Higher-derivative Field Theories 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.

Calculations in induced gravity from higher-derivative field theories

In this paper, we investigate Einstein’s gravity induced from higher-derivative scalar field theories. We develop an approach utilizing an effective theory of multiple fields for the higher-derivative theory. The expressions for induced cosmological constant and the induced gravitational constant are obtained in the present scenario of induced gravity in [Formula: see text] dimensions. We also show that finite values for the induced constants can be extracted in certain infinite-derivative theories.

Benign ghosts in higher-derivative systems

A brief review of the physics of systems including higher derivatives in the Lagrangian is given. All such systems involve ghosts i.e. the spectrum of the Hamiltonian is not bounded from below and the vacuum ground state is absent. Usually this leads to collapse and loss of unitarity. In certain special cases, this does not happen, however: ghosts are benign. This happens, in particular, in exactly solvable higher-derivative theories, but exact solvability seems to be a sufficient but not a necessary condition for the benign nature of the ghosts.We speculate that the Theory of Everything is a higher-derivative field theory, characterized by the presence of such benign ghosts and defined in a higher-dimensional bulk. Our Universe represents then a classical solution in this theory, having the form of a 3-brane embedded in the bulk.

Conservation laws and stability of higher derivative extended Chern-Simons

The higher derivative field theories are notorious for the stability problems both at classical and quantum level. Classical instability is connected with unboundedness of the canonical energy, while the unbounded energy spectrum leads to the quantum instability. For a wide class of higher derivative theories, including the extended Chern-Simons, other bounded conserved quantities which provide the stability can exist. The most general gauge invariant extended Chern-Simons theory of arbitrary finite order n admits (n − 1)-parameter series of conserved energy-momentum tensors. If the 00-component of the most general representative of this series is bounded, the theory is stable. The stability condition requires from the free extended Chern-Simons theory to describe the unitary reducible representation of the Poincaré group. The unstable theory corresponds to nonunitary representation.

Stable interactions in higher derivative field theories of derived type

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.

Unitary Issues in Some Higher Derivative Field Theories

We analyze the unitarity properties of higher derivative quantum field theories which are free of ghosts and ultraviolet singularities. We point out that in spite of the absence of ghosts most of these theories are not unitary. This result confirms the difficulties of finding a consistent quantum field theory of quantum gravity.

Classical and quantum dynamics of higher-derivative systems

A brief review of the physics of systems including higher derivatives in the Lagrangian is given. All such systems involve ghosts, i.e. the spectrum of the Hamiltonian is not bounded from below and the vacuum ground state is absent. Usually, this leads to collapse and loss of unitarity. In certain special cases, this does not happen, however, ghosts are benign. We speculate that the Theory of Everything is a higher-derivative field theory, characterized by the presence of such benign ghosts and defined in a higher-dimensional bulk. Our Universe then represents a classical solution in this theory, having the form of a 3-brane embedded in the bulk.

Supersymmetric Galileons and auxiliary fields in 2+1 dimensions

In this work we study various aspects of supersymmetric three-dimensional higher-derivative field theories. We classify all possible models without derivative terms in the auxiliary field of the fermionic sector and find that scalar field theories of the form $P(X,\phi)$, where $X=-(\partial\phi)^2/2$, belong to this kind of models. A ghost-free supersymmetric extension of Galileon models is found in three spacetime dimensions. Finally, the auxiliary field problem is discussed.

Higher derivative field theories: degeneracy conditions and classes

We provide a full analysis of ghost free higher derivative field theories with coupled degrees of freedom. Assuming the absence of gauge symmetries, we derive the degeneracy conditions in order to evade the Ostrogradsky ghosts, and analyze which (non)trivial classes of solutions this allows for. It is shown explicitly how Lorentz invariance avoids the propagation of “half” degrees of freedom. Moreover, for a large class of theories, we construct the field redefinitions and/or (extended) contact transformations that put the theory in a manifestly first order form. Finally, we identify which class of theories cannot be brought to first order form by such transformations.

The Unruh effect for higher derivative field theory

We analyse the emergence of the Unruh effect within the context of a field Lagrangian theory associated with the Pais–Uhlenbeck fourth order oscillator model. To this end, we introduce a transformation that brings the Hamiltonian bounded from below and is consistent with -symmetric quantum mechanics. We find that, as far as we consider different frequencies within the Pais–Uhlenbeck model, a particle together with an antiparticle of different masses are created and may be traced back to the Bogoliubov transformation associated with the interaction between the Unruh–DeWitt detector and the higher derivative scalar field. In contrast, whenever we consider the equal frequencies limit, no particle creation is detected as the pair particle/antiparticle annihilate each other. Further, following Moschella and Schaeffer, we construct a Poincaré invariant two-point function for the Pais–Uhlenbeck model, which in turn allows us to perform the thermal analysis for any of the emanant particles.

Duality and gauge invariance of non-commutative spacetime Podolsky electromagnetic theory

The interest in higher derivative field theories has its origin mainly in their influence concerning the renormalization properties of physical models and to remove ultraviolet divergences. In this paper, we have introduced the non-commutative (NC) version of the Podolsky theory and we investigated the effect of the non-commutativity over its original gauge invariance property. We have demonstrated precisely that the non-commutativity spoiled the primary gauge invariance of the original action under this primary gauge transformation. After that we have used the Noether dualization technique to obtain a dual and gauge invariant action. We have demonstrated that through the introduction of a Stueckelberg field in this NC model, we can also recover the primary gauge invariance. In this way, we have accomplished a comparison between both methods.

Exorcising the Ostrogradsky ghost in coupled systems

The Ostrogradsky theorem implies that higher-derivative terms of a single mechanical variable are either trivial or lead to additional, ghost-like degrees of freedom. In this letter we systematically investigate how the introduction of additional variables can remedy this situation. Employing a Lagrangian analysis, we identify conditions on the Lagrangian to ensure the existence of primary and secondary constraints that together imply the absence of Ostrogradsky ghosts. We also show the implications of these conditions for the structure of the equations of motion as well as possible redefinitions of the variables. We discuss applications to analogous higher-derivative field theories such as multi-Galileons and beyond Horndeski.

Tulczyjew triples in higher derivative field theory

The geometrical structure known as Tulczyjew triple has been used with success in analytical mechanics and first order field theory to describe a wide range of physical systems including Lagrangian/Hamiltonian systems with constraints and/or sources, or with singular Lagrangian. Starting from the first principles of the variational calculus we derive Tulczyjew triples for classical field theories of arbitrary high order, i.e. depending on arbitrary high derivatives of the fields. A first triple appears as the result of considering higher order theories as first order ones with configurations being constrained to be holonomic jets. A second triple is obtained after a reduction procedure aimed at getting rid of nonphysical degrees of freedom. This picture we present is fully covariant and complete: it contains both Lagrangian and Hamiltonian formalisms, in particular the Euler-Lagrange equations. Notice that, the required Geometry of jet bundles is affine (as opposed to the linear Geometry of the tangent bundle). Accordingly, the notions of affine duality and affine phase space play a distinguished role in our picture. In particular the Tulczyjew triples in this paper consist of morphisms of double affine-vector bundles which, moreover, preserve suitable presymplectic structures.

Linear higher derivative field theories, entanglement entropy and quantum phase transition

Quantum systems described by linear, higher order spatial derivatives of the field exhibit a new kind of phase transition as the coupling strength of the higher order terms varies.This essentially happens because a fundamental change in the ratio of ground state energy eigenvalues, between linear and non-linear regime, takes place which leads to excitation of higher modes. The new phase has inverse relation with area.

GEOMETRIC HAMILTON–JACOBI FIELD THEORY

I review my proposal about how to extend the geometric Hamilton–Jacobi theory to higher derivative field theories on fiber bundles.

Gauge invariances of higher derivative Maxwell-Chern-Simons field theory: A new Hamiltonian approach

A new method of abstracting the independent gauge invariances of higher derivative systems, recently introduced in [1], has been applied to higher derivative field theories. This has been discussed taking the extended Maxwell-Chern-Simons model as an example. A new Hamiltonian analysis of the model is provided. This Hamiltonian analysis has been used to construct the independent gauge generator. An exact mapping between the Hamiltonian gauge transformations and the U(1) symmetries of the action has been established.

Generalized Lee-Wick formulation from higher derivative field theories

We study a higher derivative (HD) field theory with an arbitrary order of derivative for a real scalar field. The degree of freedom for the HD field can be converted to multiple fields with canonical kinetic terms up to the overall sign. The Lagrangian describing the dynamics of the multiple fields is known as the Lee-Wick (LW) form. The first step to obtain the LW form for a given HD Lagrangian is to find an auxiliary field (AF) Lagrangian which is equivalent to the original HD Lagrangian up to the quantum level. Until now, the AF Lagrangian has been studied only for $N=2$ and 3 cases, where $N$ is the number of poles of the two-point function of the HD scalar field. We construct the AF Lagrangian for arbitrary $N$. By the linear combinations of AF fields, we also obtain the corresponding LW form. We find the explicit mapping matrices among the HD fields, the AF fields, and the LW fields. As an exercise of our construction, we calculate the relations among parameters and mapping matrices for $N=2$, 3, and 4 cases.

Testing symmetries in effective models of higher derivative field theories

Higher derivative field theories with interactions raise serious doubts about their validity due to severe energy instabilities. In many cases the implementation of a direct perturbation treatment to excise the dangerous negative-energies from a higher derivative field theory may lead to violations of Lorentz and other symmetries. In this work we study a perturbative formulation for higher derivative field theories that allows the construction of a low-energy effective field theory being a genuine perturbations over the ordinary-derivative theory and having a positive-defined Hamiltonian. We show that some discrete symmetries are recovered in the low-energy effective theory when the perturbative method to reduce the negative-energy degrees of freedom from the higher derivative theory is applied. In particular, we focus on the higher derivative Maxwell-Chern-Simons model which is a Lorentz invariant and parity-odd theory in $2+1$ dimensions. The parity violation arises in the effective action of ${\mathrm{QED}}_{3}$ as a quantum correction from the massive fermionic sector. We obtain the effective field theory which remains Lorentz invariant, but parity invariant to the order considered in the perturbative expansion.

Dirac quantization of the Pais-Uhlenbeck fourth order oscillator

As a model, the Pais-Uhlenbeck fourth order oscillator with equation of motion $d^4q/dt^4+(\omega_1^2+\omega_2^2)d^2q/dt^2 +\omega_1^2\omega_2^2 q=0$ is a quantum-mechanical prototype of a field theory containing both second and fourth order derivative terms. With its dynamical degrees of freedom obeying constraints due to the presence of higher order time derivatives, the model cannot be quantized canonically. We thus quantize it using the method of Dirac constraints to construct the correct quantum-mechanical Hamiltonian for the system, and find that the Hamiltonian diagonalizes in the positive and negative norm states that are characteristic of higher derivative field theories. However, we also find that the oscillator commutation relations become singular in the $\omega_1 \to \omega_2$ limit, a limit which corresponds to a prototype of a pure fourth order theory. Thus the particle content of the $\omega_1 =\omega_2$ theory cannot be inferred from that of the $\omega_1 \neq \omega_2$ theory; and in fact in the $\omega_1 \to \omega_2$ limit we find that all of the $\omega_1 \neq \omega_2$ negative norm states move off shell, with the spectrum of asymptotic in and out states of the equal frequency theory being found to be completely devoid of states with either negative energy or negative norm. As a byproduct of our work we find a Pais-Uhlenbeck analog of the zero energy theorem of Boulware, Horowitz and Strominger, and show how in the equal frequency Pais-Uhlenbeck theory the theorem can be transformed into a positive energy theorem instead.

Benign vs. malicious ghosts in higher-derivative theories

Interacting theories with higher-derivatives involve ghosts. They correspond to instabilities that display themselves at the classical level. We notice that comparatively “benign” mechanical higher-derivative systems exist, where the classical vacuum is stable with respect to small perturbations and the problems appear only at the nonperturbative level. We argue the existence of benign higher-derivative field theories which are stable with respect to small fluctuations with nonzero momenta. A particular example is the 6D N = 2 higher-derivative SYM theory, which is finite and unitary at the perturbative level. The instability with respect to small fluctuations of zero-momentum modes is always present, however.