# Living with Ghosts

###### Abstract

Perturbation theory for gravity in dimensions greater than two requires higher derivatives in the free action. Higher derivatives seem to lead to ghosts, states with negative norm. We consider a fourth order scalar field theory and show that the problem with ghosts arises because in the canonical treatment, and are regarded as two independent variables. Instead, we base quantum theory on a path integral, evaluated in Euclidean space and then Wick rotated to Lorentzian space. The path integral requires that quantum states be specified by the values of and . To calculate probabilities for observations, one has to trace out over on the final surface. Hence one loses unitarity, but one can never produce a negative norm state or get a negative probability. It is shown that transition probabilities tend toward those of the second order theory, as the coefficient of the fourth order term in the action tends to zero. Hence unitarity is restored at the low energies that now occur in the universe.

## I Introduction

In standard, second order theory the Lagrangian is a function of the fields and their first derivatives. The path integral is calculated by perturbation theory, with the part of the action that contains quadratic terms in the fields and their first derivatives regarded as the free field action, and the remaining terms as interactions. One then calculates Feynman diagrams, using the interactions as vertices, and the propagator defined by the free part of the action. This is equivalent to calculating the expectation value of the interactions in the Gaussian measure defined by the free action. One would therefore expect perturbation theory to make sense, when and only when, the interaction action is bounded by the free action.

This is born out by the examples we know. In two dimensions, the free action of a scalar field ,

(1) |

is the first Sobolev norm^{1}^{1}1For
a function , , and an integer
, the Sobolev norm is defined [1] as

In four dimensions on the other hand, the first Sobolev norm does not bound the pointwise value of , but only the volume integral of . This means that the free action bounds the interactions only for theories with quartic interactions, like , or Yang–Mills. Indeed, these are the quantum field theories that are renormalizable in four dimensions. Note that even Yang–Mills is not renormalizable in dimensions higher than four, because the interactions are not bounded by the free action. Similarly, Born–Infeld is not renormalizable in dimensions higher than two.

When one does perturbation theory for gravity, one writes the metric as , where is a background metric that is a solution of the field equations. The terms quadratic in are again regarded as the free action, and the higher order terms are the interactions. The latter include terms like , multiplied by powers of . The volume integral of such an interaction is not bounded by the free action and perturbation theory breaks down for gravity, which is not renormalizable [2]. Even if all the higher loop divergences canceled by some miracle in a supergravity theory, one couldn’t trust the results, because one is using perturbation theory beyond its limit of validity; can be much larger than locally for only a small free action. In other words, there are large metric fluctuations below the Planck scale.

The situation is different however if one adds curvature squared terms to the Einstein–Hilbert action. The action is now quadratic in second derivatives of , so one takes the free action to be the quadratic terms in , and its first and second derivatives. This means that it is the second Sobolev norm of , which bounds the pointwise value of . Hence the free action bounds the interactions, and perturbation theory works. This is reflected in the fact that the theory is renormalizable [3], and in fact asymptotically free [4]. However, higher derivatives seem to lead to ghosts, states with negative norm, which have been thought to be a fatal flaw in any quantum field theory (see e.g. [5]).

In the next section we review why higher derivatives appear to give rise to ghosts. The existence of ghosts would mean that the set of all states would not form a Hilbert space with a positive definite metric. There would not be a unitary S matrix, and there would apparently be states with negative probabilities. These seemed sufficient reasons to dismiss any quantum field theory, such as Einstein gravity, that had higher derivative quantum corrections and ghosts. However, we shall show that one can still make sense of higher derivative theories, as a set of rules for calculating probabilities for observations. But one can not prepare a system in a state with a negative norm, nor can one resolve a state into its positive and negative norm components. So there are no negative probabilities, and no non unitary S matrix.

Although gravity is the physically interesting case, in this paper we consider a fourth order scalar field theory, which has the same ghostly behaviour, but doesn’t have the complications of indices or gauge invariance. We show explicitely that the higher derivative theory tends toward the second order theory, as the coefficient of the fourth order term in the action tends to zero. Hence the departures from unitarity for higher derivative gravity are very small at the low energies that now occur in the universe.

## Ii Higher Derivative Ghosts

We consider a scalar field with a fourth-order Lagrangian in Lorentzian signature,

(3) |

where . Defining

(4) |

the Lagrangian can be rewritten as

(5) |

The action of has the wrong sign. Classically it means that the energy of the field is negative, while that of is positive. If there were no interaction term, this negative energy wouldn’t matter because each of the fields, and , would live in its own world and the two worlds would not communicate with each other. However, if there is an interaction term, like , it will couple and together. Energy can then flow from one to the other, and one can have runaway solutions, with the positive energy of and the negative energy of both increasing exponentially.

In quantum theory, on the other hand, one is in trouble even in the absence of interactions, as can be seen by looking at the free field propagator for . In momentum space, this is the inverse of a fourth order expression in , which can be expanded as

(6) |

This is just the difference of the propagators for and . The important point is that the propagator for appears with a negative sign. This would mean that states with an odd number of particles, would have a negative norm. In other words, particles are ghosts. There wouldn’t be a positive definite Hilbert space metric, nor a unitary S matrix.

If there weren’t any interactions, the situation wouldn’t be too serious. The state space would be the direct sum of two Hilbert spaces, one with positive definite metric and the other negative. There wouldn’t be any physically realized operators that connected the two Hilbert spaces, so ghost number would be conserved by a superselection rule. A interaction however, would allow particles to be created or destroyed. As in the classical theory, there will be instabilities, with runaway production of and particles. These instabilities show up in the fact that interactions tend to shift the ghost poles in the two point function for into the complex -plane, where they represent exponentially growing and decaying modes [6, 7].

It seems to add up to a pretty damning indictment of higher derivative theories in general, and quantum gravity and quantum supergravity in particular. However, the problem with ghosts arises because in the canonical treatment, and are regarded as two independent variables, although they are both determined by . We shall show that, by basing quantum theory on a path integral over the field, evaluated in Euclidean space and then Wick rotated to Lorentzian space, one can obtain a sensible set of rules for calculating probabilities for observations in higher derivative theories.

## Iii Euclidean Path Integral

According to the canonical approach, one would perform the path integral over all and . The path integral over will converge, but the path integral over is divergent, because the free action for is negative definite. However, one shouldn’t do the path integrals over and separately because they are not independent fields, they are both determined by . The fourth order free action for is positive definite, thus the path integral over all in Euclidean space should converge, and should define a well determined Euclidean quantum field theory.

One way to compute the path integral for a fourth order theory, is to expand in eigenfunctions of the differential operator in the action. One then integrates over the coefficients in the harmonic expansion, which gives . Another way is to use time slicing, by dividing the period into a number of short time steps and approximating the derivatives by

(7) |

One then integrates over the values of on each time slice. In a second order theory, where the action depends on and but not on , the path integral will depend on the values of on the initial and final surfaces. However, in a fourth order theory, the use of three neighbor differences means that one has to specify on the initial and final surfaces as well.

One can also see what needs to be specified on the initial and final surfaces as follows. In classical second order theory, a state can be defined by its Cauchy data on a spacelike surface, i.e. the values of and on the surface. In a canonical 3+1 treatment, these are regarded as the position of the field and its conjugate momentum. In quantum theory, position and momentum don’t commute, so instead one describes a state by a wave function in either position space or momentum space. In ordinary quantum mechanics, the position and momentum representations are regarded as equivalent: one is just the Fourier transform of the other. However, with path integrals, one has to use wave functions in the position representation. This can be seen as follows. Imagine using the path integral to go from a state at to a state at , and then to a state at . In the position representation, the amplitude to go from a field on , to at , is given by a path integral over all fields with the given boundary values. Similarly, the amplitude to go from at , to at , is given by another path integral. These amplitudes obey a composition law,

(8) |

The composition law holds, only because one can join a field from to to a field from to , to obtain a field from to . Although in general will be discontinuous at , the field will still have a well defined action,

(9) |

On the other hand, if one would use the momentum representation and wave functions in terms of , the composition law would no longer hold, because the discontinuity of at would make the action infinite. Thus in second order theories, one should use wave functions in terms of rather than .

In a fourth order theory, a classical state is determined by the values of and its first three time derivatives on a spacelike surface. In a canonical treatment, and are usually taken to be independent coordinates. For the scalar field theory (3) we then have the conjugate momenta

(10) |

This suggests that in quantum theory, one should describe a state by a wave functional on a surface. Indeed, this is closely related to using the fields and that we introduced earlier. These were linear combinations of and , thus taking the wave function to depend on and , is equivalent to it depending on and . However, if one does the path integral between fixed values of and , one gets in trouble with the composition law, because the values of on the intermediate surface at are not constrained, Hence will be in general discontinuous at , which implies that will have a delta-function when one joins the fields above and below . In a second order action appears linearly, thus the delta-function can be integrated by parts and the action of the combined field is finite. But in a fourth order action appears, rendering the action of the combined field infinite if is a delta-function.

Therefore, the path integral requires that quantum states be specified by and in order to get the composition law for amplitudes in a fourth order theory. In the next section we show how one can obtain transition probabilities for observations from the Euclidean path integral over .

## Iv Higher Derivative Harmonic Oscillator

### iv.1 Ground State Wave Function

To illustrate how probabilities can be calculated, we consider a higher derivative harmonic oscillator, for which in Euclidean signature we take the action

(11) |

For , this is very similar to our scalar field model, since in the latter we can take Fourier components so that spatial derivatives behave like masses. The general solution to the equation of motion is given by

(12) |

where and are given by (27). For small , and .

The fourth order action for is positive definite, thus it gives a well defined Euclidean quantum field theory. In this theory, one can calculate the amplitude to go from a state at time , to a state at time . In particular, one can calculate the ground state wave function, the amplitude to go from zero field in the infinite Euclidean past, up to the given values at . This yields (see Appendix A)

(13) |

where

(14) |

and is a normalization factor.

Similarly, one can calculate the Euclidean conjugate ground state wave function , the amplitude to go from the given values at , to zero field in the infinite Euclidean future. This conjugate wave function is equal to the original ground state wave function, with the opposite sign of . The probability that a quantum fluctuation in the ground state gives the specified values and on the surface , is then given by

(15) |

The probability dies off at large values of and and is normalizable, thus the probability distribution in the Euclidean theory is well-defined. However if one Wick rotates to Minkowski space, picks up a minus sign. The probability distribution becomes unbounded for large Lorentzian and can no longer be normalized. This is another reflection of the same problem as the ghosts. You can’t fully determine a state on a spacelike surface, because that would involve specifying and Lorentzian , which doesn’t have a physically reasonable probability distribution.

Although one can not define a probability distribution for and Lorentzian on a spacelike surface, one can calculate a probability distribution for alone, by integrating out over Euclidean . This integral converges because the probability distribution is damped at large values of Euclidean . This is just what one would calculate in a second order theory. So the moral is, a fourth order theory can make sense in Lorentzian space, if you treat it like a second order theory. The normalized probability distribution that a ground state fluctuations gives the specified value on a spacelike surface is then given by,

(16) |

As the coefficient of the fourth order term in the action tends to zero, this becomes

(17) |

which tends toward the result for the second order theory.

### iv.2 Transition Probabilities

In this section we compute the Euclidean transition probability, to go from a specified value at time , to at time , for the higher derivative harmonic oscillator.

In a second order theory, a state can be described by a wave function that depends on the values of on a spacelike surface. Thus a transition amplitude is given by a path integral from an initial state on , to a final state on . To calculate the probability to go from the initial state to the final, one multiplies the amplitude by its Euclidean conjugate. This can be represented as the path integral from a third surface, at , back to . Because the path integral in a second order theory depends only on on the boundary, what happens above and below doesn’t matter. Furthermore, the path integrals above and below can be calculated independently, which implies the probability to go from initial to final, can be factorized into the product of an S matrix and its adjoint. The S matrix is unitary, because probability is conserved.

Now let us calculate the probability to go from an initial to a final state in the fourth order theory (11). The path integral requires quantum states to be specified by and . The transition amplitude to go from a state at time , to a state at time , reads

(18) |

This is evaluated in Appendix A, by writing , where obeys the equation of motion with the given boundary conditions on both surfaces.

The result is

(19) | |||||

The coefficient functions in the exponent are given by (A), and is a normalization factor.

Again, one can construct a three layer ’sandwich’ to calculate the probability to go from the initial state to the final. However, in contrast with the second order theory the path integral now depends on both and on the boundaries. This has two important implications for the calculation of the transition probability. Firstly, as we just showed, one can’t observe Lorentzian because it has an unbounded Lorentzian probability distribution. Therefore one should take to be continuous on the surfaces and integrate over all values, fixing only the values of on the surfaces. Because the path integrals above and below both depend on , the probability to observe the initial and final specified values of does not factorize into an S matrix and its adjoint. Instead, there is loss of quantum coherence, because one can not observe all the information that characterizes the final state.

After multiplying by the Euclidean conjugate amplitude and integrating out over we obtain

(20) |

Another consequence of the dependence of the path integral on is that what goes on outside the sandwich, now affects the result. The most natural choice, would be the vacuum state above and below . In other words, one takes the path integral to be over all fields that have the given values on the three surfaces, and that go to zero in the infinite Euclidean future and past. This means that to obtain the transition probability we also ought to multiply by the appropriately normalized ground state wave function and its Euclidean conjugate. The probability is then given by

(21) | |||||

Here is the coefficient in the exponent of the ground wave function (13) and is a normalization factor. In the limit , this reduces to

(22) |

Hence the probability given by the sandwich tends toward that of the second order theory, as the coefficient of the fourth order term in the action tends to zero. This is important, because it means that fourth order corrections to graviton scattering can be neglected completely at the low energies that now occur in the universe. On the other hand, in the very early universe, when fourth order terms are important, we expect the Euclidean metric to be some instanton, like a four sphere. In such a situation, one can not define scattering or ask about unitarity. The only quantities we have any chance of observing are the n-point functions of the metric perturbations, which determine the n-point functions of fluctuations in the microwave background. With Reall we have shown that Starobinsky’s model of inflation [8], in which inflation is driven by the trace anomaly of a large number of conformally coupled matter fields, can give a sensible spectrum of microwave fluctuations, despite the fact it has fourth order terms and ghosts[9]. Moreover, the fourth order terms can play an important role in reducing the fluctuations to the level we observe.

Finally, in order to obtain the Minkowski space probability, one analytically continues to future infinity in Minkowski space, and and to past infinity, keeping their Euclidean time values fixed. This gives the Minkowski space probability, to go from an initial value to a final value .

## V Runaways and Causality

The discussion in Section II suggests that even the slightest amount of a fourth order term will lead to runaway production of positive and negative energies, or of real and ghost particles. The classical theory is certainly unstable, if one prescribes the initial value of and its first three time derivatives. However, in quantum theory every sensible question can be posed in terms of vacuum to vacuum amplitudes. These can be defined by Wick rotating to Euclidean space and doing a path integral over all fields that die off in the Euclidean future and past. Thus the Euclidean formulation of a quantum field theory implicitly imposes the final boundary condition that the fields remain bounded. This removes the instabilities and runaways, like a final boundary condition removes the runaway solution of the classical radiation reaction force. The price one pays for removing runaways with a final boundary condition, is a slight violation of causality. For instance, with the classical radiation reaction force, a particle would start to accelerate before a wave hit it. This can be seen by considering a single electron which is acted upon by a delta-function pulse [10]. The equation of motion for the -component reduces to

(23) |

with . This has the solution

(24) |

The integrand has two singularities, at and . The final boundary condition that should tend to a finite limit, implies one must choose an integration contour that stays close to the real axis, going below the second singularity. This yields

(25) | |||||

which is without runaways, but acausal.

However, this pre-acceleration is appreciable only for a period of time comparable with the time for light to travel the classical radius of the electron, and thus practically unobservable.

Similarly, if we would add an interaction term to the higher derivative scalar field theory (3), the imposition of a final boundary condition to eliminate the runaway solutions, would lead to acausal behaviour on the scale of , where is the mass of the ghost particle. However, in the context of quantum gravity, one could again never detect a violation of causality, because the presence of a mass introduces a logarithmic time delay , where is the impact parameter. Thus there is no standard arrival time, one can always arrive before any given light ray by taking a path which stays a sufficiently large distance from the mass.

## Vi Concluding Remarks

We conclude that quantum gravity with fourth order corrections can make sense, despite apparently having negative energy solutions and ghosts. In doing this, we seem to go against the convictions of the last 25 years, that unitarity and causality are essential requirements of any viable theory of quantum gravity. Perturbative string theory has unitarity and causality, so it has been claimed as the only viable quantum theory of gravity. But the string perturbation expansion does not converge, and string theory has to be augmented by non perturbative objects, like D-branes. One can have a world-sheet theory of strings without higher derivatives, only because two dimensional metrics are conformally flat, meaning perturbations don’t change the light cone. Still, we live either on a 3-brane, or in the bulk of a higher dimensional compactified space. The world-sheet theory of D-branes with greater than one has similar non-renormalizability problems to Einstein gravity and supergravity. Thus string theory effectively has ghosts, though this awkward fact is quietly glided over.

To summarize, we showed that perturbation theory for gravity in dimensions greater than two required higher derivatives in the free action. Higher derivatives seemed to lead to ghosts, states with negative norm. To analyze what was happening, we considered a fourth order scalar field theory. We showed that the problem with ghosts arises because in the canonical approach, and are regarded as two independent coordinates. Instead, we based quantum theory on a path integral over , evaluated in Euclidean space and then Wick rotated to Lorentzian space. We showed the path integral required that quantum states be specified by the values of and on a spacelike surface, rather than and as is usually done in a canonical treatment. The wave function in terms of and is bounded in Euclidean space, but grows exponentially with Minkowski space . This means one can not observe but only . To calculate probabilities for observations one therefore has to trace out over on the final surface, and lose information about the quantum state. One might worry that integrating out would break Lorentz invariance. However, is conjugate to so tracing over is equivalent to not observing . Since, according to eq.(4), and are linear combinations of and , this means that one only considers Feynman diagrams whose external legs are . You don’t observe the other linear combination, .

Because one is throwing away information, one gets a density matrix for the final state, and loses unitarity. However, one can never produce a negative norm state or get a negative probability. We illustrated with the example of a higher derivative harmonic oscillator that probabilities for observations tend toward those of the second order theory, as the coefficient of the fourth order term in the action tends to zero. This means that the departures from unitarity for higher derivative gravity will be very small at the low energies that now occur in the universe. On the other hand, the higher derivative terms will be important in the early universe, but there unitarity can not be defined.

Acknowledgements

It is a pleasure to thank David Gross, Jim Hartle and Edward Witten for helpful discussions. We would also like to thank the ITP at Santa Barbara and the Department of Physics at Caltech, where some of this work was done, for their hospitality. TH is Aspirant FWO, Belgium.

## Appendix A Transition Amplitude

We compute the Euclidean transition amplitude, to go from an initial state on a spacelike surface at , to a final state at , for the higher derivative harmonic oscillator (11). The general solution to the equation of motion is given by

(26) |

where

(27) |

The transition amplitude is given by a path integral,

(28) |

This can be evaluated by separating out the ’classical’ part of . If we write , where obeys the equation of motion with the required boundary conditions on both surfaces and , then the amplitude becomes

(29) |

The classical action is

(30) | |||||

where

(31) | |||||

with

(32) |

The pre-exponential factor in (29) can be derived from the classical action alone [11], it is basically the Jacobian of the change of variables . Because the Lagrangian is quadratic, the prefactor is independent of the values specifying the initial and final states, and the transition amplitude (29) is exact. It is given by

(33) | |||||

The normalization factor is independent of to first order. It is determined by taking in (33) and requiring that the amplitude tends toward the product of two normalized ground state wave functions and .

For small , and , hence the transition amplitude becomes

(34) | |||||

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