Discussion Overview
The discussion revolves around the eigenvalues of the annihilation operator in the context of coherent states, exploring the implications of infinite-dimensional matrices versus finite-dimensional matrices, particularly in relation to bosons and fermions.
Discussion Character
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant asserts that the annihilation operator's matrix has zero as its only eigenvalue, referencing the equation \(\hat{a}|n>=\sqrt{n}|n-1>\).
- Another participant challenges this claim, stating that zero is not the only eigenvalue of the operator.
- There is a discussion about the infinite dimensionality of the matrix and how this affects the process of finding eigenvalues compared to finite dimensions.
- One participant suggests that the determinant is not necessary for finding eigenvalues in this context and provides a method to derive the eigenvalues based on the matrix structure.
- Participants note that for fermions, the corresponding matrix is finite dimensional, which leads to the conclusion that coherent states do not exist for fermions, although some still explore the concept using anti-commuting numbers.
Areas of Agreement / Disagreement
Participants express differing views on the eigenvalues of the annihilation operator, with no consensus reached regarding the claim that zero is the only eigenvalue. The discussion also highlights the distinction between infinite and finite dimensional matrices, indicating a lack of agreement on the implications of this difference.
Contextual Notes
The discussion includes assumptions about the nature of the matrices involved and the definitions of coherent states, which may not be universally agreed upon. The implications of using anti-commuting numbers in the context of fermionic coherent states are also mentioned but not fully explored.
