1. SU(3) gauge symmetry of collective rotations
George Rosensteel and Nick Sparks, Tulane University
G. Rosensteel and N. Sparks, J. Phys. A (2015)

3. GL(3,R) action A = diag(a1,a2.a3)=Stretching R = Rotation R A S
GL(3,R) =  = R A S = SO(3) x Diagonal x G = Gauss decomposition
G = group of 3x3 orthogonal matrices = Vortex Group
Orientation and Shape:
b =  T = R A2 R T = symmetric 3x3 positive-definite matrix
= independent of vortex motion S

4. Reducible vs Irreducible
Fock Space operators make a reducible GCM(3) representation
Irreducible representations create GCM(3) models
Irreps indexed by the Kelvin Circulation C
Kelvin circulation quantized to integral units of ℏ.
Legacy Bohr-Mottelson model is C=0 irrep.
G. Rosensteel and D.J. Rowe Ann. Phys. (N.Y.) 96, 1 (1976).
O.L. Weaver, R.Y. Cusson and L.C. Biedenharn Ann.Phys.(N.Y.) 102, 493 (1976).

6. G. Rosensteel, J. Phys. A34, L169-L178 (2001).

8. Questions:
What value or values of C are physical?
What are the moments of inertia?
What are the connection coefficients?
Answers:
Reciprocity theorem for symplectic irreps.
Determined by metric on GL(3,R).
Determined by Yang-Mills equation.

9. Sp(3,R) symplectic irrep
C=circulation
GCM(3)
I=ang. mom.
SU(3)

10. Reciprocity Theorem Fix an Sp(3,R) irrep: N0 (l0,m0).
In the decomposition of this irrep into GCM(3) irreps:
The number of GCM(3) irreps labeled by C
= the number of angular momentum values I in the SU(3) irrep (l0,m0).
e.g., 20Ne, (l0,m0)=(8,0). I=0,2,4,6.8. Hence, C=0,2,4,6,8.
D.J. Rowe and J. Repka 1998 J. Math. Phys. 39, 6214 (1998),

19. Workshop Speculations
Gauge theory = method to combine 2 different sets of (collective) degrees of freedom. (a) pairing + quadrupole, SU(2) structure group (b) single-particle + collective, U(n).
How does a system respond to changes? Cranking model.
Yang-Mills equation for connection. E&M: 4-potential = A  dA = F = Faraday tensor  Maxwell eqs.. Given sources, solve Maxwell eqs to find connection A.
Direct experimental measurement of collective current in deformed nuclei. Transverse E2 form factor  g.
C=I  Intrinsic wave function is changing in momentum space.
Physical interpretation of circulation SU(3). Quadurpole deformation in momentum space.
2nd quantization.

20. I, KI’
C’, KC’ Y
C, KC
I, KI

21. Conclusions
Very different physical interpretation of Bohr-Mottelson and Elliott SU(3).
Wave functions Y(R;bg) take vector-values in an SU(3) irrep space. The SU(3) irrep describes the Kelvin circulation, not the angular momentum.
Yrast states are created by a coupling of I and C due to the covariant derivative. I=C for yrast states.

22. Conclusions Thank you. 4. Differential geometry of bundles is crucial.
5. The connection coefficients A should be determined from the Yang-Mills equation.
6. All operators in the bundle theory are microscopic one-body observables.
Thank you.