Equations 11.58 and 11.69 in Zwiebach seem contradictory

[ehrenfest]

Homework Statement

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?

Homework Equations

The Attempt at a Solution

 

[Jimmy Snyder]

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?

Actually, they are distinguished in the text. Above equation (11.58) it says:

In the Lorentz covariant quantization of the point particle, ...

while below equation (11.60) it says:

This is an elegant result, but it is by no means clear that it carries over to our light-cone gauge quantization.

Then at the top of page 199, it says:

Equations (11.65), (11.66), and (11.67) show that $p^-$ does not generate the expected transformations.

He might have added equation (11.69) to the list.

 

[ehrenfest]

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?

 

[ehrenfest]

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?

What am I missing? Is the lc gauge not a constraint like I am thinking of it?

 

[ehrenfest]

jimmy, i promise no more questions for 5 days if you answer this one :)

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?