Discussion Overview
The discussion centers on the computation of the variation of the determinant of the metric in the context of varying Polyakov's action with respect to the metric on the world sheet. Participants explore mathematical expressions and seek clarification on the derivation process, particularly in relation to the square root of the determinant.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant presents an expression for the variation of the determinant, suggesting that
δ(h)=2 h h_{αβ}δ(h^{αβ}) is a starting point.
- Another participant challenges this expression, stating it is inaccurate and providing an alternative formulation:
δ(h^2)=2h δh = 2 h^2 h^{αβ}δh_{αβ}, explaining the derivation involves considerations from general relativity and mathematics.
- A third participant mentions deriving the result for the 2D case and requests a general proof or hint for broader applicability.
- A fourth participant suggests starting from the relation
det = exp Tr log as a potential approach to the problem.
Areas of Agreement / Disagreement
Participants express differing views on the initial expressions for the variation of the determinant, indicating a lack of consensus on the correct formulation. The discussion remains unresolved as participants continue to explore various approaches and seek clarification.
Contextual Notes
Some participants reference established methods from general relativity and mathematics, but the discussion does not resolve the assumptions or dependencies involved in the derivations presented.