Does Berry's Phase for State Vectors Tell us the Behaviour of Density Matrices?

K. Datta

Carl A. Brannen

Carlo Rovelli's ``relational interpretation'' of quantum mechanics tells us that our understanding of quantum states is limited to their interactions with other quantum states. This implies that we have no understanding of the symmetry properties of a state vector except when considered in relation to at least one other state vector. The SU(3)xSU(2)xU(1) symmetry of the Standard Model is derived from their interactions and therefore cannot necessarily be applied to a single state vector. Steven Weinberg showed that mixed density matrices can have symmetries that are not available to state vectors. We explore the symmetry properties of finite symmetry groups for mixed density matrices. Just as mixed density matrices can define mixed states that require multiple state vectors, we find that the symmetries of mixed density matrices define ``mixed symmetries'' that are similar in structure to the symmetry of the Standard Model. We explore the point group symmetries and show one that gives the Standard Model symmetry. With this symmetry, we can generalize the Pauli spin matrices to a set that has irreducible representations matching the Standard Model plus a dark matter candidate.

G. N. Brodimas, A. D. Jannussis, N. Patargias

Zheng-Chuan Wang

This study presents the generalization of geometric phases in density matrices. We show that the extended sub-geometric phase has an unified expression during the adiabatic or nonadiabatic process and establish the relations between them and the usual Berry or Aharonov-Anandan phases. We also demonstrate the influence of sub-geometric phases on the physical observables. Finally, the above treatment is used to investigate the geometric phase in a mixed state.

Demosthenes Ellinas, Stephen Mark Barnett, M.‐A. Dupertuis

We examine the role of Berry's geometrical phase [Proc. R. Soc. London, Ser. A 392, 45 (1984)] in the coherent excitation of atoms. The dynamics of a two-level atom prepared in a superposition of its eigenstates will be modified in a nontrivial fashion by the Berry phase. We extend Berry's formalism to treat the Liouville equation for the density matrix. This extension allows us to discuss statistical mixtures and to incorporate Berry's phase within the well-known Bloch-vector description of optical resonance. Furthermore, our density-matrix formalism provides the means of investigating Berry's phase in open, dissipative systems.