Ostrogradsky Formalism Research Articles

Gauging the higher derivative field theories in Ostrogradsky formalism

The gaugings of the higher derivative fields theory are analyzed in Ostrogradsky formalism with the aid of BRST charge. Treating the terms of higher-order time derivatives of fields as independent dynamic variables, we are able to convert the higher derivative Lagrangian into the more suitable Ostrogradski Hamiltonian formalism. Using the knowledge of local BRST-cohomology, it is possible for us to work out the deformed functional equations of the BRST charge as well as the corresponding BRST-invariant Hamiltonian. Moreover, by inserting these deformed quantities into the original system and utilizing the canonical equations of motion of phase variables, we gain the usual local gauge symmetries of the resulting interacting theory directly through the standard Legendre transformations.

Hamiltonian formulation of a class of constrained fourth-order differential equations in the Ostrogradsky framework

We consider a class of Lagrangians that depend not only on some configurational variables and their first time derivatives, but also on second time derivatives, thereby leading to fourth-order evolution equations. The proposed higher-order Lagrangians are obtained by expressing the variables of standard Lagrangians in terms of more basic variables and their time derivatives. The Hamiltonian formulation of the proposed class of models is obtained by means of the Ostrogradsky formalism. The structure of the Hamiltonians for this particular class of models is such that constraints can be introduced in a natural way for eliminating expected instabilities of the fourth-order evolution equations. Moreover, canonical quantization of the constrained equations can be achieved by means of Dirac’s approach to generalized Hamiltonian dynamics.

Instability states and Ostrogradsky formalism

Contemporary physics, both Classical and Quantum, requires a notion of inertial reference frames. However, how to find a physical inertial frame in reality where there always exist random weak forces? We suggest a description of the motion in instability frames by means of inclusion of higher time derivatives. They may play a role of non-local hidden variables in a more general description complementing both classical and quantum mechanics.

On the canonical formalism of f(R)-type gravity using Lie derivatives

We present a canonical formalism of the $f(R)$-type gravity using the Lie derivatives instead of the time derivatives by refining the formalism of our group. The previous formalism is a direct generalization of the Ostrogradski's formalism. However the use of the Lie derivatives was not sufficient in that Lie derivatives and time derivatives are used in a mixed way, so that the expressions are somewhat complicated. In this paper, we use the Lie derivatives and foliation structure of the spacetime thoroughly, which makes the procedure and the expressions far more concise.

A quantum cure for the Ostrogradski instability

Interacting fourth order quantum mechanics is in the Ostrogradski formalism afflicted by an instability involving the decay of the vacuum. When treating such systems as 1 + 0 -dimensional Euclidean field theories in the transfer operator formalism the ‘instability problem’ and the ‘unitarity problem’ are distinct and decoupled. The instability problem is shown to be absent: a stable ground state always exists and is typically normalizable and strictly positive. The generator H of the transfer operator replaces the Ostrogradski Hamiltonian and is non-Hermitian but selfadjoint with respect to a Krein structure, which also ensures consistency with the Lagrangian functional integral. The case of a scalar quartic derivative interaction is treated in detail. Variational perturbation theory, a strong coupling expansion, and direct diagonalization of matrix truncations are used to compute the spectrum of H in this case.

Moyal algebra: relevant properties, projective limits and applications in noncommutative field theory

From the definition of the Moyal ∗-product in terms of projective limits of the ring of polynomials of vector fields, the Moyal configuration space of Schwartzian functions, equipped with the ∗-product, is built as a formal power series ring with elements assimilated to free indeterminates. We then define the projector on the ideal depending on a fixed indeterminate, which allows to use the definition of algebraic derivations with respect to any order of field derivative. As a consequence and in a direct manner, Euler-Lagrange equations of motion, in the framework of both the noncommutative scalar and gauge induced Dirac fields, are deduced from the nonlocal Lagrange function. A connection of this theory to a generalized Ostrogradski's formalism is also discussed here.

Fractional embedding of differential operators and Lagrangian systems

This paper is a contribution to the general program of embedding theories of dynamical systems. Following our previous work on the stochastic embedding theory developed with Darses [C. R. Acad. Sci. Ser. I: Math 342, 333 (2006); (preprint IHES 06/27, p. 87, 2006)], we define the fractional embedding of differential operators and ordinary differential equations. We construct an operator combining in a symmetric way the left and right (Riemann-Liouville) fractional derivatives. For Lagrangian systems, our method provides a fractional Euler-Lagrange equation. We prove, developing the corresponding fractional calculus of variations, that such equation can be derived via a fractional least-action principle. We then obtain naturally a fractional Noether theorem and a fractional Hamiltonian formulation of fractional Lagrangian systems. All these constructions are coherents, i.e., the embedding procedure is compatible with the fractional calculus of variations. We then extend our results to cover the Ostrogradski formalism. Using the fractional embedding and following a previous work of Riewe [Phys. Rev. E 53, 1890 (1996); Phys. Rev. E 55, 3581 (1997)], we obtain a fractional Ostrogradski formalism which allows us to derive nonconservative dynamical systems via a fractional generalized least-action principle. We also discuss the Whittaker equations and obtain a fractional Lagrangian formulation. Last, we discuss the fractional embedding of continuous Lagrangian systems. In particular, we obtain a fractional Lagrangian formulation of the classical fractional wave equation introduced by Schneider and Wyss [J. Math. Phys. 30, 134 (1989)] as well as the fractional diffusion equation.

Inverse variational problem and canonical structure of Burgers equations

It is demonstrated that Burgers equations, which are often believed to describe dissipative systems, are non-Lagrangian. Following Bateman’s analysis of damped harmonic oscillator an action is defined to look for a Lagrangian representation for equations in the Burgers hierarchy. The associated higher-order Lagrangian densities are found to be degenerate such that the Hamiltonian structure could be studied by a repackaging of Ostrogradski formalism and Dirac’s theory of constraints.

Ostrogradski formalism for higher-derivative scalar field theories

We carry out the extension of the Ostrogradski method to relativistic field theories. Higher-derivative Lagrangians reduce to second differential order with one explicit independent field for each degree of freedom. We consider a higher-derivative relativistic theory of a scalar field and validate a powerful order-reducing covariant procedure by a rigorous phase-space analysis. The physical and ghost fields appear explicitly. Our results strongly support the formal covariant methods used in higher-derivative gravity.

Finding the Hamiltonian for Cosmological Models in Fourth-Order Gravity Theories Without Resorting to the Ostrogradski or Dirac Formalism

The Hamilton formalism of cosmological models in fourth-order theories of gravity is considered. An approach to constructing the Hamilton function is presented which starts by replacing the second order derivatives of configuration space coordinates by functions depending on these coordinates, its first order derivatives, and additional variables playing the role of configuration space coordinates. This formalism, which does not resort to the Ostrogradski or Dirac formalism, is elucidated and applied to examples. For a special class of Lagrange functions, it is demonstrated that the canonical coordinates of the considered formalism and of the Ostrogradski formalism are related via a canonical transformation. The canonical transformation is a transformation of the configuration space coordinates and a transformation of momentum components induced by the transformation of the configuration space coordinates for a special element of the class of Lagrange functions mentioned. The Wheeler-DeWitt equations belonging to this Lagrange function are related via minisuperspace coordinate transformations.

BRST FORMALISM FOR SYSTEMS WITH HIGHER ORDER DERIVATIVES OF GAUGE PARAMETERS

For a wide class of mechanical systems, invariant under gauge transformations with arbitrary higher order time derivatives of gauge parameters, the equivalence of Lagrangian and Hamiltonian BRST formalisms is proved. It is shown that the Ostrogradsky formalism establishes the natural rules to relate the BFV ghost canonical pairs with the ghosts and antighosts introduced by the Lagrangian approach. Explicit relation between corresponding gauge-fixing terms is obtained.

Effective Lagrangians with higher derivatives and equations of motion.

The problems that are connected with Lagrangians which depend on higher order derivatives (namely additional degrees of freedom, unbound energy from below, etc.) are absent if effective Lagrangians are considered because the equations of motion may be used to eliminate all higher order time derivatives from the effective interaction term. The application of the equations of motion can be realized by performing field transformations that involve derivatives of the fields. Using the Hamiltonian formalism for higher order Lagrangians (Ostrogradsky formalism), Lagrangians that are related by such transformations are shown to be physically equivalent (at the classical and at the quantum level). The equivalence of Hamiltonian and Lagrangian path integral quantization (Matthews's theorem) is proven for effective higher order Lagrangians. Effective interactions of massive vector fields involving higher order derivatives are examined within gauge noninvariant models as well as within (linearly or nonlinearly realized) spontaneously broken gauge theories. The Stueckelberg formalism, which relates gauge noninvariant to gauge invariant Lagrangians, becomes reformulated within the Ostrogradsky formalism.

Quantization of the radiation-damped harmonic oscillator

The Abraham-Loreentz equation for the harmonic oscillator is reconsidered. Quantization of the system is performed by applying Ostrogradsky's formalism of generalized momenta to a Lagrangian which contains higher than first derivatives in time. The final six-dimensional phase-space does not allow the interpretation as phase-space of a three-boson-system but only as one of one boson, its antiboson and a “ghost”.