Stable Interactions Between the Extended Chern-Simons Theory and a Charged Scalar Field with Higher Derivatives: Hamiltonian Formalism

Abstract

We consider constrained multi-Hamiltonian formulation for the extended Chern-Simons theory with higher derivatives of arbitrary finite order. The order n extension of the theory admits (n − 1)-parametric series of conserved tensors. The 00component of any representative of the series can be chosen as Hamiltonian. The theory admits a series of Hamiltonian formulations, including the canonical Ostrogradski formulation. The Hamiltonian formulations with different Hamiltonians are not connected by canonical transformations. Also, we demonstrate the inclusion of stable interactions with charged scalar field that preserves one specified Hamiltonian from the series. IntroductionThe Hamiltonian formulations for theories with higher derivatives have been discussed once and again for decades since the work of Ostrogradski [1]. The procedure for constructing such Hamiltonian formulation for degenerate theories was originally proposed in [2]. This procedure and its modifications can be applied to study different gauge theories, including gravity (see [3,4]). The canonical Ostrogradski Hamiltonian used in all these methods is not bounded. It leads to the wellknown stability problem and difficulties with constructing of quantum theory [5][6][7].The alternative Hamiltonian formulation was first introduced in the Pais-Uhlenbeck theory [8,9]. In the work [10], it was noticed that a wide class of theories with higher derivatives admits the series of Hamiltonians and the Poisson brackets that are not connected with each other by canonical transformations. The canonical Ostrogradski Hamiltonian is included in the series. The series of non-canonical Hamiltonian formulations was constructed explicitly in [11,12] for the extended Chern-Simons theory [13] of the third and fourth order.In the present work, we consider the Hamiltonian formulation for the extended Chern-Simons of arbitrary finite order. At free level, the theory of order n admits (n − 1)-parametric series of Hamiltonian formulations with Hamiltonian being the 00-component of any conserved tensor from the series [14]. We demonstrate that non-Lagrangian interaction vertices with * abakumova@phys.tsu.ru † dsc@phys.tsu.ru ‡ sll@phys.tsu.ru

Stable interactions in higher derivative field theories of derived type

“…The constrained Hamiltonian formalism for n-th order derived theory with the stable interactions constructed in Sec. IV is developed in the recent work [38]. 6 The "gauging" schemes mean that the gauge symmetry of self-interacting gauge fields can change upon inclusion of the non-gauge matter fields.…”

“…The invariance of action (34) with respect to the gauge transformations (38) is equivalent to the condition…”

“…The following facts ensure consistency of interaction (56): (i) the equations of motion are preserved by the gauge transformation (38), while transformation law for the equations of motion reads…”

“…As we can see, the quantity J μ ðϕ; βΦÞ is the charge current of complex scalar field. The infinitesimal gauge transformation (38) of the action functional (69) reads…”

Stability in the higher derivative Abelian gauge field theory

“…After that, a more general and systematic method was carried out as a guide to investigate the stabilities in a wide class of higher derivative systems named derived type theories [40][41][42][43][44]. Generally speaking, these derived theories are based on simpler free primary models whose equations of motion only involve first and second order differential operators without higher derivatives.…”