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 Poincaré group in field theories with higher-order derivatives and in non-local field theories. We consider an example of non-local charged scalar field equations with broken C (charge conjugation) and CPT (charge conjugation, parity, and time reversal) symmetries. For this case, we find simple analytical expressions for the conserved currents.

According to Noether’s theorem [

Quantum field theories with higher derivatives are used for intermediate regularization procedures (see, e.g., [

The charge conjugation, parity, and time reversal (CPT) theorem states 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 [

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

In the remainder of this paper, we use a system of units such that

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:

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

For the infinitesimal parameters

The intrinsic symmetry generates variation

Returning to Equation (

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 Equation (

The first term has the form of a divergence, whereas in the second term, the derivative

Finally, the last term in the recursion can be obtained by shifting over

Thus, the result of this procedure for the right-hand side of Equation (

Finally, Equation (

Using Noether’s theorem one can find the conserved currents with accuracy within an arbitrary factor. In Equations (

We remark that a complete rotor, whose divergence is identically zero, can always be added to the conserved Noether current to achieve another conserved current, e.g.,

In 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 Equations (

We consider an example of a non-local charged scalar field described by the Lagrangian

The particles follow a relativistic dispersion law

Let us check whether CPT invariance holds in the non-local field theory defined by (

For the complex conjugate scalar field, one has

Together with the sign reversal of the charge

One can easily check that the Lagrangian given in (

The Lagrangian expressed in (

We will work in terms of a power series over the derivatives. Expanding

One can easily find the zeroth component of the conserved currents as the Lagrangian expressed in (

We turn to momentum space, substituting into Equations (

To find the spatial components of the conserved currents, one must specify the action of the derivatives in expressions (

Following the rules listed in

Let us write Equation (

By performing contractions of the indices and with the aid of Equation (

It is useful to rewrite the vector current in the momentum space. By substituting the plane waves

The variational derivative of the action functional

The field

Equations (

An analysis that is fundamentally identical to that presented in the previous section leads to the conserved energy-momentum tensor. Considering that

Using Equation (

The conservation of angular momentum arises from the invariance of the system with respect to rotation. Taking

The first terms of the series expansion are

By performing contraction of the indices in Equation (

The arguments presented in

For

The conserved currents defined by Equation (

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

Equations (

Among the possible applications of the considered formalism, transformations related to the conformal symmetry group are of particular interest.

Both authors have contributed equally to the work. All authors have read and agreed to the published version of the manuscript.

M.I.K. was partially supported by the Russian Foundation for Basic Research (RFBR) Grant No. 16-02-01104, Grant No. HLP-2015-18 of the Heisenberg-Landau Program. AT acknowledges the support from Votruba-Blokhintsev Cooperation Program in Theoretical Physics and the Program of target financing of the Ministry of Education and Science of the Republic of Kazakhstan, Grant No. BR05236277 and AP05133630.

A.T. wishes to acknowledge the kind hospitality of Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia.

The authors declare no conflict of interest.

The following abbreviations are used in this manuscript:

In this section, we consider algebraic rules for the manipulation of the field’s derivatives in a Minkowski space. The proofs are valid, however, in the general case of

After the replacements

The factor

The arrows over △ indicate the direction in which the differentiation acts. The series can be summed up using the factorization formula

This formula allows the simplification of the expression in brackets of Equation (

In the transition to the second line, we use the fact that

On the way we proved a useful formula

The sum over

Here, the indices are those of tensors in Minkowski space (e.g.,

Equation (

By writing the complex conjugate part of the expression explicitly, one can simplify the above equation using the formula as expressed in (

Finally, substituting the expression given in (

The minimal substitution provides a gauge invariance of theory. After the minimal substitution the Lagrangian takes the form

The variation of

The current (