7.Continuous Groups
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7. Continuous Groups
Rotations in 2-D : 2 ; 0,2SO R 1
det 1T
R
R RSpecialOrthogonal
Rotations in 3-D : 3 , , ; , , 0,2SO R
Rotations in n-D : ; 0,2 ; 1, , 1 / 2iSO n R i n n α
SO(n) = Lie group of order n(n1)/2.
{ R() } = Fundamental representation
( indep. elements in nn SO matrix )
Generalization to complex vector space:
= Lie group of order n21.
SpecialUnitary 2; 0,2 ; 1, , 1iSU n U m i n α 1
det 1
U
U U
( indep. real parameters in nn SU matrix )
Used in classification of elementary particles
For elements close to I, Sj = generators
Lie Groups & Their Generators
Lie group of order n = group that is also an n-D differentiable manifold.
( group elements have local 1-1 map to region in Rn.)
~ group with continuous parameters over finite n-D region(s).
1
n
j jj
I i
U φ S
1
lim
Nn
jjN
j
I iN
U φ Sj
j N
1
expn
j jj
i
U φ S
& jj
i
φ 0
US
for the identity component of G.
sign chosen to make S = L
Example 17.7.1. SO(2) Generator
Rotations about a fixed axis : cos sin
sin cos
U active point of view.( eq.17.38 is the passive version )
x
y
r U r
cos sin
sin cos
x
y
cos sin
sin cos
x y
x y
1
1
U0 1
1 0I
1
n
j jj
I i
U φ S
1
expn
j jj
i
U φ S
0 1
1 0i
S 2σ
2exp i U σ exp cos sink ki i σ I σ
§ 2.2, Euler identity :
cos sin
sin cos
sin cos
cos sin
U
0
0 1
1 0
U2i σ i S
SO(n) & SU(n) 1
n
j jj
I i
U φ S 1
expn
j jj
i
U φ S
1
1
expn
j jj
i
U φ S
1
expn
j jj
i
U φ S
j jS S Sj are hermitianU unitary
det ii
U
Let i be the eigenvalues of U :
exp ln ii
exp lnTr U
1
expn
j jj
Tr i
S
det 1 j U 0jTr S Sj are traceless
1
expn
j jj
i Tr
S
exp ln ii
Set
Let expj j j ji U S 2 2 31
2j j j jI i O S S
2 2 2 2 31
2k j k k j j k k j j k j k jI i O U U S S S S S S
j kl
j k lj k
f
&
1 1 3,k j k j j k j kI O U U U U S S
1
n
j jj
I i
U φ S
1 1 2 2 2 2 31
2k j k k j j k k j j k j k jI i O U U S S S S S S
1 expj j j ji U S 2 2 31
2j j j jI i O S S
multiplication is closed 2
1
njk j k
l ll
I i O
U θ S
1
,n
j k jk l ll
i f
S S Sf j k l = structure constants
P
jk lP jk lf f ( f j k l is antisymmetric in its indices.)
Can be used to define “identity component” of G.
rank of G = max # of mutually commuting independent generators.
Basis of IRs of G are labelled using the eigenvalues of such set of generators.
E.g., SO(n) & SU(n) ~ generated by generalized angular momenta
rank of G = # of indices needed to label the basis of an IR.
For SO(3), rank = 1 IR label = ML .
For SU(2), rank = 1 IR label = MS .
For SU(3), rank = 2 IR label = ( I3 , Y ) .
Casmir operator = operator that commutes with all generators of G.
For SO(3), L2 is the Casmir operator.
IRs of G are labelled using the eigenvalues of the Casmir operator(s).
§16.4 : see next page
SO(2) & SO(3)
For SO(2) 2
0
0
i
i
S σ
For SO(3) 3
0 0
0 0
0 0 0
i
i
S 3
cos sin 0
sin cos 0
0 0 1
U
1
1 0 0
0 cos sin
0 sin cos
U 1
1
0
0 0 0
0 0
0 0
di i
i
US
2
cos 0 sin
0 1 0
sin 0 cos
U 2
0 0
0 0 0
0 0
i
i
S
,j k j k l li S S S i iK S
Alternatively,
Basis { x, y }
x
y
r U r
cos sin
sin cos
x
y
cos sin
sin cos
x y
x y
Using functions { x, y } as basis :(passive point of view)
cos sin, ,
sin cosx y x y
Basis = { i , j }(active point of view)
,x y V
generator for V is 2exp i V σ 1 U0 1
1 0i
S 2σ
zL i x yy x
, ,z zL x L y i y ix 0,
0
ix y
i
2
0
0z
i
i
L σ
1R f f r U r
exp zi V L
V f r exp zi L f r
f y x fx y
r r 1 0 1
1 0
U I
S
QM rotation op. ( = 1 )
Orthonormality :
Example 17.7.2. Generators Depend on Basis
SO(3) with Y1m (Cartesian rep) as basis :
1 1, ,
2 2x i y z x i y
ψ 3
4i j i j i jx x d x x
xL i y zz y
, , 0 , ,x x xL x L y L z iz i y
1 2 3, ,x x x xL L L L ψ 1 1, ,
2 2z i y z
1 2 3
10 0
21 1
, , 02 2
10 0
2
3 1
1
2x 3 1
2
iy 2z
0 1 0
11 0 1
20 1 0
xL
Similarly:
0 01
02
0 0y
i
L i i
i
1 0 01
0 0 02
0 0 1zL
SU(2) & SU(2)-SO(3) Homomorphism
# of generators: SO(3) = 3 SU(2) = 3 SU(3) = 8
complex matrices general H / SU H & tr=0
# of indep. elements n2 n (n+1) / 2
n = 2 4 3
n = 3 9 6
# of indep. real params. 2n2 n2 1
n = 2 8 4 3
n = 3 18 9 8
# of independent real parameters for nn complex matrices :
SU(2) :1
1,2,32i i i S σ ,j k j k l li S S S
expj j j ji U SRotation operator (passive) : cos sin2 2
j jji
I σ
SO(3) SU(2)
Generators { Lx ,Ly , Lz } { sx , sy , sz }
Basis { Ylm , m=l,...,l } spinors
Dim. of IR 2l+1 ; l = 0,1,2,... 2s+1 ; s = 0, ½, 1, ...
U ( ,, [0,2) ) single -valued double -valued
cos sin2 2
j jj j ji
U I σ
2j U I 4j U I 0j U I j ji U σ
SU(2) SO(3) is a 2-1 homomorphism.
SU(3)
p - n behaves nearly identically in strong interaction.
Heisenberg : p - n is a doublet [ 2-D IR of SU(2) ] (approximate symmetry )
Isospin : 1,2,3j j j τ σ 3
1 / 2
1 / 2
pI
n
Gell-Mann : is an octet [ 8-D rep of SU(3) ] 0 0, , , , , , ,n p
m (MeV) Y I3 S
1321 1 1/2 2
0 1315 1 +1/2 2
1197 0 1 1
0 1193 0 0 1
+ 1189 0 +1 1
0 1116 0 0 1
N n 940 1 1/2 0
p 938 1 +1/2 0
Y = 2 ( Q I3 ) = Hypercharge
S = (ns ns ) = Strangeness
ns = # of strange quarks
ns = # of strange antiquarks
Pre-quark def:S = +1 for anti-partcleS = 1 for partcle
SU(3) : order = # of generators = 8, rank = IR labels = 2
11, ,8
2i i i S λ i = Gell-Mann matrices
1
0 1 0
1 0 0
0 0 0
λ 2
0 0
0 0
0 0 0
i
i
λ
7
0 0 0
0 0
0 0
i
i
λ
3
1 0 0
0 1 0
0 0 0
λ
4
0 0 1
0 0 0
1 0 0
λ 5
0 0
0 0 0
0 0
i
i
λ
8
1 0 01
0 1 03
0 0 2
λ6
0 0 0
0 0 1
0 1 0
λ
GMMs for SU(2) subgroup:
{ 1 , 2 , 3}, { 6 , 7 , 3 }, { 4 , 5 , 3 }.
3 8 3
0 0 0
3 0 1 0
0 0 1
λ λ λ 3 8 3
1 0 0
3 0 0 0
0 0 1
λ λ λ
I3 Y
eigen-
values of S3 ( 2 /3 ) S8
Example 17.7.3.Quantum Numbers of Quarks
3 3
1 0 01 1
0 1 02 2
0 0 0
S λ 8 8
1 0 02 1 1
0 1 033 3
0 0 2
S λ
Quark model :
Basis : { u, d, s } quarks
I3 Y = 2 ( Q I3 )
eigen-
values of S3 ( 2 /3 ) S8
3
1 0 0
0 1 0
0 0 0
λ 8
1 0 01
0 1 03
0 0 2
λ
I3 Y Q = I3 + Y/2
u 1/2 1/3 2/3
d 1/2 1/3 1/3
s 0 2/3 1/3
Commutation Rules
1 1
0 1 01
1 0 02
0 0 0
S I
2 2
0 01
0 02
0 0 0
i
i
S I 7 2
0 0 01
0 02
0 0
i
i
S U
3
1 0 01
0 1 02
0 0 0
S4 1
0 0 11
0 0 02
1 0 0
S V
5 2
0 01
0 0 02
0 0
i
i
S V 8
1 0 02 1
0 1 033
0 0 2
S
6 1
0 0 01
0 0 12
0 1 0
S U
Ladder operators : 1 2i X X X , , orX I U V
3 , S I I 3
1,
2 S U U 3
1,
2 S V V
8 , S I 0 8
3,
2 S U U 8
3,
2 S V V
Mathematica
8 3 8 3 3
3, , , ,
2I Y I Y Y I Y S U S U U
3 3 3 3, ,I Y I I Y S 8 3 3
3, ,
2I Y Y I Y S
3
3,
2I Y U
8 3 3
3, 1 ,
2I Y Y I Y S U U
3 , S I I 3
1,
2 S U U 3
1,
2 S V V
8 , S I 0 8
3,
2 S U U 8
3,
2 S V V
3 3 3 3, ,I Y I I Y S
8 3 3
3, ,
2I Y Y I Y S
8 3 3
3, 1 ,
2I Y Y I Y S U U
3 3 3 3 3 3, , , ,I Y I Y I I Y S U S U U
3 3 3 3
1, ,
2I Y I I Y
S U U
3
1,
2I Y U
3 3
1, , 1
2UI Y C I Y
U
Similarly : 3 3, 1 ,II Y C I Y I
3 3
1, , 1
2VI Y C I Y
V
Y
I3
I+
U+ V+
Example 17.7.4. Quark LaddersI3 Y
u 1/2 1/3
d 1/2 1/3
s 0 2/3
Mathematica
11 1
, 02 3
0
u
01 1
, 12 3
0
d
02
0 , 03
1
s
0 1 0
0 0 0
0 0 0
I
0 0 1
0 0 0
0 0 0
U
0 0 0
1 0 0
0 0 0
I
0 0 0
0 0 1
0 0 0
V
0 0 0
0 0 0
1 0 0
U
0 0 0
0 0 0
0 1 0
V
I+ I U+ U V+ V
u 0 d 0 0 0 s
d u 0 0 s 0 0
s 0 0 d 0 u 0
3 3
1, , 1
2UI Y C I Y
U
3 3, 1 ,II Y C I Y I
3 3
1, , 1
2VI Y C I Y
V
I3 Y
u 1/2 1/3
d 1/2 1/3
s 0 2/3
I+ I U+ U V+ V
u 0 d 0 0 0 s
d u 0 0 s 0 0
s 0 0 d 0 u 0
u (1/2 , 1/3 )
s ( 0 , 2/3 )
d (1/2 , 1/3 )
V
V+
I
I+
U
U+
Y
I3
I+I
V+
V
U+
U
(1/2 , 1 )
(1/2 , 1 )
(1/2 , 1 )
(1/2 , 1 )
(1 , 0 ) (1 , 0 )
Root Diagram :Effects of operators
Conversion between quarks
Baryons
Quark model: Each baryon consists of 3 quarks.
# of basis functions = 3 3 3 = 27
Decomposing into IR bases : 27 = 10 + 8 + 8 + 1
3 3 3 10 8 10 1
Standard tool for the task is the Young tableaux (see Tung).
3 3 3 10 8 8 1Short hand :
Rep :
Here, we’ll use the ladder operators ( see root diagram ).
Example 17.7.5. Generators for Direct Products
Group operation on products of basis functions :
1 2 1 2i j i jR U R U R
1 21 2i S i Si je e
1 2 1 2i S S
i je
1 21 2 i S Si S i Se e e i.e.,
e.g., 1 2 3 I I I I
group elements generators
Lie group Lie algebra
1 2 3 1 2 3 1 2 3 1 2 3u u u C d u u u d u u u d I
uuu C duu udu uud IShort hand :
3 3 3 31 2 3 1 2 3 1 2 3 1 2 3u u u u u u u u u u u u S S S S
31 2 3
2u u u
3
1 0 01
0 1 02
0 0 0
S
shorthand: 3
3
2u u u u u uS
8
1 0 02 1
0 1 033
0 0 2
S
3
1 1 1
2 2 2u u d u u d
I
1
2u u d
uuu has I3 = 3/2.
uud has I3 = 1/2.
8
2 1 2 2
3 3 33d s s d s s
S d s s dss has Y = 1.
Example 17.7.6. Decomposition of Baryon Multiplets
( I3 , Y ) values of the 27 possible 3-quark products :
( 3/2 , 1 ) ( 1/2 , 1 ) ( 1/2 , 1 ) ( 3/2 , 1 )
uuu uud , udu, duu udd , dud, ddu ddd
( 1 , 0 ) ( 0 , 0 ) ( 1 , 0 )
uus , usu , suu uds , dus , usd , dsu , sud , sdu
dds , dsd , sdd
( 1/2 , 1 ) ( 1/2 , 1 )
uss , sus , ssu dss , sds , ssd
( 0 , 2 )
sss
( 3/2 , 1 ) ( 1/2 , 1 ) ( 1/2 , 1 ) ( 3/2 , 1 )
uuu uud , udu, duu udd , dud, ddu ddd
( 1 , 0 ) ( 0 , 0 ) ( 1 , 0 )
uus , usu , suu uds , dus , usd , dsu , sud , sdu
dds , dsd , sdd( 1/2 , 1 ) ( 1/2 , 1 )
uss , sus , ssu dss , sds , ssd
( 0 , 2 )
sss
Baryon decuplet : generated from uuu. Baryon Octet : generated from [duu].
Mathematica
[...] means appropriate symmetrized linear combination of ... .
Mass SplittingParticles in a multiplet actually have slightly different masses ( SU(3) symmetry only approximate ).
This is caused by the weak & EM forces that break the symmetry of the strong force.
8. Lorentz Group
Physical laws should be the same for all observers.
Mathematically, this means equations of physical laws must be covariant, i.e.,
General relativity : Their forms are unchanged under any space-time coordinate (observer) transformations.
Special relativity : Their forms are unchanged under any transformations between moving inertial space-time coordinate systems (observers).
( Lorentz transformations ; Lorentz group )
Galilean relativity : Their forms are unchanged under any spatial coordinate transformations between moving inertial systems (observers)
Inertial system: System travelling with constant velocity w.r.t. a standard reference system (the distant stars).
Transformations between stationary inertial systems:
Translational invariance Conservation of (linear) momentum
Rotational invariance Conservation of angular momentum
Homogeneous Lorentz Group
Lorentz transformations : Transformations of space-time coordinatesbetween moving inertial systems.
Space-time is homogeneous & isotropic symmetries in coordinate transformations
Lorentz transformations ~ (homogeneous) Lorentz group
Lorentz transformations + space-time translations
Inhomogeneous Lorentz group( Poincare group )~
Special relativity : Space-time = linear 4-D space with Minkowski metric.( This makes velocity of light = c for all inertial observers )
1 2 3, , , ,x ct ct x x x x 0,1,2,3
2 2 22d dx d x c d t d x
Lorentz group : All transformations that keep 2 2d d
event interval
Let x be moving in +z direction with small velocity v :
z z vt
2
1a
c
1 2 3, , , ,x ct ct x x x x Same event as recorded by observer travelling with velocity v
t t avz a indep of v.
2 2 2 22 2c dt dz c dt avdz dz vdt 2 22 2 22 1c dt dz v ac O v
vct ct z
c
1
1
ct ct
z z
v
c
small v only
1 2 3, , , ,x ct ct x x x x
Comparing with the actual z-boost :
1
1
ct ct
z z
z
ct cti
z z
I S 0 1
1 0z i
S 1i σ
expz zi U S 1cos sini i i σ 1exp i i σ
cos sinkike i σ σ
1cosh sinh σ
cosh sinh
sinh coshz
U
Generator for z-boost
Operator for z-boost
2
z z vt
vzt t
c
tanhv
c
2 2
1
1 /v c
cosh sinh
= rapidity
Successive Boosts cosh sinh
sinh coshz
Utanh
v
c
cosh sinh cosh sinh
sinh cosh sinh coshz z
U U
cosh cosh sinh sinh sinh cosh cosh sinh
sinh cosh cosh sinh cosh cosh sinh sinh
cosh sinh
sinh cosh
z z z U U U z z U U, not , is the group parameter
Successive boosts in different directions give boost + rotation Thomas precession ( crucial in SO coupling )
11 1 1 1e e e e e σσ σ σ σ
Example 17.8.1.Addition of Collinear Velocities
tanhv
c
Successive z-boosts:
Resultant velocity: tanh tanh tanh
1 tanh tanh
1
0 , 1 , 0 & finite 0 1
1 1
11
Minkowski Space
Metric tensor :1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
g g
x g x d x g d x
x U x i S x
Boost
9. Lorentz Covariance of Maxwell’s Equations
Let B At
A
E SI units
,Ac
A F A A x
,
c t
0F
0 1ii iA
Fc t c
F F
1iiA
c t c
1 iEc
j ii j
i j
A AF
x x
i j k kB
1 2 3
1 3 2
2 3 1
3 2 1
0
010
0
E E E
E cB cBF
E cB cBc
E cB cB
F
i i j kii j k
EF cdt dx B dx dx
c
iiB B
see E.g.4.6.2
Lorentz Transformation of E & B
cosh sinh
sinh coshz
U
1 2 3
1 3 2
2 3 1
3 2 1
0
010
0
E E E
E cB cB
E cB cBc
E cB cB
F
cosh sinh
0 0
0 1 0 0
0 0 1 0
0 0
zU
U
F U U F
TF U F U U F U T U U
1 2 2 1 3
1 2 3 2 1
1 2 3 1 2
3 2 1 1 2
0
010
0
E cB E cB E
E cB cB cB E
E cB cB cB Ec
E cB E cB E
F
Mathematica
E E v B
2
1
c
B B v E
/ / / / E E
/ / / / B B
Example 17.9.1. Transformation to Bring Charge to Rest
Charge q moving with velocity v is at rest in frame boosted by v.
In boosted frame q F E
In original frame q F E v B
q E v B Lorentz force
10. Space Groups
Perfect crystal = basis of atoms / molecules placed on each point of a Bravais lattice.
Bravais lattice = points given by1
d
i ii
n
b h all integersin dimension
of space
d
hi = unit lattice vectors
For d = 3 , there’re
14 possible Bravais lattices, &
32 compatible crystallographic point groups,
which give rise to 230 space groups.
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