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I'm reading Zweibach's introduction to string theory, and don't understand one of his claims.

He defined the mass shell to be the set of points in momentum space s.t. $p^2+m^2 = 0$. Then the physical mass shell is the subset of points where $E=p^0>0$. He claims that in light cone coordinates this may be parameterised by the transverse momenta $\vec{p}_T$ and the light cone momenta $p^+$ for which $p^+>0$.

Why the condition $p^+>0$? Surely $p^0>0$ doesn't immediately give us this? Shouldn't there be some $p^1$ dependence in the mix somewhere?

Many thanks in advance!

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up vote 2 down vote accepted

Zwiebach explains the inequality $p^{\pm}>0$ e.g. in Section 2.5 of his book A First course in String Theory:

$$\sqrt{2} p^{\pm}~=~p^0\pm p^1~=~\sqrt{\vec{p}\cdot\vec{p}+(mc)^2}\pm p^1~>~|\vec{p}|\pm p^1~\geq~0~ $$

if $m>0$.

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Thanks hadn't spotted that! – Edward Hughes Aug 20 '12 at 16:51

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