Why Do Zwiebach's Light-Cone Gauge Equations for X^+ and X^- Differ?

[ehrenfest]

Homework Statement

One thing that has been bothering me about the light-cone 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 coordinate-system dependent, how can physics depend on which way we choose our coordinate system?

Homework Equations

The Attempt at a Solution

 

[Jimmy Snyder]

One thing that has been bothering me about the light-cone 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 coordinate-system dependent, how can physics depend on which way we choose our coordinate system?

$X^+$ and $X^-$ are not 'physics', they are coordinates. $L_0^{\perp}$ on the other hand, is physics. Equation (9.77) shows that changing $X^1$ with $-X^1$ has no effect on the physics.

 

[ehrenfest]

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".

 

[Jimmy Snyder]

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".

I'm not sure if this is the answer you are looking for, but here goes anyway. The assumption he made was just above equation (9.61) where he defines the light-cone gauge as imposing conditions (9.27, page 155) with a vector n that gives $n\cdot X = X^+$, i.e. $n = (\frac{1}{\sqrt2}, \frac{1}{\sqrt2}, 0, ...)$.
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 $n\cdot X = X^-$, i.e. $n = (\frac{1}{\sqrt2}, \frac{-1}{\sqrt2}, 0, ...)$? 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.

 

[ehrenfest]

OK.

The sentence above 9.61 says that "Selecting the light-cone 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 light-cone 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 light-cone guage?

 

[Jimmy Snyder]

The sentence above 9.61 says that "Selecting the light-cone 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?

Yes. He was defining the meaning of the phrase 'light-cone gauge'.

If the sentence had read:

"Selecting the light-cone 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 light-cone guage?

Edit - What follows is garbage. I will correct it in a later post.
I don't see how it could. I think this is the first place in the book that the light-cone gauge is mentioned. The index gives no earlier reference. Again, this is a definition, not an observation.

 

[Jimmy Snyder]

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 $n\cdot X = X^-$ as the light-cone gauge, he would be using $n = (\frac{1}{\sqrt2}, \frac{-1}{\sqrt2}, 0, ...)$ and this n is spacelike. I'm sorry for wasting your time in this speculation.

 

[ehrenfest]

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 $n\cdot X = X^-$ as the light-cone gauge, he would be using $n = (\frac{1}{\sqrt2}, \frac{-1}{\sqrt2}, 0, ...)$ and this n is spacelike. I'm sorry for wasting your time in this speculation.

I'm confused? I thought both of the light-cone vectors were light-like i.e. null?

 

[Jimmy Snyder]

I'm confused? I thought both of the light-cone vectors were light-like i.e. null?

I speak of the vector n, it must be timelike or null. I'm not sure what you mean by both of the light-cone vectors.

 

[ehrenfest]

I was thinking that because the light-cone was the boundary between spacelike and time-like 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 that's not right though because equation 2.8 has deltas.

 

[ehrenfest]

I speak of the vector n, it must be timelike or null. I'm not sure what you mean by both of the light-cone vectors.

Can you show me explicitly why is is timelike or null and why the n for X^+ is spaceline?

 

[ehrenfest]

I think it actually comes from even earlier in the book. On page 21, he says that "we will take x^+ to be the light-cone time coordinate" and he admits that this is completely arbitrary in the paragraph above.

Another weird thing, though. On page 27, he says "In light-cone coordinates, p_+ appears together with the light-cone time x^+"

I am not sure what "appears together" means?

 

[Jimmy Snyder]

I think it actually comes from even earlier in the book. On page 21, he says that "we will take x^+ to be the light-cone time coordinate" and he admits that this is completely arbitrary in the paragraph above.

Another weird thing, though. On page 27, he says "In light-cone coordinates, p_+ appears together with the light-cone time x^+"

I am not sure what "appears together" means?

He means that in equation (2.85) the two are factors in the same term, just as $p_0$ and $x^0$ are in equation (2.84).
I think though, that you are confusing light-cone coordinates with the light-cone gauge. The first sentence in section 9.5 on page 160 says:

The light-cone solution of the equations of motion involves using light-cone coordinates to represent the motion of strings, and imposing a set of conditions that defines the light-cone gauge.

Later on the page he explains what 'selecting the light-cone gauge' means.