor T f g with three components

T T
f12g

f11g

= q 1q

1

1 = p (q 1 q 2 + q 2 q 1 ) 2 T f22g = q2q2 p and wehaveintroduced 1= 2 to normalize this componentto 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 1 (1.17) T 12] = p (q1q2 ; q2q1) p j("# ; #")i 2 2 p and wehaveintroduced 1= 2 to normalize this componentto unity. Let us for example form the product of the spinor q and its conjugate spinor q whose basic vectors could be taken as tworows (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 =(q q ; 1 q q )+ 1 q q T + I 2 2 15

j ""i 1 p2 j("# + #")i j ##i


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 f g 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 = (1.18) = 1 (qiqj qk ; qj qiqk ; qiqk qj + qj qk qi ; qk qj qi + qk qiqj )+ 4 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

t

j A

and tj is just the IR corresponding to one A spin 1/2 state of three 1/2 states (the two The state with the sz =+1=2 is just 1 1 tA = p (q1q2 ; q2q1)q1 2 (Here q1 =", q2 =#.) 16

of two possible constructions of of them being antisymmetrized). 1 p2 j "#" ; #""i: (1.19)


The last example would be to form a spinor IR from the product of the symmetric tensor T fikg and a spinor qj .

T

fij g

qk =

1 = 4 (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 +3n +2) + n(n 3; 1) : 6 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 TSj is just the IR corresponding state of three 1/2 states (with with the sz =+1=2isjust 1 TS1 = p (e122q 6 1 p6 j

fikgj

TSk :

to the 2nd possible construction of spin 1/2 two of them being symmetrized). The state
112

q q + e21(q2q1 + q1q2)q

1

(1.21)

2

""# ; "#" ; #""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 classi cation 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 wehave seen 2-dimensional spinor as a basis of representation. Let us introduce now a group of isotopic transformations SU (2)I . Now de ne nucleon as a state with the isotopic spin I =1=2 with two pro jections ( 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)

! p N= n what means that proton and neutron are de ned as ! ! 1 0: p= 0 n= 1 Representation of the dimension 2 is realized byPauli matrices 2 2 k k = 1 2 3 ( instead of symbols i i =1 2 3 whichwe reserve for spin 1/2 in usual space). Note that isotopic operator + = 1=2( 1 + i 2) transforms neutron into proton , while ; =1=2( 1 ; i 2) 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 de ne isotopic vector ~ =( 1 2 3) as a basis (where real pseudoscalar elds 1 2 are related to charged pions byformula = 1 i 2, and 0 = 3), generators Al l =1 2 3 as the algebra representation and matrices Rl l =1 2 3asthe group representation with angles k de ned 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
a b

=2 = ( + i )=p2 ( 1 2
3

p

1

; i p=p ) ;=2
2 3

2

!

1 p

2

0

;

; ;

+ 1 p2 0

!

where charged pions are described by complex elds =( 1 i 2)= 2. So, pions can be given in isotopic formalism as 2-dimensional matrices:
+

p

= 01 00

!

;

= 00 10

!

0

=

1 p

0

2

0

!

1 p2

which form basis of the representation of the dimension 3 whereas the representation itself is given by the unitary unimodular matrices 2 2U. 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:
1 p2 ; 0

;

+ 1 p2 0

!

:

(1.22)

Among particles with half-integer spin note, for example, isotriplet of strange 0 hyperons found in early 60's with the spin 1/2 and masses 1192 MeV which can be writen in the SU (2) basis as
1 p 2 0

;

;

+ 1 p2 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 de ned as anumber 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 0 (1385) ! 0 0 (88 are 'elementary' was discussed intensively) 2%) 0 or (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 di erent charges) : a quartet of nucleon resonances of spin J = 3=2 ++ (1232), +(1232), 0(1232), ; (1232), decaying into nucleon and pion (measured mass di erence M + -M 0 =2,7 0,3 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 whichwe 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 + 1 Y: 2 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 componentofisovector current and of the hypercharge currentwhich 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 magni cient 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 1 010 0 ;i 0 B C B 0 0C (1.24) A 1=@ 1 0 0 A 2=@ i 000 000 0 1 1 0 001 100 C B C B (1.25) 4=@ 0 0 0 A 3 = @ 0 ;1 0 A 000 100 0 1 0 1 0 0 ;i 000 B B C 0C (1.26) A 5=@ 0 0 6=@ 0 0 1 A i00 010 0 1 0 1 00 0 10 0 1B B C 0C (1.27) A 7 = @ 0 0 ;i A 8=p @ 0 1 3 0 0 ;2 0i 0 1 1 ]= if 1 ij k k 2i2j 2 1 1 1 1 1 f123 =1 f147 = 2 f156 = ; 2 f246 = 2 f257 = 2 f346 = 1 f367 = ; 2 f458 = 2 p p 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 10 10 1 B 1 C B 0 C B 0 C: @0A @1A @0A 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 B0C B1C B0C B0C BC BC BC BC B0C B0C B1C B0C BC BC BC BC BC BC BC BC B0C B0C B0C B C x2 = B C x3 = B C :x4 = B 1 C BC x1 = B 0 C B0C B0C B0C BC BC BC BC BC BC BC BC B0C B0C B0C B0C BC BC BC BC B0C B0C B0C B0C @A @A @A @A 0 0 0 0 001 001 001 001 B0C B0C B0C B0C BC BC BC BC B0C B0C B0C B0C BC BC BC BC BC BC BC BC B0C B0C B0C B C x6 = B C :x7 = B C x8 = B 0 C BC x5 = B 1 C B0C B0C B0C BC BC BC BC BC BC BC BC B0C B1C B0C B0C BC BC BC BC B0C B0C B1C B0C @A @A @A @A 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 any8-vector in SU (3) X =(x1 ::: x8) can be put into the form of the 3 3 matrix: 1X (1.28) X = p 8=1 k xk = 2k 1 0 x3 + p13 x8 x1 ; ix2 x4 ; ix5 C 1B p2 B x1 + ix2 ;x3 + p13 x8 x6 ; ix7 C : A @ 2 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 = 1 fq q + q q g + 1 q q ; q q ] T f g + T ]: (1.29) 2 2 22


Symmetric tensor of the 2nd rank has dimension dn = n(n +1)=2 and for S n =3 d3 =6 which is seen from its matrix representation: S

T

fg

and wehavetaken into accountthat T fikg = T fkig T ik (i 6= k i k =1 2 3). Antisymmetric tensor of the 2nd rank has dimension dn = n(n ; 1)=2 A and for n =3 d3 = 3 which is also seen from its matrix representation: A

0 BT =@ T T

11 12 13

T T T

12 22 23

1 T 13 T 23 C A T 33

T

]

0 1 0 t12 t13 = B ;t12 0 t23 C @ A ;t13 ;t23 0
ik
]

and wehavetaken into accountthat T and T 11] = T 22] = T 33] =0. In terms of dimensions it would be

= ;T

ki

]

tik (i 6= k i k =1 2 3)

n n = n(n +1)=2

j

SS

+ n(n ; 1)=2

jAA

(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) 1 1 q q =(q q ; n q q )+ n 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 pointwe nish for a moment with a group formalism and makea transition to the problem of classi cation of particles along the representation of the group SU (3) and to some consequencies of it. 23