Nambu – Goto action, especially for the benefit of readers without much knowledge of string theory. Experts can safely skip this section. See [@12; @13] for more details.\
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Let us in fact consider the bosonic $p-$brane with $p\ge 2$. Its action is defined in terms of an embedding $X:\;M\to T;\;y\mapsto X^\mu(y)$ from a $p-$dimensional worldsheet $M$ to a $D-$dimensional target space $T$ and a target space metric tensor field $X\mapsto \eta_{\mu\nu}(X)$ of Minkowskian signature $(-1,1,..,1)$ (the case of Euclidean signature is treated in the companion paper [@11]). Here we take the worldsheet coordinates $y^\alpha,\;\alpha=0,..,p-1$ to be dimension – free while the target space coordinates $X^\mu,\mu=0,..,D-1$ have dimension of length. The bosonic $p-$brane action is nothing else than the volume of $M$ as measured by $\eta$, that is, \[2.1\] S\[X\]:=-\_M d\^py Here $\alpha'$ is called the $p-$brane tension and its has dimensions such that $\hbar\alpha'=:\ell_s^p$ has dimensions of cm$^p$. In order that $X$ be an embedding, the vector fields $\partial/\partial 
y^\alpha$ must be everywhere linearly independent on $M$. Furthermore, the pull-back metric on $M$ given by $X^\ast \eta$ is supposed to have everywhere Minkowskian signature.

We proceed to the canonical analysis. We assume that $M=\Rl\times \sigma$ where $\sigma$ is a 