-symmetry, Cartan decompositions, Lie triple systems and Krein space-related Clifford algebras
@article{Gnther2010symmetryCD, title={-symmetry, Cartan decompositions, Lie triple systems and Krein space-related Clifford algebras}, author={Uwe G{\"u}nther and Sergii Kuzhel}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2010}, volume={43}, pages={392002} }
Gauged quantum mechanics (PTQM) and corresponding Krein space setups are studied. For models with constant non-Abelian gauge potentials and extended parity inversions compact and noncompact Lie group components are analyzed via Cartan decompositions. A Lie-triple structure is found and an interpretation as -symmetrically generalized Jaynes–Cummings model is possible with close relation to recently studied cavity QED setups with transmon states in multilevel artificial atoms. For models with…
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References
SHOWING 1-10 OF 86 REFERENCES
General aspects of PT -symmetric and P-self-adjoint quantum theory in a Krein space
- Mathematics
- 2006
In our previous work, we proposed a mathematical framework for -symmetric quantum theory, and in particular constructed a Krein space in which -symmetric operators would naturally act. In this work,…
Darboux transformations of the Jaynes-Cummings Hamiltonian
- Physics, Mathematics
- 2004
A detailed analysis of matrix Darboux transformations under the condition that the derivative of the superpotential be self-adjoint is given. As a consequence, a class of symmetries associated with…
On the Lie-Algebraic Origin of Metric 3-Algebras
- Mathematics
- 2009
Since the pioneering work of Bagger–Lambert and Gustavsson, there has been a proliferation of three-dimensional superconformal Chern–Simons theories whose main ingredient is a metric 3-algebra. On…
PT-symmetric brachistochrone problem, Lorentz boosts, and nonunitary operator equivalence classes
- Physics
- 2008
The $\mathcal{P}\mathcal{T}$-symmetric (PTS) quantum brachistochrone problem is re-analyzed as a quantum system consisting of a non-Hermitian PTS component and a purely Hermitian component…
Gauging non-Hermitian Hamiltonians
- Physics
- 2009
We address the problem of coupling non-Hermitian systems, treated as fundamental rather than effective theories, to the electromagnetic field. In such theories the observables are not the x and p…
J-self-adjoint operators with -symmetries: an extension theory approach
- Mathematics, Physics
- 2009
A well-known tool in conventional (von Neumann) quantum mechanics is the self-adjoint extension technique for symmetric operators. It is used, e.g., for the construction of Dirac–Hermitian…
n-ary algebras: a review with applications
- Mathematics
- 2010
This paper reviews the properties and applications of certain n-ary generalizations of Lie algebras in a self-contained and unified way. These generalizations are algebraic structures in which the…
Extended Jaynes?Cummings models and (quasi)-exact solvability
- Physics
- 2006
The original Jaynes–Cummings model is described by a Hamiltonian which is Hermitian and exactly solvable. Here, we extend this model by several types of interactions leading to a non-Hermitian…
Cohomology of Lie triple systems and Lie algebras with involution
- Mathematics
- 1961
A Lie triple system is a subspace of a Lie algebra closed under the ternary composition [[xy]z]; equivalently, it may be defined as the subspace of elements mapped into their negatives by an…