Download - 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
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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
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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
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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
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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.
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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).
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§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
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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 )
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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
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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
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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.
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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
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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
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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
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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
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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+
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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
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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
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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 ).
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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 :
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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.
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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
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( 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 ... .
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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.
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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
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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 )~
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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
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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
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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
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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
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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
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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
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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
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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
![Page 35: 7.Continuous Groups](https://reader034.vdocuments.us/reader034/viewer/2022051517/56815848550346895dc59dc1/html5/thumbnails/35.jpg)
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.