Higher Derivative Scalar Quantum Field Theory in Curved Spacetime

TL;DR

This paper investigates higher derivative scalar quantum fields in curved spacetime, deriving energy-momentum tensors, exploring quantisation issues, and establishing non-existence of static solutions, with applications to black holes and cosmology.

Contribution

It provides a detailed analysis of higher derivative scalar fields in curved backgrounds, including energy-momentum tensors, quantisation currents, and no-hair theorems, with novel insights into stability in cosmological settings.

Findings

Derived energy-momentum tensors for higher derivative scalar fields.

Established non-existence of static scalar solutions around black holes.

Showed Hubble friction stabilizes Pais-Uhlenbeck fields in de Sitter space.

Abstract

We study a free scalar field $\phi$ in a fixed curved background spacetime subject to a higher derivative field equation of the form $F(\Box)\phi =0$, where $F$ is a polynomial of the form $F(\Box)= \prod_i (\Box-m_i^2)$ and all masses $m_i$ are distinct and real. Using an auxiliary field method to simplify the calculations, we obtain expressions for the Belinfante-Rosenfeld symmetric energy-momentum tensor and compare it with the canonical energy-momentum tensor when the background is Minkowski spacetime. We also obtain the conserved symplectic current necessary for quantisation and briefly discuss the issue of negative energy versus negative norm and its relation to Reflection Positivity in Euclidean treatments. We study, without assuming spherical symmetry, the possible existence of finite energy static solutions of the scalar equations, in static or stationary background geometries. Subject to various assumptions on the potential, we establish non-existence results including a no-scalar-hair theorem for static black holes. We consider Pais-Uhlenbeck field theories in a cosmological de Sitter background, and show how the Hubble friction may eliminate what would otherwise be unstable behaviour when interactions are included.

Equations

$$\delta L-\delta t\frac{\partial L}{\partial t}$$

$$=$$

$$\frac{\partial L}{\partial x}-\frac{d}{dt}\frac{\partial L}{\partial \dot{x}}+\frac{d^2}{d{t}^2}\frac{\partial L}{\partial \ddot{x}}=0.$$

$$\frac{d}{dt}\Big{(}L-\dot{x}\frac{\partial L}{\partial\dot{x}}-\ddot{x}\frac{\partial L}{\partial\ddot{x}}+\dot{x}\frac{d}{dt}\frac{\partial L}{\partial\ddot{x}}\Big{)}=\frac{\partial L}{\partial t}\,.$$

$$E=\dot{x}\frac{\partial L}{\partial \dot{x}}+\ddot{x}\frac{\partial L}{\partial \ddot{x}}-\dot{x}\frac{d}{dt}\frac{\partial L}{\partial \ddot{x}}-L,$$

$$E=\dot{x}\,\Big{(}\frac{\partial L}{\partial\dot{x}}-\frac{d}{dt}\frac{\partial L}{\partial\ddot{x}}\Big{)}+\ddot{x}\,\frac{\partial L}{\partial\ddot{x}}-L\,.$$

$$p_x=\frac{\partial L}{\partial \dot{x}}-\frac{d}{dt}\frac{\partial L}{\partial \ddot{x}},$$

$${\dot{p}}_x=\frac{\partial L}{\partial x},$$

$$E=\dot{x}{p}_x+\ddot{x}\frac{\partial L}{\partial \ddot{x}}-L.$$

$$y=\dot{x},p_y=\frac{\partial L}{\partial \ddot{x}}.$$

$$\frac{{\partial }^{2}L}{\partial {\ddot{x}}^2}\ne 0$$

$$\ddot{x}=a(x,y,p_y).$$

$$H=p_{x}y+p_{y}a(x,y,p_y)-L(x,y,a(x,y,p_y)).$$

$$L=\frac{1}{2}{\dot{x}}^2-\frac{1}{2}\beta {\ddot{x}}^2-\frac{1}{2}{\Omega }^2{x}^2.$$

$$\frac{{\partial }^{2}L}{\partial {\ddot{x}}^2}=-\beta .$$

$$\ddot{x}+\beta \stackrel{....}{x}+{\Omega }^{2}x=0,$$

$$p_y=-\beta \ddot{x},\Rightarrow \ddot{x}=-\frac{1}{\beta }{p}_y=a.$$

$$p_x=\dot{x}+\beta \stackrel{...}{x},$$

$${\dot{p}}_x=-{\Omega }^{2}x.$$

$${\dot{p}}_x=-\frac{dV}{dx}.$$

$${\omega }^4-\frac{1}{\beta }{\omega }^2+\frac{{\Omega }^2}{\beta }=0,$$

$$\omega^{2}=\frac{1}{2\beta}\bigl{(}1\pm\sqrt{1-4\beta\,\Omega^{2}}\bigr{)}\,.$$

$${\omega }_1^2+{\omega }_2^2=\frac{1}{\beta },{\omega }_1^2{\omega }_2^2=\frac{{\Omega }^2}{\beta }.$$

$$\stackrel{....}{x}+({\omega }_1^2+{\omega }_2^2)\ddot{x}+{\omega }_1^2{\omega }_2^{2}x=0.$$

$$x=A{e}^{-i{\omega }_{1}t}+B{e}^{-i{\omega }_{2}t}+\bar{A}{e}^{i{\omega }_{1}t}+\bar{B}{e}^{i{\omega }_{2}t},$$

$$x(0)$$

$$\dot{x}(0)$$

$$\ddot{x}(0)$$

$$\stackrel{...}{x}(0)$$

$$E=\frac{1}{2}(\dot{x}{)}^2-\frac{1}{2}\beta (\ddot{x}{)}^2+\beta \dot{x}\stackrel{...}{x}+\frac{1}{2}{\Omega }^2{x}^2,$$

$$H$$

$${\dot{p}}_x=-{\Omega }^{2}x,$$

Full text

1. 1 Introduction

2. 2 A Higher Derivative Point Particle

3. 2.1 Energy and and Momentum Conservation

4. 2.2 Ostrogradsky’s Hamiltonian

5. 2.3 Quantisation of the Pais-Uhlenbeck Oscillator

6. 3 Pais-Uhlenbeck Field Theory in Minkowski Spacetime

7. 3.1 The Energy Momentum Tensor

8. 3.2 Second Quantisation of the Pais-Uhlenbeck Field Theory

9. 3.2.1 Symplectic Current for the Pais-Uhlenbeck Field Theory

10. 4 Pais-Uhlenbeck Field Theory in Curved spacetime

11. 4.1 Simplest model

12. 4.1.1 Auxiliary field formulation

13. 4.1.2 Energy-momentum tensor

14. 4.2 Auxiliary Fields

15. 4.2.1 Powers of Laplace/d’Alembert operator

16. 4.2.2 Auxiliary Fields for General Higher-Order Operators

17. 4.3 Symplectic Current in Higher-Derivative Scalar Theory

18. 5 Green Functions and Euclidean Formulation

19. 5.1 Reflection Positivity in the 4th-Order Case

20. 5.2 $2N$’th-Order Case

21. 5.3 Higher Dimensional Green Functions

22. 6 No-Hair Theorems for Static and Stationary Solutions

23. 6.1 Pais-Uhlenbeck Field Theories

24. 6.2 Higher-Order Theories with a General Potential

25. 6.3 Virial Theorem

26. 6.4 Stationary Metrics

27. 7 Stabilisation By Hubble Friction

28. 8 Conclusion

29. A Canonical Noether Energy-Momentum Tensor

1 Introduction

Higher-derivative field theories have received a considerable amount of attention over the years for a variety of reasons, not least because of the realisation that theories incorporating standard Einstein gravity inevitably suffer from problems of non-renormalisability. Although it was shown by ’t Hooft and Veltman [1] that pure Einstein gravity itself was finite at one loop, this does not persist at higher loop order [2], nor if (generic) matter is included. The demonstration by Stelle [3] in the 1970s that the non-renormalisability problem itself could be overcome by adding quadratic curvature terms to the action opened the door to many investigations of such theories, although the realisation that renormalisability could only thereby be achieved at the price of introducing ghost states of negative norm, or energies that are unbounded below, dampened the enthusiasm for the idea. Subsequently, with the development of string theory, the idea of higher-derivative corrections to gravity acquired a new impetus, but now in the framework of an effective field theory in which quadratic curvature corrections would represent just the start of an infinite sequence of higher-order terms. Within this framework, a focus on any particular lower-order term or terms in the infinite sequence of corrections would seem to be unjustified, since in circumstances where such a term or terms leads to substantial modifications to the solutions or spectrum of the theory, yet-higher order terms that are being ignored would have at least as important an effect.

Although the conviction that the ghost states or unbounded negative energies of a finite-order higher derivative theory are sufficient to rule them out of consideration is widespread, it is perhaps still worthwhile to study them in more detail, and to investigate whether the ostensible problems really are as severe as is commonly believed. One of the earliest investigations of higher-derivative field theories was in the classic work of Pais and Uhlenbeck [4], who studied scalar field theories in Minkowski spacetime. Much of their focus was on differential operators of infinite order, such as the exponential of the d’Alembertian, but they did also consider operators of finite order, such as powers of the d’Alembertian or products of massive Klein-Gordon operators. Many of the features that are encountered in theories such as these will have counterparts in more complicated theories such as higher-derivative theories of gravity.

As a preliminary to studying higher-derivative field theories, much insight can be gained in the simpler case of a classical higher-derivative particle mechanical theory, and its quantum mechanical extension. A simple and much studied example is the Pais-Uhlenbeck oscillator, $(D^{2}-\omega_{1}^{2})(D^{2}-\omega_{2}^{2})x(t)=0$, where $D=d/dt$. Its Hamiltonian, constructed using the Ostrogradsky procedure [5], which corresponds essentially to the difference between standard oscillator Hamiltonians for particles with frequencies $\omega_{1}$ and $\omega_{2}$, is unbounded both above and below. At the level of the free theory this is not a problem, but a widespread intuition is that once interactions are included, instabilities will inevitably set in since the energies in the negative energy modes and the positive energy modes can both increase (in magnitude) while the total energy remains constant.111Although this intuition is commonly attributed to Ostrogradsky, either as the “Ostrogradsky Instability” or “Ostrogradsky’s Instability Theorem,” it appears that Ostrogradsky himself never actually discussed the question of instabilities in higher-derivative theories. A number of studies have show that this need not in fact be the case [7, 8, 9, 6]. For example, for given initial conditions, an added $\lambda x^{4}$ interaction term will not give instabilities provided $\lambda$ is sufficiently small [6]. And if one instead adds a bounded potential, such as $\lambda\sin^{4}x$, the evolution of initial data is unconditionally stable [6]. For a consideration of of the Pais-Uhlenbeck oscillator from the point of view of statistical physics, see [10].

It has often been argued (see, for example, [11]) that instability problems in a higher-derivative field theory will be much more severe than in a particle mechanical theory, because of the diverging entropy associated with the infinity of unobserved high-momentum states, and that this would lead to an instantaneous decay of the vacuum. Countering this, it has been argued that the divergent entropy results from integrating over the phase space of unobserved decay products, and that in a closed system such as our universe there exist no external observers who would integrate over this infinite phase space. Actual observers, inside the universe, would implicitly have measured the momenta of the particles, and so there would be no diverging phase space integrals leading to instantaneous vacuum decay [12]. It has, further, been argued that the experience with adding interactions to a higher-derivative particle mechanics model, where potentials that are bounded both above and below do not give rise to any instabilities, suggests that similarly bounded potentials in a higher-derivative field theory may not in any case give rise to instabilities [12]. There have also been proposals, such as that by Hawking [13], that one might impose boundary conditions to eliminate ghost excitations, at least in asymptotic states, since the concomitant acausality could be at sufficiently short time scales that it may not be observable in a Wheeler-DeWitt framework. More recently, Hawking and Hertog [14] argued that in a path-integral treatment in a Euclidean framework, one can obtain well-defined rules for calculating probabilities, after continuing to Lorentzian spacetime, even in a higher-derivative field theory.

In the light of some of these investigations, we shall take the view that it is still of considerable interest to study the properties of higher-derivative field theories, and that scalar theories of this kind may be useful as models for more complicated systems such as higher-derivative gravities. Most of the previous work on higher-derivative scalar field theories has been in a flat Minkowski spacetime background. Our main purpose, in this paper, is to consider scalar theories of the kind studied by Pais and Uhlenbeck, but now within the framework of curved spacetime backgrounds. We shall show how the calculation of the Belinfante-Rosenfeld energy-momentum tensor can be greatly simplified by introducing an auxiliary field formalism, and we shall also make comparisons with the canonical flat spacetime energy-momentum tensor calculated using Noether methods. We also obtain some general results for the symplectic Noether currents that can be used in order to construct the norms on states in the theories. One further motivation for studying higher-derivative scalar field theories in curved backgrounds is that, as we shall discuss later, this can actually help to mitigate some of the instabilities that might otherwise occur in flat spacetime.

The organisation of the paper is as follows. We begin in section 2 with a review of some basic aspects of the classical and quantum properties of a point particle moving in one dimension, governed by a higher-derivative equation of motion. Our discussion includes a review of the Ostrogradsky [5, 15] construction of the Hamiltonian describing the system, and an elegant analysis by Smilga [9] of the quantum theory of a 4th-order Pais-Uhlenbeck oscillator. In section 3 we review some properties of a scalar Pais-Uhlenbeck field theory in a Minkowski background, including the construction of the canonical energy-momentum tensor and the conserved symplectic current. In section 4 we extend our discussion to the scalar Pais-Uhlenbeck field theory in a curved spacetime background, showing how one can use an auxiliary field formulation in order to facilitate the construction of the Belinfante-Rosenfeld energy-momentum tensor, which allows a consistent coupling to the gravitational field. By this means, one avoids the necessity of varying the metrics in high powers of the covariant derivative.

In section 5 we discuss the quantisation of higher derivative scalar field theories using Euclidean methods. This may be done if the background metric admits an analytic continuation containing a real section on which the metric is positive definite, as it does for real tunnelling metrics [16]. It may also arise in a full blown Euclidean quantum gravity path integral calculation when a gravitational instanton saddle point admits a reflection map, i.e. an involutive isometry fixing a separating hypersurface. We generalise previous arguments [13] showing that if one adopts the Osterwalder and Schrader prescription [17] for constructing the quantum mechanical Hilbert space, then Reflection Positivity is not satisfied despite the Euclidean action of the scalar field being positive definite. In other words, negative norm states are inevitable.

In section 6, subject to various assumptions about the scalar potential, if present, we prove non-existence results for static solutions of the equations of motion in static or stationary spacetimes. The background could be Minkowski spacetime, in which case these results rule out stable solitons and unstable sphaleron type solutions. They also hold in globally stationary backgrounds. They are easily extended to cover the case of static black hole backgrounds with regular event horizons.

In section 7, we consider Pais-Uhlenbeck field theories in a de Sitter spacetime background, and show how Hubble friction can eliminate instabilities that would otherwise occur when nonlinear couplings are present. These examples serve to illustrate the fact that the often-claimed instabilities of higher-order field theories can sometimes be illusory.

2 A Higher Derivative Point Particle

To orient ourselves, we begin by recalling some facts about theories of particles moving in one spatial dimension governed by a Lagrangian $L=L(x,\dot{x},\ddot{x},t)$ depending upon the position $x$, velocity $\dot{x}$ and acceleration $\ddot{x}$. Varying the Lagrangian gives

[TABLE]

The equation of motion is thus

[TABLE]

2.1 Energy and and Momentum Conservation

If the equation of motion holds, then

[TABLE]

Thus, if $L$ does not depend upon $t$, there is a conserved energy

[TABLE]

which may be written as

[TABLE]

If the Lagrangian is translationally invariant, $\frac{\partial L}{\partial x}=0$, we expect that momentum should be conserved. With an eye on the equation of motion (2.4), we define the momentum $p_{x}$ by

[TABLE]

so that equation of motion (2.4) takes the form

[TABLE]

and hence if the Lagrangian is translationally invariant the momentum $p_{x}$ is conserved. In terms of $p_{x}$, the energy is given by

[TABLE]

2.2 Ostrogradsky’s Hamiltonian

Following [5, 15, 4]222For a geometrical treatment, see [18]., we define

[TABLE]

If the “nondegeneracy” condition

[TABLE]

holds, then we may solve for the acceleration as a function of $(x,y,p_{y})$:

[TABLE]

To obtain the Hamiltonian, we take the Legendre transform of the Lagrangian and obtain

[TABLE]

As a concrete example, consider the Pais-Uhlenbeck oscillator, described by the Lagrangian

[TABLE]

Clearly the non-degeneracy condition holds holds as long as $\beta\neq 0$, since

[TABLE]

The equations of motion are

[TABLE]

i.e.,

[TABLE]

We have

[TABLE]

and so the equation of motion implies

[TABLE]

In a general potential $V(x)$ we would have

[TABLE]

Thus Newton’s second law, expressed in terms of momentum is obeyed.

Making the ansatz $x=\Re Ae^{-i\omega t}$ we find

[TABLE]

and thus

[TABLE]

If $0<4\beta\,\Omega^{2}<1$, then all four roots $\omega=(\pm\omega_{1},\pm\omega_{2})$ will be real, distinct and non-vanishing . Useful relations are

[TABLE]

The equation of motion may be written in terms of $\omega_{1}$ and $\omega_{2}$ as

[TABLE]

We write the general solution as

[TABLE]

and therefore

[TABLE]

Since $\omega_{1}\neq\omega_{2}$, there exits a unique solution for $A$ and $B$ for all initial data $(x(0),\dot{x}(0),\ddot{x}(0)$ and $\dddot{x}(0))$. Moreover, this solution is bounded for all time.

Thus the theory is predictive, the initial value problem has a solution for all time and it is well posed: it depends continuously on the initial data.

It is, however, true that the conserved energy

[TABLE]

or equivalently in terms of the canonically conjugate variables, the Hamiltonian

[TABLE]

depends linearly on $p_{x}$, and it is the sum of two positive and two negative squares (if $\beta$ is positive, as discussed above). In fact Hamilton’s equations

[TABLE]

in this case are linear, and one may verify that they have the same characteristic frequencies.

As mentioned in the introduction, it is less obvious what happens if an interaction term is added, rendering the equations of motion non-linear. An example would be a potential term of the form $\lambda x^{4}$, where $\lambda$ is a constant. A widely held intuition is that some sort of instability will result, but, as shown in [6], for given initial conditions the evolution may remain bounded if the $\lambda$ interaction is sufficiently small. The question may also depend crucially on the definition of stability. The motion may always remain within a bounded neighbourhood of the unperturbed motion, $x=0$ for example, even though it may not return asymptotically to $x=0$. Moreover the time scale for any instability may, in the cosmological context, be sufficiently long, for small enough $\lambda$, as not to lead to any observable effect within the relevant time scale.

2.3 Quantisation of the Pais-Uhlenbeck Oscillator

An obvious procedure would be to consider wave functions of the form $\Psi=\Psi(x,y)$, in which case

[TABLE]

leading to

[TABLE]

More revealing would be to consider wave functions of the form $\Psi=\Psi(x,p_{y})$ in which case

[TABLE]

leading to

[TABLE]

The first two terms are the wave equation in two-dimensional Minkowski spacetime with the flat metric

[TABLE]

in which $x$ is a lightlike coordinate.

Smilga [9] succeeded in finding a canonical transformation which clarifies what is going on. Starting with the (rescaled) Lagrangian

[TABLE]

Ostrogradsky’s Hamiltonian becomes

[TABLE]

Now let

[TABLE]

which implies

[TABLE]

This is a canonical transformation, since

[TABLE]

Pulling back the Hamiltonian one finds

[TABLE]

This may be quantised in the obvious way, with a positive norm on the Hilbert space, by introducing annihilation and creation operators in the standard way, with the non-vanishing commutators $[a_{X},a_{X}^{\dagger}]=[a_{Y},a_{Y}^{\dagger}]=1$. We have

[TABLE]

Assuming that the “ground state” satisfies

[TABLE]

all norms are positive, but the spectrum of the Hamitonian is unbounded below.

Alternatively, we could assume that the vacuum is defined by

[TABLE]

in which case the spectrum of the Hamitonian is bounded below but the norm on states would be indefinite. For example, the state $a_{Y}\,|0\rangle$ now has norm

[TABLE]

3 Pais-Uhlenbeck Field Theory in Minkowski Spacetime

In this section we recall some aspects of the simplest non-interacting higher derivative scalar field theories in a Minkowski background spacetime. The field equations are of the form [4]

[TABLE]

where333Throughout the paper we adhere to the same $-+++$ signature convention as [4].

[TABLE]

and $F(u)$ is a real valued function of $u$.

A suitable action for a real scalar $\phi$, up to factors and possible boundary terms, is

[TABLE]

The Lagrangian models studied by Ostrogradsky are obtained by replacing $\square\phi$ in (3.1) by $-D^{2}$, where $D=\frac{d}{dt}$.

A simple non-trivial example is

[TABLE]

where, unless otherwise stated, we assume that the masses $m_{1}$ and $m_{2}$ are real and distinct. The case when $m_{1}^{2}=m_{2}^{2}$ has some different properties, and is referred to as di-polar. If $\square=-D^{2}$ we obtain the so-called Pais-Uhlenbeck oscillator.

As we shall see, one may always introduce a symplectic current $J^{\mu}(\phi_{1},\phi_{2})=-J^{\mu}(\phi_{2},\phi_{1})$ such that

[TABLE]

Thus on shell,

[TABLE]

and hence a conserved (pre)-symplectic form on solutions $\phi_{1}$ and $\phi_{2}$ of (3.1) is given by

[TABLE]

where $\Sigma$ is any Cauchy surface. (The prefix “pre” would apply if the symplectic form were degenerate.)

The equation of motion (3.1) is clearly linear and if $F(\square)$ is a polynomial of degree $N$, then of necessity quasi-linear,444In other words, all highest derivative terms occur linearly. and hence its characteristic surfaces $S={\rm constant}$ must, by standard theory, satisfy

[TABLE]

In other words the characteristic surfaces are null hypersurfaces. Thus (3.1) satisfies the Einstein-Causality Principle.

Since (3.1) is linear we may substitute in it the Ansatz

[TABLE]

where $A$ and $k_{\mu}$ are constant, and hence find that

[TABLE]

where $k_{\mu}=(-\omega,{\bf k})$. If the zeros of $F(u)$ are at $u=m^{2}_{i}$ with real $m_{i}$, then we have several branches to the dispersion relation:

[TABLE]

3.1 The Energy Momentum Tensor

In the simple case (3.4), an energy-momentum tensor is presented in [19]. Making allowance for the conversion from the $(+---)$ spacetime signature used in that reference, and also our choice for the overall sign for the Pais-Uhlenbeck action, this symmetric tensor is [19]

[TABLE]

It obeys

[TABLE]

and thus is conserved on shell. Moreover if $\phi=\phi(t)$ then

[TABLE]

which agrees with the energy of the Pais-Uhlenbeck oscillator derived using Noether’s theorem.

The derivative terms in (3.12) differ from those in the standard energy-momentum tensor for a scalar field that one obtains by the Belinfante-Rosenfeld [20, 21] procedure (i.e. minimally coupling the action to a metric $g_{\mu\nu}$ and taking the variational derivative). Later we shall construct an energy-momentum tensor using the Belinfante-Rosenfeld procedure, and in appendix A we show how it is related to (3.12).

3.2 Second Quantisation of the Pais-Uhlenbeck Field Theory

Recall that for a free theory we first construct the one-particle Hilbert space and then construct the full (non-separable) Hilbert space using the Fock construction. The standard approach is to complexify the Cauchy data and introduce a (symmetric) sesquilinear form

[TABLE]

where now $\phi_{1}$ and $\phi_{2}$ are complex valued, and

[TABLE]

In the case of just two time derivatives in the equations of motion one has

[TABLE]

If $\phi_{1}=\phi_{2}=Ae^{-i\omega t}$ we have

[TABLE]

Thus positive norm states have $\omega>0$ and so we identify the one-particle Hilbert space with the space of positive frequency Cauchy data.

3.2.1 Symplectic Current for the Pais-Uhlenbeck Field Theory

When $m_{1}^{2}\neq m_{2}^{2}$, the general solution of the Pais-Uhlenbeck equation $(\square-m_{1}^{2})(\square-m_{2}^{2})\phi=0$ is given by

[TABLE]

where $\chi_{1}$ and $\chi_{2}$ are solutions of $(\square-m_{1}^{2})\chi_{1}=0$ and $(\square-m_{2}^{2})\chi_{2}=0$, and $a$ and $b$ are constants. Thus $\chi_{1}$ and $\chi_{2}$ provide a basis for the general solution. Without loss of generality, we shall assume $m_{1}^{2}>m_{2}^{2}$ in what follows.

We now write the field equation as

[TABLE]

In Minkowski spacetime we have the following identities

[TABLE]

Thus if

[TABLE]

then on shell

[TABLE]

Now suppose that $\phi_{1}$ and $\phi_{2}$ are any two solutions of

[TABLE]

where each of $\mu_{1}^{2}$ and $\mu_{2}^{2}$ can be either $m_{1}^{2}$ or $m_{2}^{2}$. Then

[TABLE]

Evidently if $\phi_{1}$ and $\phi_{2}$ have different masses (i.e. if $\mu_{1}^{2}=m_{1}^{2}$ and $\mu_{2}^{2}=m_{2}^{2}$, or else if $\mu_{1}^{2}=m_{2}^{2}$ and $\mu_{2}^{2}=m_{1}^{2}$), then $J^{\mu}=0$.

If $\phi_{1}$ and $\phi_{2}$ have equal mass-squared $\mu_{1}^{2}=\mu_{2}^{2}=m_{1}^{2}$ then

[TABLE]

while if $\phi_{i}$ and $\phi_{2}$ have equal mass-squared $\mu_{1}^{2}=\mu_{2}^{2}=m_{2}^{2}$ then

[TABLE]

If we choose the complex modes with mass $m_{1}$ to have positive frequency and the complex modes with mass $m_{2}$ to to have negative frequency, the Hermitian form $h(\cdot\,,\cdot)$ will be positive definite and diagonalised in this basis.

Thus, despite the fact that we associate one creation operator with a positive frequency mode and the other with a negative frequency mode, we obtain a positive norm on our Hilbert space. There are no states with negative norm (sometimes called “ghosts”). Of course if $m_{2}^{2}>m_{1}^{2}$ we must reverse the convention.

4 Pais-Uhlenbeck Field Theory in Curved spacetime

4.1 Simplest model

We now replace $\partial_{\mu}$ by $\nabla_{\mu}$, the Levi-Civita covariant derivative, and so $\square=\nabla_{\mu}\nabla^{\mu}$, the generally-covariant d’Alembertian. The field equation is

[TABLE]

The field equation may be derived from the action

[TABLE]

from which one finds that under a variation of $\phi$,

[TABLE]

The operator $E$ may also be written as

[TABLE]

The identities

[TABLE]

imply the formal self-adjointness of $E$ since, from them, one deduces that

[TABLE]

where

[TABLE]

If $\chi$ and $\phi$ are two solutions of the equation of motion (4.1) then one deduces that

[TABLE]

and hence that the (pre-)symplectic form $\omega(\chi,\phi)$ on the space of solutions of (4.1), given by

[TABLE]

where $\Sigma$ is a Cauchy surface, is conserved.

Now we suppose that $\chi=\phi_{1}$ satisfies $\square\,\phi_{1}=\mu_{1}^{2}\,\phi_{1}$ and $\phi=\phi_{2}$ satisfies $\square\,\phi_{2}=\mu_{2}^{2}\,\phi_{2}$, where each of $\mu_{1}^{2}$ and $\mu_{2}^{2}$ can each be either $m_{1}^{2}$ or $m_{2}^{2}$. Then

[TABLE]

Thus the space of Cauchy data and the symplectic form splits as a direct sum of two summands, labelled by the mass-squared $m_{1}^{2}$ and $m_{2}^{2}$.

4.1.1 Auxiliary field formulation

A convenient way to handle the higher-derivative terms in the Pais-Uhlenbeck equation is by introducing auxiliary fields. In particular, as we shall see below, this provides a simple way to calculate the contributions to the energy-momentum tensor from the higher-derivative terms. We shall employ two related, but slightly different, approaches in this paper. For now, we shall handle the various higher-derivative terms that arise in a general Pais-Uhlenbeck theory separately, by developing an auxiliary-field construction for a Lagrangian ${\cal L}_{n}=-{1\over 2}\sqrt{-g}\,\phi\,(-\square)^{n}\,\phi$. The general Pais-Uhlenbeck Lagrangian is then given as a linear combination of such terms with certain constant coefficients. The energy-momentum tensor for the PU theory is then similarly given by the analogous linear combination of the associated energy-momentum tensors. In appendix B, we discuss a slightly different approach, in which the Lagrangian for the full PU field theory is directly rewritten by introducing auxiliary fields.

In the present simplest example we are considering, for which the PU action is given by (4.2), there is just the one higher-derivative term, namely

[TABLE]

We now introduce an auxiliary field $\psi$ and replace (4.12) by

[TABLE]

The variation of (4.13) with respect to $\psi$ yields

[TABLE]

and upon substituting this back into (4.13) one indeed recovers (4.12). The variation of (4.13) with respect to $\phi$ gives

[TABLE]

whence we obtain the field equation $\square^{2}\,\phi=0$ for $\phi$, which is indeed the same as the equation that arises directly from (4.12).

4.1.2 Energy-momentum tensor

Up to an integration by parts the action (4.13) is equivalent to

[TABLE]

Now the associated Belinfante-Rosenfeld energy-momentum tensor $T^{{\scriptscriptstyle(2)}}_{\mu\nu}$ is given by

[TABLE]

Substituting (4.14) in (4.18) therefore gives

[TABLE]

as the contribution to the energy-momentum tensor of the Pais-Uhlenbeck theory from the $(\square\,\phi)^{2}$ term in (4.2). One may verify that

[TABLE]

and so indeed this contribution to $T_{\mu\nu}$ is covariantly conserved if the field equation $\square^{2}\,\phi=0$ holds.

Combining the contribution $T_{\mu\nu}^{{\scriptscriptstyle(2)}}$ with the contributions from the remaining terms in (4.2), which are straightforward to calculate in the standard way, we find that the total energy-momentum tensor for this PU theory is given by

[TABLE]

However it should be noted that $T_{\mu\nu}$ is different, in its flat-spacetime limit, from the expression for $\bar{T}_{\mu\nu}$ given by (3.12). In appendix A we discuss a general derivation of the canonical energy-momentum tensor derived by the Noether method in flat spacetime, and its relation to the energy-momentum tensor that we obtained in (4.21) by using the Belinfante-Rosenfeld procedure.

Interestingly, although the flat space limit of (4.21) is not the same as (3.12), it gives the same expression (3.14) if one specialises to $\phi=\phi(t)$.

4.2 Auxiliary Fields

4.2.1 Powers of Laplace/d’Alembert operator

Consider now the $n$’th power of the d’Alembert operator $\square=\nabla^{2}$. By introducing auxiliary fields $\chi_{k}$ and Lagrange multipliers $\eta_{k}$, for $k=1,\ldots,n-1$, we can re-write

[TABLE]

Indeed, varying with $\eta_{k}$ we find

[TABLE]

On the other hand, varying with respect to $\chi_{k}$ we find that

[TABLE]

This shows that we may identify

[TABLE]

This identification reduces the number of extra fields by 2. We have, for instance,

[TABLE]

If we also define $\chi_{0}\equiv 0$ and $\chi_{n}\equiv\phi$, then after making the identifications (4.25) we see that

[TABLE]

Note that combined with the solutions (4.25) for the auxiliary fields, we have

[TABLE]

Consider the Lagrangian

[TABLE]

whose equations of motion are

[TABLE]

As established above, this can be written, after integrations by parts, as

[TABLE]

with the auxiliary fields given by

[TABLE]

Calculating the energy-momentum tensor $T_{\mu\nu}=-(2/\sqrt{-g})\,\delta{\cal L}/\delta g^{\mu\nu}$, we thus find

[TABLE]

and hence, using (4.32),

[TABLE]

(Note that the first term is in fact automatically symmetric in $\mu$ and $\nu$, in view of the summation over $k$.) The first few examples, for $n=1$, 2 and 3, are:

[TABLE]

4.2.2 Auxiliary Fields for General Higher-Order Operators

Consider the higher-derivative Lagrangian

[TABLE]

where the operators $A_{i}$ are formally self-adjoint and need not necessarily be mutually commuting. An example for which they are mutually commuting is

[TABLE]

Paralleling the procedure for introducing auxiliary fields for the Lagrangian $L_{n}=\phi\Delta^{n}\phi$, consider here

[TABLE]

Varying the $\eta_{i}$ fields gives

[TABLE]

Varying the $\chi_{i}$ fields gives

[TABLE]

Unlike in the case of the pure $\square^{n}$ Lagrangian, we cannot equate the set of $\eta_{i}$ fields with the set of $\chi_{i}$ fields.

It is easily seen that the Lagrangian (4.38) gives rise to (4.36) after plugging in the solutions for the $\chi_{i}$ fields, given in (4.39). Note that if we define

[TABLE]

then the Lagrangian (4.38) can be written as

[TABLE]

Note that together with the solutions (4.39) and (4.40) for the auxiliary fields, we have555The discussion we gave in section 4.2, where we introduced auxiliary fields for the Lagrangian $\phi\,\square^{n}\,\phi$, is a special case of the our discussion here, in which $A_{k}=\square$ for all $k$. As can be seen from (4.39) and (4.40), in this special case one can equate $\eta_{k}=\chi_{n-k}$ as in section 4.2. But in the more general case discussed in this section, one cannot equate the $\eta$ and $\chi$ fields, even in the case that the operators $A_{i}$ commute. If, however, the operators $A_{i}$ take the form given in (4.37), where the differential operator parts of all the $A_{i}$ are the same, then there exist purely algebraic relations between the set of $\eta_{k}$ fields and the set of $\chi_{k}$ fields, as can be seen from (4.43). In this case, therefore, one needs only $\phi$ and either the set of $\chi_{k}$ or the set of $\eta_{k}$ as auxiliary fields.

[TABLE]

If we consider the example where $A_{i}$ is given by (4.37), the energy-momentum tensor that follows from (4.42) is

[TABLE]

4.3 Symplectic Current in Higher-Derivative Scalar Theory

Consider first the Lagrangian ${1\over 2}\phi\square^{n}\phi$, whose equation of motion is

[TABLE]

By extracting derivatives in sequence from

[TABLE]

we obtain

[TABLE]

where we have defined

[TABLE]

Thus, if $\phi$ and $\psi$ are any solutions of $\square^{n}\phi=0$ and $\square^{n}\psi=0$, we have a conserved current

[TABLE]

Consider now the Lagrangian, and consequent equations of motion,

[TABLE]

If we assume the masses are all unequal, the general solution is a linear combination of “elementary solutions” satisfying

[TABLE]

for any $i$ in the range $1\leq i\leq N$.

We now consider the conserved current $J_{\Delta_{N}}^{\mu}$ for the operator $\Delta_{N}$ defined in (4.50). If we write the operator as

[TABLE]

then this current is given by

[TABLE]

where $J^{\mu}_{\square^{n}}(\psi,\phi)$ is defined in (4.49).

There are two cases of particular interest to consider. The first is when $\psi$ and $\phi$ are two “elementary” solutions of $\Delta_{N}$, as defined in (4.51), corresponding to two different $m_{i}^{2}$ values. Without loss of generality we may take $\psi=\phi_{1}$ and $\phi=\phi_{2}$, that is to say they satisfy

[TABLE]

Writing the operator $\Delta_{N}$ as

[TABLE]

with

[TABLE]

and noting that

[TABLE]

we see that

[TABLE]

Collecting the terms in the square brackets as a sum $\sum_{p=0}^{k-1}$ together with the additional terms from the first two summations we immediately find that the total vanishes for each $k$. Thus we find

[TABLE]

whenever $\psi$ and $\phi$ are elementary solutions with different mass values.

The other case of interest is when $\psi$ and $\phi$ are solutions with the same mass value. Without loss of generality, we may consider the case where $\psi$ and $\phi$ satisfy

[TABLE]

This means that $J^{\mu}_{\square^{n}}(\psi,\phi)=m_{1}^{2n-2}\,J^{\mu}_{\square}(\psi,\phi)$. Writing the operator $\Delta_{N}$ as

[TABLE]

with

[TABLE]

we find that

[TABLE]

In other words, when $\psi$ and $\phi$ are solutions with the same mass, as in (4.60), the conserved current is given by

[TABLE]

where $J^{\mu}(\psi,\phi)$ is the usual current

[TABLE]

In the general case when $\square\psi=y^{2}\psi$ and $\square\phi=x^{2}\phi$, we have

[TABLE]

and

[TABLE]

From (4.49) we therefore have

[TABLE]

and hence we have

[TABLE]

5 Green Functions and Euclidean Formulation

Pais and Uhlenbeck solved for the time-independent Green function $G^{(4)}({\bf r})$ for the 4th-order operator in flat space, defined by

[TABLE]

where ${\bf r}={\bf r}_{1}-{\bf r}_{2}$, by using Fourier transforms. Since

[TABLE]

the formal solution666Pais and Uhlenbeck omit the denominator $(m_{2}^{2}-m_{1}^{2})$. is

[TABLE]

where $r=|{\bf r}|$. $G^{(4)}({\bf r})$ goes to a constant value at the origin, and it admits a Taylor expansion in powers of $r$. However, because the expansion includes odd powers of $r$ it is not differentiable with respect to the Cartesian coordinates at the origin $r=0$, and indeed for this reason it does in fact satisfy (5.1) with the delta function on the right-hand side. This may be easily verified by integrating (5.1) over a small spherical volume centred on the origin, and using the divergence theorem to turn the derivative terms into surface integrals. Note that $G^{(4)}({\bf r})$ is symmetric under the interchange of $m_{1}$ and $m_{2}$. Both $(-\nabla^{2}+m_{1}^{2})$ and $(-\nabla^{2}+m_{2}^{2})$ are positive operators, and their Green functions $\frac{e^{-m_{1}r}}{4\pi r}$ and $\frac{e^{-m_{2}r}}{4\pi r}$ are both positive. The operator $(-\nabla^{2}+m_{1}^{2})(-\nabla^{2}+m_{2}^{2})$ is positive and, as one may check, $G^{(4)}({\bf r})$ is also positive.

The Green function (5.3) is unique, since there are no non-singular solutions of the homogeneous equation

[TABLE]

that fall off sufficiently fast at infinity. This is easily established by multiplying (5.4) by $\phi$ and integrating over ${E}^{3}$. Integrating by parts gives

[TABLE]

whence $\phi=0$.

Similar conclusions hold in four dimensional Euclidean space, but the explicit form of the Green function is more complicated since it involves Bessel functions. Concretely, the Euclidean Green function $G_{E}(r)$ for the 2nd-order Klein-Gordon operator, obeying $(-\nabla^{2}+m^{2})G_{E}(r)=\delta({\bf r})$, is given by

[TABLE]

where $K_{\nu}(x)$ is the modified Bessel function of the second kind, which has the integral representation

[TABLE]

At small $r$ the Green function $G_{E}({\bf r})$ has the expansion

[TABLE]

where $\gamma$ is the Euler-Mascheroni constant. For the 4th-order Pais-Uhlenbeck operator, the corresponding Green function $G_{E}^{(4)}(r)$ obeys (5.1) (with $\nabla^{2}$ now understood to be the four-dimensional Euclidean Laplacian), and is given by

[TABLE]

One may easily verify, for example by using (5.7), that both (5.8) and (5.9) are positive. (Note that the $r^{-2}$ singularity in (5.8) is cancelled in (5.9).) Because of the positivity of $G_{E}^{(4)}({\bf r})$, one might be led to believe that the Osterwalder-Schrader construction of the quantum field theory could be carried out with no negative norm states. This however is not the case [13], as we shall discuss below.

5.1 Reflection Positivity in the 4th-Order Case

If we work on a Riemannian manifold the Euclidean action functional

[TABLE]

is positive definite. Thus the Euclidean functional integrals should make sense and the associated two-point function or Green function

[TABLE]

being the inverse of the positive operator, will be positive.

In general, given a Riemannian manifold there is no Lorentzian manifold associated with it by analytic continuation. If there is, one may analytically continue the two-point function and ask what, if any, is its physical significance. In the case of Euclidean space $\mathbb{E}^{4}$ it is well known that one obtains the Wightman function associated with the vacuum state. If one periodically identifies one of the coordinates with period $\beta$, one obtains expectation values in the Gibbs state at temperature $T=\frac{1}{\beta}$. In fact using a construction due to Osterwalder and Schrader one need not depart from Euclidean space in order to construct the quantum mechanical Hilbert space. This will be possible as long as the Green function satisfies the requirement of Reflection Positivity [17].777For a helpful introduction the reader may consult [22] It has been suggested by a number of authors [23, 24, 25, 26, 27, 28, 29, 30, 31] that this construction may be generalised to a restricted class of Riemannian manifolds, including globally static spacetimes, which more generally may be characterised as “Real Tunnelling Geometries” [16].

One requires that the Riemann manifold admit a reflection symmetry $\theta$, an involutive isometry which leaves fixed a two-sided hypersurface $\Sigma$ such that

[TABLE]

where $\sqcup$ denotes the disjoint union. Let $V$ be the space of complex valued functions that vanish outside $M_{+}$. If $|f\rangle\in V$ and ${\bar{f}}^{\star}_{\theta}={\bar{f}}(\theta(x))$ is the pull-back of its complex conjugate under $\theta$, we define

[TABLE]

If the right hand side of (5.13) is positive definite then the left hand side provides a Hermitian inner product on $V$ which may be identified with the one-particle Hilbert space.

If

[TABLE]

then this reflection positivity condition is satisfied [30]. However if

[TABLE]

then reflection positivity is not satisfied [13]. To see this, suppose $m_{1}^{2}>m_{2}^{2}$ and let

[TABLE]

where $h\in V$. Then

[TABLE]

since ${\bar{f}}(\theta x)$ and $h$ have disjoint support. Thus states of the form $|(-\nabla^{2}+m_{2}^{2})h\rangle$ have negative norm squared.

5.2 $2N$’th-Order Case

For the $2N$’th-order operator we have

[TABLE]

where

[TABLE]

Proceeding as before, and considering

[TABLE]

we shall have

[TABLE]

This can have either sign, depending on the sign of $C_{i}$, and this depends upon the relative values for the masses $m_{i}$. For example, if we order the masses so that

[TABLE]

then $C_{N}$ is positive, and the signs of the $C_{i}$ alternate as $i$ decreases. Thus there will always be states of negative norm as well as states of positive norm.

It is interesting to note that if we construct the static Green function $G^{(2N)}({\bf r})$ generalising (5.3) to the $2N$’th-order operator, we obtain

[TABLE]

and that this is finite at $r=0$ for all $N\geq 2$. In fact, one can see that888These results can be proved by considering the contour integrals

$\displaystyle\oint_{C}dz\,z^{p}\,\prod_{i=1}^{N}\frac{1}{(z-m_{i}^{2})}$

for $0\leq p\leq N-1$, where the contour $C$ is taken to be a circle of radius $R$ centred on the origin, in the limit where $R$ is sent to infinity.

[TABLE]

and that

[TABLE]

This implies that the Taylor expansion of $G^{(2N)}({\bf r})$ in (5.24) has the form

[TABLE]

where the constant $a_{k}$ coefficients depend on the masses $m_{i}$. The non-analyticity of the Green function, as a function of the Cartesian coordinates, is associated with the occurrence of odd-integer powers of $r$ in the Taylor expansion (5.27). The first such term, for the $2N$’th-order operator, occurs at the order $r^{2N-3}$.

This softening of the Green function at short distance is analogous to the softening encountered in Pauli-Villars regularisation [32].

The Euclidean Green function $G_{E}^{(2N)}({\bf r})$ for the $2N$’th-order Pais-Uhlenbeck operator can also be calculated using the same formalism as above. From (5.6), we see that

[TABLE]

The identities (5.25) and (5.26) again imply that the non-analytic behaviour of $G_{E}^{(2N)}({\bf r})$ (in the form of the $\log(m_{i}\,r/2)$ terms) is deferred to an increasingly higher order in a small-$r$ expansion as $N$ increases.

5.3 Higher Dimensional Green Functions

The Green function $G_{E}({\bf r})$ in $d$-dimensional Euclidean space, obeying $(-\nabla^{2}+m^{2})G_{E}({\bf r})=\delta({\bf r})$, is given by

[TABLE]

The cases we discussed previously correspond to $d=3$, for the static Green function in four-dimensional spacetime, and $d=4$, for the four-dimensional Euclidean Green function. The analysis of the Euclidean Green functions for the higher-order Pais-Uhlenbeck operators proceeds in the same manner as in the cases we discussed previously. As in those cases, the onset of non-analytic behaviour in the Green functions for the higher-order Pais-Uhlenbeck operators, namely

[TABLE]

is deferred to increasingly higher powers in the small-$r$ expansion as $N$ increases.

6 No-Hair Theorems for Static and Stationary Solutions

In this section, we address the question of whether there could exist regular static solutions to certain classes of higher-derivative field theories. First, we consider field theories of the Pais-Uhlenbeck type. Then, we consider a class of $2N$ order field theories with a general potential $V(\phi)$.

6.1 Pais-Uhlenbeck Field Theories

Our discussion here applies rather generally to the situation where we have a static asymptotically flat background metric, regular outside a regular event horizon, in which a scalar field which is also static satisfies the Pais-Uhlenbeck equations

[TABLE]

For concreteness, we shall first consider the fourth-order example (4.1). If we integrate over the domain $D$ obtained by taking an initial spacelike surface $\Sigma_{(i)}$ extending from infinity to the horizon, and moving it up the orbits of the static Killing vector field $K^{\mu}$ which coincides on the horizon with its null generators, to a final surface $\Sigma_{(f)}$, and then use (4.6) with $\chi=\phi$, we find that

[TABLE]

Now $\partial D$ has four boundary components: One is at infinity, $\partial D_{\infty}$; one is on the horizon, $\partial D_{H}$; one is the initial surface $\partial D_{i}=\Sigma_{(i)}$; and the fourth is the final surface $\partial D_{f}=\Sigma_{(f)}$. The integrals over $\partial D_{i}$ and $\partial D_{f}$ cancel one another, since, by construction, $\Sigma_{(f)}$ is obtained by carrying $\Sigma_{(i)}$ along the orbits of the Killing vector $K^{\mu}$. The integrand on $\partial D_{H}$ vanishes because $d\Sigma_{\mu}$ on $H$ is proportional to $K^{\mu}$ and $K^{\mu}\partial_{\mu}\phi=K^{\mu}\partial_{\mu}\nabla^{2}\phi=0$. The integral over $\partial D_{\infty}$ vanishes by the boundary conditions on $\phi$.

Thus the left-hand side of (6.2) vanishes. But, unless $m_{1}^{2}m_{2}^{2}=0$, the integrand is positive definite. Thus in this case $\phi=0$. If it happens that $m_{1}^{2}m_{2}^{2}=0$ but $m_{1}^{2}+m^{2}_{2}\neq 0$, we may only deduce that $\phi={\rm constant}$. If $m_{1}^{2}+m^{2}_{2}=0$ we deduce that $\nabla^{2}\phi=0$ and hence, by a standard argument, that $\phi={\rm constant}$ in this case as well. Thus we have shown that there can exist no regular, static solutions for the scalar field $\phi$ in a static black hole background. A special case of this result is when there is no black hole, and the spacetime is just flat Minkowski spacetime.

The above argument generalises immediately to the $2N$-order Pais-Uhlenbeck theory (6.1), since by appropriate integrations by parts, the analogue of the left-hand side in (6.2) will consist of a sum of squares with positive coefficients. The analogue of the surface terms on the right-hand side will again vanish, by virtue of the assumption of staticity and the imposition of an appropriate boundary condition on $\phi$ at infinity. Thus again, one concludes that there can be no non-trivial static solutions for $\phi$ in flat space or a regular static black hole background.

The result also immediately generalises to higher spacetime dimensions.

One may also consider adding a scalar potential $V(\phi)$. If this is such that $\phi\,V^{\prime}(\phi)$ is everywhere non-negative, then one will again be able to show that $\phi$ must vanish for a static solution.

6.2 Higher-Order Theories with a General Potential

In this section, we suppose that $\phi$ is static and obeys

[TABLE]

in a curved $d$-dimensional spacetime. Differentiating with respect to $\partial_{\mu}$ and then contracting with $\partial^{\mu}\phi$ implies

[TABLE]

By extracting derivatives sequentially, this can be written in the form

[TABLE]

if $n$ is odd, or as

[TABLE]

if $n$ is even.

If we assume that the solution $\phi$ is static, and that the metric is static, then we can proceed as in section 6.1, by integrating (6.5) or (6.6) over a portion of spacetime bounded by initial and final surfaces $\Sigma_{(i)}$ and $\Sigma_{(f)}$; an horizon $H$ and a boundary at infinity. For the same reasons as in section 6.1, the boundary contributions on $\Sigma_{(i)}$ and $\Sigma_{(f)}$ will cancel, the term on the horizon will vanish, and the contribution at infinity will give zero subject to appropriate fall-off conditions for $\phi$. We may then conclude that if potential is convex, i.e. $V^{\prime\prime}\geq 0$ everywhere, then $\phi$ must be a constant.

By a straightforward extension of the argument above, we can similarly conclude that there are no non-trivial static solutions to the equation

[TABLE]

where

[TABLE]

6.3 Virial Theorem

A different technique for ruling out static solutions under certain assumptions about the potential is to consider the virial theorem.

Calculating the spatial trace $T_{ii}$ of $T_{\mu\nu}$, given by (4.34), in a $d$-dimensional Minkowski background for a purely static field, we find that

[TABLE]

where t.d. denotes total derivatives, and

[TABLE]

(Because of the assumption that $\phi$ is static, $\square$ is just $\partial_{i}\partial_{i}$ here.) Similarly, the energy density is given by

[TABLE]

Note that the total energy $\int d^{d-1}x\,T_{00}$ is always positive provided that $V$ is non-negative.

Because, for a general energy-momentum tensor, $\partial_{i}T_{ij}=0$ for a static field in Minkowski spacetime, we have $\partial_{i}(x_{j}\,T_{ij})=T_{ii}$. Integrating this over the $(d-1)$-dimensional space, and assuming that the fall-off conditions for $\phi$ are such that the surface term gives zero, it follows that we must have

[TABLE]

and hence, from (6.9),

[TABLE]

(The same result may be obtained by using Derrick’s scaling argument [33], and demanding that the total energy $\int d^{d-1}x\,T_{00}$ be extremised for any static solution.)

We see from (6.13) that if $V$ is positive and $d>2n+1$, there can be no static solutions. Similarly, if $V$ is negative and $d<2n+1$, there can again be no static solutions.

6.4 Stationary Metrics

The results of sections 6.1 and 6.2 remain valid for globally stationary metrics; that is, metrics admitting an everywhere timelike Killing vector field $K^{\mu}$. This is because the condition $K^{\mu}\,\partial_{\mu}\phi=0$ implies that $\partial_{\mu}\phi$ is spacelike, or zero, and so $g^{\mu\nu}\,\partial_{\mu}\phi\,\partial_{\nu}\phi\geq 0$. In the case that $K^{\mu}$ is not everywhere timelike, in other words an ergoregion is present, the argument just given will not necessarily be valid. However, if the metric is both stationary and axisymmetric, and $\phi$ is assumed to be both stationary and axisymmetric, the argument will go through.

7 Stabilisation By Hubble Friction

Following the work of [34], we have studied the behaviour of the Pais-Uhlenbeck oscillator in a background de Sitter universe. We choose coordinates in which the metric is given by

[TABLE]

where $H=\sqrt{\Lambda/3}$ and $\Lambda$ is the cosmological constant. The non-linear 4th-order Pais-Uhlenbeck equation of motion for a spatially homogeneous scalar field $\phi(t)$ with a potential $V(\phi)$ is

[TABLE]

In the absence of the potential, we have two uncoupled damped simple harmonic oscillators, with the solution

[TABLE]

as long as $m_{i}^{2}>9H^{2}/4$.

If we first set $H=0$ and consider, as an example, a potential $V=\frac{1}{4}\lambda\,\phi^{4}$, we can verify Pavsic’s observation [6] that for given initial data, the solution in Minkowski spacetime remains bounded provided that $\lambda$ is sufficiently small. If we choose a larger value of $\lambda$ such that the solution becomes unstable, we can then verify, by numerical analysis, that turning on $H$ can render it stable again, and in fact $\phi$ then decays to zero at large $t$ (i.e. within a few Hubble times).

One may also consider inhomogeneous solutions of the free 4th-order equation, with wave vector ${\bf k}$. The equation of motion is obtained by replacing $m_{i}^{2}$ by $m_{i}^{2}+{\bf k}^{2}\,e^{-2Ht}$ in (7.2). The additional term decays to zero rapidly, and so we would expect that our results in the non-linear situation will remain valid. A complete treatment would require further investigation going beyond the ODEs we studied here.

Our results will be qualitatively similar for all the higher-order Pais-Uhlenbeck oscillators.

8 Conclusion

In this paper we have studied a scalar field $\phi$ coupled to a fixed background metric $g_{\mu\nu}$, for which the scalar equation of motion contains derivatives of arbitrary order. In all the examples considered, the equations of motion are linear in $\phi$ apart from non-linearities $V^{\prime}(\phi)$ introduced by a possible potential term $V(\phi)$. It follows that classically the propagation is causal, that is, the characteristic surfaces across which discontinuities may propagate are null hypersurfaces with respect to the metric $g_{\mu\nu}$. This is despite the fact that the various energy-momentum tensors, such as the canonical or the Belinfante-Rosenfeld energy-momentum tensor, do not satisfy the dominant energy condition.

We have obtained general expressions for both the canonical and the symmetric Belinfante-Rosenfeld energy-momentum tensors, in the latter case by introducing auxiliary fields. This formulation greatly simplifies the calculations, since it eliminates the need to vary the metrics in high-order covariant derivatives of the scalar field. The Belinfante-Rosenfeld energy-momentum tensor is required for a consistent coupling of the scalar field to the gravitational field. We have also obtained general expressions for the conserved symplectic current used in quantising the scalar field. Both exhibit the well known difficulty that the energy-momentum tensors will not not satisfy the usual energy conditions, and the sesquilinear inner product defined using the symplectic current will not be positive definite.

Thus, if the standard definition of “positive frequency” is adopted, the quantum mechanical norm on states will be indefinite. On the other hand, swapping the sign in the definition of “positive frequency” for the modes carrying negative energy results in physical states carrying negative energy.

To investigate this further we have considered Euclidean formulations of the quantised theory in which the metric $g_{\mu\nu}$ is taken to be positive definite. Assuming that the Riemannian manifold admits a reflection map decomposing the manifold into two disjoint domains symmetric about a separating hypersurface, we have investigated whether Osterwalder and Schrader’s Reflection Positivity condition holds. We show that this is never the case for the theories we study, confirming earlier studies indicating that the Euclidean approach to these theories leads inevitably to the presence of “ghosts,” that is states of negative norm.

Finally, we studied, without assuming spherical symmetry, the possible existence of non-singular finite energy static solutions of our equations on static or stationary backgrounds, and were able to rule out such solutions subject to various assumptions on the potential $V(\phi)$. These results encompass the important cases of soliton solutions in flat space, bounce solutions inducing the decay of false vacuua, and scalar hair for black holes.

Although the implications of higher-derivative theories for quantum gravity at the microscopic level remain unclear, the work of [7, 8, 9, 6, 12], and the cosmological studies of [34], indicate that there is still much to be understood about the behaviour of such higher-derivative theories at the macroscopic, classical, level. After all, the discoveries of Slipher, Friedmann, Lemaître and Hubble, Riess et al. [35, 37, 36, 38, 39] clearly indicate that the universe at the largest scale is definitely not in a state of static equilibrium. Moreover, as we illustrated in this paper, the phenomenon of Hubble friction clearly mitigates against the instability, in the expanding state, of modes that are unstable in a static configuration.

Acknowledgements

We thank Stam Nicolis, Michael Volkov and Claude Warnick for informative discussions on higher-derivative theories. G.W.G. is grateful for the award of a LE STUDIUM Chair held at the LMPT of the University of Tours under the auspices of which some of the work described in this paper was carried out. The work of C.N.P. is supported in part by DOE grant DE-FG02-13ER42020. The research of S.S. was partially supported by the ERC-AdG-2015 grant 694896. C.N.P. is grateful the University of Tours for hospitality during the course of some of this work. S.S. would like to thank M. Shaposhnikov for warm hospitality at EPFL during some stages of this project.

Appendix A Canonical Noether Energy-Momentum Tensor

In this appendix, we discuss the construction of the canonical conserved energy-momentum tensor for a scalar field with a higher-derivative equation of motion, in a Minkowski spacetime background. We show also in a some example cases how the canonical energy-momentum tensor is related to the conventional energy-momentum that derived by the Belinfante-Rosenfeld procedure of coupling the scalar to gravity and then varying the action with respect to the metric. In particular, we show in these examples how the canonical energy-momentum tensor can be derived by applying the Belinfante-Rosenfeld procedure to an action in which an appropriate non-minimal coupling of the scalar field to gravity has been added.

Consider a scalar field $\phi$ in Minkowski spacetime, described by a Lagrangian

[TABLE]

where $\phi_{\nu_{1}}=\partial_{\nu_{1}}\phi$, $\ \ \phi_{\nu_{1}\nu_{2}}=\partial_{\nu_{1}}\partial_{\nu_{2}}\phi$, etc. The Euler-Lagrange equation is

[TABLE]

Using this to substitute for $\partial{{\cal L}}/\partial\phi$ in

[TABLE]

then after some straightforward manipulations one finds that

[TABLE]

and hence that $T_{\mu}{}^{\nu}$ defined by

[TABLE]

is conserved, $\partial_{\nu}\,T_{\mu}{}^{\nu}=0$, where

[TABLE]

and so on. Note that $-W_{\mu}^{{\scriptscriptstyle(n)}\,\nu}$ gives the contribution to the canonical energy-momentum tensor associated with the dependence of the Lagrangian on the $n$th-order spacetime derivatives of $\phi$. As always with the canonically-defined energy-momentum tensor coming from the Noether symmetries of the Lagrangian (i.e. the fact that the Lagrangian has no explicit dependence on the spacetime coordinates $x^{\mu}$), $T_{\mu}{}^{\nu}$ may not necessarily be symmetric in $\mu$ and $\nu$ after lowering the $\nu$ index. In the event that it is not, one can construct a conserved symmetric 2-index tensor from it by making use of the freedom to add a term $\partial_{\sigma}\,\psi_{\mu}{}^{\nu\sigma}$ to $T_{\mu}{}^{\nu}$, where $\psi_{\mu}{}^{\nu\sigma}$ is any tensor that is antisymmetric in $\nu$ and $\sigma$,

[TABLE]

Here we present examples of the canonical energy-momentum tensors for some simple Lagrangians. For the standard kinetic term for a scalar field we obtain

[TABLE]

which is the same as the conventional Belinfante-Rosenfeld energy-momentum tensor that one would obtain from varying the metric in ${{\cal L}}=-{1\over 2}\sqrt{-g}\,(\partial\phi)^{2}$ and then specialising to a Minkowski background. If instead we were to integrate the Lagrangian by parts before the calculation of the canonical energy-momentum tensor, we would obtain instead the energy-momentum tensor $\widetilde{T}_{\mu}{}_{\nu}$:

[TABLE]

The difference between the two is

[TABLE]

and in fact $\widetilde{T}_{\mu}{}^{\nu}$ is the conventional Belinfante-Rosenfeld energy-momentum tensor one would obtain by varying the metric in the non-minimally coupled Lagrangian

[TABLE]

and then specialising to the Minkowski spacetime background.

If we consider now a 4th-order Lagrangian, we obtain

[TABLE]

We have denoted this tensor with a prime, because it is not symmetric in $\mu$ and (the lowered) $\nu$ indices. We can obtain a symmetric conserved energy-momentum tensor by adding a term $\partial_{\sigma}\,\psi_{\mu}{}^{\nu\sigma}$ with

[TABLE]

leading to the energy-momentum tensor $T_{\mu}{}^{\nu}=T^{\prime}_{\mu}{}^{\nu}+\partial_{\sigma}\,\psi_{\mu}{}^{\nu\sigma}$, where

[TABLE]

The tensors (A.9) and (A.14) are precisely of the form of the 2nd-order and 4th-order contributions in the energy-momentum tensor (3.12) given in [19] for the 4th-order Pais-Uhlenbeck theory.

The canonical energy-momentum tensor $T_{\mu}{}^{\nu}$ for the $-{1\over 2}(\square\phi)^{2}$ Lagrangian, given by (A.14), is different from the flat-space specialisation of the expression (4.19) that we obtained by the Belinfante-Rosenfeld metric-variation method. In fact the difference between the two energy-momentum tensors can be generated as the metric variation of a certain non-minimal coupling of the scalar field to gravity. Specifically, we find that the canonical energy-momentum tensor in (A.14) can be obtained by varying the metric in the Lagrangian

[TABLE]

and then specialising to a Minkowski-spacetime background. The non-minimal coupling term in (A.15) is similar in form to a Horndeski coupling [40] that has been studied in other contexts in the literature, namely a term $\sqrt{-g}(R_{\mu\nu}-{1\over 2}Rg_{\mu\nu})\,\partial^{\mu}\phi\,\partial^{\nu}\phi$.