Understanding Zwiebach's Problem 12.8: Case Analysis of m,n>0 and m≠n

[ehrenfest]

Homework Statement

Why does Zwiebach say the we only look at the case where m,n>0 and m not equal to n in this problem?

Homework Equations

The Attempt at a Solution

 

[nrqed]

Homework Statement

Why does Zwiebach say the we only look at the case where m,n>0 and m not equal to n in this problem?

Homework Equations

The Attempt at a Solution

Are you sure it's problem 12.8??

 

[ehrenfest]

Yes. Look at the attachment when it is approved.

 

[nrqed]

Yes. Look at the attachment when it is approved.

then it's a different edition than my books (the problem is called Reparametrizations generated by Virasoro operators in my book and there is no mention of m or n indices)

 

[ehrenfest]

That is what it is called in my edition also. The m and n indices arise when you verify that the generators form a Virasoro algebra. They are not in the problem but they are in the attached solution.

 

[ehrenfest]

The attachment was approved.

 

[Jimmy Snyder]

His wording in the problem is slightly off. He asks:

Show that the generators of these reparameterizations form a subalgebra of the VIrasoro algebra.

Well, that is as trivial a problem as you could want since all the operators L_m - L_{-m} are in the Virasoro algebra. What he obviously means is:

Show that the generators of these reparameterizations form a proper subalgebra of the VIrasoro algebra.

The set
$$\{L_m^{\perp} - L_{-m}^{\perp}: m = 1, 2, 3, ...\}$$
is not the subalgebra, it is a generating set. The subalgebra he is interested in is the smallest algebra that contains this set. He wants to show that it excludes something, anything, in V, the Virasoro algebra. He can do this in two steps. First show that the product of any two generators is in the vector space span of the generators. Then show that there is a Virasoro operator that is not in the span. To do this, there is no need to use the larger generating set
$$\{L_m^{\perp} - L_{-m}^{\perp}: m = 0, \pm 1, \pm 2, \pm 3, ...\}$$
because the generated algebra is exactly the same. He does need to cover the case m = n, but it is quite trivial, I suppose he forgot to mention it.