# -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}
}
• Published 6 June 2010
• Mathematics, Physics
• Journal of Physics A: Mathematical and Theoretical
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…
21 Citations
These notes describe some links between the group SL2(R), the Heisenberg group and hypercomplex numbers - complex, dual and double numbers. Relations between quantum and classical mechanics are
• Mathematics
• 2011
We consider one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to
We revise the construction of creation/annihilation operators in quantum mechanics based on the representation theory of the Heisenberg and symplectic groups. Besides the standard harmonic oscillator
• Mathematics
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
• 2013
The paper is devoted to the development of the theory of self-adjoint operators in Krein spaces (J-self-adjoint operators) involving some additional properties arising from the existence of
• Mathematics, Physics
• 2012
UDC 517.98 We develop a general theory of PT -symmetric operators. Special attention is given to PT -symmetric quasiself-adjoint extensions of symmetric operator with deficiency indices h2;2i: For
In the spirit of geometric quantisation we consider representations of the Heisenberg(–Weyl) group induced by hypercomplex characters of its centre. This allows to gather under the same framework,
• Physics
• 2011
The dynamical aspects of a spin-1/2 particle in Hermitian coquaternionic quantum theory are investigated. It is shown that the time evolution exhibits three different characteristics, depending on
• Physics
Journal of Physics: Conference Series
• 2019
In the so called crypto-Hermitian formulation of quantum theory (incorporating, in particular, the P T -symmetric quantum mechanics as its special case) the unitary evolution of a system is known to
• Mathematics, Physics
• 2012
Generalized -symmetric operators acting on a Hilbert space are defined and investigated. The case of -symmetric extensions of a symmetric operator S is investigated in detail. The possible

## References

SHOWING 1-10 OF 86 REFERENCES

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,
• 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
• 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
• 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
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
• 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
• 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
• 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
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