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A scalar field has $\delta {\varphi }_a=0$ for a pure change of spacetime argument. With $\delta x^k={ϵ}^k$ and $\delta x^0=0$, its variation at a fixed argument is $\bar{\delta }{\varphi }_a=-{ϵ}^k{\partial }_k{\varphi }_a$. The supplied Noether charge becomesSince $\delta Q/\delta {\pi }_a=-{ϵ}^k{\partial }_k{\varphi }_a$, solution and antisymmetry of the Poisson bracket giveCanonical quantization replaces the Poisson bracket by a commutator divided by $i\hslash$:With this sign convention one may use $U=e^{-i\hat{Q}/\hslash }$ and ${\hat{\varphi }}_a^{\prime }=U^{†}{\hat{\varphi }}_{a}U$. Because $Q=-{ϵ}^k{P}_k$, the negative shift of the field argument follows with the required sign.

For $\delta x^0={ϵ}^0$ the Noether charge is $Q=-{ϵ}^{0}H$, with the Hamiltonian density $\mathcal{H}={\pi }_a{v}_a-\mathcal{L}(\varphi ,v,\nabla \varphi )$ and $v_a={\dot{\varphi }}_a$. Invertibility of the momentum-velocity relation allows $v_a=v_a(\varphi ,\pi ,\nabla \varphi )$. When taking a functional derivative at fixed $\varphi$, the implicit dependence of $v$ cancels in the Legendre transform:It follows thatThis proves the same generator relation for time translations. In a constrained theory the assumed inversion fails, so the appropriate reduced variables or Dirac brackets are needed instead. That is precisely the issue encountered for the Proca field in Question 4.

For a complex scalar field with global $U(1)$ invariance and Lagrangian $\mathcal{L}=\frac{1}{2}|{\partial }_t\varphi {|}^2-\frac{1}{2}|\nabla \varphi {|}^2-U(|\varphi |)$, a Q-ball is a localized phase-rotating configuration with conserved Noether charge $Q=\int \mathrm{Im}(\bar{\varphi }{\varphi }_t)$. At fixed charge its real profile can minimizeSuch solutions require a suitable potential and charge range; the complex nature of a field alone does not ensure existence. A common condition is ${\inf }_{f>0}2U(f)/f^2<U^{\prime \prime }(0)$.

A scalar-field shift symmetry acts by $\varphi (x)\mapsto \varphi (x)+c$ for constant $c$. Its Noether charge is the spatial integral of the canonical momentum, up to the normalization of $c$, and interactions invariant under the symmetry contain derivatives of the scalar field.

This nonnegative potential has its only vacuum at $s=0$, since $U(s)=s[(s-1/2{)}^2+3/4]$. For the complex kinetic normalization $\frac{1}{2}|\partial \varphi {|}^2$, the small-amplitude mass squared is $2U^{\prime }(0)=2$, whereas ${\min }_{s>0}2U(s)/s=3/2$. The radial Q-ball equation has a positive localized profile for frequencies $3/2<{\omega }^2<2$, for example in three spatial dimensions. It is a standard non-topological soliton potential: the effective profile potential $U(f^2)-{\omega }^2{f}^2/2$ is negative somewhere, while its quadratic coefficient near zero is positive. The conserved Noether charge, not vacuum winding, supplies the stabilization mechanism.

The odd particle variables form an even antisymmetric tensor through their bilinear. Adding it to orbital angular momentum gives a conserved Lorentz Noether charge. The particle and odd-variable equations make the two time derivatives cancel. After spinning-particle Dirac quantization, the bilinear becomes the spinor Lorentz generator rather than an ordinary commuting classical spin vector.

Translations and Lorentz transformations give Noether charges ${\mathcal{P}}_m=\int P_{m}d\sigma$ and ${\mathcal{J}}_{mn}=\int (X_m{P}_n-X_n{P}_m)d\sigma$. Their time derivatives are boundary fluxes, which vanish for a closed string or a free-ended open string.

For a smooth Skyrme model field $U:{\mathbb{R}}^3\to SU(2)$ with $U(\infty )=1$, compactification gives a map $S^3\to S^3$. With $L_i=U^{†}{\partial }_{i}U$, its degree of a map between oriented manifolds is $B=-(24{\pi }^2{)}^{-1}\int {ϵ}_{ijk}\mathrm{tr}(L_i{L}_j{L}_k)d^{3}x$, with sign chosen so the standard decreasing hedgehog profile has $B=1$. The integrand is the pullback of the normalized volume form on $SU(2)$. Hence it is integer-valued and unchanged by smooth finite-energy homotopies. In the rational map approximation for Skyrmions, a degree-$N$ angular map and profile $f(0)=\pi ,f(\infty )=0$ give $B=-(2N/\pi )\int f^{\prime }{\sin }^{2}fdr=N$. This is a topological conservation law, not the Noether charge of isospin.