the remaining three light-cone coordinates $T=(\tilde{n}x)=(t+x_n)/2$, and $\bm{x}_{\perp}=(x_{\perp,1},x_{\perp,2})=-((xa_1),(xa_2))=(\bm{x}\cdot\bm{a}_1,\bm{x}\cdot\bm{a}_2)$. Analogously, the light-cone coordinates of an arbitrary four-vector $v^{\mu}=(v^0,\bm{v})$ will be indicated as $v_-=(nv)=v^0-v_n$, with $v_n=\bm{n}\cdot \bm{v}$, $v_+=(\tilde{n}v)=(v^0+v_n)/2$, and $\bm{v}_{\perp}=(v_{\perp,1},v_{\perp,2})=-((va_1),(va_2))=(\bm{v}\cdot\bm{a}_1,\bm{v}\cdot\bm{a}_2)$. Since we will employ the operator technique, it is convenient to also introduce the momenta operators $P_{\phi}=-i\partial_{\phi}=-(\tilde{n}P)=-(i\partial_t-i\partial_{x_n})/2$, $P_T=-i\partial_T=-(nP)=-(i\partial_t+i\partial_{x_n})$, and $\bm{P}_{\perp}=(P_{\perp,1},P_{\perp,2})=-i(\bm{a}_1\cdot\bm{\nabla},\bm{a}_2\cdot\bm{\nabla})$. These operators are the momenta conjugated to the light-cone coordinates in the sense that the commutator between the operator corresponding to each light-cone coordinate and the associated momentum operator is equal to the imaginary unit (all other possible commutators vanish): $[\phi,P_{\phi}]=[T,P_T]=i$ and $[X_{\perp,j},P_{\perp,k}]=i\delta_{jk}$, which are equivalent to the commutation relations $[X^{\mu},P^{\nu}]=-i\eta^{\mu\nu}$, with $P^{\mu}=i\partial^{\mu}$.

The commutation relations $[X^{\mu},P^{\nu}]=-i\eta^{\mu\nu}$ imply that $[P^{\mu},f(X)]=i\partial_X^{\mu}f(X)$, where $f(X)$ is an arbitrary function of the four-position operator that can be expanded in Taylor series and $\partial_X^{\mu}=\partial/\partial X_{\mu}$. Analogously, it can easily be shown that $\exp[if(X)]P^{\mu}\exp[-if(X)]=P^{\mu}+\partial^{\mu}f(X)$ and then formally that $\exp[if(X)]g(P)\exp[-if(X)]=g(P+\partial f(X))$, where $g(P)$ is a function of the four-momentum that can be expanded in Taylor series \[this identity has to be intended to apply to the Taylor series expansion of the function $g(P)$\]. The same commutation relations imply that $\exp[ig(P)]X^{\mu}\exp[-ig(P)]=X^{\mu}-\partial_P^{\mu}g(P)$ and that $\exp[ig(P)]f(X)\exp[-ig(P)]=f(X-\partial_Pg(P))$, where $\partial_P^{\mu}=\partial/\partial P_{\mu}$ \[as above, this identity has to be intended to apply to the Taylor 