Discussion Overview
The discussion revolves around the displacement operator as presented in Dirac's text, specifically focusing on the equality involving the limit of the expression [D*exp(iy)-1]/δx and its relationship to [D-1+iy]/δx. Participants explore the implications of the Taylor expansion of the exponential function and the assumptions made regarding the limits as δx approaches zero.
Discussion Character
- Technical explanation
- Debate/contested
- Conceptual clarification
Main Points Raised
- One participant questions how the equality [D*exp(iy)-1] = [D-1+iy] holds, suggesting that the Taylor expansion becomes exact if y tends to zero.
- Another participant expresses concern that there should be a D in front of the iy and iax, implying a possible misprint in Dirac's text.
- A different participant mentions that they encountered the same expression in a lecture by Dr. Fitzpatrick, suggesting a potential reliance on Dirac's work.
- One participant critiques Dirac's treatment of the infinitesimal displacement operator as unrigorous and expresses disappointment.
- Another participant offers their interpretation of the equality, emphasizing the irrelevance of the phase factor exp(iy) and questioning the assumption that y tends to zero as δx approaches zero.
- There is a request for clarification on the Taylor expansion for the exponential and the reasoning behind the assumption regarding y.
Areas of Agreement / Disagreement
Participants express differing views on the correctness of Dirac's formulation and the assumptions made in the derivation. There is no consensus on whether the treatment is rigorous or if the expressions are correctly represented.
Contextual Notes
Participants note potential limitations in Dirac's treatment, including assumptions about the behavior of y and the validity of the Taylor expansion. The discussion highlights the need for careful consideration of the definitions and conditions under which the limits are taken.