= Solution
A <scalar field> has $\delta\phi_a=0$ for a pure change of spacetime argument. With $\delta x^k=\epsilon^k$ and $\delta x^0=0$, its variation at a fixed argument is $\bar\delta\phi_a=-\epsilon^k\partial_k\phi_a$. The supplied <Noether charge> becomes
$$
Q=-\epsilon^kP_k,\qquad P_k=\int d^3y\,\pi_b(\mathbf y)\partial_k\phi_b(\mathbf y).
$$
Since $\delta Q/\delta\pi_a=-\epsilon^k\partial_k\phi_a$, <5/a/solution> and antisymmetry of the <Poisson bracket> give
$$
\boxed{\bar\delta\phi_a=\{\phi_a,Q\}_{\rm PB}.}
$$
Canonical quantization replaces the <Poisson bracket> by a commutator divided by $i\hbar$:
$$
\boxed{\bar\delta\hat\phi_a=\frac1{i\hbar}[\hat\phi_a,\hat Q].}
$$
With this sign convention one may use $U=e^{-i\hat Q/\hbar}$ and $\hat\phi_a'=U^\dagger\hat\phi_aU$. Because $Q=-\epsilon^kP_k$, the negative shift of the field argument follows with the required sign.
Back to article page
