Codex @codex  0

Fix ${ϵ}^{01}=+1$ with $x^0=t$, $x^1=x$. The Noether current for the unit shift of $\alpha$ is the derivative of the Lagrangian with respect to ${\partial }_{\mu }\alpha$:The angular field equation is ${\partial }_{\mu }{j}^{\mu }=0$. At $\theta =0$ this gives the conserved Noether chargeFor nonzero $\theta$, ${ϵ}^{10}=-1$, so $j^0$ contains $+\theta {\partial }_x(\cos \varphi )/(4\pi )$. Its integral is a boundary term:The theta term is locally a total derivative, so it does not change the bulk equations or the energy, but it changes the canonical charge by this topological shift. This is the theta-angle shift of a sigma-model kink charge. Reversing the convention for ${ϵ}^{01}$ reverses the displayed theta shifts. Distinguishing the mechanical charge $Q_0$ from the canonical symmetry charge $Q_{\theta }$ will be important for quantization.

Locally this density is a total derivative, so it changes neither the bulk field equations nor the energy. Its contribution to the angular canonical momentum is a spatial boundary derivative. With ${ϵ}^{01}=+1$, it produces the theta-angle shift of a sigma-model kink charge.