8.%20The%20Group%20SU(2)%20and%20more%20about%20SO(3)

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8. The Group SU(2) and more about SO(3)

• SU(2) Group of 2?2 unitary matrices with unit
  determinant.
• Simplest non-Abelian Lie group.
• Locally equivalent to SO(3) share same Lie
  algebra.
• Compact simply connected ? All IRs are
  single-valued.
• Is universal covering group of SO(3).

• Ref Y.Choquet, et al, "Analysis, manifolds
  physics"
• ( Y, f ) is a universal covering space for X if
  it is a covering space Y is simply connected.
• A covering space for X is a pair ( Y, f )
  where Y is connected locally connected space
  f Y ? X is a homeomorphism ( bi-continuous
  bijection ) if restricted to each connected
  component of f 1(N(x)) ? neighborhood N(x) of
  every point x?X.
• X is simply connected if every covering space
  (Y,f) is isomorphic to (X,Id)

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8.1 The Relationship between SO(3) and SU(2)
8.2 Invariant Integration 8.3 Orthonormality
and Completeness Relations of 8.4 Projection
Operators and Their Physical Applications 8.5
Differential Equations Satisfied by the D j
Functions 8.6 Group Theoretical Interpretation
of Spherical Harmonics 8.7 Multipole Radiation
of the Electromagnetic Field
U(n) Number of real components 2 n2 Number of
real constraints n 2 (n2n)/2 n2 ?
Dimension n2
Dimension of SU(n) n2 1
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8.1. The Relationship between SO(3) and SU(2)
Proved in 7.3
Converse is also true. Proof ( of Theorem 8.1)
Let
Unitarity condition
i.e.
Ansatz
?
4
must hold ?? ?, ?
?
?
n, m integers
?
( m 0 only )
There's no loss of generality in setting n 0.
?
or
Ansatz
Theroem 8.1 U(2) matrices 4-parameters
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Corollary SU(2) matrices 3-parameters
?
?
?
However, this range of ? ? covers twice the
area covered by ?? ?. One compromise, chosen by
Tung, is to set 0 lt ? lt 2?.
SU(2) matrices form a double-valued rep of SO(3)
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Cartesian parametrization of SU(2) matrices
with
Group manifold 4D spherical surface of radius
1. Compact simply-connected.
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SU(2) ? SO(3)
Let

where ?i are the Pauli matrices
?
Let
Since X is hermitian traceless, so is X'.
i.e.,
?
?
Mapping
with
is 2-to-1 ( ??A to same R )
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Let ( r1, r2, r3 ) be the independent parameters
in the Cartesian parametrization.

i.e.
?
Near E, we have
k 1,2,3
?
i.e., ?k is a basis of the Lie algebra su(2).
Since
? su(2) so(3) are the same if we set
Since SU(2) is simply-connected, all IRs of su(2)
are also single-valued IRs of SU(2)
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• Higher dim rep's can be generated using tensor
  techniques of Chap 5
• IRs are generated by irred tensors belonging to
  symm classes of Sn.
• Totally symmetric tensors of rank n form an
  (n1)-D space for the j n/2 IR of SU(2) See
  Example 2, 5.5

• Explicit construction of

Let
where
Spinor
Under rotation
i.e.
Totally symmetric tensor in tensor space V2n
( n1 possible values )
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n1 independent ??'s in (convenient) normalized
form
?m transforms as the canonical components
of the j n/2 IR of su(2)
c.f. Problem 8.5
?
Derivation see Hamermesh, p.353-4
Correctness of Eq(8.1-25)
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8.2. Invariant Integration
?
?
Also holds for different parametrizations of same
group element
Specific method for SU(2) to find ??
Let A, A', B be prarametrized by
resp.
e.g.,
with
?? ri ? r'i is orthogonal ??
?
( r ? r' is linear )
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Since
where
?
where
Integrate over r0 ?
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Switching to ?, ?, ? parametrization
?
?
Integrate over r ?
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Switching to ?, ?, ? parametrization
?
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Theorem 8.2 Invariant Integration Measure Let
A(?) be a parametrization of a compact Lie group
G define
by
where J? are the generators of the Lie
algebra g.
Then
with
Proof Let A(?) be another parametrization.
Consider any point under different
parametrizations. We have
?
( as required )
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Let ?i be the local coordinates at A. For a
fixed element B, the coordinates at BA is
?
i.e.,
?
In case another parametrization ?i is used
at BA, we have
QED
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Another choice of generators J'? can always
be expressed as a linear combination of the old
generators J? , i.e.,
where S is independent of coordinates.
?
Example SU(2) with Euler angle parametrization (
?, ?, ? )
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Example SU(2) with Euler angle parametrization (
?, ?, ? )
With the help of Mathematica, we get
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C is an arbitrary constant
?
Normalized invariant measure
Group volume
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Rearrangement lemma for SU(2)
( Left invariant )
Left right invariant measures coincide for
compact groups. See Gilmore or Miller for proof.
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8.3. Orthonormality and Completeness Relations
of D j

• The existence of an invariant measure,
• which is true for every compact Lie group,
• establishes the validity of the rearrangement
  theorem,
• which in turn guarantees that
• Every IR is finite-dimensional.
• Every IR is equivalent to some unitary
  representation.
• A reducible representation is decomposable.
• The set of all inequivalent IRs are orthogonal
  complete.

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Theorem 8.3 Orthonormality of IRs for SU(2)

• d?A is normalized
• nj 2j1

In the Euler angle parametrization scheme dj(?)
real
?
( no sum over n, m )
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Theorem 8.4 Completeness of DR
(Peter-Weyl) The IR D?? (A)mn form a complete
basis in L2(G). L2(G) ( Hilbert ) space of
(Lebesgue) square integrable functions defined
on the group manifold of a compact Lie group G.
i.e.,
For G SU(2),
?
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( completeness )
?
?

• Comment
• C.f. Fourier theorem in functional analysis.
• f(A) can be vector- or operator- valued.

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Bosons Fermions
Bosons Fermions
?
i.e.
?
?
both cases
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Often,
with ?? n, or, n ½
For ? 0
Setting (?,?) ? (?,?) gives
i.e., the spherical harmonics Ylm forms an
orthonormal basis for square integrable functions
on the unit sphere.
Peter-Weyl
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8.4. Projection Operators their Physical
Applications
Transfer operator
c.f. Chap 4
if non-vanishing, transforms like the IBVs j
m ?? under SU(2) / SO(3)
i.e.,
(error in eq8.4-2)
Henceforth, indices within ? or ?? are
exempted from summation rules
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8.4.1. Single Particle State with Spin
Intrinsic spin s ? states of particle in
rest frame are eigenstates of J2 with
eigenvalue s(s1) .
Denote these states by
with
Task Find p ?? 0, ? ?
i.e., find X ??
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Alternatively, treating J P as the generators
of rotation translation, resp,
(eq 9.6-5)
?
( Theorem 9.12 )
Prove it !
Similarly,
? JP , P , J2, J3 share the same eigenstates
Let
be the "standard state" . Then
L3 0 since motion is along z
( Helicity ? )
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Let
where
and
?
( Problem 8.1 )
i.e., helicity of a particle is the same in all
inertial frames.
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States with definite angular momentum (J, M)
? ? excluded from summation
?
?
c.f. Peter-Weyl Thm, eq(8.3-4)
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For a spinless particle, s ? 0
c.f. 7.5.2
where
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p J M ? ? ? fixed is complete for
1-particle states ?
can be inverted using
to give
Standard state
D diagonal
Traditional description eigenstates of P2, L2,
L3, S3
with
Difficulty L3, S3 not conserved
Helicity is preferred
Partial remedy
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8.4.2. Two Particle States with Spin
Group theoretical methods essential to avoid
complications such as the LS jj coupling
schemes.
Standard state C.M. frame,
,
,
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General plane-wave states with
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p J M ?1, ?2 ? ?j fixed is complete
for 2-particle states ?
See Jacob Wick, Annals of Physics (NY) 7, 401
(59)

• Advantages of the helicity states
• All quantum numbers are measurables.
• Relation between linear- angular- momentum
  states is direct there is no need for the
  coupling-schemes.
• Well-behaved under discrete symmetries.
• Applicable to zero-mass particles.
• Simplifies application to scattering decay
  processes.

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8.4.3. Partial Wave Expansion for 2-Particle
Scattering with Spin
Initial state
Final state

• All known interactions are invariant under SO(3).
• Scattering matrix preserves J.

WignerEckart theorem
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( General partial wave expansion for 2-particle
scattering ) c.f. 7.5.3, 11.4, 12.7
Static spin version would involve multiple CGCs.
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8.5. Differential Equations Satisfied by the D
j Functions
1-D translation (6.6)
?
?
?
Functions e i x p are IRs of Lie group T1.
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Alternate version is in SU(2)_New.ppt
R_New.nb
?
The following derivations are Mathematica
assisted. See R.nb
?
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Using
?
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(1)
(2)
(3)
(1) i sin? (2) cos? (3)
(1) i sin? (2) cos? (3)
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?
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(No Transcript)
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The J? eqs give the recurrence relations
The ?J3 equation is an identity.
?
Since the J's are independent of ?,?, ?, we
have
?
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(Mathematica R.nb )
?
?
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For m 0, j must be an integer D j is
independent of ?. Let ( j, m') ( l, m ) (
?,?) (?,?), we have
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d j is related to the Jacobi polynomials by
Eq(8.5-13) is wrong
where (Mathematica R.nb )
In particular, setting ( j,n,m ) ? ( l,m,0 ) gives
?
For n m 0
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8.6. Group Theoretical Interpretation of
Spherical Harmonics
Special functions Group representation
functions

• Roles played by Ylm(?,?)
• They are matrix elements of the IRs of SO(3).
• They are transformation coefficients between
  bases ? ? ? l m ?.

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8.6.1. Transformation under Rotation
Let
?
c.f. 7.6
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8.6.2. Addition Theorem
For m 0
?
( Addition Theorem )
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8.6.3. Decomposition of Products of Ylm with
the Same Arguments
From 7.7
eq (8.6-4) is wrong
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8.6.4. Recursion Formulas
Exercise show that
Hint start with the adjoint of the eqs
and use
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Using the CGCs in App V, we have
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8.6.6. Orthonormality and Completeness
Theorem 8.3
?
Orthonormality
Theorem 8.4 (Peter-Weyl, for j integer l)
c.f. eqs(8.314,15)
?
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8.6.7. Summary Remarks

• Geometric interpretations were given for
• Differential eqs
• Recursion formulae
• Addition theorem
• Orthonormality completeness relations
• Further development generalization of Fourier
  analysis to functions on manifold of any compact
  Lie group (for which the Peter-Weyl theorem
  holds).
• The D-functions, e.g., Ylm, are also natural
  bases for Hilbert space vectors (tensor)
  operators (see 7.5, 8.4 8.7).

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8.7. Multipole Radiation of the Electromagnetic
Field
Plane wave photon state of helicity ?
Photon state with angular momentum specified by
J,M (c.f. 8.4.1)
The creation operators a(k,?) a( k, J, M,
?) are defined by
where 0 ? is the (vacuum) state of no photons.
?
Using the half-integer case of Peter-Weyl theorem
(see eqs(8.311,12)
we get
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Annihilation operators
Vector potential in a sourcefree region is given
by
?
where
Electromagnetic fields ( ?potential ? 0 )
?
?
are the multipole wave functions
where
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Evaluation of AJMk?(x)
where
See Jackson 16.8
( Addition theorem )
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From 7.7
?
?
Comparing with the inverse of
i.e.,
we have
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where
c.f. Prob 8.10
Vector spherical harmonics
Electric and magnetic multipoles ( of definite
parities )
See Chap 11
Note The above results are derived with no
explicit reference to the Maxwell eqs.
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Example Photo-Absorption
1st order perturbation transition probability
amplitude
Using the Wigner-Eckart theorem, we have