Steps for

$($a$)$ verify that for Noether theorm hat$(N{)}_{}^{}$ given by

hat$(N{)}_{}^{}=i(hat({)}^{+}de{l}^{}(hat())-(hat())de{l}^{}$hat$({)}^{+})$

the corresponding reduces to the form

hat$(ox)={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{d{k}^3}{(2{)}^3}[hat(a{)}^{+}(k)(hat(a))(k)-$hat$(b{)}^{+}(k)(hat(b))(k)]$

and that, with hat$(H)$ given by

hat$(H)={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{d^{3}k}{(2{)}^3}[hat(a{)}^{+}(k)(hat(a))(k)+$hat$(b{)}^{+}(k)(hat(b))(k)]$

$[(hat(H)),(hat(N_{}))]=0$

$($b$)$ verify equation

hat$(U)()hat()hat(U{)}^{-1}()=e^{-i}$hat$()=$hat$({)}^{\text{'}}$

with

hat$(U)()=e^{ihat(N{)}_{}}$