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For a free massless real scalar field with wave speed set to one and metric $(+,-)$, define $\mathcal{E}=({\varphi }_t^2+{\varphi }_x^2)/2$ and $p=-{\varphi }_t{\varphi }_x$. Its symmetric stress-energy tensor is $T^{\mu \nu }=\left(\begin{array}{cc}\mathcal{E}& p\\ p& \mathcal{E}\end{array}\right)$. The Lorentz current has density $tp-x\mathcal{E}$ and flux $t\mathcal{E}-xp$. Its conserved charge is the displayed expression when boundary flux vanishes. In one spatial dimension it generates boosts, not spatial rotations; there is no independent purely spatial angular-momentum component. Its conservation says that the center of energy moves uniformly. This is a symmetry of the wave-field model and need not survive boundaries or additional material effects.

Lorentz invariance gives the Lorentz currentAn infinitesimal rotation through angle $\theta$ around the $z$-axis hascorresponding, up to the stated index convention, to the only nonzero components ${\omega }^{12}=-{\omega }^{21}=\theta$. Its conserved angular momentum is

A continuous symmetry is a family of transformations depending continuously on parameters for which the action is unchanged, possibly up to a spacetime boundary term. For an infinitesimal field variation $\delta \varphi =ϵ\Delta \varphi$, a Noether current $j^{\mu }$ obeyswhen the Euler-Lagrange field equation holds. Its Noether chargeis conserved provided the flux ${\int }_{S_{\infty }}{j}^i{n}_{i}dS$ vanishes.

Noether theorem gives the implication from a differentiable global variational symmetry to an on-shell conserved current. A conserved current gives a conserved charge only under suitable boundary and convergence conditions. Conversely, a conserved charge generates a continuous symmetry through Poisson brackets classically or a commutator quantum mechanically when a regular Hamiltonian formulation exists. Currents can be changed by identically conserved improvement terms, and inverse Noether statements require regularity and the exclusion of such trivial currents, so the three notions are related but not literally in one-to-one correspondence.

For spacetime translations, the canonical stress-energy tensor isForraising the second index gives the symmetric tensorTranslation invariance gives four conserved currents $T^{\mu \nu }$, one for each fixed $\nu$, and the four conserved charges are the four-momentum$P^0$ is the energy and $\mathbf{P}$ is the spatial momentum.

The Lorentz transformation variation of a scalar is generated solely by its argument. The corresponding Lorentz current isUsing ${\partial }_{\mu }{T}^{\mu \nu }=0$ and symmetry of $T^{\rho \sigma }$,In particular, conservation of the boost chargeimpliesThe quantity denoted $V^i$ in the question is therefore the conserved momentum component $P^i$, assuming the same vanishing boundary flux.

An infinitesimal Lorentz transformation is $\delta X^{\mu }={\omega }^{\mu }{}_{\nu }{X}^{\nu }$ with ${\omega }_{\mu \nu }=-{\omega }_{\nu \mu }$. Applying Noether theorem to this continuous symmetry gives the conserved Lorentz currentIts Noether charge is $M^{\mu \nu }={\int }_0^{\pi }d\sigma J_{\tau }^{\mu \nu }$. Substituting the open-string mode expansion and using orthogonality of the cosine modes yieldsThe first term is orbital angular momentum; the sum is the contribution of the string oscillators.

Use $x^{\mu }=(t,x)$ and Minkowski metric ${\eta }^{\mu \nu }=\mathrm{diag}(1,-1)$, with the wave speed equal to one. Regarding $\varphi$ as a scalar field, the symmetric canonical stress-energy tensor isIt obeys ${\partial }_{\mu }{T}^{\mu \nu }=0$ on the wave equation. For a scalar field the intrinsic spin current vanishes. The total orbital Lorentz current and its integrated angular-momentum tensor areBecause there is only one spatial dimension, its only independent possibly nonzero component iswhere $P=\int pdx$. The corresponding flux is ${\mathcal{J}}^{101}=t\mathcal{E}-xp$. Symmetry and conservation of $T$ giveThus $J^{01}$ is conserved provided the boundary flux vanishes and the integrals exist. For example, localized wave packets on an infinite line have this property. One can also check directly that $d\int x\mathcal{E}dx/dt=P$, so the two terms in $J^{01}$ have cancelling derivatives.

The significance is the boost charge of a one-dimensional scalar field. There is no independent spatial rotation or ordinary spatial angular-momentum component in one dimension. Instead, the action has the mathematical Lorentz invariance of a massless wave field with invariant speed $c_s=1$, and $J^{01}$ generates its boost. Its conservation says that the center of energy moves uniformly. This is a symmetry of the ideal wave-field model, rather than an assertion of fundamental vacuum Lorentz symmetry for a material rod. Fixed boundaries can break it, and a general solution with nonvanishing boundary flux need not conserve the integrated charge.