SU(3) gauge symmetry of collective rotations

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Presentation transcript:

SU(3) gauge symmetry of collective rotations George Rosensteel and Nick Sparks, Tulane University G. Rosensteel and N. Sparks, J. Phys. A48 445203 (2015)

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

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).

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

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.

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

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),

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.

I, KI’ C’, KC’ Y C, KC I, KI

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.

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.