Noether's theorem in non-local field theories

Explicit expressions are constructed for a locally conserved vector current associated with a continuous internal symmetry and for energy-momentum and angular-momentum density tensors associated with the Poincar\'e group in field theories with higher-order derivatives and in non-local field theories. An example of non-local charged scalar field equations with broken C and CPT symmetries is considered. For this case, we find simple analytical expressions for the conserved currents.


Introduction
According to Noether's theorem [1], the invariance of the Lagrangian of a physical systemwith respect to continuous transformations leads to the existence of conserved charges. In its standard form, Noether's theorem refers to local field theories with derivatives of no higher than second order in the field equations.
Quantum field theories with higher derivatives are used for intermediate regularization procedures (see, e.g., [2]). The low-energy regime of QCD is known to be successfully described by chiral perturbation theory based on a power-series expansion in derivatives [3,4]. Field theories with higher derivatives are also discussed in the context of general relativity [5].
The CPT theorem tells that the CPT symmetry violation can be related to non-local interactions. Low-energy nuclear and atomic experiments provide strict constraints on the scale of a possible violation of CPT symmetry. A simple classification of the effects of the violation of the C, P and T symmetries and their combinations is presented by Okun [6]. A class of inflationary models is based on a non-local field theory [7].
In this paper, the question of whether one can generalize Noether's theorem to non-local field theory is discussed.
As an initial step, we consider a Lagrangian that contains, along with a field Ψ = (φ, φ * ),its higher derivatives ∂ µ 1 . . . ∂ µn Ψ up to order n ≥ 1: L = L(Ψ, ∂ µ 1 Ψ, . . . , ∂ µ 1 . . . ∂ µn Ψ). (1.1) The Lagrangian given in (1.1) is still local because it is a function of the field and its finite-order derivatives evaluated at a single point in space-time. To obtain a non-local field theory, one must include in (1.1) a dependence on an infinite number of field derivatives, i.e., by considering the limit n → ∞.
In the remainder of this paper, we use a system of units such that = c = 1. Indices µ, ν, ..., denoted by Greek letters from the middle of the alphabet, run from 0 to 3. Indices α, β, ... denote the spatial components of tensors and run from 1 to 3. We use a time-like metric g µν = diag(+1, −1, −1, −1), and indices enumerating members of internal-symmetry multiplets are suppressed.

Symmetries and the conserved currents
All observable quantities can be expressed in terms of fields and their combinations. The fields that appear in the Lagrangian, in general, belong to a representation space of the internal symmetry group. Linear transformations of 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 internalsymmetries, one can write the transformation matrix as follows where the ω 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.
In the general case, along with the internal symmetry of a system, one must consider the existence of external symmetries 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 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 µ ν = δ µ ν + ε µ· ·ν , where ε µν 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: where Σ µν is a matrix defined by the transformation properties of the field. Thus, the complete transformations of the coordinates and the field corresponding to the internal and external symmetries can be expressed in matrix notation as follows: where the notation used in Eq. (2.3) dictates a particular order of the transformations, namely, translation is performed after Lorentz transformation. In the opposite case, one must use x ′ = a(x + b). In the particular case of a scalar field, we obtain the simple expression where φ is a scalar with respect to the internal group of symmetries and with respect to the Lorentz transformations. The field Ψ(x) in general belongs to a nontrivial representation of the internal symmetry group and a Poincaré group representation.
For the infinitesimal parameters ω a , ε µν and b µ , the variation of the field takes the form (2.5) The intrinsic symmetry generates variation δΨ = (δφ, δφ * ) with δφ = −iω a t a φ and δφ * = iω ata φ * . We thus use T a = (t a , −t a ). The spin generators act as δφ = − i 2 ε µν σ µν φ and δφ * = i 2 ε µν σ µν * φ * . The rotation operators are defined by To derive the expression for the conserved current, one must use the generalized higher-order Euler-Lagrange equation By replacing the first term on the right-hand side of Eq. (2.6) with the corresponding expression from the Euler-Lagrange equation (2.7), one can rewrite the right-hand side of Eq. (2.6) as follows: The purpose is to present the above expression in the form of the divergence of some quantity.
The n-th order term under the first summation symbol in Eq. (2.8) can be rewritten in the (2.9) The first term has the form of a divergence, whereas in the second term, the derivative ∂ µn is shifted to the right and acts on δΨ. By rewriting the second term of (2.9) in the same way, we again obtain a divergence and one more derivative of δΨ in the second term. This implies that through such recursion, one can shift the derivative to the right until it lies immediately before δΨ. With each such procedure, the second term in the rewritten part of the expression changes sign and the first term has the form of a divergence.
Finally, the last term in the recursion can be obtained by shifting over n derivatives; this term will have an additional sign (−1) n , and consequently, it will have a sign opposite to that of the second term of Eq. (2.8) and will therefore vanish.
Thus, the result of this procedure for the right-hand side of Eq. (2.6) has the form (2.11) Finally, Eq. (2.6) can be written fully in the form of a divergence as follows: where δΨ in the parentheses is given by Eq. (2.5). The terms that are linear in the parameter ω a determine the set of conserved currents J aσ related to the internal symmetry group. The terms that are proportional to the vector −b σ determine the conserved second-rank tensor that can be identified with the energy-momentum tensor T σ µ . Finally, the terms that are proportional to the tensor ε µν determine the conserved third-rank tensor M σ µν . The spatial components of this tensor correspond to the total angular momentum density of the system. In the case of n = 1, we obtain the standard results. The set of conserved currents takes the form Noether's theorem allows us to find the conserved currents accurately to within an arbitrary factor. In Eqs. (2.13) -(2.15), the factors are chosen in such a way that the quantity T 0 0 coincides with the energy density defined by the Legendre transform of the Lagrangian. The quantity M 0 αβ then coincides with the angular momentum density of the system for the spatial indices α and β. Equation (2.14) is in agreement with Ref. [5].
In a non-local field theory, we expand non-local operators of the Lagrangian in an infinite power series over the differential operators. The conserved currents are then given by Eqs. (2.13) -(2.15), with the summation over n extended to +∞. This method is applied below to construct the conserved currents in a non-local charged scalar field theory.

Non-local charged scalar field
We consider an example of a non-local charged scalar field described by the Lagrangian The particles follow a relativistic dispersion law E(p) = p 2 + m 2 . Because of the absence of negative-frequency solutions, the particles do not have antiparticles.
Let us check whether CPT invariance holds in the non-local field theory defined by (3.1).
First consider the charge-conjugation operation, C. In the momentum space given by p α = −i∇ = −(p α ) * , with α = 1, 2, 3, we replace the particles' momenta in (3.1) with the generalized momenta, p µ → p µ − eA µ . The evolution equation in an external electromagnetic field takes the form For the complex conjugate scalar field, we obtain Together with the sign reversal of the charge e in Eq. (3.3), 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 (3.1) is invariant under the parity transformation P: φ(t, x) → φ(t, −x). By the same analysis, one can check that the time-reversal symmetry, T: φ(t, x) → φ * (−t, x), is conserved as well. Thus, the Lagrangian of (3.1) 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 (3.1) is explicitly invariant under global phase rotations of φ, which may imply the existence of a conserved vector current. The Lagrangian given in (3.1) 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 (3.1) 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 ǫ 0α = 0, ǫ αβ = 0.
We will work in terms of a power series over the derivatives. Expanding L, one can rewrite it as follows: derivatives. This implies that the series in Eqs. (2.13) -(2.15) are truncated at the first term of the sum. Thus, the charge density J 0 , the energy density T 0 µ and the angular momentum density M 0 αβ take the following simple forms:

Time-like components
We turn to momentum space, substituting into Eqs. (3.7)-(3.9) plane waves for outgoing and incoming particles with momenta p ′ and p. The four-momentum operator in coordinate space is given by p µ = (E, p) = (i∂ t , −i∇) . In terms of the transition matrix elements, the conserved currents (3.7)-(3.9) take the forms J 0 (p ′ , p) = 1, (3.10) To find the spatial components of the conserved currents, one must specify the action of the derivatives in expressions (2.13) -(2.15). The rules that are useful for deriving the expressions for the conserved currents are given in Appendix A.
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 Eq. (B.2), we obtain . (3.17) The detailed derivation of Eq. (3.16) is given in Appendix B. In terms of the four-dimensional operator iD σ ≡ (1, iD α ), the four-dimensional vector current can be written as It is useful to rewrite the vector current in the momentum space. By substituting the plane waves φ * (x) ∼ e ip ′ x and φ(x) ∼ e −ipx with momenta p ′ and p into Eq. (3.18) and omitting the exponential factors from the final expression, we obtain .
On the mass shell, the vector current is conserved: The variational derivative of the action functional S = d 4 xL with respect to the vector

Energy-momentum tensor
An analysis that is fundamentally identical to that presented in the previous section leads to the conserved energy-momentum tensor. Considering that δ α µ L = 0 for the fields that satisfy the equations of motion, one can rewrite Eq. (2.14) with the Lagrangian given in (3.1) k=2,4,6,...
The lowest-order l = 1 term of the expansion yields By performing contractions of the indices in Eq. (3.23), we obtain k=2,4,6,...
where J σ (p ′ , p) is given by Eq. (3.19). Using Eq. (3.22), we obtain the conservation condition for the energy-momentum tensor on the mass shell:

Angular momentum tensor
The conservation of angular momentum arises from the invariance of the system with respect to rotation. Taking Σ µν = 0 for the charged scalar field and substituting δ α µ L = 0 into Eq. (2.15), one can write the expression for the angular momentum density in the following form: k=2,4,6,...
The first terms of the series expansion are By performing contraction of the indices in Eq. (3.29), we obtain k=2,4,6,... (3.31) The arguments presented in Appendix B enable the summation of the series in Eq. (3.31), For σ = 0, we recover Eq. (3.9). R αβ is not diagonal in the momentum representation, so the momentum-space representation of M γ αβ offers no significant advantages. Using the equations of motion, one can verify that The conserved currents defined by Eq. (3.32) correspond to the space-like components of the parameter ε αβ , which describe a rotation; thus, the conserved charges are the components of the angular momentum tensor.

Conclusion
In non-local field theory with an internal symmetry and symmetries of the Poincaré group there exist conserved vector current and energy-momentum and angular momentum tensors.

A Field derivatives
In this section, we consider algebraic rules for the manipulation of the field's derivatives in Minkowski space. The proofs are valid, however, in the general case of R m,n . The fields Ψ and their derivatives are not assumed to be smooth; therefore, the sequence of the differentiation operations matters. As a result, is not necessarily symmetric under the permutation of indices. The conserved currents (2.13)-(2.15) are then calculated using the following formulas: . . .
In particular, and After the replacements → ∆ and g µν → δ αβ , these formulas also hold in Euclidean space. The

B Series summation
The factor 1/2 in Eq.
Finally, substituting the expression given in (3.17) into the above equation and combining the result with Eq. (3.9), we obtain (3.32).

C Vector current from the minimal substitution
The minimal substitution provides a gauge invariance of theory. After the minimal substitution the Lagrangian takes the form Based on the equation (3.21), we can immediately write J 0 = φ * φ.
The variation of S under variation of A can be found using the arguments similar to those of Appendix B: where we integrated by parts to remove derivatives from δA. The bottom line is obtained using