hawking, steven, living with ghosts

8/9/2019 Hawking, Steven, Living With Ghosts

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arXiv:hep-th/0107088v22

7Jul2001

Living with Ghosts

S.W. Hawking, Thomas Hertog

DAMTPCentre for Mathematical Sciences

Wilberforce Road, Cambridge, CB3 0WA, UK.(June 21, 2002)

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 2 are regarded as two independent variables. Instead, we base quan-

tum 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 obser-

vations, 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 theirfirst derivatives. The path integral is calculated by perturbation theory, with the part ofthe action that contains quadratic terms in the fields and their first derivatives regarded asthe free field action, and the remaining terms as interactions. One then calculates Feynmandiagrams, using the interactions as vertices, and the propagator defined by the free partof the action. This is equivalent to calculating the expectation value of the interactions inthe Gaussian measure defined by the free action. One would therefore expect perturbation

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theory to make sense, when and only when, the interaction action is bounded by the freeaction.

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

S =

dx2

2 + m2

2

, (1)

is the first Sobolev norm1 2,1 of the field . In two dimensions, the first Sobolev normbounds the pointwise value of , thus it also bounds the volume integral of any entirefunction of. This means that the free action bounds any interaction action, so perturbationtheory should work. Indeed one finds that in two dimensions, any quantum field theory isrenormalizable.

In four dimensions on the other hand, the first Sobolev norm does not bound the point-wise value of, but only the volume integral of 4. This means that the free action boundsthe interactions only for theories with quartic interactions, like 4, or YangMills. Indeed,these are the quantum field theories that are renormalizable in four dimensions. Note thateven YangMills is not renormalizable in dimensions higher than four, because the interac-tions are not bounded by the free action. Similarly, BornInfeld is not renormalizable indimensions higher than two.

When one does perturbation theory for gravity, one writes the metric as g0 + g, whereg0 is a background metric that is a solution of the field equations. The terms quadratic in gare again regarded as the free action, and the higher order terms are the interactions. Thelatter include terms like (g)2, multiplied by powers of g. The volume integral of suchan interaction is not bounded by the free action and perturbation theory breaks down forgravity, which is not renormalizable [2]. Even if all the higher loop divergences canceled bysome miracle in a supergravity theory, one couldnt trust the results, because one is using

perturbation theory beyond its limit of validity; g can be much larger than g0 locally foronly a small free action. In other words, there are large metric fluctuations below the Planckscale.

The situation is different however if one adds curvature squared terms to the EinsteinHilbert action. The action is now quadratic in second derivatives ofg, so one takes the freeaction to be the quadratic terms in g, and its first and second derivatives. This means thatit is the second Sobolev norm g2,2 of g, which bounds the pointwise value of g. Hencethe free action bounds the interactions, and perturbation theory works. This is reflected inthe fact that the R + R2 theory is renormalizable [3], and in fact asymptotically free [4].

1For a function f C(M), 1 p < , and an integer k 0, the Sobolev norm is defined [1] as

fp,k =

M

0jk

|Djf|pg1/p

, (2)

where |Djf| is the pointwise norm of the jth covariant derivative and g is the Riemannian volumeelement.

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However, higher derivatives seem to lead to ghosts, states with negative norm, which havebeen 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. Theexistence of ghosts would mean that the set of all states would not form a Hilbert spacewith 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 dismissany quantum field theory, such as Einstein gravity, that had higher derivative quantumcorrections and ghosts. However, we shall show that one can still make sense of higherderivative theories, as a set of rules for calculating probabilities for observations. But onecan not prepare a system in a state with a negative norm, nor can one resolve a state intoits positive and negative norm components. So there are no negative probabilities, and nonon unitary S matrix.

Although gravity is the physically interesting case, in this paper we consider a fourthorder scalar field theory, which has the same ghostly behaviour, but doesnt have the com-plications of indices or gauge invariance. We show explicitely that the higher derivativetheory 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 gravityare 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,

L = 12

2 m21

2m22

4 (3)

where m2 > m1. Defining

1 =(2 m22)

[2(m22 m21)]1/22 =

(2m21) [2(m22 m21)]1/2

(4)

the Lagrangian can be rewritten as

L =1

21

2m21

1 1

22

2m22

2 4

(m22 m21)2(1 2)4 (5)

The action of 2 has the wrong sign. Classically it means that the energy of the 2field is negative, while that of 1 is positive. If there were no interaction term, this negative

energy wouldnt matter because each of the fields, 1 and 2, would live in its own world andthe two worlds would not communicate with each other. However, if there is an interactionterm, like 4, it will couple 1 and 2 together. Energy can then flow from one to the other,and one can have runaway solutions, with the positive energy of1 and the negative energyof 2 both increasing exponentially.

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

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G(p) =1

(m22 m21)

1

(p2 + m21) 1

(p2 + m22)

, (6)

This is just the difference of the propagators for 1 and 2. The important point is thatthe propagator for 2 appears with a negative sign. This would mean that states with an

odd number of 2 particles, would have a negative norm. In other words, 2 particles areghosts. There wouldnt be a positive definite Hilbert space metric, nor a unitary S matrix.If there werent any interactions, the situation wouldnt be too serious. The state space

would be the direct sum of two Hilbert spaces, one with positive definite metric and theother negative. There wouldnt be any physically realized operators that connected thetwo Hilbert spaces, so ghost number would be conserved by a superselection rule. A 4

interaction however, would allow 2 particles to be created or destroyed. As in the classicaltheory, there will be instabilities, with runaway production of 1 and 2 particles. Theseinstabilities show up in the fact that interactions tend to shift the ghost poles in the twopoint function for into the complex p-plane, where they represent exponentially growingand 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 withghosts arises because in the canonical treatment, and 2 are regarded as two independentvariables, although they are both determined by . We shall show that, by basing quantumtheory on a path integral over the field, evaluated in Euclidean space and then Wick rotatedto Lorentzian space, one can obtain a sensible set of rules for calculating probabilities forobservations in higher derivative theories.

III. EUCLIDEAN PATH INTEGRAL

According to the canonical approach, one would perform the path integral over all 1and 2. The path integral over 1 will converge, but the path integral over 2 is divergent,because the free action for 2 is negative definite. However, one shouldnt do the pathintegrals over 1 and 2 separately because they are not independent fields, they are bothdetermined by . The fourth order free action for is positive definite, thus the path integralover all in Euclidean space should converge, and should define a well determined Euclideanquantum field theory.

One way to compute the path integral for a fourth order theory, is to expand ineigenfunctions of the differential operator O in the action. One then integrates over thecoefficients in the harmonic expansion, which gives (det O)1/2. Another way is to use timeslicing, by dividing the period into a number of short time steps and approximating thederivatives by

, (n+1 n)

, , (n+2 2n+1 + n)2

(7)

One then integrates over the values of on each time slice. In a second order theory, wherethe action depends on and , but not on , , the path integral will depend on the valuesof on the initial and final surfaces. However, in a fourth order theory, the use of three

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neighbor differences means that one has to specify , on the initial and final surfaces aswell.

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 spacelikesurface, 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 dont commute, so instead one describes a state by a wave functionin either position space or momentum space. In ordinary quantum mechanics, the positionand momentum representations are regarded as equivalent: one is just the Fourier transformof the other. However, with path integrals, one has to use wave functions in the positionrepresentation. This can be seen as follows. Imagine using the path integral to go from astate at 1 to a state at 2, and then to a state at 3. In the position representation, theamplitude to go from a field 1 on 1, to 2 at 2, is given by a path integral over all fields with the given boundary values. Similarly, the amplitude to go from 2 at 2, to 3 at 3,is given by another path integral. These amplitudes obey a composition law,

G(3, 1) =

d2G(3, 2)G(2, 1) (8)

The composition law holds, only because one can join a field from 1 to 2 to a fieldfrom 2 to 3, to obtain a field from 1 to 3. Although in general , will be discontinuousat t2, the field will still have a well defined action,

S(3, 1) = S(3, 2) + S(2, 3) (9)

On the other hand, if one would use the momentum representation and wave functions interms of,, the composition law would no longer hold, because the discontinuity of at 2would make the action infinite. Thus in second order theories, one should use wave functionsin terms of rather than ,.

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

= ,+ (m21 + m22 22), , , = , (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 1 and 2 thatwe introduced earlier. These were linear combinations of and 2, thus taking the wavefunction to depend on 1 and 2, is equivalent to it depending on and ,. However,if one does the path integral between fixed values of and ,, one gets in trouble withthe composition law, because the values of , on the intermediate surface at 2 are notconstrained, Hence , will be in general discontinuous at 2, which implies that , willhave a delta-function when one joins the fields above and below 2. In a second order action, appears linearly, thus the delta-function can be integrated by parts and the action of thecombined field is finite. But in a fourth order action (,)

2 appears, rendering the actionof the combined field infinite if , is a delta-function.

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

IV. HIGHER DERIVATIVE HARMONIC OSCILLATOR

A. Ground State Wave Function

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

S =

d

2

22, +

1

22, +

1

2m22

(11)

For 2 > 0, this is very similar to our scalar field model, since in the latter we can takeFourier components so that spatial derivatives behave like masses. The general solution tothe equation of motion is given by

() = A sinh 1 + B cosh 1 + Csinh 2 + D cosh 2, (12)

where 1 and 2 are given by (A2). For small , 1 m and 2 1/.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(1, 1,) at time 1, to a state (2, 2,) at time 2. In particular, one can calculate theground state wave function, the amplitude to go from zero field in the infinite Euclideanpast, up to the given values (0, 0,) at = 0. This yields (see Appendix A)

0(0, 0,) = N exp

F

20, +

m

20

+

2m2 m/(2 1)2 00,

(13)

where

F =(1 4m22)

22(1 + 2)(2 1)2 (14)

and N(, m) is a normalization factor.Similarly, one can calculate the Euclidean conjugate ground state wave function 0, the

amplitude to go from the given values at = 0, to zero field in the infinite Euclidean future.This conjugate wave function is equal to the original ground state wave function, with theopposite sign of 0,. The probability that a quantum fluctuation in the ground state givesthe specified values 0 and 0, on the surface = 0, is then given by

P(0, 0,) = 00 = N

2 exp2F

20, +

m

20

(15)

The probability dies off at large values of and , and is normalizable, thus the prob-ability distribution in the Euclidean theory is well-defined. However if one Wick rotates

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to Minkowski space, 2, picks up a minus sign. The probability distribution becomes un-bounded for large Lorentzian , and can no longer be normalized. This is another reflectionof the same problem as the ghosts. You cant fully determine a state on a spacelike surface,because that would involve specifying and Lorentzian ,t, which doesnt have a physicallyreasonable probability distribution.

Although one can not define a probability distribution for and Lorentzian ,t on aspacelike surface, one can calculate a probability distribution for alone, by integrating outover Euclidean ,. This integral converges because the probability distribution is dampedat large values of Euclidean ,. This is just what one would calculate in a second ordertheory. So the moral is, a fourth order theory can make sense in Lorentzian space, if youtreat it like a second order theory. The normalized probability distribution that a groundstate fluctuations gives the specified value 0 on a spacelike surface is then given by,

P(0) =

2Fm

1/2exp

2mF

20

(16)

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

P(0) =

m

1/2(1 +

m

2) exp[m(1 + m)20], (17)

which tends toward the result for the second order theory.

B. Transition Probabilities

In this section we compute the Euclidean transition probability, to go from a specifiedvalue 1 at time 1, to 2 at time 2, for the higher derivative harmonic oscillator.

In a second order theory, a state can be described by a wave function that depends on thevalues of on a spacelike surface. Thus a transition amplitude is given by a path integralfrom an initial state 1 on 1, to a final state 2 on 2. To calculate the probability to gofrom 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 3, back to 2. Becausethe path integral in a second order theory depends only on on the boundary, what happensabove 3 and below 1 doesnt matter. Furthermore, the path integrals above and below 2can 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 fourthorder theory (11). The path integral requires quantum states to be specified by and ,.The transition amplitude to go from a state (1, 1,) at time 1 = T, to a state (2, 2,)at time 2 = 0, reads

(2, 2,; 0)|(1, 1,;T) =(2,2,)(1,1,)

d[()] exp[S()] (18)

This is evaluated in Appendix A, by writing = cl + , where cl obeys the equation of

motion with the given boundary conditions on both surfaces.

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The result is

(2, 2,; 0)|(1, 1,;T) =(1 + N)H

22

1/2exp

E(21 + 22) F(21, + 22,)

G1,2, + H12 K(2,2 1,1) L(2,1 1,2)] (19)

The coefficient functions in the exponent are given by (A6), and N 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 pathintegral now depends on both and , on the boundaries. This has two important im-plications for the calculation of the transition probability. Firstly, as we just showed, onecant observe Lorentzian , because it has an unbounded Lorentzian probability distribu-tion. Therefore one should take , to be continuous on the surfaces and integrate over allvalues, fixing only the values of on the surfaces. Because the path integrals above andbelow 2 = 0 both depend on 2,, the probability P(2, 1) to observe the initial and finalspecified 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 characterizesthe final state.

After multiplying by the Euclidean conjugate amplitude and integrating out over 2, weobtain

(1 + N)H22

2F

1/2exp

2E(21 + 22) 2F 21, + 2H12 +

G2

2F21,

(20)

Another consequence of the dependence of the path integral on , is that what goes onoutside the sandwich, now affects the result. The most natural choice, would be the vacuumstate above 3 = T and below 1 = T. In other words, one takes the path integral to be overall fields that have the given values on the three surfaces, and that go to zero in the infiniteEuclidean future and past. This means that to obtain the transition probability we alsoought to multiply by the appropriately normalized ground state wave function 0(1, 1,)and its Euclidean conjugate. The probability P(2, 1) is then given by

P(2, 1) =

d[1,]00

d[2,](1, 1,)|(2, 2,)(2, 2,)|(1, 1,)

=

2(1 + N)2H2

22(4F(F + F) G2)

1/2exp

2E(21 + 22) 2

mF

21 + 2H12

(21)

Here F(, m) is the coefficient in the exponent of the ground wave function ( 13) and N is

a normalization factor. In the limit 0, this reduces to

P(2, 1) =m

2 sinh mTexp

m cosh mT(

21 +

22) 2m12

sinh mT m21

(22)

Hence the probability given by the sandwich tends toward that of the second ordertheory, as the coefficient of the fourth order term in the action tends to zero. This isimportant, because it means that fourth order corrections to graviton scattering can beneglected completely at the low energies that now occur in the universe. On the other hand,

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in the very early universe, when fourth order terms are important, we expect the Euclideanmetric to be some instanton, like a four sphere. In such a situation, one can not definescattering or ask about unitarity. The only quantities we have any chance of observing arethe n-point functions of the metric perturbations, which determine the n-point functions offluctuations in the microwave background. With Reall we have shown that Starobinskys

model of inflation [8], in which inflation is driven by the trace anomaly of a large number ofconformally 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 termscan 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 2to future infinity in Minkowski space, and 1 and 3 to past infinity, keeping their Euclideantime values fixed. This gives the Minkowski space probability, to go from an initial value 1to a final value 2.

V. RUNAWAYS AND CAUSALITY

The discussion in Section II suggests that even the slightest amount of a fourth orderterm will lead to runaway production of positive and negative energies, or of real and ghostparticles. The classical theory is certainly unstable, if one prescribes the initial value of andits first three time derivatives. However, in quantum theory every sensible question can beposed in terms of vacuum to vacuum amplitudes. These can be defined by Wick rotating toEuclidean space and doing a path integral over all fields that die off in the Euclidean futureand past. Thus the Euclidean formulation of a quantum field theory implicitly imposesthe final boundary condition that the fields remain bounded. This removes the instabilitiesand runaways, like a final boundary condition removes the runaway solution of the classicalradiation 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 reactionforce, a particle would start to accelerate before a wave hit it. This can be seen by consideringa single electron which is acted upon by a delta-function pulse [10]. The equation of motionfor the x-component reduces to

x,tt = x,ttt + (t), (23)

with = 2e2

3mc3. This has the solution

x(t) =

d

2exp[it] 1

2

i3

. (24)

The integrand has two singularities, at = 0 and = i1. The final boundary conditionthat x,t should tend to a finite limit, implies one must choose an integration contour thatstays close to the real axis, going below the second singularity. This yields

x(t) = exp[t/], t < 0

= t + , t > 0 (25)

which is without runaways, but acausal.

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

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 m12 , where m2 is the mass of the ghost particle.

However, in the context of quantum gravity, one could again never detect a violation ofcausality, because the presence of a mass introduces a logarithmic time delay t m log b,where b is the impact parameter. Thus there is no standard arrival time, one can alwaysarrive before any given light ray by taking a path which stays a sufficiently large distancefrom 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 againstthe convictions of the last 25 years, that unitarity and causality are essential requirements ofany viable theory of quantum gravity. Perturbative string theory has unitarity and causal-ity, so it has been claimed as the only viable quantum theory of gravity. But the stringperturbation expansion does not converge, and string theory has to be augmented by nonperturbative objects, like D-branes. One can have a world-sheet theory of strings withouthigher derivatives, only because two dimensional metrics are conformally flat, meaning per-turbations dont change the light cone. Still, we live either on a 3-brane, or in the bulk of ahigher dimensional compactified space. The world-sheet theory of D-branes with p greaterthan 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 greaterthan two required higher derivatives in the free action. Higher derivatives seemed to leadto ghosts, states with negative norm. To analyze what was happening, we considered afourth order scalar field theory. We showed that the problem with ghosts arises because inthe canonical approach, and 2 are regarded as two independent coordinates. Instead,we based quantum theory on a path integral over , evaluated in Euclidean space and thenWick rotated to Lorentzian space. We showed the path integral required that quantumstates 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 , isbounded in Euclidean space, but grows exponentially with Minkowski space ,. This means

one can not observe , but only . To calculate probabilities for observations one thereforehas 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, , isconjugate to , so tracing over , is equivalent to not observing 2. Since, according toeq.(4), 1 and 2 are linear combinations of and 2, this means that one only considersFeynman diagrams whose external legs are 1 2. You dont observe the other linearcombination, m221 m212.

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

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probability. We illustrated with the example of a higher derivative harmonic oscillator thatprobabilities for observations tend toward those of the second order theory, as the coefficientof the fourth order term in the action tends to zero. This means that the departures fromunitarity for higher derivative gravity will be very small at the low energies that now occurin 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 dis-cussions. We would also like to thank the ITP at Santa Barbara and the Department ofPhysics at Caltech, where some of this work was done, for their hospitality. TH is AspirantFWO, Belgium.

APPENDIX A: TRANSITION AMPLITUDE

We compute the Euclidean transition amplitude, to go from an initial state (1, 1,) ona spacelike surface at = T, to a final state (2, 2,) at = 0, for the higher derivativeharmonic oscillator (11). The general solution to the equation of motion is given by

() = A sinh 1 + B cosh 1 + Csinh 2 + D cosh 2, (A1)

where

1 =122

(1

1 4m22) , 2 = 1

22

(1 +

1 4m22) (A2)

The transition amplitude is given by a path integral,

(2, 2,; 0)|(1, 1,;T) =(2,2,)(1,1,)

d[()] exp[S()] (A3)

This can be evaluated by separating out the classical part of . If we write = cl + ,

where cl obeys the equation of motion with the required boundary conditions on bothsurfaces = 0 and = T, then the amplitude becomes

(2, 2,; 0)|(1, 1,;T) = exp[Scl(1, 1,, 2, 2,)](0,0)(0,T)

d[()] exp[S()] (A4)

The classical action is

Scl =T0

d

2

22cl, +

1

22cl, +

1

2m22cl

= E(21 + 22) + F(

21, +

22,) + G1,2, H12

+K(2,2 1,1) + L(2,1 1,2) (A5)

where

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E =m(1 4m22)23(22 21)P2

2m

(cosh 2T cosh 1T)(1 sinh 1T + 2 sinh 2T)

+ sinh 1Tsinh 2T(1(222 +

1

2) cosh 2Tsinh 1T 2(221 +

1

2)sinh 2T cosh 1T)

F =

(1

4m22)

22(22 21)P2 2m

(cosh 2T cosh 1T)(2 sinh 1T + 1 sinh 2T)+ sinh 1Tsinh 2T(2(2

21 +

1

2) cosh 2Tsinh 1T 1(222 +

1

2)sinh 2T cosh 1T)

G =(1 4m22)

2(22 21)P2

2m

(cosh 2Tcosh 1T 1)

12

sinh 1Tsinh 2T

(1 sinh 2T 2 sinh 1T)

H =m(1 4m22)3(22 21)P2

2m

(cosh 2Tcosh 1T 1)

1

2 sinh 1Tsinh 2T

(1 sinh 1T 2 sinh 2T)K =

1

P2

m

4m2 +

1

2

sinh 1Tsinh 2T(1 cosh 1 cosh 2T)

+2m2

2(2 3(cosh2 1T + cosh2 2T))

L =m(1 4m22)3(22 21)P2

2m

(cosh 2Tcosh 1T 1)

12

sinh 1Tsinh 2T

(cosh 2T cosh 1T) (A6)with

P = (21 + 22)sinh 1T sinh 2T + 212(1 cosh 1T cosh 2T) (A7)

The pre-exponential factor in (A4) can be derived from the classical action alone [11], it isbasically the Jacobian of the change of variables (1, 1) (2, 1). Because the Lagrangianis quadratic, the prefactor is independent of the values specifying the initial and final states,and the transition amplitude (A4) is exact. It is given by

(2, 2,; 0)|(1, 1,;T) =(1 + N)H

22

1/2exp

E(21 + 22) F(21, + 22,)

G1,2, + H12 K(2,2 1,1) L(2,1 1,2)] (A8)

The normalization factor N is independent of to first order. It is determined bytaking T + in (A8) and requiring that the amplitude tends toward the product of twonormalized ground state wave functions 0(1, 1,) and 0(2, 2,).

For small , 1 m and 2 1/, hence the transition amplitude becomes

(2, 2,; 0)|(1, 1,;T) =

m

22 sinh mT

1/2exp

m cosh mT(

21 +

22)

2 sinh mT

2(21, +

22,)

+m cosh mT(2,2 1,1)

sinh mT m(2, + 2)(1, 1)

sinh mT

(A9)

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[4] E.S. Fradkin and A.A. Tseytlin, Nucl. Phys. B201, 469 (1982).[5] S.W. Hawking, in Quantum Field Theory and Quantum Statistics:Essays in Honor ofthe 60th Birthday of E.S. Fradkin, eds. A. Batalin, C.J. Isham and C.A. Vilkovisky,Hilger, Bristol, UK (1987).

[6] E. Tomboulis, Phys. Lett. 70B, 361 (1977).[7] D.A. Johnston, Nucl. Phys. B297, 721 (1987).[8] A.A. Starobinsky, Phys. Lett. 91B, 99 (1980).[9] S.W. Hawking, T. Hertog, H.S. Reall, Phys. Rev. D63, 083504 (2001).

[10] S. Coleman in Theory and Phenomenology in Particle Physics, ed. A. Zichichi, NewYork (1969).

[11] M.S. Marinov, Phys. Rep. 60, 1 (1980).

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hawking, steven, living with ghosts

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