We continue the study of non-commutative operator graphs generated by resolutions of identity covariant with respect to unitary actions of the circle group and the Heisenber-Weyl group as well. It is shown that the graphs generated by the circle group has the system of unitary generators fulfilling permutations of basis vectors. For the graph generated by the Heisenberg-Weyl group the explicit formula for a dimension is given. Thus, we found a new description of the linear structure for the operator graphs introduced in our previous works.
Grötschel, Lovász, and Schrijver generalized the Lovász $\vartheta$ function by allowing a weight for each vertex. We provide a similar generalization of Duan, Severini, and Winter's $\tilde{\vartheta}$ on non-commutative graphs. While the classical theory involves a weight vector assigning a non-negative weight to each vertex, the non-commutative theory uses a positive semidefinite weight matrix. The classical theory is recovered in the case of diagonal weight matrices. Most of Grötschel, Lovász, and Schrijver's results generalize to non-commutative graphs. In particular, we generalize the in…
We study quantum equivalents of non-commutative operators in quantum mechanics. Any matrix "$B$" satisfying the non-commuting relation $[A,B]\neq 0$ with "$A$", can be used via $B^{-1} AB$ to reproduce eigenvalues of "$A$". This universality relation is also equally valid for any matrix in any branch of physical or social science and also any operator involving co-ordinate$(x)$ or momentum$(p)$. Pictorially this is represented in fig. 1. Many interesting models including logarithmic potential have been considered.
It is well known that an (in general, non-commutative) set of non-Hermitian operators $Λ_j$ with real eigenvalues need not necessarily represent observables. We describe a specific class of quantum models in which these operators plus the underlying physical Hilbert-space metric $Θ$ are all represented in terms of an auxiliary operator $(N+1)-$plet $Z_k$, $k=0,1,\ldots,N$. Our formalism degenerates to the ${\cal PT}-$symmetric quantum mechanics at $N=2$, with metric $Θ=Z_2Z_1$, parity ${\cal P}=Z_2$, charge ${\cal C}=Z_1$ and Hamiltonian $H=Z_0$.
