Abstract
We consider the general higher derivative field theories of derived type. At free level, the wave operator of derived-type theory is a polynomial of the order $n\geq 2$ of another operator $W$ which is of the lower order. Every symmetry of $W$ gives rise to the series of independent higher order symmetries of the field equations of derived system. In its turn, these symmetries give rise to the series of independent conserved quantities. In particular, the translation invariance of operator $W$ results in the series of conserved tensors of the derived theory. The series involves $n$ independent conserved tensors including canonical energy-momentum. Even if the canonical energy is unbounded, the other conserved tensors in the series can be bounded, that will make the dynamics stable. The general procedure is worked out to switch on the interactions such that the stability persists beyond the free level. The stable interaction vertices are inevitably non-Lagrangian. The stable theory, however, can admit consistent quantization. The general construction is exemplified by the order $N$ extension of Chern-Simons coupled to the Pais-Uhlenbeck-type higher derivative complex scalar field.
Highlights
The higher derivative field theories are notorious for the stability problems at interacting level, and in quantum theory, see [1,2,3,4] for discussions and further references
Some other models with similar properties can be found in the Refs. [10,11,12]. The stability of this exceptional class of higher derivative theories is related to the fact that the canonical energy is bounded because of strong second class constraints
Level, the field equations of the derived theory are defined by the higher derivative wave operator M being a polynomial of another differential operator W. The latter is supposed to be of the first or second order. This class of systems admits, under certain conditions, inclusion of stable interactions, and the stability persists at quantum level
Summary
The higher derivative field theories are notorious for the stability problems at interacting level, and in quantum theory, see [1,2,3,4] for discussions and further references. As is noticed in the paper [26], even the simplest derived system, with the wave operator M being the second-order polynomial of another differential operator W, admits two different Lagrange anchors This explains the existence of one more conserved quantity connected to time-independence besides the canonical energy. To construct the stable interactions between two derived theories, we identify the series of conserved tensors at free level such that every item is connected to the space-time translation symmetry. We define the free derived theories and elaborate on the series of symmetries and conserved quantities connected to each symmetry of the primary wave operator W. In the higher derivative scalar field model the maximal number of independent conserved quantities is connected with the space-time translation invariance [26]. The Podolsky’s electrodynamics and the extended ChernSimons are consistent with this assumption, for example
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