What is the correct definition of the energy of a string ?

I suddenly get confused with the definition of the energy of a string. Considering, for instance, a bosonic open string in the light-cone gauge, We have $H = p^-=p_+$, the hamiltonian, and we have $p^0$, which appears in the expansion of the string in light-cone gauge $X^o(\sigma, \tau) = x^0 + \frac{p^0}{p^+}\tau + ...$.

a) In Zwiebach's First course in string theory (1), there is a calculus of the entropy of a string, which begins (Chapter $16.2$ p $354$) by a function partition for a non relativistic string with fixed points (the quantum violin string). For a given energy $E = N \hbar w_0$, where $N$ is the level, the entropy is computed (this is direcly linked to the number of partitions p(n)). More precisely, an expression for the entropy, $\sim T$, and the energy ,$\sim T^2$, are obtained, se we get a $S \sim \sqrt{E}$, and plugin the $E \sim N$ expression, we get a $S \sim \sqrt{N}$.

"Suddenly", in chapter $16.3$ page $361$, it is said that "we now return to relativist strings that carry no spatial momentum, and this happens if the open string endpoints end on a D-0 brane, so the energy levels are given by the rest mass of its quantum states". Of course, we have $m^2 \sim N$ (for large $N$), so $m\sim \sqrt{N}$, so if the energy ($p^0 ?$) is identified with $m$, we have $S \sim E$
Now, if we take the hamiltonian $H$, for large $N$, we have $H \sim N$, so $S \sim \sqrt{H}$

So, this confused me, because I have $2$ law expressions for the entropy, depending on the definition of the energy.

b) An other problem, is saying that the energy of the string is proportionnal to its length. If we look at $H$, we have $H=p^- \sim \frac{1}{l}$ (with zero trans verse momenta $p^i$), while $p^+ \sim \frac{l}{\alpha'}$. So, may I write that the correct energy is $p^o \sim (p^+ + p^-)?$

This post imported from StackExchange Physics at 2014-03-12 15:51 (UCT), posted by SE-user Trimok