# Noether's Theorem in Non-Local Field Theories
## 3 Non-Local Charged Scalar Field
### Vector Current  
Following the rules listed in Appendix A, we find the spatial components of the vector current as follows:  
\[ \begin{split}\mathfrak{J}^{\alpha}=&\frac{i}{2}\sum_{l= 1}^{\infty}f_{l}(m)\sum_{k=1,3,5,\ldots}^{2l-1}\,(\partial_{\alpha_{k-1}} \ldots\partial_{\alpha_{1}}\phi^{*})\delta^{\alpha_{1}\alpha_{2}}\ldots\delta^ {\alpha_{k+1}}\ldots\delta^{\alpha_{2l-1}\alpha_{2}}\partial_{\alpha_{k+1}} \ldots\partial_{\alpha_{2l}}\phi\\ -&\frac{i}{2}\sum_{l=1}^{\infty}f_{l}(m)\sum_{k=2,4,6, \ldots}^{2l}\,(\partial_{\alpha_{k-1}}\ldots\partial_{\alpha_{1}}\phi^{*}) \delta^{\alpha_{1}\alpha_{2}}\ldots\delta^{\alpha_{k-1}\alpha}\ldots\delta^{ \alpha_{2l-1}\alpha_{2}}\partial_{\alpha_{k+1}}\ldots\partial_{\alpha_{2l}} \phi\\ &+\text{c.c.},\end{split}  \tag{29}\]  
where \(f_{l}(m)\) is given by Equation (21) and \(\delta\phi=-i\phi\) and \(\delta\phi^{*}=i\phi^{*}\) for the \(U(1)\) symmetry group. Let us write Equation (29) in the lowest-order approximation. Equation (21) yields \(f_{1}(m)=-1/(2m)\). The space-like component of the vector current \(\mathfrak{J}^{\sigma}\) reduces to the standard expression  
\[\mathfrak{J}^{\alpha}=\frac{1}{2m}\phi^{*}i\overset{\leftrightarrow}{ \partial}\phi+\ldots. \tag{30}\]  
By performing contractions of the indices and with the aid of Equation (A5) from Appendix A, we obtain  
\[ \begin{split}\mathfrak{J}^{\alpha}=&-\frac{i}{2} \sum_{l=1}^{\infty}f_{l}(m)\sum_{k=1,3,5,\ldots}^{2l-1}\,(\triangle^{(k-1)/2} \phi^{*})\,\triangle^{l-(k+1)/2}\,\partial^{\alpha}\phi\\ &+\frac{i}{2}\sum_{l=1}^{\infty}f_{l}(m)\sum_{k=2,4,6,\ldots}^{2 }\,(\triangle^{(k-2)/2}\partial^{\alpha}\phi^{*})\,\triangle^{l-k/2}\,\phi+ \text{c.c.}\end{split}  \tag{31}\]  
The sum of the first two terms is real, so adding the complex conjugate expression doubles the result. After some simple algebra and with the use of Equation (A7), we obtain  
\[\mathfrak{J}^{\alpha}=\phi^{*}i\mathcal{D}^{\alpha}\phi, \tag{32}\]  
where  
\[\mathcal{D}^{\alpha}=\frac{\overset{\leftrightarrow}{\partial}\alpha}{\sqrt{ m^{2}-(\overset{\leftrightarrow}{\triangle})}+\sqrt{m^{2}-(\overset{ \rightarrow}{\triangle})}}. \tag{33}\]  
The detailed derivation of Equation (32) is given in Appendix B. In terms of the four-dimensional operator \(i\mathcal{D}^{\sigma}\equiv(1,i\mathcal{D}^{\alpha})\), the four-dimensional vector current can be written as  
\[\mathfrak{J}^{\sigma}=\phi^{*}i\mathcal{D}^{\sigma}\phi. \tag{34}\]  
It is useful to rewrite the vector current in the momentum space. By substituting the plane waves \(\phi^{*}(x)\sim e^{i\mathcal{p}^{\prime}x}\) and \(\phi(x)\sim e^{-ipx}\) with momenta \(p^{\prime}\) and \(p\) into Equation (34) and omitting the exponential factors from the final expression, we obtain  
\[\mathfrak{J}^{\sigma}(p^{\prime},p)=\left(1,\frac{\mathbf{p}^{\prime}+\mathbf{ p}}{E(\mathbf{p}^{\prime})+E(\mathbf{p})}\right). \tag{35}\]  
On the mass shell, the vector current is conserved:  
\[\partial_{\sigma}\mathfrak{J}^{\sigma}=0. \tag{36}\]The variational derivative of the action functional \(S=\int d^{4}x\mathcal{L}\) with respect to the vector field \(A_{\sigma}(x)\),  
\[\mathfrak{J}^{\sigma}(x)=-\frac{\delta S}{\delta A_{\sigma}(x)}, \tag{37}\]  
introduced into \(\mathcal{L}\) with the use of minimal substitution, is associated for \(A_{\sigma}(x)=0\) with a vector current. Current (37) is defined off the mass shell and it coincides with the Noether current (34) on the mass shell, as shown in Appendix C. Multiplying (35) with \((p^{\prime}-p)_{\sigma}\) yields the result  
\[(p^{\prime}-p)_{\sigma}\mathfrak{J}^{\sigma}(p^{\prime},p)=G^{-1}(p^{\prime}) -G^{-1}(p), \tag{38}\]  
where \(G(p)=(p^{0}-\sqrt{m^{2}+\mathbf{p}^{2}})^{-1}\) is the particle propagator. This equation can be recognized as the Ward identity. The field \(\phi_{\mathrm{s}}(x)\), which behaves like a true scalar under Lorentz transformations, may be defined by the equation \(\phi(x)=\left(i\partial_{t}+\sqrt{-\Delta+m^{2}}\right)^{1/2}\phi_{\mathrm{s} }(x)\).