We’ve found an exponential expression for operators

1. 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

2. 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

3. O(n) set of all orthogonal UT=U-1 (therefore real) matrices of dimension n×n SO(n) “special” subset of the above: unimodular, i.e., det(U)=1 group of all rotations in a space of n-dimensions Rotations in 3-dim space SO(3) ALL known “external” space-time symmetries in physics 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  0 0 0 1 U(n) set of all n×n UNITARY matrices U†=U-1 i.e. U†U=I new “internal” symmetries (beyond space-time) SU(n) “special” unimodular subset of the above det(U)=1

4. SO(3) cos3sin3 0 cos20 -sin2 1 0 0 do not commute R(1,2,3)= -sin3cos3 0 0 cos1sin1 0 1 0 sin2 0 cos2 0 -sin1cos1 0 0 1 cos3cos2+sin3sin2cos1sin3-sin1sin2sin3sin1sin3-cos1sin2cos3 = -cos2sin3 cos1cos3-sin1sin2sin3sin1cos3-cos1sin2sin3 sin2-sin1cos2cos1cos2 do commute Contains SO(2) subsets like: acting on vectors like NOTICE: all real and orthogonal cossin 0 vx vy vz v = Rz() = -sincos 0 in the i, j, k basis 0 0 1 ^ ^ ^

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

6. An Eigenvalue Problem cos-l sin 0 -sin cos-l0 = 0 0 0 1-l = (1-l)[cos2-2lcos+l2+sin2]=0 (1-l)[1 - 2lcos + l2]=0 l=1 Eigenvalues:l=1, cos  + isin , cos -isin 

7. To find the eigenvectors cos sin 0 aa -sin cos 0 b=lb 0 0 1 cc forl=1 acos + b sin = a -asin + b cos = b c = c a(1-cos) = bsin b(1-cos) = -asin a/b = -b/a ?? a=b=0 acos  + b sin  = a(cos+isin) -asin  + b cos  = b(cos+isin) c = c(cos+isin) forl=cos+isin b =i a, c = 0 since a*a + b*b = 1 a=b= forl=cos-isin b =-i a, c = 0 since a*a + b*b = 1 a=b=

8. With < v | R | v > cos sin 0 0 -sin cos 0 0 0 0 1 0 1 0 URU† eigenvectors cos+isin 0 0 = 0 1 0 0 0 sin-icos

9. cos+isin 0 0 = 0 1 0 0 0 sin-icos < v | R | v > and under a transformation to this basis (where the rotation operator is diagonalized) vectors change to: v1(v1+iv2)/ Uv = Uv2 = v3 v3(v1-iv2)/

10. SO(3) cos3cos2+sin3sin2cos1sin3-sin1sin2sin3sin1sin3-cos1sin2cos3 R(1,2,3) = -cos2sin3 cos1cos3-sin1sin2sin3sin1cos3-cos1sin2sin3 sin2-sin1cos2cos1cos2 Contains SO(2) subsets like: acting on vectors like cossin 0 vx vy vz v = Rz() = -sincos 0 in the i, j, k basis 0 0 1 ^ ^ ^ which we just saw can beDIAGONALIZED: e+i0 0 0 1 0 0 0 -ie-i Rv =

11. Block diagonal form means NO MIXINGof components! e+i0 0 0 1 0 0 0 -ie-i Rv = Reduces to new “1-dim” representation of the operator acting on a new “1-dim” basis: e+i + -ie-i -i -

12. R(1) R(2)= R(1+2) UNITARYnow! (not orthogonal…) ei  is the entire set of all 1-dim UNITARYmatrices, U(1) obeying exactly the same algebra as SO(2) SO(2)is ISOMORPHIC toU(1)

13. SO(2) is supposed to be the group of allORTHOGONAL 22 matrices withdet(U) = 1 a b c d a c b d a2+b2 ab+bd ac+bd c2+d2 = a2 + b2= 1 ac = -bd c2 + d2= 1 and det(U) = ad – bc = 1 along with: abd – b2c = b -a2c – b2c = b -c(a2 + b2) = b -c = b which means: ac = -(-c)d a = d

14. So all matrices have the SAME form: SO(2) a b -b a a2 + b2= 1 with i.e., the set of all rotations in the space of 2-dimensions is the complete SO(2) group!

15. det(A)  n1 n2 n3···nN An11 An22 An33 …AnNN N n1,n2,n3…nN completely antisymmetric tensor (generalized Kroenicker  ) since these are just numbers some properties det(AB) = (detA)(detB) = (detB)(detA) = det(BA) which means Determinant values do not change under similarity transformations! 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)  …(n1 n2 n3···nN An11)An22 An33 …AnNN N N N N nN n3 n2 n1 Only A11 term0 only diagonal terms survive, here that’s A22 det(A)  123…NA11 A22 A33 …ANN = l1l2l3...lN

16. If A and B are both diagonal* another useful property det(A+B) = (A+B)11(A+B)22(A+B)33…(A+B)NN = (A11+B11)(A22+B22)(A33+B33)…(ANN+BNN)  P(Akk+Bkk)  P(lAk+lBk) N N k=1 k=1 *or are commuting Hermitian matrices so P(Akk+Bkk+Ckk+Dkk+…) N det(A+B+C+D+…) = k=1

17. We define the “trace” of a matrix as the sum of its diagonal terms Tr(A)  Aii = A11+A22 +A33…+ANN N i Notice: Tr(AB) =(AB)ii = AijBji= BjiAij N N N N N i i j j i =(BA)jj = Tr(BA) Traces, like determinants are invariant under basis transformations Which automatically implies: Tr(UAU†) = Tr(AU†U) = Tr(A) So…IFAis HERMITIAN (which means it can always be diagonalized) Tr(A) = l1+l2+l3+... +lN

18. What does this digression have to do with the stuff WE’VE been dealing with?? Operators like † † † if unitaryU: 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=eA is diagonal (whenever A is)!

19. detU=eiaTr(A) If U=eiaA For SU(n)…unitary transformation matrices with det=1 detU= 1 Tr(A)=0

20. SO(3) is a set of operators (namely rotations) on the basis 0 1 0 0 0 1 1 0 0 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)

21.  - 1951 p Look!   p +  - m=1115.6 MeV mp=938.27 MeV

22. By 1953 +  p +  0 m=1115.6 MeV -   +  - m=1321.0 MeV  p +  -

23. 0 1 0 0 0 1 1 0 0 Where a general state (particle) could be expressed where for some set of generators (we have yet to specify)

24. A model that considered p’s paired composites of these 3 eigenstates (n+ n) pn pn and successfully accounted for the existence, spin, and mass hierachy of  +,  0,  - K+, K0, K-, K0   , ,  unfortunately also predicted the existence of states like: ppp ppn pnn nnn ppp ppn pnn nnn

25. none the less for some set of generators (we have yet to specify) and SU(3) meansUNITARY The Gi must all be HERMITIAN The Gi must all be TRACELESS and detU= 1 As an example consider

26. 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 generatorsHERMITIAN? TRACELESS? cover ALL possible Hermitian 2×2 matrices? Does In other words:Are they linearly independent? Do they span the entire space?

27. What’s the most general tracelessHERMITIAN 22 matrices? a-ib c a+ib -c and check out: c a-ib a+ib -c 0 1 1 0 0 -i i 0 1 0 0 -1 = a +b +c They do completely span the space! Are they orthogonal (independent)? You can’t make one out of any combination of the others!