The discussion revolves around equations 11.58 and 11.69 from Zwiebach's text, focusing on potential contradictions between these equations in the context of Lorentz covariant and light-cone gauge quantization. Participants express confusion regarding the use of similar notation for different coordinate systems and the implications of these equations.
The conversation is ongoing, with participants exploring the implications of the equations and seeking clarification on the distinctions between coordinate systems. Some have noted specific text references that may clarify the context, while others express uncertainty about the implications of the light-cone gauge.
There is mention of a potential lack of clarity in the text regarding the use of Greek letters for different types of coordinates, which contributes to the confusion. Participants are also grappling with the justification provided for equation 11.58 and its applicability under different constraints.
Actually, they are distinguished in the text. Above equation (11.58) it says:ehrenfest said:I am confused about equation 11.58.
Doesn't that contradict equation 11.69 which should be -i by equation 11.58?
Is one of them using lc coordinates and the other one using the lc gauge? That is amazingly confusing for me if they are since they are not distinguished in any way?
while below equation (11.60) it says:Zwiebach said:In the Lorentz covariant quantization of the point particle, ...
Then at the top of page 199, it says:Zwiebach said:This is an elegant result, but it is by no means clear that it carries over to our light-cone gauge quantization.
He might have added equation (11.69) to the list.Zwiebach said:Equations (11.65), (11.66), and (11.67) show that [itex]p^-[/itex] does not generate the expected transformations.
ehrenfest said:OK. So the commutation relation 11.58 is true only when those indices are 0,1,2,3 . I still think it is confusing that he uses the same greek letters mu and nu to represent Lorentz covariant coordinates and light-cone coordinates. When I go back to those equation I have no idea if they are true for lorentz coordinates, and/or lc coordinates, and/or the lc gauge.
Another thing that bothers me is that it seems like Zwiebach's only justification for 11.58 is that "it seems reasonable". We know it is true for the spatial part from QM, but how can you just write down an equation because it "seems reasonable"?
EDIT: wait. so apparently 11.58 is true for lc coordinates as he discusses in the paragraph underneath. in that case i just do not understand why it would not be true in the lc gauge because the lc gauge uses light cone coordinates and it is only a constraint on the lc coordinates so if 11.58 is true for lc coordinates shouldn't it be true for the constrained lc coordinates in the lc gauge?