# Noether's Theorem in Non-Local Field Theories
## 3 Non-Local Charged Scalar Field  
We consider an example of a non-local charged scalar field described by the Lagrangian  
\[\mathcal{L}=\frac{1}{2}\phi^{*}\left(i\partial_{t}-\sqrt{-\Delta+m^{2}}\right) \phi+\text{c.c.} \tag{17}\]  
The particles follow a relativistic dispersion law \(E(\mathbf{p})=\sqrt{\mathbf{p}^{2}+m^{2}}\). Because of the absence of negative-frequency solutions, the particles do not have antiparticles, which leads to a violation of the Lorentz invariance. After quantization of the system (17), the field commutator \([\phi(x),\phi(y)]\) does not disappear at space-like intervals \((x-y)^{2}<0\), so signals can propagate at speeds above the speed of light, ultimately violating causality. The condition \([\phi(x),\phi(y)]=0\) for \((x-y)^{2}<0\) is also required for the Lorentz invariance of the \(T\) product of the field operators. The system described by [PERSON] (17) is interesting from methodological and historical points of view, since the corresponding evolution equation was considered in the past as a possible relativistic generalization of the Schrodinger equation.  
Let us check whether CPT invariance holds in the non-local field theory defined by (17). First we consider the charge-conjugation operation, C. In the momentum space given by \(p^{\alpha}=-i\
abla=-(p^{\alpha})^{*}\), with \(\alpha=1,2,3\), we replace the particle's momenta in (17) with the generalized momenta, \(p_{\mu}\to p_{\mu}-eA_{\mu}\). Equating the functional derivative \(\delta\mathcal{L}/\delta\phi^{*}\) to zero, we obtain the evolution equation in an external electromagnetic field:  
\[(i\partial_{t}-eA_{0})\phi=\sqrt{(\mathbf{p}-e\mathbf{A})^{2}+m^{2}}\,\phi. \tag{18}\]  
For the complex conjugate scalar field, one has  
\[(i\partial_{t}+eA_{0})\phi^{*}=-\sqrt{(\mathbf{p}-e\mathbf{A})^{2}+m^{2}}\, \phi^{*}. \tag{19}\]  
Together with the sign reversal of the charge \(e\) in Equation (19), a negative sign appears at the root. Obviously, the charge-conjugation symmetry is broken. Violation of the C symmetry means that the properties of a particle and its corresponding antiparticle are different or, as in our case, the corresponding antiparticles do not exist.  
One can easily check that the Lagrangian given in (17) is invariant under the parity transformation, P: \(\phi(t,\mathbf{x})\to\phi(t,-\mathbf{x})\). By the same analysis, one can check that the time-reversal symmetry, T: \(\phi(t,\mathbf{x})\to\phi^{*}(-t,\mathbf{x})\), is conserved as well. Thus, the Lagrangian of (17) is symmetric under P and T transformations; whereas, the C symmetry is broken. The combined CPT symmetry is therefore broken, which is consistent with the fact that the theory is non-local.  
The Lagrangian expressed in (17) is explicitly invariant under global phase rotations of \(\phi\), which may imply the existence of a conserved vector current. The Lagrangian given in (17) is also explicitly invariant under space-time translations and three-dimensional rotations. We thus expect the existence of conserved energy-momentum and angular momentum tensors. The dispersion law takes a relativistic form; therefore, the field theory of (17) is apparently invariant under boost transformations. This symmetry is, however, implicit, and we do not discuss its consequences here. We thus restrict ourselves to the case of \(\epsilon^{0\alpha}=0,\epsilon^{\alpha\beta}\
eq 0\).  
We will work in terms of a power series over the derivatives. Expanding \(\mathcal{L}\), one can rewrite it as follows:  
\[\mathcal{L}=\frac{1}{2}\left(\phi^{*}i\partial_{t}\phi-\sum_{l=0}^{\infty}f_{l }(m)\phi^{*}\Delta^{l}\phi\right)+\text{c.c.}, \tag{20}\]  
where  
\[f_{l}(m)=(-1)^{l}\frac{\Gamma(\frac{3}{2})}{l!\ \Gamma(\frac{3}{2}-l)\ m^{2l-1}}, \tag{21}\]  
such that  
\[\sum_{l=0}^{\infty}f_{l}(m)x^{l}=\sqrt{m^{2}-x}. \tag{22}\]