Higher derivative theories with constraints: exorcising Ostrogradski's ghost

Chen, Tai-jun; Fasiello, Matteo; Lim, Eugene A.; Tolley, Andrew J.

doi:10.1088/1475-7516/2013/02/042

Cited by 146 publications

(184 citation statements)

References 45 publications

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“…This fact is usually referred as the existence of implicit or "hidden" constraints. However, such approach is not fully satisfactory from an action point of view and one may be eager to reveal the "hidden" constraint with the help of a Lagrange multiplier [44]. In that sense, as shown in Sec.…”

Section: Summary and Discussionmentioning

confidence: 99%

Derivative-dependent metric transformation and physical degrees of freedom

et al. 2015

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We study metric transformations which depend on a scalar field φ and its first derivatives and confirm that the number of physical degrees of freedom does not change under such transformations, as long as they are not singular. We perform a Hamiltonian analysis of a simple model in the gauge φ = t. In addition, we explicitly show that the transformation and the gauge fixing do commute in transforming the action. We then extend the analysis to more general gravitational theories and transformations in general gauges. We verify that the set of all constraints and the constraint algebra are left unchanged by such transformations and conclude that the number of degrees of freedom is not modified by a regular and invertible generic transformation among two metrics. We also discuss the implications on the recently called "hidden" constraints and on the case of a singular transformation, a.k.a. mimetic gravity.

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Section: Summary and Discussionmentioning

confidence: 99%

Derivative-dependent metric transformation and physical degrees of freedom

et al. 2015

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“…The use of implicit (constraint) relations to remove the higher time derivatives of the dynamical variables is related to a recent result found in [83], which allows to "exorcise Ostrogradski's ghost in higher order derivatives with constraints". It states that theories with constraints which reduce the dimensionality of the phase space of the system do not have a linear instability in the Hamiltonian, even if the original Lagrangian includes second or higher derivatives with respect to time.…”

Section: A Loophole In Horndeski's Theorem: Hidden Constraintsmentioning

confidence: 93%

Transforming gravity: From derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian

2014

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We study the structure of scalar-tensor theories of gravity based on derivative couplings between the scalar and the matter degrees of freedom introduced through an effective metric. Such interactions are classified by their tensor structure into conformal (scalar), disformal (vector) and extended disformal (traceless tensor), as well as by the derivative order of the scalar field. Relations limited to first derivatives of the field ensure second order equations of motion in the Einstein frame and hence the absence of Ostrogradski ghost degrees of freedom. The existence of a mapping to the Jordan frame is not trivial in the general case, and can be addressed using the Jacobian of the frame transformation through its eigenvalues and eigentensors. These objects also appear in the study of different aspects of such theories, including the metric and field redefinition transformation of the path integral in the quantum mechanical description. Although sane in the Einstein frame, generic disformally coupled theories are described by higher order equations of motion in the Jordan frame. This apparent contradiction is solved by the use of a hidden constraint: the contraction of the metric equations with a Jacobian eigentensor provides a constraint relation for the higher field derivatives, which allows one to express the dynamical equations in a second order form. This signals a loophole in Horndeski's theorem and allows one to enlarge the set of scalar-tensor theories which are Ostrogradski-stable. The transformed Gauss-Bonnet terms are also discussed for the simplest conformal and disformal relations.PACS numbers: 04.50. Kd, 98.80.Cq, 95.36.+x,

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“…The theory we consider is the Pais-Uhlenbeck oscillator [1008], a simple classical-mechanical example of a higher-derivative quadratic Lagrangian Recall that the equation of motion (C-XII) depended on .... q , indicating the Lagrangian is nondegenerate, and we see here that we can trivially solve forq in terms of the canonical coordinates and P 2 . We can then construct the Hamiltonian as [1009][1010][1011] H PU = P 1 Q 2 + P 2 2 − L PU (Q 1 , Q 2 , P 2 ) = P 1 Q 2 + 1 2 P 2 2 + 1 2 (m 2 1 + m 2 2 )Q 2 2 − 1 2 m 2 1 m 2 2 Q 2 1 . (C-XIV)…”

Section: (C-v)mentioning

confidence: 99%

“…A powerful theorem, due to Ostrogradsky [220] tells us that, in most cases, if the equations of motion are higher than second order in time derivatives, the theory will have a ghost instability (we review this theorem in Appendix C). 63 For proposals on how to deal with the ghost in this model, see [1010,1011]. In theories with a ghostly field, the vacuum is unstable to rapid pair production of ghost particles and healthy particles, causing the theory to be ill-defined (See [96,1012]).…”

Section: D1 Ghostsmentioning

confidence: 99%

Beyond the cosmological standard model

et al. 2015

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After a decade and a half of research motivated by the accelerating universe, theory and experiment have a reached a certain level of maturity. The development of theoretical models beyond Λ or smooth dark energy, often called modified gravity, has led to broader insights into a path forward, and a host of observational and experimental tests have been developed. In this review we present the current state of the field and describe a framework for anticipating developments in the next decade. We identify the guiding principles for rigorous and consistent modifications of the standard model, and discuss the prospects for empirical tests.We begin by reviewing recent attempts to consistently modify Einstein gravity in the infrared, focusing on the notion that additional degrees of freedom introduced by the modification must "screen" themselves from local tests of gravity. We categorize screening mechanisms into three broad classes: mechanisms which become active in regions of high Newtonian potential, those in which first derivatives of the field become important, and those for which second derivatives of the field are important. Examples of the first class, such as f (R) gravity, employ the familiar chameleon or symmetron mechanisms, whereas examples of the last class are galileon and massive gravity theories, employing the Vainshtein mechanism. In each case, we describe the theories as effective theories and discuss prospects for completion in a more fundamental theory. We describe experimental tests of each class of theories, summarizing laboratory and solar system tests and describing in some detail astrophysical and cosmological tests. Finally, we discuss prospects for future tests which will be sensitive to different signatures of new physics in the gravitational sector.The review is structured so that those parts that are more relevant to theorists vs. observers/experimentalists are clearly indicated, in the hope that this will serve as a useful reference for both audiences, as well as helping those interested in bridging the gap between them.

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Higher derivative theories with constraints: exorcising Ostrogradski's ghost

Abstract: Abstract. We prove that the linear instability in a non-degenerate higher derivative theory, the Ostrogradski instability, can only be removed by the addition of constraints if the original theory's phase space is reduced.

Cited by 146 publications

References 45 publications

Derivative-dependent metric transformation and physical degrees of freedom

Derivative-dependent metric transformation and physical degrees of freedom

Transforming gravity: From derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian

Beyond the cosmological standard model

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