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Zwiebach string theory quesiton 
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#1
Oct1207, 12:37 AM

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1. The problem statement, all variables and given/known data
One thing that has been bothering me about the lightcone gauge that Zwiebach uses is that we have equations such as 9.63 and 9.71 in which X^+ and X^ are given by completely different expressions. I am not sure why this makes sense physically since we can interchange X^+ and X^ just by moving to coordinate system in which X^1' = X^1. So, shouldn't it be arbitrary which coordinate is X^+ and X^ and if it is arbitrary and coordinatesystem dependent, how can physics depend on which way we choose our coordinate system? 2. Relevant equations 3. The attempt at a solution 


#2
Oct1207, 03:38 AM

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#3
Oct1207, 01:14 PM

P: 1,996

OK. I see your point.
But I guess my point is that there is no reason why the light cone +'s and 's in 9.63 and 9.71 should not be reversed. Somewhere Zwiebach must have just made some assumption that arbitrarily distinguished X^ from X^+ . Obviously he is free to do this but I am just looking for the point where X^+ and X^ "diverge". 


#4
Oct1207, 01:53 PM

P: 2,179

Zwiebach string theory quesiton
Edit  What follows is garbage. I will correct it in a later post. So what would happen if he went the other way and imposed the conditions (9.27) with a vector n that gives [itex]n\cdot X = X^[/itex], i.e. [itex]n = (\frac{1}{\sqrt2}, \frac{1}{\sqrt2}, 0, ...)[/itex]? I don't know because I haven't worked it out. However, the easy guess is that nothing would happen except for swapping + and  signs here and there. 


#5
Oct1207, 02:31 PM

P: 1,996

OK.
The sentence above 9.61 says that "Selecting the lightcone gauge means imposing the conditions (9.27) with a vector n^mu that gives nX = X^+" So, is this where he arbitrarily chose a + sign instead of a minus sign? If the sentence had read: "Selecting the lightcone gauge means imposing the conditions (9.27) with a vector n^mu that gives nX = X^" would that have contradicted something he said previously about the lightcone guage? 


#6
Oct1207, 02:58 PM

P: 2,179

I don't see how it could. I think this is the first place in the book that the lightcone gauge is mentioned. The index gives no earlier reference. Again, this is a definition, not an observation. 


#7
Oct1207, 04:28 PM

P: 2,179

I'm sorry Ehrenfest, but my responses have been wrong. Note that on page 150 he discusses the properties of the vector n. It needs to be either timelike, or null so that the string is spacelike. In order to define [itex]n\cdot X = X^[/itex] as the lightcone gauge, he would be using [itex]n = (\frac{1}{\sqrt2}, \frac{1}{\sqrt2}, 0, ...)[/itex] and this n is spacelike. I'm sorry for wasting your time in this speculation.



#8
Oct1207, 04:45 PM

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#9
Oct1207, 04:53 PM

P: 2,179




#10
Oct1207, 05:33 PM

P: 1,996

I was thinking that because the lightcone was the boundary between spacelike and timelike vectors, every vector on the light cone must be null.
Is that a contravariant expression for n? If you plug that into equation 2.8, it seems like it is null as well as the + version of n. Maybe thats not right though because equation 2.8 has deltas. 


#11
Oct1907, 07:29 PM

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#12
Oct2007, 12:23 PM

P: 1,996

I think it actually comes from even earlier in the book. On page 21, he says that "we will take x^+ to be the lightcone time coordinate" and he admits that this is completely arbitrary in the paragraph above.
Another weird thing, though. On page 27, he says "In lightcone coordinates, p_+ appears together with the lightcone time x^+" I am not sure what "appears together" means? 


#13
Oct2007, 05:02 PM

P: 2,179

I think though, that you are confusing lightcone coordinates with the lightcone gauge. The first sentence in section 9.5 on page 160 says: 


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