Abstract
We consider a gauge theory of vector fields in 3D Minkowski space. At the free level, the dynamical variables are subjected to the extended Chern–Simons (ECS) equations with higher derivatives. If the color index takes n values, the third-order model admits a 2n-parameter series of second-rank conserved tensors, which includes the canonical energy–momentum. Even though the canonical energy is unbounded, the other representatives in the series have a bounded from below the 00-component. The theory admits consistent self-interactions with the Yang–Mills gauge symmetry. The Lagrangian couplings preserve the energy–momentum tensor that is unbounded from below, and they do not lead to a stable non-linear theory. The non-Lagrangian couplings are consistent with the existence of a conserved tensor with a 00-component bounded from below. These models are stable at the non-linear level. The dynamics of interacting theory admit a constraint Hamiltonian form. The Hamiltonian density is given by the 00-component of the conserved tensor. In the case of stable interactions, the Poisson bracket and Hamiltonian do not follow from the canonical Ostrogradski construction. Particular attention is paid to the “triply massless” ECS theory, which demonstrates instability even at the free level. It is shown that the introduction of extra scalar field, serving as Higgs, can stabilize the dynamics in the vicinity of the local minimum of energy. The equations of motion of the stable model are non-Lagrangian, but they admit the Hamiltonian form of dynamics with a Hamiltonian that is bounded from below.
Highlights
The higher-derivative theories are well known for their better convergency properties at classical and quantum levels and wider symmetry
The extended Chern–Simons (ECS) theory Equation (16) is the first higher-derivative model with a non-abelian gauge symmetry admitting an alternative Hamiltonian formulation with a bounded Hamiltonian. This means that the concept of the stabilization of dynamics by means of an alternative Hamiltonian formalism applies beyond the linear level
In the remaining part of this section, we address an issue of the construction of the constrained Hamiltonian formalism with a bounded Hamiltonian for the higher-derivative Equation (49)
Summary
The higher-derivative theories are well known for their better convergency properties at classical and quantum levels and wider symmetry. It has been recognized that the higher-derivative dynamics can be stable at the classical and quantum levels even if the canonical energy of the model is unbounded. The model admits the inclusion of stable non-Lagrangian interactions with scalar, fermionic and gravitational fields that preserve a selected representative in the series of conserved quantities of free model [38,43,44]. The inclusion of non-Lagrangian interactions can solve the issue of the dynamic stability at the interacting level, because such couplings preserve bounded conserved quantities. The general model in the class model preserves a selected conserved tensor of the free theory, which can be bounded or unbounded from below depending on the values of coupling constants. The concluding section discusses the potential applications of the model that are considered in the article
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