# Conservation Laws and Stability of Field Theories of Derived Type

## Abstract

**:**

## 1. Introduction

## 2. Symmetries, Characteristics, and Conserved Quantities of Linear Systems

^{μ}, μ = 0, 1, …, d − 1, we consider the set of fields φ

^{i}(x). The multi-index i includes all the tensor, spinor, and isotopic indices, which label the fields. We assume the existence of an appropriate constant metric which can be used to raise and lower the multi-indices. This gives rise to the inner product of fields,

_{μ}= ∂/∂x

^{μ}.

_{M}labels the operator M, which is involved in the left-hand side of Equation (5).

^{α}= ε

^{α}(x) are functions of space-time coordinates, and summation over the repeated index α is assumed. The matrix differential operators R

_{α}are called gauge generators.

^{α}are some matrix differential operators. These symmetries are present in every theory and do not contain any valuable information about the dynamics of the model. In what follows, we systematically ignore them.

^{†}. Trivial variational symmetries have the form

## 3. Lagrange Anchor and Generalization of the Noether Theorem

^{α}are arbitrary operators. The trivial Lagrange anchors do not contain valuable information about the dynamics of the theory, and they are also useless for the connection of symmetries and conserved quantities. We systematically ignore them.

_{A}(∂)U

^{A}(∂) = 0 implies V(∂)Z

_{A}(∂)U

^{A}(∂) = R

_{α}(∂)U

^{α}(∂) for some U

^{α}, summation over repeated indices is implied.

## 4. Higher-Derivative Theories of Derived Type

^{i}, we introduce the variational primary model without higher-derivatives

_{0},…,α

_{n}distinguish different representatives in the class. The equations of motion (33) come from the least-action principle for the functional

_{a}, a = 1,…,s with these symmetries. The conserved currents are linearly independent if the symmetry generators are linearly independent (modulo trivial symmetries). In what follows in this section, we assume the linear independence of the generators L

_{a}(37). The operators L

_{a}(37) can form a Lie algebra, at least in some models. The Poincare symmetry in the relativistic theories is one of the relevant examples of such kind. We do not specify the structure of the algebra of symmetries because we are mostly interested in the correspondence between the linear spaces of symmetries and conserved quantities.

_{p}, p = 0, …, n − 1, are constant parameters. For p = 0, this series includes the symmetries (37) of the primary model (33). The other entries in (38) are higher-derivative operators whose origin is a consequence of the derived structure (34) of the equations of motion (2). The symmetries in the series (38) are usually independent, even though it is not a theorem. The general argument is that if all the powers of the primary operator are independent initial data and L

_{a}’s are transitive operators, the characteristic (38) cannot be zero on-shell. The exceptions are possible in the class of gauge theories, where some entries of the series (38) can trivialize on the mass-shell with account of gauge symmetries.

^{0}’s represents the canonical conserved quantities of the derived model (4), (34) that are connected with the original symmetry (37) of the primary model (33). The other conserved currents, j

^{p}, p = 1, …, n − 1, come from the higher-symmetries in the set (38). The conditions of linear independence of conserved currents are the same as that of symmetries.

_{s}are additional generators of trivial symmetries. In principle, the wave operator and M

_{s}can be dependent in some models. We should ignore M in the formulas below in these circumstances. The analogue of relations (44), (45) reads

_{s}, have no common roots. This condition is less restrictive than in the case of non-gauge models.

## 5. Stability of Higher-derivative Dynamics

^{μ}∂

_{μ}.

_{k}and m

_{k}label different roots and their multiplicities. The integer r is the total number of different roots. We assume that all the quantities λ

_{p}are real numbers. We do not consider complex roots because the corresponding derived theory is always unstable.

^{k;mk−1}are the canonical energy-momentum currents of the components, while the others are independent quantities. It is obvious that (55) are linear combinations of conserved tensors (39).

^{k;l})

^{μν}are defined by the rule

^{+}and M if, and only if, these polynomials have no common roots. On the other hand, each multiple root is common for M and N

^{+}. Hence, the bounded-conserved quantity can only be connected with space-time translations if all the roots of the characteristic equation are simple. The Pais–Uhlenbeck oscillator and higher-derivative scalar field models again serve as demonstrations for this observation.

## 6. Examples

#### 6.1. Fourth-order Pais–Uhlenbeck Oscillator

_{1}, ω

_{2}are parameters of the model. The Euler–Lagrange equation for the action functional is of derived type,

_{1}β

_{2}is nonzero. This gives an alternative proof of the stability of the Pais–Uhlenbeck oscillator with different frequencies.

_{1}= ω

_{2}= ω, the Pais–Uhlenbeck oscillator has two conserved quantities, only one of which is bounded from below,

^{1}. The characteristic (62) for this conserved quantity reads

^{2}with the characteristic polynomial of the wave operator (67). This means that the bounded integral of motion cannot be connected with the time translation symmetry in the model with resonance.

#### 6.2. Higher-derivative Scalar Field

_{p}determine the spectrum of masses in the theory. The equations of motion belong to derived type, with the primary operator being d’Alembertian,

_{p}, p = 1, …, n − 2 are usual scalar fields, while φ

_{n-1}obeys higher-derivative equations. For simple roots the conserved tensors have the form (78), we do not repeat the expressions for them. Two conserved tensors are associated with the multiple root,

^{2}with the characteristic polynomial of the theory. In doing so, the bounded-conserved quantity cannot be connected with the time translations. This demonstrates that the free higher-derivative scalar field theory with different masses is stable, while the multiple roots indicate the instability of the model.

#### 6.3. Extended Chern–Simons Model

_{μ}(x)dx

^{μ}on 3d Minkowski space with the action functional

_{1},…,α

_{n}. The action of the Chern–Simons operator on the vector field is determined by the relation

_{012}= 1. The Euler–Lagrange equations for the model have the form

^{p})

^{μν}, p = 0, …, n − 2, are independent, while

_{k}are non-zero.

_{k}− 1. For l = m

_{k}− 1, expressions (98) are exact with no dotted terms, and the corresponding conserved quantities are bounded.

^{k;}

^{0}in the conserved-quantity series are bounded. For simple zero roots the corresponding conserved quantity is the Chern–Simons energy, which is trivial. The additional conserved currents are associated with multiple roots. In the case of non-zero root, all these conserved quantities are independent and unbounded. In the case of zero roots, the additional quantity has a bounded 00-component (It is the canonical energy of 3d electrodynamics, which is known to be bounded), while all the other independent entries are independent and unbounded. This means that all of the additional conserved quantities in the set (56) are unbounded if the multiple zero root has a multiplicity greater than two. The subseries (61) of conserved quantities with the bounded 00-component has the form

_{0}, …, γ

_{n}− 1 being real numbers. All the entries of the series are non-trivial. The Lagrange anchor (101) takes the characteristic (100) into a symmetry by the rule (27). This symmetry is the space-time translation if the condition (42) is satisfied. The general trivial symmetry in the considered case reads

^{+}and M

^{−}have no common roots. The following restriction on the multiplicity of roots is implied:

^{+}and the characteristic polynomial of the derived theory, which has the place in case of non-gauge systems.

## 7. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{p}(z) denote characteristic polynomials of operators (50). By construction, Λ

_{p}(z) form the basis in the space of polynomials of order n − 1 in the variable z. In this setting, the chosen ansatz for V(γ;z) is a different parameterization of Lagrange anchor series (40).

_{p}> 0, and Λ

_{p}(−μ

_{p}

^{2})’s are non-zero. Moreover, we observe that

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Kaparulin, D.S.
Conservation Laws and Stability of Field Theories of Derived Type. *Symmetry* **2019**, *11*, 642.
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Kaparulin DS.
Conservation Laws and Stability of Field Theories of Derived Type. *Symmetry*. 2019; 11(5):642.
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Kaparulin, Dmitry S.
2019. "Conservation Laws and Stability of Field Theories of Derived Type" *Symmetry* 11, no. 5: 642.
https://doi.org/10.3390/sym11050642