Ramsey theory enables re-shaping of the basic ideas of the quantum mechanics. Quantum observables, represented by linear Hermitian operators are seen as the vertices of the graph. Relation of commutation define coloring of the edges linking the vertices: if the operators commute, they are connected with the red link; if they do not commute they are connected with the green link. Thus, a bi-colored, complete, Ramsey graph emerges. According to the Ramsey theorem, complete, bi-colored graph built of six vertices, will inevitably contain at least one monochromatic triangle; in other words, the Ramsey number R(3,3)=6. In our interpretation, this triangle represents the triad of observables, which could or, alternatively, could not be established simultaneously in a given quantum system. The Ramsey approach to the quantum mechanics is illustrated with the numerous examples, including the motion of a particle in a centrally symmetrical field.Ramsey theory enables re-shaping of the basic ideas of the quantum mechanics. Quantum observables, represented by linear Hermitian operators are seen as the vertices of the graph. Relation of commutation define coloring of the edges linking the vertices: if the operators commute, they are connected with the red link; if they do not commute they are connected with the green link. Thus, a bi-colored, complete, Ramsey graph emerges. According to the Ramsey theorem, complete, bi-colored graph built of six vertices, will inevitably contain at least one monochromatic triangle; in other words, the Ramsey number R(3,3)=6. In our interpretation, this triangle represents the triad of observables, which could or, alternatively, could not be established simultaneously in a given quantum system. The Ramsey approach to the quantum mechanics is illustrated with the numerous examples, including the motion of a particle in a centrally symmetrical field.

arXiv:2411.02082Oct 11, 20240 citations

On quantum multi-hamiltonian systems

Mich�le Irac-Astaud, Guy Rideau

Letters in Mathematical PhysicsNov 1, 19938 citations

Thermodynamics of an Open Two-Level $\mathcal{PT-}$ Symmetric Quantum SystemThermodynamics of an Open Two-Level $\mathcal{PT-}$Symmetric Quantum System

Baibhab Bose, Devvrat Tiwari, Subhashish Banerjee

For a subclass of a general $\mathcal{PT}-$ symmetric Hamiltonian obeying anti-commutation relation with its conjugate, a Hermitian basis is found that spans the bi-orthonormal energy eigenvectors. Using the modified projectors constructed from these eigenvectors, the generalized density matrix of the $\mathcal{PT}-$ symmetric evolution is calculated, and subsequently, ergotropy for a closed system is obtained. The $\mathcal{PT}-$ symmetric system, in an open system scenario, is studied to understand ergotropy under different regimes of non-Hermiticity of the Hamiltonian. The consistency of the three laws of thermodynamics for the $\mathcal{PT}-$ symmetric system in an open system scenario is also analyzed.For a subclass of a general $\mathcal{PT}-$symmetric Hamiltonian obeying anti-commutation relation with its conjugate, a Hermitian basis is found that spans the bi-orthonormal energy eigenvectors. Using the modified projectors constructed from these eigenvectors, the generalized density matrix of the $\mathcal{PT}-$symmetric evolution is calculated, and subsequently, ergotropy for a closed system is obtained. The $\mathcal{PT}-$symmetric system, in an open system scenario, is studied to understand ergotropy under different regimes of non-Hermiticity of the Hamiltonian. The consistency of the three laws of thermodynamics for the $\mathcal{PT}-$symmetric system in an open system scenario is also analyzed.

quant-pharXiv:2512.03935Journal of Physics A Mathematical and TheoreticalDec 4, 20250 citations

Thermodynamics of an Open Two-Level

\mathcal{PT-}

Symmetric Quantum System

For a subclass of a general

\mathcal{PT}-

symmetric Hamiltonian obeying anti-commutation relation with its conjugate, a Hermitian basis is found that spans the bi-orthonormal energy eigenvectors. Using the modified projectors constructed from these eigenvectors, the generalized density matrix of the

\mathcal{PT}-

symmetric evolution is calculated, and subsequently, ergotropy for a closed system is obtained. The

\mathcal{PT}-

symmetric system, in an open system scenario, is studied to understand ergotropy under different regimes of non-Hermiticity of the Hamiltonian. The consistency of the three laws of thermodynamics for the

\mathcal{PT}-

symmetric system in an open system scenario is also analyzed.