Discussion Overview
The discussion revolves around the concept of commutation in quantum mechanics, specifically addressing why commutation is not transitive. Participants explore the implications of non-abelian groups and the diagonalizability of operators in relation to the Hamiltonian.
Discussion Character
- Technical explanation
- Conceptual clarification
- Debate/contested
Main Points Raised
- One participant questions the transitivity of commutation, citing the case of the Hamiltonian and non-abelian groups.
- Another participant clarifies that a basis is determined by a complete set of commuting observables, indicating that multiple bases can exist.
- A participant confirms that two operators commuting with the Hamiltonian can split its eigen-subspaces differently, leading to a lack of shared diagonalizability.
- It is noted that while additional operators may be diagonalizable together, not all eigenvectors of one operator are necessarily eigenvectors of the other, particularly in the context of non-abelian symmetries.
Areas of Agreement / Disagreement
Participants generally agree on the non-transitive nature of commutation and the implications for diagonalizability, but there is ongoing exploration of specific cases and conditions under which these properties hold.
Contextual Notes
The discussion highlights the dependence on definitions of diagonalizability and the structure of the Hilbert space, as well as the role of non-abelian symmetries in the context of commuting operators.