In these lectures on group theory and its application to the particle physics


 Spencer Blake
 1 years ago
 Views:
Transcription
1 hapter Introduction into the group theor. Introduction In these lectures on group theor and its application to the particle phsics there have been considered problems of classication of the particles along representations of the unitar 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 theor and theor of group representations. In some datails there are given unitar groups SU(2) and SU(3) which pla eminent role in modern particle phsics. 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() of the Salam Weinberg model. The group SU(3) from the other side is the base of the model of unitar smmetr with three avours as well as the group of colour that is on it stas the whole edice of the quantum chromodnamics. In order to aknowledge a reader on simple eamples with the group theor formalism mass formulae for elementar particles would be analed in detail. Other important eamples would be calculations of magnetic moments and aialvector weak coupling constants in unitar smmetr 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
2 current of the GlashowSalamWeinberg model has been constructed. The notion of colour has been introduced and simple eamples with it are given. Introduction of vector bosons as gauge elds are eplained. uthor would tr to write lectures in such a wa as to enable an eventual reader to perform calculations of man properties of the elementar particles b oneself..2 Groups and algebras. asic notions. Denition of a group Let be a set of elements Gg g 2 :::g n :, with the following properties:. There is a multiplication law g i g j = g l, and if g i g j 2 G, then g i g j = g l 2 G, i j l = 2 ::: n. 2. There is an associative law g i (g j g l )=(g i g j )g l. 3. There eists a unit element e, eg i = g i, i = 2 ::: n. 4. There eists an inverse element g i ;, g i ; g i = e, i = 2 ::: n. Then on the set G eists the group of elements g g 2 ::: g n : s a simple eample 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.. Multiplication law is just a summation of angles: + 2 = 3 2 : 2. ssociative law is written as ( + 2 )+ 3 = +( ): 3. The unit element is the rotation on the angle (+2n): 4. The inverse element is the rotation on the angle ;(+2n): Thus rotations on the plane about some ais perpendicular to this plane form the group. Let us consider rotations of the coordinate aes, which dene Decartes coordinate sstem in the 3dimensional space at the angle 3 on the plane around the ais : = cos 3 sin 3 ;sin 3 cos 3 = R 3 ( 3 ) (.) Let be an innitesimal rotation. Let is epand the rotation matri R 3 () 4
3 in the Talor series and take onl terms linear in : R 3 () =R 3 () + dr 3 + O( 2 )=+i 3 + O( 2 ) (.2) d = where R 3 () is a unit matri and it is introduced the matri 3 = ;i dr 3 = d = ;i i (.3) which we name 'generator of the rotation around the 3rd ais' (or ais). Let us choose = 3 =n then the rotation on the angle 3 could be obtained b ntimes application of the operator R 3 (), and in the limit we have R 3 ( 3 ) = lim n [R 3 ( 3 =n)] n = lim n [ + i 3 3 =n] n = e i 3 3 : (.4) Let us consider rotations around the ais U: = cos 2 sin 2 ;sin 2 cos 2 = R 2 ( 2 ) where a generator of the rotation around the ais U is introduced: Repeat it for the ais : = 2 = ;i dr 2 = d = cos sin ;sin cos ;i i = R ( ) (.5) : (.6) where a generator of the rotation around the ais U is introduced: = ;i dr = d = 5 ;i i (.7) (.8)
4 Now we can write in the 3dimensional space rotation of the Descartes coordinate sstem on the arbitrar angles, for eample: R ( )R 2 ( 2 )R 3 ( 3 )=e i e i 2 2 e i 3 3 However, usuall one denes rotation in the 3space in some other wa, namel b using Euler angles: R( ) =e i 3 e i 2 e i 3 functions). (Usuall abibbokobaashimaskawa matriischosen as V KM = = c 2 c 3 s 2 c 3 s 3 e ;i 3 ;s 2 c 23 ; c 2 s 23 s 3 e i 3 c 2 c 23 ; s 2 s 23 s 3 e i 3 s 23 c 3 s 2 s 23 ; c 2 c 23 s 3 e i 3 ;c 2 s 23 ; s 2 c 23 s 3 e i 3 c 23 c 3 Here c ij = cos ij s ij = sin ij (i j = 2 3) while ij  generalied abibbo angles. How to construct it using Eqs.(.,.5,.7) and setting 3 =?) Generators l l = 2 3 satisf commutation relations i j ; j i =[ i j ]=i ijk k i j k = 2 3 where ijk is absolutel antismmetric tensor of the 3rd rang. Note that matrices l l= 2 3 are antismmetric, while matrices R l are orthogonal, that is, R T i R j = ij, where inde T means 'transposition'. Rotations could be completel dened b generators l l = 2 3. In other words, the group of 3dimensional rotations (as well as an continuous Lie group up to discrete transformations) could be characteried b its algebra,that is b denition of generators l l = 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 2 L the law ofmultiplication to numbers from the set K is dened), (ii) For 2L the commutator is dened as [ ], and[ ] has the following properties: [ ] =[ ] [ ] =[ ] at 2 K and [ + 2 ]= [ ]+[ 2 ], [ + 2 ]=[ ]+[ 2 ] for all 2L [ ]=for all 2L [[ ]]+[[]]+[[]]= (Jacobi identit). 6 :
5 .3 Representations of the Lie groups and algebras efore a discussion of representations we should introduce two notions: isomorphism and homomorphism. Denition of isomorphism and homomorphism Let be given two groups, G and G. Mapping f of the group G into the group G is called isomorphism or homomorphis If f(g g 2 )=f(g )f(g 2 ) for an g g 2 2 G: This means that if f maps g into g and g 2 W g 2, then f also maps g g 2 into g g 2 Ḣowever if f(e) maps e into a unit element ing, the inverse in general is not true, namel, e from G is mapped b theinverse transformation f ; into f ; (e ), named the core (orr nucleus) of the homeomorphism. If the core of the homeomorphism is e from G such onetoone 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 toever element g in the group G put in correspondence linear operator T (g) in the space L in such a wa that the following conditions are fullled: () T (g g 2 )=T (g )T (g 2 ) for all g g 2 2 G (2) T (e) = where 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 the map onetoone L on L. ecause of that the propert () 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 onetoone correspondence mapping of L in L). If the space L is nitedimensional its dimension is called dimension of the representation T and named as n T. In this case choosing in the space L abasise e 2 ::: e n 7
6 current of the GlashowSalamWeinberg model has been constructed. The notion of colour has been introduced and simple eamples with it are given. Introduction of vector bosons as gauge elds are eplained. uthor would tr to write lectures in such a wa as to enable an eventual reader to perform calculations of man properties of the elementar particles b oneself..2 Groups and algebras. asic notions. Denition of a group Let be a set of elements Gg g 2 :::g n :, with the following properties:. There is a multiplication law g i g j = g l, and if g i g j 2 G, then g i g j = g l 2 G, i j l = 2 ::: n. 2. There is an associative law g i (g j g l )=(g i g j )g l. 3. There eists a unit element e, eg i = g i, i = 2 ::: n. 4. There eists an inverse element g i ;, g i ; g i = e, i = 2 ::: n. Then on the set G eists the group of elements g g 2 ::: g n : s a simple eample 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.. Multiplication law is just a summation of angles: + 2 = 3 2 : 2. ssociative law is written as ( + 2 )+ 3 = +( ): 3. The unit element is the rotation on the angle (+2n): 4. The inverse element is the rotation on the angle ;(+2n): Thus rotations on the plane about some ais perpendicular to this plane form the group. Let us consider rotations of the coordinate aes, which dene Decartes coordinate sstem in the 3dimensional space at the angle 3 on the plane around the ais : = cos 3 sin 3 ;sin 3 cos 3 = R 3 ( 3 ) (.) Let be an innitesimal rotation. Let is epand the rotation matri R 3 () 4
7 in the Talor series and take onl terms linear in : R 3 () =R 3 () + dr 3 + O( 2 )=+i 3 + O( 2 ) (.2) d = where R 3 () is a unit matri and it is introduced the matri 3 = ;i dr 3 = d = ;i i (.3) which we name 'generator of the rotation around the 3rd ais' (or ais). Let us choose = 3 =n then the rotation on the angle 3 could be obtained b ntimes application of the operator R 3 (), and in the limit we have R 3 ( 3 ) = lim n [R 3 ( 3 =n)] n = lim n [ + i 3 3 =n] n = e i 3 3 : (.4) Let us consider rotations around the ais U: = cos 2 sin 2 ;sin 2 cos 2 = R 2 ( 2 ) where a generator of the rotation around the ais U is introduced: Repeat it for the ais : = 2 = ;i dr 2 = d = cos sin ;sin cos ;i i = R ( ) (.5) : (.6) where a generator of the rotation around the ais U is introduced: = ;i dr = d = 5 ;i i (.7) (.8)
8 Now we can write in the 3dimensional space rotation of the Descartes coordinate sstem on the arbitrar angles, for eample: R ( )R 2 ( 2 )R 3 ( 3 )=e i e i 2 2 e i 3 3 However, usuall one denes rotation in the 3space in some other wa, namel b using Euler angles: R( ) =e i 3 e i 2 e i 3 functions). (Usuall abibbokobaashimaskawa matriischosen as V KM = = c 2 c 3 s 2 c 3 s 3 e ;i 3 ;s 2 c 23 ; c 2 s 23 s 3 e i 3 c 2 c 23 ; s 2 s 23 s 3 e i 3 s 23 c 3 s 2 s 23 ; c 2 c 23 s 3 e i 3 ;c 2 s 23 ; s 2 c 23 s 3 e i 3 c 23 c 3 Here c ij = cos ij s ij = sin ij (i j = 2 3) while ij  generalied abibbo angles. How to construct it using Eqs.(.,.5,.7) and setting 3 =?) Generators l l = 2 3 satisf commutation relations i j ; j i =[ i j ]=i ijk k i j k = 2 3 where ijk is absolutel antismmetric tensor of the 3rd rang. Note that matrices l l= 2 3 are antismmetric, while matrices R l are orthogonal, that is, R T i R j = ij, where inde T means 'transposition'. Rotations could be completel dened b generators l l = 2 3. In other words, the group of 3dimensional rotations (as well as an continuous Lie group up to discrete transformations) could be characteried b its algebra,that is b denition of generators l l = 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 2 L the law ofmultiplication to numbers from the set K is dened), (ii) For 2L the commutator is dened as [ ], and[ ] has the following properties: [ ] =[ ] [ ] =[ ] at 2 K and [ + 2 ]= [ ]+[ 2 ], [ + 2 ]=[ ]+[ 2 ] for all 2L [ ]=for all 2L [[ ]]+[[]]+[[]]= (Jacobi identit). 6 :
9 .3 Representations of the Lie groups and algebras efore a discussion of representations we should introduce two notions: isomorphism and homomorphism. Denition of isomorphism and homomorphism Let be given two groups, G and G. Mapping f of the group G into the group G is called isomorphism or homomorphis If f(g g 2 )=f(g )f(g 2 ) for an g g 2 2 G: This means that if f maps g into g and g 2 W g 2, then f also maps g g 2 into g g 2 Ḣowever if f(e) maps e into a unit element ing, the inverse in general is not true, namel, e from G is mapped b theinverse transformation f ; into f ; (e ), named the core (orr nucleus) of the homeomorphism. If the core of the homeomorphism is e from G such onetoone 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 toever element g in the group G put in correspondence linear operator T (g) in the space L in such a wa that the following conditions are fullled: () T (g g 2 )=T (g )T (g 2 ) for all g g 2 2 G (2) T (e) = where 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 the map onetoone L on L. ecause of that the propert () 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 onetoone correspondence mapping of L in L). If the space L is nitedimensional its dimension is called dimension of the representation T and named as n T. In this case choosing in the space L abasise e 2 ::: e n 7
10 it is possible to dene operators T (g) b matrices of the order n: t(g) = t t 2 ::: t n t 2 t 22 ::: t 2n ::: ::: ::: ::: t n t n2 ::: t nn T (g)e k = X t ij (g)e j t(e) = t(g g 2 )=t(g )t(g 2 ): The matri t(g) is called a representation matri T: If the group G itself consists from the matrices of the ed order, then one of the simple representations is obtained at T (g) = g ( identical or, better, adjoint representation). Such adjoint representation has been alread considered b usabove and is the set of the orthogonal 33 matrices of the group of rotations O(3) in the 3dimensional space. Instead the set of antismmetrical matrices i i= 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). the transformations of similarit PODOIQ T (g) = ; T (g) it is possible to obtain from T (g) representation T (g) =g which is equivalent to it but, sa, more suitable (for eample, representation matri can be obtained in almost diagonal form). Let us dene a sum of representations T (g) =T (g)+t 2 (g) and sa that a representation is irreducible if it cannot be written as such a sum ( For the Lie group representations is denition is sucientl correct). For search and classication of the irreducible representation (IR) Schurr's lemma plas an important role. Schurr's lemma: Let be given two IR's, t (g) and t (g), ofthegroup G. n matri w, such that wt (g) =t (g) for all g 2 G either is equal to (if t (g)) and t (g) are notequivalent) or KRTN is proportional to the unit matri I. Therefore if 6= I eists which commutes with all matrices of the given representation T (g) it means that this T (g) is reducible. Reall,, if T (g) is reducible and has the form T (g) =T (g)+t 2 (g) = T (g) T 2 (g) 8
11 then = I 6= I 2 I 2 and [T (g)]=: For the group of rotations O(3) it is seen that if [ i ]=i= 2 3 then [R i ]=,i.e., for us it is sucient to nd a matri commuting with all the generators of the given representation, while eigenvalues of such matri operator can be used for classications of the irreducible representations (IR's). This is valid for an Lie group and its algebra. So, wewould like to nd all the irreducible representations of nite dimension of the group of the 3dimensional rotations, which can be be reduced to searching of all the sets of hermitian matrices J 23 satisfing commutation relations [J i J j ]=i ijk J k : There is onl one bilinear invariant constructed from generators of the algebra (of the group): ~J 2 = J 2 + J 2 + J 2 for which [ 2 3 ~J 2 J i ]=i = 2 3: So IR's can be characteried b the inde 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 ndimensional space L n and could be realied b nn matrices where n is the dimension of the irreducible representation. In this linear space ndimensional vectors ~v are dened and an vector can be written as a linear combination of n arbitraril chosen linear independent vectors ~e i, ~v = P n i= v i ~e i. In other words, the space L n is spanned on the n linear independent vectors ~e i forming basis in L n. For eample, for the rotation group O(3) an 3vector can be denned, as we have alread seen b the basic vectors as ~ = e = e = e = or ~ = e +e +e : nd the 3dimensional representation ( adjoint in this case) is realied b the matrices R i i= 2 3. We shall write it now in a dierent wa. 9
12 Our problem is to nd matrices J i of a dimension n in the basis of n linear independent vectrs, and we know, rst, commutation reltions [J i J j ]= i ijk J k, and, the second, that IR's can be characteried b J ~ 2. esides, it is possible to perform similarit transformation of the Eqs.(.8,.6,.3,) in such awa that one of the matrices, sa J 3, becomes diagonal. Then its diagonal elements would be eigenvalues of new basical vectors. U = p 2 ;i +i ;i U 2 U ; =2 U ; = U( + 3 )U ; =2 p 2 ; U( ; 3 )U ; =2 ;i i ;i i i ; i i In the case of the 3dimensional representation usuall one chooses p J = p 2 p p 2 J 2 = 2 Let us choose basic vectors as j+>= J 3 = ;i p 2 i p 2 ;i p 2 i p 2 j >= ; (.9) (.) (.) : (.2) j ; >=
13 and J 3 j+>= +j+> J 3 j >= j+> J 3 j ; >= ;j+>: In the theor of angular momentum these quantities form basis of the representation with the full angular momentum equal to. ut the could be identied with 3vector in an space, even hpothetical one. For eample, going a little ahead, note that triplet of mesons in isotopic space could be placed into these basic vectors: + ; j >= j > j >= j >: Let us also write in some details matrices for J = 2, i.e., for the representation of the dimension n =2J +=5: J 2 = J = q 3=2 q q 3=2 3=2 q 3=2 ;i i ;iq 3=2 q iq 3=2 ;i 3=2 iq 3=2 ;i i J 3 = 2 ; ;2 (.3) (.4) (.5) s it should be these matrices satisf commutation relations [J i J j ]=i ijk J k ijk = 2 3, i.e., the realie representation of the dimension 5 of the Lie algebra corresponding to the rotation group O(3).
14 asic vectors can be chosen as: j+2>= j+>= j >= j ; >= j ; 2 >= : J 3 j2 +2 >= +2j2 +2 > J 3 j2 + >= +j2 + > J 3 j2 >= j2 > J 3 j2 ; >= ;j2 ; > J 3 j2 ;2 >= ;2j2 ;2 >: Now let us formall put J = =2 although strictl speaking we could not do it. The obtained matrices up to a factor =2 are well known Pauli matrices: J = 2 J 2 = 2 ;i i J 3 = 2 ; These matrices act in a linear space spanned on two basic 2dimensional vectors : There appears a real possibilit to describe states with spin (or isospin) /2. ut in a correct wa itwould be possible onl in the framework of another group which contains all the representations of the rotation group O(3) plus PLS representations corresponding to states with halfinteger spin (or isospin, for mathematical group it is all the same). This group is SU(2). 2
15 .4 Unitar unimodular group SU(2) Now after learning a little the group of 3dimensional rotations in which dimension of the minimal nontrivial representation is 3 let us consider more comple group where there is a representation of the dimension 2. For this purpose let us take a set of 2 2 unitar unimodular U,i.e., U U = detu =. Such matri U can be written as U = e i ka k k k = 2 3 being hermitian matrices, k = k,chosen in the form of Pauli matrices = 2 = ;i i 3 = ; and a k k = 2 3 are arbitrar real numbers. The matrices U form a group with the usual multipling law for matrices and realie identical (adjoint) represenation of the dimension 2 with two basic 2dimensional vectors. Instead Pauli matrices have the same commutation realtions as the generators of the rotation group O(3). Let us tr to relate these matrices with a usual 3dimensional vector ~ =( 2 3 ). For this purpose to an vector ~ let us attribute SOPOSTWIM a quantit X = ~~ : Xb a = 3 ; i 2 a b = 2: (.6) + i 2 ; 3 Its determinant isdetx a b = ;~2 that is it denes square of the vector length. Taking the set of unitar unimodular matrices U, U U =detu= in 2 dimensional space, let us dene X = U XU and detx = det(u XU)=detX = ;~ 2 : We conclude that transformations U leave invariant the vector length and therefore corresponds to rotations in 3
16 the 3dimensional space, and note that U correspond to the same rotation. orresponding algebra SU(2) is given b hermitian matrices k k = 2 3 with the commutation relations [ i j ]=2i ijk k where U = e i ka k: nd in the same wa as in the group of 3rotations O(3) the representation of lowest dimension 3 is given b three independent basis vectors,for eample,,, in SU(2) 2 dimensional representation is given b two independent basic spinors q, = 2 which could be chosen as q = q 2 The direct product of two spinors q and q can be epanded into the sum of two irreducible representations (IR's) just b smmetriing and antismmetriing the product: q q = 2 fq q + q q g + 2 [q q ; q q ] T fg + T [] : Smmetric tensor of the 2nd rank has dimension d n SS = n(n +)=2 and for n =2d 2 SS =3which is seen from its matri representation: T fg = T T 2 T 2 T 22 and we have taken into account thatt f2g = T f2g. ntismmetric tensor of the 2nd rank has dimension d n = n(n ; )=2 and for n =2d 2 =which is also seen from its matri representation: T [] = T 2 ;T 2 and we have taken into account thatt [2] = ;T [2] and T [] = ;T [22] =. Or instead in values of IR dimensions: 2 2=3+: : 4
17 ccording to this result absolutel antismmetric tensor of the 2nd rank ( 2 = ; 2 = ) 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 : 2 = u u u 2 u 2 2 =(u u 2 2 ; u 2 u 2) 2 = DetU 2 = 2 as DetU =. (The same for 2.).5 SU(2) as a spinor group ssociating q with the spin functions of the entities of spin /2, q j"i and q 2 j #i being basis spinors with +/2 and /2 spin projections, correspondingl, (barons of spin /2 and quarks as we shall see later) we can form smmetric tensor T fg with three components T fg = q q j ""i T f2g = p 2 (q q 2 + q 2 q ) p 2 j("# + #")i T f22g = q 2 q 2 j ##i and we have introduced = p 2 to normalie this component tounit. Similarl for antismmetric tensor associating again q with the spin functions of the entities of spin /2 let us write the onl component ofa singlet as T [2] = p (q q 2 ; q 2 q ) p j("# ; #")i (.7) 2 2 and we have introduced = p 2 to normalie this component tounit. Let us for eample form the product of the spinor q and its conjugate spinor q whose basic vectors could be taken as two rows ( ) and ( ). Now epansion into the sum of the IR's could be made b subtraction of a trace (remind that Pauli matrices are traceless) q q =(q q ; 2 q q )+ 2 q q T + I 5
18 where T is a traceless tensor of the dimension d V =(n 2 ;) corresponding to the vector representation of the group SU(2) having at n = 2 the dimension 3 I being a unit matri corresponding to the unit (or scalar) IR. Or instead in values of IR dimensions: 2 2=3+: 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 eample of epansion of the product of two IR'sisgiven b the product T [ij] q k = = 4 (qi q j q k ; q j q i q k ; q i q k q j + q j q k q i ; q k q j q i + q k q i q j )+ (.8) 4 (qi q j q k ; q j q i q k + q i q k q j ; q j q k q i + q k q j q i ; q k q i q j )= or in terms of dimensions: = T [ikj] + T [ik]j n(n ; )=2 n = n(n2 ; 3n +2) 6 + n(n2 ; ) : 3 For n =2antismmetric tensor of the 3rd rank is identicall ero. So we are left with the miedsmmetr tensor T [ik]j of the dimension 2 for n =2, that is, which describes spin /2 state. It can be contracted with the the absolutel anismmetric tensor of the 2nd rank e ik to give e ik T [ik]j t j and t j is just the IR corresponding to one of two possible constructions of spin /2 state of three /2 states (the two of them being antismmetried). The state with the s =+=2 isjust t = p 2 (q q 2 ; q 2 q )q p 2 j "#" ; #""i: (.9) (Here q =", q 2 =#.) 6
19 The last eample would be to form a spinor IR from the product of the smmetric tensor T fikg and a spinor q j. T fijg q k = = 4 (qi q j q k + q j q i q k + q i q k q j + q j q k q i + q k q j q i + q k q i q j )+ (.2) 4 (qi q j q k + q j q i q k ; q i q k q j ; q j q k q i ; q k q j q i ; q k q i q j )= or in terms of dimensions: = T fikjg + T fikgj n(n +)=2 n = n(n2 +3n +2) 6 + n(n2 ; ) : 3 Smmetric tensor of the 4th rank with the dimension 4 describes the state of spin S=3/2, (2S+)=4. Instead tensor of mied smmetr describes state of spin /2 made of three spins /2: e ij T fikgj T k S : T j S is just the IR corresponding to the 2nd possible construction of spin /2 state of three /2 states (with two of them being smmetried). The state with the s =+=2 isjust T S = p 6 (e 2 2q q q 2 + e 2 (q 2 q + q q 2 )q (.2) p 6 j2 ""# ; "#" ; #""i: 7
20 .6 Isospin group SU(2)I Let us consider one of the important applications of the group theor and of its representations in phsics of elementar particles. We would discuss classication of the elementar particles with the help of group theor. s a simple eample let us consider proton and neutron. It is known for ears that proton and neutron have quasi equal masses and similar properties as to strong (or nuclear) interactions. That's wh Heisenberg suggested to consider them one state. ut for this purpose one should nd the group with the (lowest) nontrivial representation of the dimension 2. Let us tr (with Heisenberg) to appl here the formalism of the group SU(2) which has as we have seen 2dimensional 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 ==2 with two projections ( proton with I 3 =+=2 and neutron with I 3 = ;=2 ) in this imagined 'isotopic space' practicall in full analog with introduction of spin in a usual space. Usuall basis of the 2dimensional representation of the group SU(2) I is written as a isotopic spinor (isospinor) N = what means that proton and neutron are dened as p = p n n = Representation of the dimension 2 is realied b Pauli matrices 2 2 k k = 2 3 ( instead of smbols i i= 2 3 which we reserve for spin /2 in usual space). Note that isotopic operator + ==2( + i 2 ) transforms neutron into proton, while ; ==2( ;i 2 ) instead transforms proton into neutron. It is known also isodoublet of cascade hperons of spin /2 ; and masses 32 MeV. It is also known isodoublet of strange mesons of spin K + and masses 49 MeV and antidublet of its antiparticles K ;. nd in what wa to describe particles with the isospin I =? Sa, triplet of mesons + ; of spin ero and negative parit (pseudoscalar mesons) with masses m( ) = MeV, m( )= MeV and practicall similar properties as to strong interactions? 8 :
21 In the group of (isotopic) rotations it would be possible to dene isotopic vector ~ =( 2 3 ) as a basis (where real pseudoscalar elds 2 are related to charged pions bformula = i 2, and = 3 ), generators l l= 2 3 as the algebra representation and matrices R l l= 2 3asthe 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 ~ =( 2 3 ) in the group SU(2) I the basis of the form a b = 3 = p 2 ( ; i 2 )= p 2 ( + i 2 )= p 2 ; 3 = p 2 p 2 + ; ; p 2 where charged pions are described b comple elds =( i 2 )= p 2. So, pions can be given in isotopic formalism as 2dimensional matrices: + = ; = p = 2 ; p 2 which form basis of the representation of the dimension 3 whereas the representation itself is given b the unitar unimodular matrices 2 2 U. In a similar wa it is possible to describe particles of an spin with the isospin I = : mong meson one should remember isotriplet of the vector (spin ) mesons with masses 76 MeV: p 2 + ; ; p 2 : (.22) mong particles with halfinteger spin note, for eample, isotriplet of strange hperons found in earl 6's with the spin /2 and masses 92 MeV which can be writen in the SU(2) basis as p 2 + ; ; p 2 : (.23) Representation of dimension 3 is given b the same matrices U. Let us record once more that eperimentall isotopic spin I is dened as anumber of particles N =(2I +) similar in their properties, that is, having the same spin, similar masses (equal at the level of percents) and practicall identical along strong interactions. For eample, at the mass close to 5 9
22 MeV it was found onl one particle of spin /2 with strangeness S=  it is hperon with ero electric charge and mass MeV. Naturall, isospin ero was ascribed to this particle. In the same wa isospin ero was ascribed to pseudoscalar meson (548). It is known also triplet of baron resonances with the spin 3/2, strangness S= and masses M( + (385)) = m\w, M( (385)) = m\w, M( ; (385)) = m\w, (resonances are elementar particles decaing due to strong interactions and because of that having ver short times of life one upon a time the question whether the are 'elementar' was discussed intensivel) (385) (88 2%) or (385) (2 2%) (one can nd instead smbol Y (385) for this resonance). It is known onl one state with isotopic spin I =3=2 (that is on eperiment itwere found four practicall identical states with dierent charges) : a quartet of nucleon resonances of spin J =3=2 ++ (232), + (232), (232), ; (232), decaing into nucleon and pion (measured mass dierence M +M =2,7,3 MeV). ( We can use also another smbol N (232).) There are also heavier 'replics' of this isotopic quartet with higher spins. In the sstem ; it was found onl two resonances with spin 3/2 (not measured et) ; with masses 52 MeV, so the were put into isodublet with the isospin I = =2. Isotopic formalism allows not onl to classif practicall the whole set of strongl interacting particles (hadrons) in economic wa in isotopic multiplet but also to relate various deca and scattering amplitudes for particles inside the same isotopic multiplet. We shall not discuss these relations in detail as the are part of the relations appearing in the framework of higher smmetries which we start to consider below. t the end let us remind GellMann{Nishijima relation between the particle charge Q, 3rd component of the isospin I 3 and hpercharge Y = S +, S being strangeness, being baron number (+ for barons,  for antibarons, for mesons): Q = I Y: s 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 ofisovector current and of the hpercharge current which is isoscalar. 2
Chapter 9 Unitary Groups and SU(N)
Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three
More informationSemiSimple Lie Algebras and Their Representations
i SemiSimple Lie Algebras and Their Representations Robert N. Cahn Lawrence Berkeley Laboratory University of California Berkeley, California 1984 THE BENJAMIN/CUMMINGS PUBLISHING COMPANY Advanced Book
More informationPQM Supplementary Notes: Spin, topology, SU(2) SO(3) etc
PQM Supplementary Notes: Spin, topology, SU(2) SO(3) etc (away from the syllabus, but interesting material, based on notes by Dr J.M. Evans) 1 Rotations and Noncontractible Loops Rotations in R 3 can
More informationIrreducible Representations of SO(2) and SO(3)
Chapter 8 Irreducible Representations of SO(2) and SO(3) The shortest path between two truths in the real domain passes through the complex domain. Jacques Hadamard 1 Some of the most useful aspects of
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationEXAMPLE 6 Find the gradient vector field of f x, y x 2 y y 3. Plot the gradient vector field together with a contour map of f. How are they related?
9 HAPTER 3 VETOR ALULU 4 _4 4 _4 FIGURE 5 EXAMPLE 6 Find the gradient vector field of f, 3. Plot the gradient vector field together with a contour map of f. How are the related? OLUTION The gradient vector
More informationOrthogonal Bases and the QR Algorithm
Orthogonal Bases and the QR Algorithm Orthogonal Bases by Peter J Olver University of Minnesota Throughout, we work in the Euclidean vector space V = R n, the space of column vectors with n real entries
More informationThe Mathematics any Physicist Should Know. Thomas Hjortgaard Danielsen
The Mathematics any Physicist Should Know Thomas Hjortgaard Danielsen Contents Preface 5 I Representation Theory of Groups and Lie Algebras 7 1 PeterWeyl Theory 9 1.1 Foundations of Representation Theory...............
More informationSpaceTime Approach to NonRelativistic Quantum Mechanics
R. P. Feynman, Rev. of Mod. Phys., 20, 367 1948 SpaceTime Approach to NonRelativistic Quantum Mechanics R.P. Feynman Cornell University, Ithaca, New York Reprinted in Quantum Electrodynamics, edited
More informationSome notes on group theory.
Some notes on group theory. Eef van Beveren Departamento de Física Faculdade de Ciências e Tecnologia P COIMBRA (Portugal) February Contents Some intuitive notions of groups.. What is a group?..............................
More informationA Modern Course on Curves and Surfaces. Richard S. Palais
A Modern Course on Curves and Surfaces Richard S. Palais Contents Lecture 1. Introduction 1 Lecture 2. What is Geometry 4 Lecture 3. Geometry of InnerProduct Spaces 7 Lecture 4. Linear Maps and the Euclidean
More informationAn Introduction to Tensors for Students of Physics and Engineering
NASA/TM 2002211716 An Introduction to Tensors for Students of Physics and Engineering Joseph C. Kolecki Glenn Research Center, Cleveland, Ohio September 2002 The NASA STI Program Office... in Profile
More informationChapter 5. Banach Spaces
9 Chapter 5 Banach Spaces Many linear equations may be formulated in terms of a suitable linear operator acting on a Banach space. In this chapter, we study Banach spaces and linear operators acting on
More informationOn parameters for the group SO(2n)
On parameters for the group SO(2n) James Arthur In memory of Ilya PiatetskiShapiro. 1. Background This article is an attempt to refine a property of automorphic representations from the monograph [A3].
More informationWHICH SCORING RULE MAXIMIZES CONDORCET EFFICIENCY? 1. Introduction
WHICH SCORING RULE MAXIMIZES CONDORCET EFFICIENCY? DAVIDE P. CERVONE, WILLIAM V. GEHRLEIN, AND WILLIAM S. ZWICKER Abstract. Consider an election in which each of the n voters casts a vote consisting of
More informationChapter 10 Electronic Wavefunctions Must Also Possess Proper Symmetry. These Include Angular Momentum and Point Group Symmetries
Chapter 10 Electronic Wavefunctions Must Also Possess Proper Symmetry. These Include Angular Momentum and Point Group Symmetries I. Angular Momentum Symmetry and Strategies for Angular Momentum Coupling
More informationTHIS CHAPTER INTRODUCES the Cartesian coordinate
87533_01_ch1_p001066 1/30/08 9:36 AM Page 1 STRAIGHT LINES AND LINEAR FUNCTIONS 1 THIS CHAPTER INTRODUCES the Cartesian coordinate sstem, a sstem that allows us to represent points in the plane in terms
More informationOPRE 6201 : 2. Simplex Method
OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2
More informationA Tutorial on Spectral Clustering
A Tutorial on Spectral Clustering Ulrike von Luxburg Max Planck Institute for Biological Cybernetics Spemannstr. 38, 7276 Tübingen, Germany ulrike.luxburg@tuebingen.mpg.de This article appears in Statistics
More informationFoundations of Data Science 1
Foundations of Data Science John Hopcroft Ravindran Kannan Version /4/204 These notes are a first draft of a book being written by Hopcroft and Kannan and in many places are incomplete. However, the notes
More informationHow many numbers there are?
How many numbers there are? RADEK HONZIK Radek Honzik: Charles University, Department of Logic, Celetná 20, Praha 1, 116 42, Czech Republic radek.honzik@ff.cuni.cz Contents 1 What are numbers 2 1.1 Natural
More informationFourier Theoretic Probabilistic Inference over Permutations
Journal of Machine Learning Research 10 (2009) 9971070 Submitted 5/08; Revised 3/09; Published 5/09 Fourier Theoretic Probabilistic Inference over Permutations Jonathan Huang Robotics Institute Carnegie
More informationALGEBRAIC NUMBER THEORY AND QUADRATIC RECIPROCITY
ALGEBRAIC NUMBER THEORY AND QUADRATIC RECIPROCITY HENRY COHN, JOSHUA GREENE, JONATHAN HANKE 1. Introduction These notes are from a series of lectures given by Henry Cohn during MIT s Independent Activities
More informationA Perfect Example for The BFGS Method
A Perfect Example for The BFGS Method YuHong Dai Abstract Consider the BFGS quasinewton method applied to a general nonconvex function that has continuous second derivatives. This paper aims to construct
More informationIEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 4, APRIL 2006 1289. Compressed Sensing. David L. Donoho, Member, IEEE
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 4, APRIL 2006 1289 Compressed Sensing David L. Donoho, Member, IEEE Abstract Suppose is an unknown vector in (a digital image or signal); we plan to
More informationCommunication Theory of Secrecy Systems
Communication Theory of Secrecy Systems By C. E. SHANNON 1 INTRODUCTION AND SUMMARY The problems of cryptography and secrecy systems furnish an interesting application of communication theory 1. In this
More informationA Case Study in Approximate Linearization: The Acrobot Example
A Case Study in Approximate Linearization: The Acrobot Example Richard M. Murray Electronics Research Laboratory University of California Berkeley, CA 94720 John Hauser Department of EESystems University
More informationAll of mathematics can be described with sets. This becomes more and
CHAPTER 1 Sets All of mathematics can be described with sets. This becomes more and more apparent the deeper into mathematics you go. It will be apparent in most of your upper level courses, and certainly
More informationA Googlelike Model of Road Network Dynamics and its Application to Regulation and Control
A Googlelike Model of Road Network Dynamics and its Application to Regulation and Control Emanuele Crisostomi, Steve Kirkland, Robert Shorten August, 2010 Abstract Inspired by the ability of Markov chains
More informationA Course on Number Theory. Peter J. Cameron
A Course on Number Theory Peter J. Cameron ii Preface These are the notes of the course MTH6128, Number Theory, which I taught at Queen Mary, University of London, in the spring semester of 2009. There
More information