# Noether's Theorem in Non-Local Field Theories
## 2 Symmetries and the Conserved Currents  
Any observable quantity can be expressed in terms of fields and their certain combinations. In general, the fields that appear in the Lagrangian belong to a representation space of the internal symmetry group. Linear transformations of the fields related to the internal symmetry group do not affect physical quantities, which is the case considered in the present paper. Thus, for infinitesimal transformations related to the internal symmetries, one can write the transformation matrix as follows:  
\[U(\omega)=1-i\omega^{a}\mathbb{T}^{a}, \tag{2}\]  
where the \(\omega^{a}\) are a set of infinitesimal real parameters and the \(T^{a}\) are generators of group transformations. If the matrix \(U\) is unitary, then the \(T^{a}\) are Hermitian matrices. For the \(U(1)\) symmetry group, \(T^{a}=1\), and for \(SU(2)\), the \(T^{a}\) are the Pauli matrices. Along with the internal symmetry of a physical system, in the general case, one must consider the existence of external symmetries that are related to the invariance of physical quantities with respect to translations and the Lorentz transformations. Invariance under space-time translations leads to energy-momentum conservation, whereas the Lorentz invariance gives rise to the conservation of angular momentum. For an infinitesimal element of the Lorentz group, coordinate transformations can be realized by means of the matrix \(a_{v}^{\mu}=\delta_{v}^{\mu}+\varepsilon_{v}^{\mu}\), where \(\varepsilon^{\mu\
u}\) is an infinitesimal antisymmetric tensor. This implies that the infinitesimal Lorentz transformation matrix in the representation space of the field can be written in the most general form as follows:  
\[S(a)=1-\frac{i}{2}\varepsilon_{\mu\
u}\Sigma^{\mu\
u}, \tag{3}\]  
where \(\Sigma^{\mu\
u}\) is a matrix defined by the transformation properties of the field. Thus, the complete transformations of the coordinates and the field corresponding to internal and external symmetries can be expressed in matrix notation as follows:  
\[x^{\prime} = ax+b, \tag{4}\] \[\Psi^{\prime}(x^{\prime}) = U(\omega)S(a)\Psi(x), \tag{5}\]  
where the notation used in Equation (4) dictates a particular order of the transformations, namely, translation is performed after the Lorentz transformation. In the opposite case, one must use \(x^{\prime}=a(x+b)\). In the particular case of a scalar field, we obtain a simple expression \(\phi^{\prime}(x^{\prime})=\phi(x)\), with \(\phi\) being a scalar with respect to the internal symmetries group and the Lorentz transformations. The field \(\Psi(x)\) in general belongs to a nontrivial representation of the internal symmetry group and a Poincare group representation. For the infinitesimal parameters \(\omega^{a}\), \(\varepsilon^{\mu\
u}\), and \(b^{\mu}\), the variation of the field takes the form  
\[ \begin{split}\delta\Psi(x)&=\Psi^{\prime}(x)-\Psi(x) \\ &=S(a)U(\omega)\Psi(a^{-1}x-b)-\Psi(x)\\ &=\left(-i\omega^{a}T^{a}\right)\Psi(x)-b^{\mu}\partial_{\mu}\Psi (x)-\frac{i}{2}\varepsilon^{\mu\
u}\left(\Lambda_{\mu\
u}+\Sigma_{\mu\
u} \right)\Psi(x).\end{split}  \tag{6}\]  
The intrinsic symmetry generates variation \(\delta\Psi=(\delta\phi,\ \delta\phi^{*})\) with \(\delta\phi=-i\omega^{a}t^{a}\phi\) and \(\delta\phi^{*}=i\omega^{a}t^{a}\phi^{*}\). We thus use \(T^{a}=(t^{a},-t^{a})\). The spin generators act as \(\delta\phi=-\frac{i}{2}\varepsilon_{\mu\
u}\sigma^{\mu\
u}\phi\) and \(\delta\phi^{*}=\frac{i}{2}\varepsilon_{\mu\
u}\sigma^{\mu\
u*}\phi^{*}\). The rotation operators are defined by \(\Lambda_{\mu\
u}=(\mathcal{R}_{\mu\
u},-\mathcal{R}_{\mu\
u})\) where \(\mathcal{R}_{\mu\
u}=x_{\mu}i\partial_{\
u}-x_{\
u}i\partial_{\mu}\) and \(\Sigma^{\mu\
u}=(\sigma^{\mu\
u},-\sigma^{\mu\
u*})\). The order of the transformations is as follows: the matrix \(U(\omega)\) is applied first, followed by the Lorentz transformations and then translation.