We’ve found an exponential expression for operators n number of dimensions of the continuous parameter  Generator G The order (dimensions) of G is the same as H

We classify types of transformations (matrix operator groups) as OrthogonalO(2)SO(2) O(3)SO(3) UnitaryU(2)SU(2) U(3)SU(3) groups in the algebraic sense: closed within a defined mathematical operation that observes the associative property with every element of the group having an inverse

O (n) set of all orthogonal U T =U  1 (therefore real) matrices of dimension n×n Rotations in 3-dim space SO(3) 4-dim space-time Lorentz transformations SO(4) Orbital angular momentum rotations SO( ℓ ) (mixing of quantum mechanical states) cos  sin  0 Rz(  ) = -sin  cos  U (n) set of all n×n UNITARY matrices U † =U  1 i.e. U † U  SU(n) “special” unimodular subset of the above det(U)=1 SO (n) “special” subset of the above: unimodular, i.e., det(U)=1 group of all rotations in a space of n -dimensions ALL known “external” space-time symmetries in physics new “internal” symmetries (beyond space-time)

SO(3) cos  3 sin  3 0 -sin  3 cos  cos  2 0 -sin  sin  2 0 cos  cos  1 sin  1 0 -sin  1 cos  1 R(  1,  2,  3 )= cos  3 cos  2 +sin  3 sin  2 cos  1 sin  3 -sin  1 sin  2 sin  3 sin  1 sin  3 -cos  1 sin  2 cos  3 -cos  2 sin  3 cos  1 cos  3 -sin  1 sin  2 sin  3 sin  1 cos  3 -cos  1 sin  2 sin  3 sin  2 -sin  1 cos  2 cos  1 cos  2 = Contains SO(2) subsets like: cos  sin  0 -sin  cos  R z (  ) = acting on vectors like v = vxvyvzvxvyvz in the i, j, k basis ^^ ^ do not commute do commute NOTICE: all real and orthogonal

Call this SO(2) cos  sin  - sin  cos  R  v = vxvyvxvy Obviously “reduces” to a 2-dim representation What if we TRIED to diagonalize it further? seek a similarity transformation on the basis set: U†xU†x ^ which transforms all vectors: UvUv  and all operators: URU†URU†

cos  - sin  0 -sin  cos  - 0 = An Eigenvalue Problem Eigenvalues: =1, cos  + isin , cos  isin  (1-  [1 - 2 cos  +   ]=0 =1 = (1-  [cos 2  -2 cos  +   sin 2  ]=0

cos  sin  0a a -sin  cos  0b = b c c To find the eigenvectors a/b =  b/a ??  a=b=0 acos  + b sin  = a  asin  + b cos  = b c = c a(  cos  ) = bsin  b(  cos  ) =  asin  for =cos  +isin  for =cos  isin  for =1 b = i a, c = 0 since a*a + b*b = 1 a = b = b =  i a, c = 0 since a*a + b*b = 1 a = b = acos  + b sin  = a(cos  +isin  )  asin  + b cos  = b(cos  +isin  ) c = c(cos  +isin  )

cos  sin  0 0 -sin  cos  With eigenvectors URU † cos  +isin  0 0 = sin  icos 

cos  +isin  0 0 = sin  icos  and under a transformation to this basis (where the rotation operator is diagonalized) vectors change to: v 1 (v 1 +iv 2 ) / U v = U v 2 = v 3 v 3 (v 1  iv 2 ) /

SO(3) R(1,2,3)R(1,2,3) cos  3 cos  2 +sin  3 sin  2 cos  1 sin  3 -sin  1 sin  2 sin  3 sin  1 sin  3 -cos  1 sin  2 cos  3 -cos  2 sin  3 cos  1 cos  3 -sin  1 sin  2 sin  3 sin  1 cos  3 -cos  1 sin  2 sin  3 sin  2 -sin  1 cos  2 cos  1 cos  2 = Contains SO(2) subsets like: cos  sin  0 -sin  cos  R z (  ) = acting on vectors like v = vxvyvzvxvyvz in the i, j, k basis ^^ ^ which we just saw can be DIAGONALIZED: Rv = e +i   ie  i 

Block diagonal form means NO MIXING of components! Rv = e +i   ie  i  Reduces to new “1-dim” representation of the operator acting on a new “1-dim” basis: e+ie+i  ie  i  +  ii

R(  1 ) R(  2 )= R(  1 +  2 ) UNITARY now! (not orthogonal…) e i   is the entire set of all 1-dim UNITARY matrices, U(1) obeying exactly the same algebra as SO(2) SO(2) is ISOMORPHIC to U(1)

SO(2) is supposed to be the group of all ORTHOGONAL 2  2 matrices with det(U) = 1 a b c d a c b d = a 2 +b 2 ab+bd ac+bd c 2 +d 2 ac =  bd a 2 + b 2 =  c 2 + d 2 =  and along with: det(U) = ad – bc = 1 abd – b 2 c = b  a 2 c – b 2 c = b  c(a 2 + b 2 ) = b  c = b which means: ac =  (  c)d a = d

a b -b a SO(2)So all matrices have the SAME form: a 2 + b 2 = 1 with i.e., the set of all rotations in the space of 2-dimensions is the complete SO(2) group!

det ( A )    n 1 n 2 n 3 ··· n N A n 1 1 A n 2 2 A n 3 3 … A n N N n1,n2,n3…nNn1,n2,n3…nN N some properties det( AB ) = (det A )(det B ) = (det B )(det A ) = det( BA ) since these are just numbers which means det( UAU † ) = det( AU † U ) = det( A ) So if A is HERMITIAN it can be diagonalized by a similarity transformation (and if diagonal) det ( A )   …  (   n 1 n 2 n 3 ··· n N A n 1 1 ) A n 2 2 A n 3 3 … A n N N nNnN N n3n3 N n2n2 N n1n1 N Only A 11 term  0 only diagonal terms survive, here that’s A 22 det ( A )   123…N A 11 A 22 A 33 … A NN =     N Determinant values do not change under similarity transformations! completely antisymmetric tensor (generalized Kroenicker  )

  ( A kk +B kk ) k=1 N another useful property det( A+B ) = ( A + B ) 11 ( A + B ) 22 ( A + B ) 33 … ( A + B ) NN = ( A 11 + B 11 )( A 22 + B 22 )( A 33 + B 33 ) … ( A NN + B NN ) If A and B are both diagonal*   ( A k + B k ) k=1 N *or are commuting Hermitian matrices det( A+B+C+D+ … ) = so  ( A kk +B kk +C kk +D kk + … ) k=1 N

Tr ( A )   A ii = A 11 +A 22 +A 33 … +A NN i N We define the “trace” of a matrix as the sum of its diagonal terms Tr ( AB ) =  (AB) ii =  A ij B ji =  B ji A ij i N i N j N j N i N =  (BA) jj = Tr ( BA ) Notice: Tr( UAU † ) = Tr( AU † U ) = Tr( A ) Which automatically implies: Traces, like determinants are invariant under basis transformations So…IF A is HERMITIAN (which means it can always be diagonalized) Tr ( A ) =       N

Operators like if unitary U : † † † † †† which has to equal = 1 G = G † The generators of UNITARY operators are HERMITIAN and those kind can always be diagonalized Since in general In a basis where A is diagonal, so is AA, AAA,… I is already! So U=e A is diagonal (whenever A is)! What does this digression have to do with the stuff WE’VE been dealing with??

If U=e i  A detU=e i  Tr(A) For SU(n)…unitary transformation matrices with det=1 detU = 1Tr( A )=0

SO(3) is a set of operators (namely rotations) on the basis such that:preserves LENGTHS and DISTANCES SU(3) NEW operators (not EXACTLY “rotations”, but DO scramble components) which also act on a 3-dim basis (just not 3-dim space vectors)

p      p +   1951 m  = MeV m p = MeV Look!

By 1953  +  p +   m  = MeV     +   m  = MeV  p +  

Where a general state (particle) could be expressed where for some set of generators (we have yet to specify)

A model that considered  ’s paired composites of these 3 eigenstates pn (n  + n  ) and successfully accounted for the existence, spin, and mass hierachy of  +,  0,   K +, K 0, K , K 0  , ,  ppp ppnpnnnnn unfortunately also predicted the existence of states like: ppp ppnpnnnnn

none the less for some set of generators (we have yet to specify) SU(3) and means U NITARYThe G i must all be HERMITIAN and det U = 1 The G i must all be TRACELESS As an example consider

SU( 2 ) set of all unitary 2×2 matrices with determinant equal to 1. I claim this set is built with the Pauli matrices as generators! U which described rotations (in Dirac space) of spinors Are these generators HERMITIAN ? TRACELESS ? Does cover ALL possible Hermitian 2×2 matrices? In other words: Are they linearly independent? Do they span the entire space?

What’s the most general traceless HERMITIAN 2  2 matrices? c a  ib a+ib  c a  ib a  ib c cc and check out: = a +b +c i i They do completely span the space! Are they orthogonal (independent)? You can’t make one out of any combination of the others!