Canonical Phase Space for the Flat String

Fix a constant-$\tau$ slice of the flat-string worldsheet. Its spatial domain is $J=\mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$ for a closed string and the interior $J=(0,\pi )$ for an open string. The unreduced canonical data are smooth maps $X:J\to {\mathbb{R}}^D$ and smooth target-covector-valued spatial densities ${\mathrm{\Pi }}_{\mu }(\sigma )d\sigma$, with target indices raised and lowered by $\eta$. Here unreduced means that equations of motion, metric constraints and boundary conditions have not yet been imposed on these equal-time data. Under an orientation-preserving spatial reparameterization $\tilde{\sigma }=f(\sigma )$, ${\tilde{\mathrm{\Pi }}}_{\mu }(\tilde{\sigma })={\mathrm{\Pi }}_{\mu }(\sigma )\mathrm{/}{f}^{\mathrm{\prime }}(\sigma )$ and $\tilde{X}(\tilde{\sigma })=X(\sigma )$.

Use units $\mathrm{\hslash }=c=1$, dimensionless $\tau ,\sigma$, and $[X]=L$. Normalize the canonical pairing by ${\mathrm{\Theta }}_{(X,\mathrm{\Pi })}(\delta X,\delta \mathrm{\Pi })={\int }_J{\mathrm{\Pi }}_{\mu }\delta X^{\mu }d\sigma$, with no additional numerical or tension factor; it has units of action, hence is dimensionless in these units. Thus $[{\mathrm{\Pi }}_{\mu }]=L^{-1}$, and ${\int }_J{\mathrm{\Pi }}_{\mu }d\sigma$, when finite, has momentum units. On the open interior this pairing is evaluated on variations with compact support. For an adopted Lagrangian density $\mathcal{L}$ depending on ${\mathrm{\partial }}_{\tau }X$ and no higher time derivatives, the physical identification of this covector is the Legendre prescription ${\mathrm{\Pi }}_{\mu }=\mathrm{\partial }\mathcal{L}\mathrm{/}\mathrm{\partial }({\mathrm{\partial }}_{\tau }{X}^{\mu })$; the kinematic bracket itself does not choose $\mathcal{L}$.

For a differentiable functional $F(X,\mathrm{\Pi })$, its variational derivatives are the smooth coefficient fields in $\delta F={\int }_J[(\delta F\mathrm{/}\delta X^{\mu })\delta X^{\mu }+(\delta F\mathrm{/}\delta {\mathrm{\Pi }}_{\mu })\delta {\mathrm{\Pi }}_{\mu }]d\sigma$. Restrict to functionals with these derivatives periodic on the circle, or compactly supported in the open interior, so the following integral exists. Define

$\{F,G\}:={\int }_J(\frac{\delta F}{\delta X^{\mu }}\frac{\delta G}{\delta {\mathrm{\Pi }}_{\mu }}-\frac{\delta F}{\delta {\mathrm{\Pi }}_{\mu }}\frac{\delta G}{\delta X^{\mu }})d\sigma \mathrm{.}$

For the closed string the fields and test functions are periodic; for the open-string interior, test functions used in smeared functionals have compact support in $(0,\pi )$, so integrations by parts have no endpoint terms. Before suppressing the action unit, $[\{F,G\}]=[F][G]\mathrm{/}[\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}]$; in the adopted units the bracket has dimension $[F][G]$. This is the classical canonical bracket convention, not a quantum commutator or a positive inner product.

Flat Bosonic String Model