Content uploaded by Marek J. Radzikowski
Author content
All content in this area was uploaded by Marek J. Radzikowski on Nov 01, 2012
Content may be subject to copyright.
arXiv:gr-qc/9603012v2 8 May 1997
Quantum Field Theory on Spacetimes with a
Compactly Generated Cauchy Horizon
Bernard S. Kay (bsk2@york.ac.uk)
Marek J. Radzikowski∗(radzikow@mail.desy.de)
Department of Mathematics, University of York, Heslington, York YO1 5DD, U.K.
(∗Present address: II. Institut f¨ur Theoretische Physik, Universit¨at Hamburg,
Luruper Chaussee 149, D-22761 Hamburg, Germany)
Robert M. Wald (rmwa@midway.uchicago.edu)
Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637, U.S.A.
Received: 14 March 1996/ Accepted: 11 June 1996
Abstract
We prove two theorems which concern difficulties in the formu-
lation of the quantum theory of a linear scalar field on a spacetime,
(M, gab ), with a compactly generated Cauchy horizon. These theo-
rems demonstrate the breakdown of the theory at certain base points
of the Cauchy horizon, which are defined as ‘past terminal accumu-
lation points’ of the horizon generators. Thus, the theorems may be
interpreted as giving support to Hawking’s ‘Chronology Protection
Conjecture’, according to which the laws of physics prevent one from
manufacturing a ‘time machine’. Specifically, we prove:
Theorem 1. There is no extension to (M, gab)of the usual field alge-
bra on the initial globally hyperbolic region which satisfies the condition
of F-locality at any base point. In other words, any extension of the
field algebra must, in any globally hyperbolic neighbourhood of any base
point, differ from the algebra one would define on that neighbourhood
according to the rules for globally hyperbolic spacetimes.
1
Theorem 2. The two-point distribution for any Hadamard state de-
fined on the initial globally hyperbolic region must (when extended to
a distributional bisolution of the covariant Klein-Gordon equation on
the full spacetime) be singular at every base point xin the sense that
the difference between this two point distribution and a local Hadamard
distribution cannot be given by a bounded function in any neighbour-
hood (in M×M) of (x, x).
In consequence of Theorem 2, quantities such as the renormalized
expectation value of φ2or of the stress-energy tensor are necessarily
ill-defined or singular at any base point.
The proof of these theorems relies on the ‘Propagation of Singu-
larities’ theorems of Duistermaat and H¨ormander.
1 Introduction
In recent years, there has been considerable interest in the question whether
it is possible, in principle, to manufacture a ‘time machine’ – i.e., whether, by
performing operations in a bounded region of an initially ‘ordinary’ space-
time, it is possible to bring about a ‘future’ in which there will be closed
timelike curves. Heuristic arguments by Morris, Thorne and Yurtsever [40]
suggested in 1988 that this might be possible with a suitable configuration of
relatively moving wormholes, and alternative ideas also have been suggested
by others (see e.g., [17]). For reviews and further references, see e.g., [49, 52].
A precise, general class of classical spacetimes, (M, gab), in which a time
machine is ‘manufactured’ in a nonsingular manner within a bounded region
of space is comprised by those with a compactly generated Cauchy horizon
[23]. By this is meant, first, that (M, gab) is time orientable and possesses
a closed achronal edgeless set S(often referred to as a partial Cauchy sur-
face) such that D+(S)6=I+(S), where D+(S) denotes the future domain of
dependence of S, and I+(S) denotes the chronological future of S. (Often
we shall refer to the full domain of dependence, D(S), of Sas the initial
globally hyperbolic region.) However, it is additionally required that there
exists a compact set Ksuch that all the past-directed null generators of the
future Cauchy horizon, H+(S), eventually enter and remain within K. It is
not difficult to see that any spacetime obtained by starting with a globally
hyperbolic spacetime and then smoothly deforming the metric in a compact
region so that the spacetime admits closed timelike curves must possess a
2
compactly generated Cauchy horizon. Conversely, as we shall discuss fur-
ther in Sect. 2, any spacetime with a compactly generated Cauchy horizon
necessarily violates strong causality, and, thus, is at least ‘on the verge’ of
creating a time machine. Thus, we will, in the following discussion, identify
the notions of ‘manufacturing a time machine’ and ‘producing a spacetime
with a compactly generated Cauchy horizon’.
It follows from the ‘Topological Censorship Theorem’ [16] that in order to
produce a spacetime with a compactly generated Cauchy horizon by means of
a traversable wormhole it is necessary to violate the weak energy condition.
More generally, the weak energy condition must be violated in any space-
time with a compactly generated Cauchy horizon with noncompact partial
Cauchy surface S[23]. Physically realistic classical matter fields satisfy the
weak energy condition, but this condition can be violated in quantum field
theory. Thus, ‘quantum matter’ undoubtedly would be needed to produce a
spacetime with a compactly generated Cauchy horizon.
The above considerations provide motivation for the study of quantum
field theory on spacetimes with a compactly generated Cauchy horizon. In
attempting to carry out such a study however, one immediately encounters
the problem that, even for linear field theories, we only have a clear and
undisputed set of rules for quantum field theory in curved spacetime in the
case that the spacetime is globally hyperbolic. In this case, there is a well-
established construction of a field algebra based on standard theorems on the
well-posedness of the corresponding classical Cauchy problem. Friedman and
Morris [14] have recently established that there exists a well defined classical
dynamics on a simple model spacetime with closed timelike curves, and they
have conjectured that classical dynamics will be suitably well posed on a
wide class of spacetimes with compactly generated Cauchy horizons. Thus
for spacetimes in this class one might expect it to be possible to mimic the
standard construction and obtain a sensible quantum field theory.
It should be noted that even if no difficulties were to arise in the formula-
tion of quantum field theory on spacetimes with compactly generated Cauchy
horizons, there still would likely be very serious obstacles to manufacturing
time machines, since it is far from clear that any solutions to the semiclassical
field equations can exist which correspond to time machine production. In
particular, not only the (pointwise) weak energy condition but the averaged
null energy condition must be violated with any time machine produced with
‘traversable wormholes’ [16]. Under some additional assumptions, violation
of the averaged null energy condition also must occur in any spacetime with
3
a compactly generated Cauchy horizon in which the partial Cauchy surface
Sis noncompact [23]. Although the averaged null energy condition can be
violated in quantum field theory in curved spacetime, there is recent evidence
to suggest that it may come ‘close enough’ to holding to provide a serious
impediment to the construction of a time machine [13, 12].
However, in the present paper, we shall not concern ourselves with issues
such as whether sufficiently strong violations of energy conditions can occur
to even create the conditions needed to produce a spacetime with a com-
pactly generated Cauchy horizon. Rather we will focus on the more basic
issue of whether a sensible, nonsingular quantum field theory of a linear field
can be defined at all on such spacetimes. There is, of course, no difficulty in
defining quantum field theory in the initial globally hyperbolic region D(S),
but there is evidence suggesting that quantum effects occurring as one ap-
proaches the Cauchy horizon must become unboundedly large, resulting in
singular behaviour of the theory. Analyses by Kim and Thorne [33] and oth-
ers [23, 52] have indicated that for all/many physically relevant states the
renormalized expectation value of the stress-energy tensor, hTabi, of a linear
quantum field, defined on the initial globally hyperbolic region D(S), must
blow up as one approaches a compactly generated Cauchy horizon.1How-
ever, these arguments are heuristic in nature, and examples recently have
been given by Krasnikov [37] and Sushkov [46, 47] of states for certain linear
quantum field models on the initial globally hyperbolic region of (two and
four dimensional) Misner space (see e.g., [24] or [23]) for which hTabiremains
finite as one approaches the Cauchy horizon. This raises the issue of whether
a quantum field necessarily becomes singular at all on a compactly generated
Cauchy horizon, and, if so, in what sense it must be ‘singular’.
The purpose of this paper is to give a mathematically precise answer to
this question. As we shall describe further in Sect. 2, every compactly gen-
erated Cauchy horizon, H+(S), contains a nonempty set, Bof ‘base points’
having the property that every generator of H+(S) approaches arbitrarily
close to Bin the past, and strong causality is violated at every x∈ B. We
shall prove the following two theorems concerning quantum field theory on
spacetimes with compactly generated Cauchy horizons. (Full statements are
given in Sect. 5. See also Sect. 6 for further discussion.):
1A first result in this direction was obtained as early as 1982 by Hiscock and Konkowski
[25] who constructed a natural quantum state for a linear scalar field on the initial globally
hyperbolic portion of four dimensional Misner space and showed that its stress energy
tensor diverges as one approaches the Cauchy horizon.
4
Theorem 1: There is no extension to (M, gab)of the usual field algebra
on the initial globally hyperbolic region D(S)which satisfies the condition
of F-locality [30]. (The F-locality condition necessarily breaks down at any
x∈ B.)
Theorem 2: The two-point distribution for any Hadamard state of the co-
variant Klein-Gordon field defined on the initial globally hyperbolic region of
a spacetime with a compactly generated Cauchy horizon must (when extended
to a distributional bisolution on the full spacetime) be singular at every x∈ B
in the sense that the difference between this two point distribution and a local
Hadamard distribution cannot be given by a bounded function in any neigh-
bourhood (in M×M) of (x, x).
Theorem 1 shows that on a spacetime with a compactly generated Cauchy
horizon the quantum field theory must be locally different from the corre-
sponding theory on a globally hyperbolic spacetime. Theorem 2 establishes
that even if hTabiremains finite as one approaches the Cauchy horizon as
in the examples of Krasnikov and Sushkov [37, 46, 47]2it nevertheless must
always (i.e., for any Hadamard state on the initial globally hyperbolic region)
be the case that hTabiis ill defined or singular at all points of B, since the
limit which defines hTabivia the point-splitting prescription cannot exist, and
in fact must diverge in some directions.
Our theorems show that very serious difficulties arise when attempting
to define the quantum field theory of a linear field on a spacetime with a
compactly generated Cauchy horizon. In particular, our results may be in-
terpreted as indicating that in order to manufacture a time machine, it would
be necessary at the very least to enter a regime where quantum effects of grav-
ity itself will be dominant. Thus, our results may be viewed as supportive
of Hawking’s ‘Chronology Protection Conjecture’ [23], although we shall re-
frain from speculating as to whether the difficulties we find might somehow
be evaded in a complete theory where gravity itself is quantized.
2Sushkov’s examples involve a mild generalization of the quantum field model discussed
here to the case of complex automorphic fields [46, 47]. While our theorems strictly don’t
apply as stated to automorphic fields, we remark that we expect everything we do to
generalize to this case. (See also the final sentence of Sect. 2.) Moreover, Cramer and Kay
[7] have recently shown directly that the conclusions of Theorem 2 are valid for the state
discussed by Sushkov in [47]. Note that, as we discuss further in Sect. 6 below, reference
[7] also points out a similar situation for the states discussed by Boulware [3] for Gott
space and Tanaka and Hiscock [48] for Grant space.
5
The proofs of the above two theorems will be based upon the ‘Propagation
of Singularities’ theorems of Duistermaat and H¨ormander [9, 28]. In particu-
lar, these theorems will allow us to conclude that the two-point distribution
of a Hadamard state is not locally in L2(M×M) in a neighbourhood of any
(x, y)∈M×Msuch that xand ycan be joined by a null geodesic. This fact
directly gives rise to the singular behaviour of Theorem 2 at B. It should be
noted that similar singular behaviour will occur in essentially all (but – see
Sect. 6 – not quite all) situations where one has a closed or ‘almost closed’
or self-intersecting null geodesic, so the results obtained here for spacetimes
with compactly generated Cauchy horizons can be generalized to additional
wide classes of causality violating spacetimes. (See Sect. 6.)
We shall begin in Sect. 2 by establishing two geometrical propositions
on spacetimes with compactly generated Cauchy horizons. In Sect. 3, after
a brief review of the essential background on distributions and microlocal
analysis required for their statement, we review the Propagation of Singular-
ities theorems of Duistermaat and H¨ormander for linear partial differential
operators. In Sect. 4 we shall briefly review some relevant aspects of linear
quantum field theory in globally hyperbolic spacetimes and discuss how one
can approach the question of what it might mean to quantize a (linear) quan-
tum field on a non-globally hyperbolic spacetime. In particular, the notion
of F-locality, introduced in [30], will be briefly reviewed there. In Sect. 5
we shall state and prove our theorems on the singular behaviour of quantum
fields. In a final discussion section (Sect. 6) we shall discuss further the sig-
nificance and interpretation of our theorems and mention some directions in
which they may be generalized.
We remark that while we explicitly treat only the covariant Klein-Gordon
equation, we expect appropriate analogues of Theorems 1 and 2 to hold for
arbitrary linear quantum field theories. Moreover, it seems possible that the
singular behaviour we find for linear quantum fields on the Cauchy horizon
may be related to pathologies found in the calculation of the S-matrix for
nonlinear fields [15].
6
2 Some Properties of Compactly Generated
Cauchy Horizons
In this section, we shall establish some geometrical properties of spacetimes
possessing compactly generated Cauchy horizons. We begin by recalling
several basic definitions and theorems, all of which can be found, e.g., in
[24, 55].
Let (M, gab ) be a time orientable spacetime, and let Sbe a closed, achronal
subset of M. We define the future domain of dependence of S(denoted,
D+(S)) to consist of all x∈Mhaving the property that every past inex-
tendible causal curve through xintersects S. The past domain of dependence
is defined similarly. It then follows that int D+(S) = I−[D+(S)] ∩I+(S),
where I−denotes the chronological past. Thus, if x∈D+(S) but x6∈ S,
then every past directed timelike curve from ximmediately enters int D+(S)
and remains in int D+(S) until it intersects S.
The future Cauchy horizon of S(denoted H+(S)) is defined by H+(S) =
D+(S)−I−[D+(S)], where D+(S) denotes the closure of D+(S). It follows
immediately that H+(S) is achronal and closed. A standard theorem [24, 55]
establishes that every x∈H+(S) lies on a null geodesic contained within
H+(S) which either is past inextendible or has an endpoint on the edge of S.
Thus, if Sis edgeless (in which case Sis referred to as a partial Cauchy surface
or slice), then H+(S) is generated by null geodesics which may have future
endpoints (i.e., they may ‘exit’ H+(S) going into the future) but cannot
have past endpoints. Note that this implies that H+(S)∩D+(S) = ∅. Since
similar results hold for H−(S), it follows that for any partial Cauchy surface
S, the full domain of dependence, D(S)≡D+(S)∪D−(S), is open.
Now, let Sbe a partial Cauchy surface. We say that the future Cauchy
horizon, H+(S), of Sis compactly generated [23] if there exists a compact
subset, K, of M, such that for each past directed null geodesic generator, λ,
of H+(S) there exists a parameter value s0(in the domain of definition of
λ) such that λ(s)∈Kfor all s > s0. In other words, when followed into the
past, all null generators of H+(S) enter and remain forever in K.
We now introduce some new terminology. Let λ:I→Mbe any con-
tinuous curve defined on an open interval, I⊂ R, which may be infinite or
semi-infinite. We say that x∈Mis a terminal accumulation point of λif for
every open neighbourhood Oof xand every t0∈Ithere exists t∈Iwith
t > t0such that λ(t)∈ O. (By contrast, xwould be called an endpoint of λif
7
for every open neighbourhood Oof xthere exists t0∈Isuch that λ(t)∈ O
for all t > t0. Thus, an endpoint is automatically a terminal accumulation
point, but not vice-versa.) Equivalently, xis a terminal accumulation point
of λif there exists a monotone increasing sequence {ti} ∈ Iwithout limit in
Isuch that λ(ti) converges to x. When λis a causal curve, we shall call xa
past terminal accumulation point if it is a terminal accumulation point when
λis parametrized so as to make it past-directed. Note that if xis a past
terminal accumulation point of a causal curve λbut xis not a past endpoint
of λ, then strong causality must be violated at x.
Let H+(S) be a compactly generated future Cauchy horizon. We define
the base set,B, of H+(S) by
B={x∈H+(S)|there exists a null generator, λ, of H+(S) such that x
is a past terminal accumulation point of λ}.(1)
Since the null generators of H+(S) cannot have past endpoints, it follows that
strong causality must be violated at each x∈ B. The following proposition
(part of which corresponds closely to the second theorem stated in Sect. V
of [23]) justifies the terminology ‘base set’:
Proposition 1. The base set, B, of any compactly generated Cauchy horizon,
H+(S), always is a nonempty subset of K. In addition, all the generators
of H+(S)asymptotically approach Bin the sense that for each past-directed
generator, λ, of H+(S)and each open neighbourhood Oof B, there exists
at0∈I(where Iis the interval of definition of λ) such that λ(t)∈ O
for all t > t0. Finally, Bis comprised by null geodesic generators, γ, of
H+(S)which are contained entirely within Band are both past and future
inextendible.
Proof. Let λbe a past directed null generator of H+(S) and let {ti}be
any monotone increasing sequence without limit on I. Then λ(ti)∈Kfor
infinitely many i. It follows immediately that {λ(ti)}must have an accumu-
lation point, x, which must lie on H+(S) since H+(S) is closed. It follows
that x∈ B, and hence Bis nonempty. The proof that B ⊂ Kis entirely
straightforward. If one could find a generator λand an open neighbourhood
Oof Bsuch that for all t0∈I, we can find a t > t0such that λ(t)6∈ O, then,
using compactness of K, we would be able to find a past terminal accumu-
lation point of λlying outside of Oand, hence, outside of B, which would
contradict the definition of B.
8
To prove the last statement in the proposition, it is useful to introduce
(using the paracompactness of M) a smooth Riemannian metric, eab, on M.
We shall parametrize all curves in Mby arc length, s, with respect to this
Riemannian metric.
Let x∈ B. We wish to show that there exists a null geodesic, γ, through
xwhich is both past and future inextendible and is contained in B. Since
x∈ B, there exists a null generator, λ, of H+(S) such that xis a past
terminal accumulation point of λ. We parametrize λso that its arc length
parameter, s, increases in the past direction. Since λis past inextendible
and Kis compact, smust extend to infinite values (even though the geodesic
affine parameter of λin the Lorentz metric gab might only extend to finite
values). Thus, there exists a sequence {si}diverging to infinity such that
λ(si) converges to x. Let (ki)adenote the tangent to λat siin the arc length
parametrization, so that (ki)ahas unit norm with respect to eab . Since the
subset of the tangent bundle of Mcomprised by points in Ktogether with
tangent vectors with unit norm with respect to eab is compact, it follows that
– passing to a subsequence, if necessary – there must exist a tangent vector
kaat xsuch that {(λ(si),(ki)a)}converges to (x, ka). Since each (ki)ais null
in the Lorentz metric gab, it follows by continuity that kaalso must be null
with respect to gab.
Let γbe the maximally extended (in M, in both future and past direc-
tions) null geodesic determined by (x, ka). We parametrize γby arc length
with respect to eab , with increasing scorresponding to going into the past
and with s= 0 at x. Let y∈γand let sdenote the arc length parameter of
y. Since the sequence {si}diverges to infinity, it follows that for sufficiently
large i, (s+si) will be in the interval of definition of λ. Since {(λ(si),(ki)a)}
converges to (x, ka), it follows by continuity of both the exponential map
(with respect to gab) and the arc length parametrization (with respect to eab)
that {λ(s+si)}converges to y. Thus, y∈ B, as we desired to show. 2
In some simple examples (see, e.g., [52]), Bconsists of a single closed null
geodesic, referred to as a ‘fountain’. However, it appears that, generically, B
may contain null geodesics which are not closed [6].
Our final result on B, which will be needed in Sect. 5, is the following.
Proposition 2. Let H+(S)be a compactly generated Cauchy horizon, and
let Bbe its base set. Let x∈ B, and let Ube any globally hyperbolic open
neighbourhood of x. Then there exist points y, z ∈ U ∩ int D+(S)such that y
and zare connected by a null geodesic in the spacetime (M, gab), but cannot
9
be connected by a causal curve lying within U.
Proof. As in the proof of the previous proposition, we introduce a smooth
Riemannian metric, eab, on Mand parametrize curves in Mby arc length,
s, with respect to eab with increasing scorresponding to going into the past.
Let λand {si}be as in the proof of the previous theorem. There cannot exist
an s0such that λ(s)∈ U for all s > s0, since otherwise strong causality would
be violated at xin the spacetime (U, gab), thereby contradicting the global
hyperbolicity of (U, gab). Thus, there exist integers i, j with si< sjsuch that
λ(si), λ(sj)∈ U but for some swith si< s < sjwe have λ(s)6∈ U. By passing
to a smaller interval around sif necessary, we may assume without loss of
generality that λis one-to-one in [si, sj] (so that, in particular, λ(si)6=λ(sj)).
It then follows from the achronality of H+(S) that any past-directed causal
curve in (M, gab) which starts at λ(si) and ends at λ(sj) must coincide with
(or contain) this segment of λ. (If not, then there would exist two distinct,
past-directed, null geodesics connecting λ(si) with λ(sj), and it would be
possible to obtain a timelike curve joining λ(si) to λ(sj+ 1).) Consequently,
λ(si) cannot be joined to λ(sj) by any past-directed causal curve contained
within U.
Since in any globally hyperbolic spacetime, the causal past of any point is
closed, it follows that there exist open neighbourhoods, Oi⊂ U and Oj⊂ U ,
of λ(si) and of λ(sj), respectively, such that no point of Oican be joined to
a point of Ojby a past-directed causal curve lying within U. Without loss
of generality, we may assume that Oiand Ojare contained in I+(S) (since
otherwise we could take their intersection with I+(S)).
Our aim, now, is to deform λsuitably to get a null geodesic in (M, gab )
which joins a point y∈ Oi∩int D+(S) to a point z∈ Oj∩int D+(S).
To do so, we choose along λ(in a neighbourhood of λ(si)), a past-directed
null vector field, la, and a spacelike vector field wasatisfying gablakb=−1,
gabwakb=gabwalb= 0, and gabwawb= 1, where kadenotes the tangent to
λ(in the arc length parametrization with respect to eab). Let ǫ > 0 and
consider the null geodesic αstarting at point λ(si+ǫ) with null tangent
ka+1
2ǫ2la+ǫwa. By continuity, for sufficiently small ǫ,y=α(ǫ) will lie in
Oiand z=α(sj−si) will lie in Oj. Since yand zcan be connected to λ(si)
by a past-directed broken null geodesic, they each lie in I−[H+(S)]. Since
they also each lie in I+(S), it follows that y, z ∈int D+(S). As shown above,
ycannot be connected to zby a past-directed causal curve lying within U.
However, since y∈int D+(S), it follows that there cannot exist any future-
10
directed causal curve, σ, in M, connecting yto z, since, otherwise we would
obtain a closed causal curve through yby adjoining σto (the time reverse
of) α. Thus, we have y, z ∈ U ∩int D+(S) such that yand zare connected by
the null geodesic αin the spacetime (M, gab ), but they cannot be connected
by any causal curve lying within U.2
To end this section, we remark, following [7], that Propositions 1 and
2 (and hence also the theorems of Sect. 5) will also clearly apply to any
spacetime (such as four dimensional Misner space) which, while having a
Cauchy horizon which is not compactly generated, arises as the product with
a flat 4 −ddimensional Euclidean space of some spacetime with compactly
generated Cauchy horizon of lower dimension d.
3 Microlocal Analysis and Propagation of Sin-
gularities
In this section, we shall review results of H¨ormander and Duistermaat and
H¨ormander [9, 27, 28] concerning the propagation of singularities in distribu-
tional solutions to partial differential equations. These results do not appear
to be widely known or used in the physics literature, and we shall attempt
to make them somewhat more accessible by stating them in the simpler case
of differential operators rather than in the more general setting of pseudod-
ifferential operators.
We begin by recalling that on an arbitrary smooth (paracompact) n-dim-
ensional manifold, M, elements of the vector space, D(M), of smooth (C∞)
(and in this paper, we shall assume real-valued) functions of compact support
are referred to as test functions. A topology is defined on D(M) as follows.
First, we introduce a Riemannian metric, eab, and a derivative operator,
∇a, on M. Next, we fix a compact set, K, and focus attention on the
subspace, DK(M), of D(M) consisting of test functions with support in K.
On this subspace, we define the family of seminorms, {|| · ||k}, by
||f||k= sup
x∈K
|∇a1...∇akf|,(2)
where by the ‘absolute value’ of the tensor appearing on the right side of this
equation we mean its norm as computed using the Riemannian metric eab.
We define the topology of DK(M) to be the weakest topology which make all
of these seminorms as well as the operations of addition and multiplication
11
by scalars continuous. It may be verified that this topology is independent
of the choices of eab and ∇a. This gives each DK(M) the structure of being a
locally convex space. Finally, we express Mas a countable union of compact
sets, Kiwhich form an increasing family (Ki⊂Ki+1) – thereby expressing
D(M) as a countable union of the DKi(M) – and take the topology of D(M)
to be given by the strict inductive limit [44] of the locally convex spaces
DKi(M). It may be verifed that the topology thereby obtained on D(M) is
independent of the choice of compact sets Ki.
A distribution, u, on M, is simply a linear map from D(M) into the
real numbers, R, which is continuous in the topology on D(M) defined in
the previous paragraph. The vector space of distributions on Mis denoted
D′(M). Denote by L1
loc(M) the collection of measurable functions on M
whose restriction to any compact set, K, is integrable with respect to a
smooth volume element ηintroduced on M. (The definition of L1
loc(M)
clearly is independent of the choice of η.) If F∈L1
loc(M), then the linear
map u:D(M)→ R given by
u(f) = ZMF f η(3)
defines a distribution on M. We remark that in the presence of a preferred
volume element, such as provided by the natural volume element
η=|det g|1/2dx1∧... ∧dxn(4)
associated with the metric gab in the case where Mis a spacetime manifold,
we may identify the function Fwith the distribution u. A distribution u∈
D′(M) will be said to be smooth if there exists a C∞function, F, on Msuch
that Eq. (3) holds. A distribution, u, will be said to be smooth at x∈Mif
there exists a test function, g, with g(x)6= 0 such that gu is smooth, where
the distribution gu is defined by
gu(f) = u(gf).(5)
The set of points y∈Mat which ufails to be smooth is referred to as the
singular support of uin M.
A key idea in the analysis of the propagation of singularities is to refine
the notion of the singular support of uin Mto a notion of the wave front
set of u, which can be thought of as describing the singular support of u
in the cotangent bundle,T∗(M), of M. This notion can be defined most
12
conveniently in terms of the Fourier transform of distributions. To begin,
let ube a distribution on Rnwhich is of compact support, i.e., there exists
a compact set Ksuch that u(f) = 0 whenever the support of fdoes not
intersect K. We define the Fourier transform, ˆuof uto be the distribution
given by
ˆu(f) = u(gˆ
f),(6)
where gis any test function such that g= 1 on K. (The obvious extension of
uto act on complex test functions is understood here; ˆuis a complex-valued
distribution, but its real and imaginary parts define real-valued distribu-
tions.) It follows that ˆuis always smooth (indeed, analytic), and the smooth
function corresponding to ˆuvia Eq. (3) (which we also shall denote by ˆu) sat-
isfies the property that it and all of its derivatives are polynomially bounded
at infinity (see Theorem IX.5 of [45]). Furthermore, it follows from the fact
that the Fourier transform maps Schwartz space onto itself that the distri-
bution uitself will be smooth if and only if for every positive integer mthere
exists a constant Cmsuch that
|ˆu(k)| ≤ Cm(1 + |k|)−m.(7)
Now let u∈ D′(M) be a distribution on an n-dimensional manifold, M.
Let x∈Mand let Obe an open neighbourhood of xwhich can be covered
by a single coordinate patch, i.e., there exists a diffeomorphism ψ:O →
U ⊂ Rn. Let gbe a test function with support contained within Osuch
that g(x)6= 0. The distribution gu may then be viewed as a distribution on
Rnwhich is of compact support. Hence, for the given choice of coordinates,
the Fourier transform, c
gu, of gu is well defined as a distribution on Rnand
satisfies the properties of the previous paragraph. We may use the local
coordinates at xto identify the cotangent space at xwith Rnby associating
with each cotangent vector pathe point in Rngiven by the components of
pain these coordinates. In this manner, we may view c
gu as a distribution on
the cotangent space at x.
Now let pabe a nonzero cotangent vector at x. We say that uis smooth
at (x, pa)∈T∗(M) if there exists a test function, gwith support contained
within Osatisfying g(x)6= 0 and there exists an open neighbourhood, Q, of
pain the cotangent space at xsuch that for each positive integer m, there
exists a constant Cmsuch that for all ρa∈ Q and all λ≥0 we have,
|c
gu(λρa)| ≤ Cm(1 + |λ|)−m.(8)
13
It can be shown that this notion of smoothness of uat (x, pa) is independent
of the choice of local coordinates at x, and, thus, is a well defined property of
u(see part (f) of Theorem IX.44 of [45]). Let S ⊂ T∗(M) denote the set of
points in T∗(M) at which uis smooth. It follows directly from its definition
that Sis open and is ‘conic’ in the sense that if pa∈ S, then λpa∈ S for
all λ > 0. The wave front set of u, denoted WF(u), is defined to be the
complement of Sin T∗(M)\0, where ‘0’ denotes the ‘zero section’ of T∗(M)
WF(u) = [T∗(M)\0] \ S .(9)
In other words, (x, pa)∈T∗(M) lies in WF(u) if and only if pa6= 0 and u
fails to be smooth at (x, pa). It can be shown (see, e.g., Theorem IX.44 of
[45]) that x∈Mis in the singular support of uif and only if there exists a
cotangent vector paat xsuch that (x, pa)∈WF(u).
It is essential that we view WF(u) to be a subset of the cotangent bundle
(rather than, e.g., the tangent bundle) in order that it be independent of
the choice of coordinates used to define Fourier transforms. Some further
insight into the meaning of the above definitions and the reason why it is the
cotangent bundle which is relevant for the definition can be obtained from
the following considerations. Let (x, pa)∈T∗(M) with pa6= 0. Then pa
determines (by orthogonality) a hyperplane in the tangent space at x. In
a sufficiently small open neighbourhood of x, we can introduce coordinates
(t, x1, ..., xn−1) so that the hypersurface of constant tpassing through xis
tangent to this hyperplane. These coordinates can be used to factorize an
open sub-neighbourhood of xas the product manifold R × Rn−1. Hence,
any distribution, u, defined on this sub-neighbourhood can be viewed as a
bi-distribution on R× Rn−1. In particular, for any test function, f, on Rn−1,
u(·, f ) defines a distribution on R. Then, it is not difficult to verify that if a
real-valued distribution uis smooth at (x, pa)∈T∗(M)\0, then there exists
ag∈ D(M) with g(x)6= 0 such that for any test function f, on Rn−1, the
distribution gu(·, f ) on Ris smooth.
The notion of smoothness or the lack of smoothness of a distribution,
u, at (x, pa)∈T∗(M)\0 can be further refined as follows. First, for any
real number s, a distribution uwill be said to lie in the local Sobolev space
Hs
loc(x) associated with a point x∈Mif there exists a test function, g, with
g(x)6= 0 and the support of gcontained within a single coordinate patch,
such that gu (viewed as a distribution of compact support on Rn) satisfies
Z(1 + |k|2)s|c
gu|2dnk < ∞.(10)
14
It is easy to see that the space so-defined is independent of the choice of
function gand of the choice of coordinate patch containing its support. When
sis a non-negative integer, this condition is equivalent to requiring that gu
and all of its (weak) derivatives up to order sare given by square integrable
functions. (It is easy to see that this result holds independently of the choice
of derivative operator and independently of the choice of (smooth) volume
element.) Note that ulies in Hs
loc(x) for all sif and only if uis smooth
at x. Following [27], we say that a distribution ulies in the local Sobolev
space Hs
loc(x, pa) associated with (x, pa)∈T∗(M)\0 if we can express u
as u=u1+u2, where u1lies in Hs
loc(x) and (x, pa)6∈ WF(u2). It can be
shown (see Theorem 18.1.31 of [27]) that we have u∈Hs
loc(x) if and only if
u∈Hs
loc(x, pa) for all nonvanishing cotangent vectors paat x. Furthermore,
we have u∈Hs
loc(x, pa) for all sif and only if uis smooth at (x, pa).
Next, we introduce some key definitions for linear partial differential op-
erators. Let Abe an arbitrary linear partial differential operator of order m
on M, so that Acan be expressed as
A=
m
X
i=0
αa1...ai
(i)∇a1...∇ai,(11)
where ∇ais an arbitrary derivative operator on Mand each αa1...ai
(i)is a
smooth tensor field. We define the principal symbol,H, of Ato be the map
H:T∗(M)→ R given by
H(x, pa) = αa1...am
(m)(x)pa1...pam.(12)
It is easily checked that His independent of the choice of derivative operator
∇a.
Now, T∗(M) has the natural structure of a symplectic manifold, so it can
be viewed as the ‘phase space’ of a classical dynamical system. By choos-
ing Hto be the Hamiltonian of this system, we thereby associate a classical
mechanics problem to each linear partial differential operator, A, on M. In
particular, associated with A, we obtain a vector field haon T∗(M) whose
integral curves correspond to solutions of Hamilton’s equations of motion
on T∗(M) with Hamiltonian H. We define the characteristic set of A, de-
noted char(A), to be the subset of T∗(M)\0 (where, again, 0 denotes the
zero-section of T∗(M)) satisfying H(x, pa) = 0. (In other words, the char-
acteristic set of Aconsists of the states in phase space with zero energy but
nonvanishing momentum.) We refer to the integral curves of hain T∗(M)\0
15
starting from points in the characteristic set as the bicharacteristics of A.
(Often, these curves are called ‘bicharacteristic strips’, and the projection of
a bicharacteristic to Mis called a ‘bicharacteristic curve’.) By ‘conservation
of energy’, all bicharacteristics are contained in the characteristic set.
We now are in a position to state the Propagation of Singularities The-
orem which will be used to prove our main results. First recall that, in the
presence of a preferred volume element η, if Ais a linear partial differential
operator, the adjoint of A, denoted A†, is defined to be the linear partial
differential operator determined by the condition that
ZgAf η=ZfA†gη(13)
for all test functions f, g. A distribution uwill be said to satisfy the equation
Au = 0 if for every test function f, we have
u(A†f) = 0 .(14)
We have the following theorem, which is obtained by restricting Theorems
26.1.1 and 26.1.4 of [28] (from the case of pseudodifferential operators) to
the simpler case of linear partial differential operators (and vanishing source
term).
Propagation of Singularities Theorem. Let Mbe an n-dimensional
manifold, with preferred volume element η. Let Abe a linear partial dif-
ferential operator of order mon Mand suppose u∈ D′(M)satisfies the
equation Au = 0. Then, we have (i) WF(u)⊂char(A)and (ii) For any
(x, pa)∈char(A), we have u∈Hs
loc(x, pa)if and only if u∈Hs
loc(x′, p′
a)for
all (x′, p′
a)lying on the same bicharacteristic as (x, pa). Thus, in particular,
if (x, pa)∈WF(u), then the entire bicharacteristic through (x, pa)lies in
WF(u).
(Part (i) of the above theorem together with the final sentence incorporate
the content of Theorem 26.1.1 of [28], while Part (ii) corresponds to Theorem
26.1.4.)
In the present paper, the partial differential operator in which we are
particularly interested is the covariant Klein-Gordon operator
A=2g−m2(15)
on a given curved spacetime (M, gab). Here, 2gdenotes the Laplace Beltrami
operator for the metric g. We remark that if we take (as we shall from now on)
16
our preferred volume element ηto be the natural spacetime volume element
(4) associated with the metric, this operator satisfies A†=A. Clearly, its
principal symbol is
H(x, p) = gab(x)papb(16)
which is well known to be a Hamiltonian for geodesics. The characteristic set
thus consists of the points of T∗(M) whose covector is null and nonvanishing,
and the bicharacteristics are curves t7→ (x(t), pa(t)) in the cotangent bundle
for which t7→ x(t) is an affinely parametrized null geodesic, and, at each
value of the parameter t,pa(t) is the cotangent vector obtained by using the
metric to ‘lower an index’ on the tangent vector to the geodesic. (Below,
we shall say that the cotangent vector pa(t) is ‘tangent’ to the geodesic.) In
other words, in this case, the bicharacteristics are the lifts to the cotangent
bundle of affinely parametrized null geodesics.
In the proof of the theorems of the present paper we shall be concerned
with certain distributional bisolutions to the covariant Klein-Gordon equa-
tion which, as we shall discuss further in the next section, occur in quantum
field theory on a curved spacetime (M, gab). The above Propagation of Sin-
gularities Theorem will be used to obtain information on the global nature
of the singularities in these two-point functions given information about the
singularities when the two points on which they depend are close together.
Roughly speaking, we shall be able to conclude from the above theorem that,
if such a distributional bisolution is singular for sufficiently nearby pairs of
points on a given null geodesic, then it will necessarily remain singular for
all pairs of points on that null geodesic. Moreover the theorem assures us
that the ‘strength’ of the singularity (as measured by the indices of the local
Sobolev spaces in which the distribution fails to lie) cannot diminish.
One may define a distributional bisolution to be a bidistribution on M
which is a distributional solution to the covariant Klein-Gordon equation
in each variable. Here, by a bidistribution Gon Mwe mean a (real or
complex valued) functional on D(M)× D(M) which is separately linear and
continuous in each variable and to say that Gis a solution to the Klein-
Gordon equation in each variable means that G((2g−m2)f, h) = 0 and
G(f, (2g−m2)h) = 0 for all f, h ∈ D(M). A bidistribution Gon Mis then
necessarily jointly continuous and arises from a distribution ˜
Gon the product
manifold M×Min the sense that G(f, g) = ˜
G(f⊗g), where, if fand gare
each test functions in D(M), f⊗gdenotes the test function in D(M×M)
with values f⊗g(x, y ) = f(x)g(y). (For the proof of these assertions, see
17
e.g., the proof of the Schwartz Kernel Theorem in Sect. 5.2 in [26].) To say
that Gis a distributional bisolution on Mmay thus be expressed by saying
that ˜
Gis a distributional solution to each of the pair of partial differential
equations A1˜
G= 0, A2˜
G= 0 on M×M, where (in an obvious notation) A1
and A2are the partial differential operators
A1= (2g−m2)⊗I, A2=I⊗(2g−m2).(17)
It is this latter way of regarding distributional bisolutions which permits
direct application of the Propagation of Singularity Theorem. (From now
on, we shall adopt an informal point of view in which we do not distin-
guish between Gand ˜
G.) Thus, a point (x, pa;y, qb) in the cotangent bundle
T∗(M×M)\0 of M×Mwill belong to the characteristic set of both A1and
A2if and only if both paand qbare null (and at least one of them is non-zero).
Thus we conclude by Part (i) of the above theorem that the wave front set
of a distributional bisolution ˜
Gmust consist of a subset of such ‘doubly null’
points. Moreover, if the wave front set of a given distributional bisolution
˜
Gincludes such a doubly null point (x, pa;y, qb) then, applying e.g., the last
sentence of the Propagation of Singularities Theorem to the operator A1we
conclude that it must also include all points (x′, p′
a;y, qb) for which (x′, p′
a)
lies on the same lifted null geodesic as (x, pa). Similarly, applying the theo-
rem to A2, we conclude that it must also include all points (x, pa;y′, q′
b) for
which (y′, q′
b) lies on the same lifted null geodesic as (y, qb).
Next, we discuss some particular distributional bisolutions to the covari-
ant Klein-Gordon equation which will play an important role both in our dis-
cussion of quantum field theory below, and in our theorems. Firstly, given any
globally hyperbolic spacetime (M, gab), then the advanced and retarded fun-
damental solutions △A(x, y), △R(x, y) of the inhomogeneous Klein-Gordon
equation exist as 2-point distributions and are uniquely defined with respect
to their support properties [38, 39, 5]. Their difference △=△A−△Ris then
a preferred distributional bisolution to the (homogeneous) covariant Klein-
Gordon equation. It is clearly antisymmetric, i.e., △(f, g) = − △ (g, f ) and,
as we discuss in the next section, plays the role of the ‘commutator func-
tion’ in the quantum theory. One may show that its wave front set consists
of all elements (x, pa;y, qb) of T∗(M×M)\0 for which xand ylie on a
single null geodesic, for which paand qbare tangent to that geodesic, and
for which pa, when parallel transported along that null geodesic from xto y
equals −qa. (One way to obtain this result is to notice that the advanced and
retarded fundamental solutions are special cases of advanced and retarded
18
‘distinguished parametrices’ in the sense of [9] from which the wave front set
may be read off. See also [41, 43].)
Secondly, for any curved spacetime (M, gab), we shall be interested in a
class of symmetric (i.e., G(f, g) = G(g, f )) distributional bisolutions Gto the
covariant Klein-Gordon equation which are what we shall call locally weakly
Hadamard. (These will arise in the quantum field theory – see Sect. 4 –
as (twice) the symmetrized two-point functions of ‘Hadamard states’.) This
notion – which is either weaker or equivalent to the various versions of the
Hadamard condition which occur in the literature on quantum field theory in
curved spacetime – is defined as follows: First, we require that Gbe locally
smooth for non-null related pairs of points in the sense that every point x
in the spacetime has a convex normal neighbourhood (see e.g., [24, 55]) Nx
such that the singular support of Gin Nx×Nxconsists only of pairs of null
related points. In consequence of this (cf. the discussion around Eq. (3)
above) on the complement, Cx, in Nx×Nxof this singular support (so Cx
consists of all pairs of non-null separated points in Nx×Nx) there will be
a smooth two-point function, which we shall denote by the symbol Gs, with
the property that, for all test functions Fsupported in Cx,
G(F) = ZCx
Gs(y, z)F(y, z)η(y)η(z),(18)
where we recall that ηdenotes the natural volume element (4) associated with
the metric. It is easy to see that, on Cx,Gsmust be a smooth bisolution
to the covariant Klein-Gordon equation. If Gis locally smooth for non-null
related pairs of points, then we say that Gis locally weakly Hadamard if for
each point xin M, on the corresponding Cx, the Gsas defined above takes
the ‘Hadamard form’ [19, 8, 53, 54, 32]. This latter condition is traditionally
expressed by demanding that there exists some smooth function won Nx×Nx
such that the following equation holds on Cx(in 4 dimensions, with similar
expressions for other dimensions)
Gs(x, y) = 1
2π2
∆1
2
σ+vln(|σ|) + w
,(19)
where σdenotes the square of the geodesic distance between xand y(which
is well defined in Cxsince Nxis a convex normal neighbourhood), ∆ 1
2is the
van Vleck-Morette determinant [8] and vis given by a power series in σwith
19
partial sums
v(n)(x, y) =
n
X
m=0
vm(x, y)σm,(20)
where each vmis uniquely determined by the Hadamard recursion relations
[19, 8]. This statement cannot be interpreted literally because the power
series defining vdoes not in general converge. We overcome this problem
and make the notion of locally weakly Hadamard precise by replacing the
above statement of Hadamard form by the demand that, for each x∈Mand
each integer n, there exists a Cnfunction w(n)on Nx×Nxsuch that, on Cx
Gs(x, y) = 1
2π2
∆1
2
σ+v(n)ln(|σ|) + w(n)
.(21)
The above notion of locally weakly Hadamard corresponds to the notion
of ‘Hadamard form’ implicit in many references (e.g., [53, 54, 1]). However,
we remark that the notion (sometimes referred to as globally Hadamard) of
‘Hadamard’ as defined (for the case of globally hyperbolic spacetimes) in
[32] (when suitably interpreted as a condition on a general symmetric distri-
butional bisolution on a globally hyperbolic spacetime) is a stronger notion
inasmuch as (a) it specifies, for each x, the local behaviour of the distribu-
tion Gon test functions supported in Nx×Nxwhose support is not confined
to Cx, by what amounts essentially to a ‘principle part’ prescription, (b) it
explicitly rules out the possible occurrence of so-called ‘non-local spacelike
singularities’. (See [32] for details.)3
Clearly, since (on any convex normal neighbourhood) σ(x, y) is smooth
and vanishes if and only if xand yare null related, the singular support
of any locally weakly Hadamard distributional bisolution G, when restricted
to Nx×Nxfor any of the neighbourhoods Nxwill consist precisely of all
pairs of null related points. Hence the wave front set of such a G, when
3Actually, it was conjectured by Kay [29, 20] and proved by Radzikowski by microlocal
analysis methods [41, 42] that if the symmetrized two point function of a quantum state
on the field algebra (see Sect. 4) for the covariant Klein-Gordon equation on a globally
hyperbolic spacetime satisfies the global Hadamard conditon of Kay and Wald locally (i.e.,
on each element of an open cover) then it is globally Hadamard. Thus, in the presence of
the positivity conditions required for a (symmetric) distributional bisolution (on a globally
hyperbolic spacetime) to be the symmetrized two-point function of a quantum state, the
strengthening of the Hadamard notion indicated in point (b) here is automatic given that
indicated in point (a).
20
restricted to T∗(Nx×Nx)\0 will include, for each such null pair, at least
one point (y, pa;z, qb), where at least one of paand qbis non-zero and where
both paand qbare null covectors which are tangent to the null geodesic
connecting yand z.4(If they were not tangent, one could get a contradiction
with the smoothness of Gat non-null related pairs of points by applying the
Propagation of Singularities Theorem.)5
In fact, one can show more than this: Namely, given any (symmetric)
locally weakly Hadamard distributional bisolution G, for each point xin M
and each pair of null related points (y, z)∈Nx×Nxwith y6=zthere exist
null covectors paat yand qbat zwhich are each tangent to the null geodesic
connecting yand zand which are not both zero (see Footnote 4) such that
Gfails to belong to L2
loc(y, pa;z, qb) (i.e., to H0
loc(y, pa;z, qb)). To prove this
result, it suffices to show that 1/σ, and hence, easily, also Gfails to belong
to L2
loc(y, z) for any pair, (y, z), of distinct null-related points in Nx×Nx,
since then we have G6∈ L2
loc(y, pa;z, qb) for some covectors paat yand qb
at zwhich are restricted by arguments of the previous paragraph. However,
the fact that G6∈ L2
loc(y, z) follows immediately from the following lemma
[where we identify Lwith M×M,hwith σ, and Xwith (y, z)]:
Lemma. Let Lbe a manifold, and let hbe a smooth function on Lwhich
vanishes at a point X∈Lbut whose gradient is nonvanishing at X. Then
1/h (or, more precisely, any distribution which agrees with 1/h for h6= 0)
fails to be locally L2at X.
(The proof is immediate once one chooses a coordinate chart around X
in which the function his one of the coordinate functions.)
4Of course, since Gis symmetric, if the point (y, pa;z, qb) is in its wave front set, then
the point (z, qb;y, pa) will also be in its wave front set.
5More is known about the wave front set of the (unsymmetrized) two-point functions
of quantum states (see Sect. 4) on globally hyperbolic spacetimes which are (globally)
Hadamard in the stronger sense of [32]: Radzikowski [41, 43] (see also [34, 4, 35, 36] for
recent further developments in this direction) has shown that the wave front set of any
such two point function consists precisely of all elements (x, pa;y, qb) of T∗(M×M)\0 for
which xand ylie on a single null geodesic, for which pais tangent to that null geodesic
and future pointing, and for which qa, when parallel transported along that null geodesic
from yto xequals −pa.
21
4 The Quantum Covariant Klein-Gordon Equa-
tion on a Curved Spacetime
In our discussion of quantum field theory, we shall restrict our interest to a
linear Hermitian scalar field, satisfying the covariant Klein-Gordon equation
(2g−m2)φ= 0 (22)
on a curved spacetime (M, gab). We shall now outline a suitable mathematical
description of this theory in terms of the algebraic approach to quantum field
theory. For further discussions of this, and closely related, approaches see
e.g., [32, 30, 56].
In the case that (M, gab ) is globally hyperbolic, we may take the field alge-
bra to be the ∗-algebra with identity Igenerated by polynomials in ‘smeared
fields’ φ(f), where franges over the space C∞
0(M) of smooth real valued
functions compactly supported on M, which satisfy the following relations
(for all f1, f2∈C∞
0(M) and for all pairs of real numbers λ1, λ2):
1. φ(f) = φ(f)∗
2. φ(λ1f1+λ2f2) = λ1φ(f1) + λ2φ(f2)
3. φ((2g−m2)f) = 0
4. [φ(f1), φ(f2)] = i △(f1, f2)I,
where △denotes the classical ‘advanced minus retarded’ fundamental solu-
tion (or ‘commutator function’) discussed in the previous section. (Of Rela-
tions (1)-(4), it is thus only Relation (4) which becomes problematic when
we attempt to go beyond the class of globally hyperbolic spacetimes. We
shall return to this point below in our discussion of ‘F-locality’.)
To be precise, what we mean by the above statement is that we regard
the set of polynomials, over the field of complex numbers, of the abstract
objects, φ(f), with f∈C∞
0(M) as a free ∗-algebra with identity and then
quotient by the ∗-ideal generated by the above relations.
We have referred above to φ(f) as a ‘smeared quantum field’. While,
in our mathematical definition above, this is to be thought of as a single
abstract object, we of course interpret it heuristically as related to the ‘field
at a point’ ‘φ(x)’ by
φ(f) = ZMφ(x)f(x)η,(23)
22
where ηis the natural volume element (4) defined in the previous section.
Of course we proceed in this way since the ‘field at a point’ is not expected
by itself to be a mathematically well defined entity. This failure of the ‘field
at a point’ to exist is of course closely related to the singular nature of the
commutator function discussed in the previous section.
Quantum states are defined to be positive, normalized (i.e., ω(I) = 1)
linear functionals on this field algebra. (Here, a state ωis said to be positive
if we have ω(A∗A)≥0 for all Abelonging to the field algebra.) A state ωis
thus specified by specifying the set of all its ‘smeared n-point functions’
ω(φ(f1). . . φ(fn)) .(24)
One expects states of interest to at least be sufficiently regular for these
smeared n-point functions to be distributions – i.e., one expects the expres-
sion above to be a continuous functional of each of the quantities f1,...,fn
when the space C∞
0(M) is topologized in the Dtopology. By the Schwartz
Kernel Theorem (see the previous section) we may equivalently regard the n
point functions as distributions on M× · · · × M.
By condition (4) above, for any state ω, twice the antisymmetric part
of the two point function, ω(φ(f1)φ(f2)), is simply i △(f1, f2), which is
smooth at all (y, z) which cannot be connected by a null geodesic. Fur-
thermore, for the reasons discussed, e.g., in [32, 30, 56] and briefly reviewed
below, we require that (twice) the symmetrized two-point function (i.e.,
ω(φ(f1)φ(f2) + φ(f2)φ(f1)) should (at least) be a locally weakly Hadamard
distributional bisolution as defined in Sect. 3. In this paper, we shall refer to
states satisfying this condition as ‘Hadamard states’. Note that this notion
only restricts the two-point function and does not restrict the other n-point
functions; we shall not need to concern ourselves here with the question of
what should be required of the short distance behaviour of other n-point
functions in order for a state to be physically realistic.
The main reason for requiring that a state satisfy this Hadamard condi-
tion is that it is necessary in order that the following ‘point-splitting pro-
cedure’ yield well-defined, finite values at each point yfor quantities such
as the renormalized expectation value in that state of φ2or of the quantum
stress-energy tensor Tab:6We define the expectation value of φ2(y) at a point
6Note also that, on a globally hyperbolic spacetime, the quasi-free Hadamard states
satisfy a number of desirable properties [51] including local quasiequivalence.
23
yby taking the neighbourhood Nyof yas in the previous section and setting
ω(φ2(y)) = lim
(x,x′)→(y,y)
1
2(ω(φ(x)φ(x′) + φ(x′)φ(x)) −H(n)(x, x′)) ,(25)
where H(n)(x, x′) is an appropriate locally constructed Hadamard parametrix,
i.e., it is a function – defined on the neighbourhood Cyconsisting of all pairs
of non-null related pairs of points in Ny– of the form (21) with a particu-
lar, locally defined algorithm used to obtain w(n)(see, e.g., [56] for further
discussion). In Eq. (25), it is understood that, before taking the limit, each
of the terms in the outer parentheses is defined initially on Cy(where they
make sense as smooth functions). Because the state is assumed to satisfy
the above Hadamard condition, the full term in parentheses will then clearly
extend to a continuous (in fact, Cn) function on Ny×Ny, thus ensuring
that the limit will be well defined. There is a similar, but more complicated,
formula corresponding to (25) for ω(Tab (y)) involving suitable (first and sec-
ond) derivatives with respect to xand x′in the terms in parentheses and
also involving the addition of a certain local correction term [54, 56]. The
resulting prescription for ω(Tab(y)) then satisfies a list of desired properties
which uniquely determines it up to certain finite renormalization ambiguities
[53, 56]. This justifies the use of the point-splitting procedure, thus leading to
the conclusion that the (locally weakly) Hadamard condition on a state must
be satisfied in order to ensure that the expectation values of the stress-energy
tensor be well-defined.7
Next we turn to consider what it might mean to quantize the covariant
Klein-Gordon equation on a spacetime (M, gab) which is not globally hyper-
bolic. Our approach will be to postulate what might be regarded as ‘rea-
sonable candidates for minimal necessary conditions’ for any such theory. In
other words, we consider statements which begin with the phrase ‘Whatever
else a quantum field theory (on a given non-globally hyperbolic spacetime)
consists of, it should at least involve ...’ We shall consider independently two
candidate conditions of this type:
7In fact, since one only requires the difference between the two point function and the
locally constructed Hadamard parametrix, H, to be C2, it would clearly suffice for the
well-definedness of expectation values of the renormalized stress-energy tensor to replace
the condition of being locally weakly Hadamard (see before (21)) by a weaker condition
where one only demands that w(n)be C2for n > 2. We remark that it is easy to see that
our Theorem 2 would continue to hold with such a further weakening of the Hadamard
condition.
24
Candidate Condition 1. Whatever else a quantum field theory consists
of, it should at least involve a field algebra satisfying F-locality [30]. In other
words (see [30] for more details) it should involve a field algebra which is
a star algebra consisting of polynomials in ‘smeared quantum fields’ φ(f)
which, just as in the globally hyperbolic case, satisfies the Relations (1), (2)
and (3) listed above. Additionally, (this is the F-locality condition of [30]) one
postulates that Relation (4) (which, as we mentioned above is the one relation
for which the assumption of global hyperbolicity is needed) should still hold in
the following local sense: Every point in Mshould have a globally hyperbolic
neighbourhood Usuch that, for all f1, f2∈C∞
0(U), Relation (4) holds with
△replaced by △U, where, by △U, we mean the advanced minus retarded
fundamental solution for the region U, regarded as a globally hyperbolic
spacetime in its own right.8
In defence of such a condition, let us simply say here (see [30] for fur-
ther discussion) that it is motivated by the philosophical bias (related to the
equivalence principle) that, on an arbitrary spacetime, the ‘laws in the small’
for quantum field theory should be the same as the familiar laws for globally
hyperbolic spacetimes. We remark that it is easy to see that the familiar laws
for globally hyperbolic spacetimes, as given above, are themselves F-local. It
is also known that there do exist some non-globally hyperbolic spacetimes
which admit field algebras satisfying F-locality. (In the language of [30], there
exist some non-globally hyperbolic F-quantum compatible spacetimes.) In
particular there do exist F-quantum compatible spacetimes with closed time-
like curves, for example the spacelike cylinder – i.e., the region of Minkowski
space (say with Minkowski coordinates (t, x, y, z)) between two times – say t1
and t2– with the (x, y, z) coordinates of opposite edges identified. (See [30]
for the case of the massless Klein-Gordon equation, and [10] for the massive
case.)
Of course, as anticipated in [30], the above philosophical bias could be
used to argue for a slightly different and possibly weaker locality notion. In
this connection, we remark that, since a first version of this paper was writ-
ten, evidence has emerged [11] that the examples of F-quantum compatible
8In [30] the F-locality condition was stated slightly differently; namely that every
neighbourhood of every point in Mshould contain a globally hyperbolic subneighbourhood
Usuch that, for all f1, f2∈C∞
0(U), Relation (4) holds with △replaced by △U. However,
it is clear from the F-locality in this latter sense of the usual field algebra on a globally
hyperbolic spacetime (see [30]) that this is equivalent to the condition given here.
25
chronology violating spacetimes mentioned above are unstable in the sense
that there are arbitrarily small perturbations of these spacetimes which fail
to be F-quantum compatible. On the other hand, as is also pointed out in
[11], if one replaces the condition of F-locality by the weaker notion of F-
locality modulo C∞9, then these examples of allowed chronology violating
spacetimes would become stable and there would be many other stable ex-
amples (i.e., of chronology violating spacetimes which admit field algebras
which are F-local modulo C∞).
Candidate Condition 2. Whatever else a quantum field theory consists of,
it should at least involve a field algebra satisfying Relations (1), (2) and (3)
listed above, and, in addition, there should exist states for which (twice) the
symmetrized two-point function is a locally weakly Hadamard distributional
bisolution G(f1, f2).
The motivation for requiring the existence of a field algebra satisying
(1)-(3) is, of course, the same as for Candidate Condition 1. One also can
motivate the requirement of the existence of Hadamard states by a philo-
sophical bias similar to that motivating the F-locality condition: The theory
should admit ‘physically acceptable states’, and these physically acceptable
states should have the same local character as in the globally hyperbolic case.
However, there is additional strong motivation for requiring the existence of
Hadamard states: Let (M, gab) and (M′, g′ab) be spacetimes – one or both of
which may be non-globally-hyperbolic – for which there exist open regions
O ⊂ Mand O′⊂M′which are isometric. Let ωbe a state on the field
algebra of (M, gab ) and let ω′be a state on the field algebra of (M′, g′ab).
Use the isometry between Oand O′to identify these two regions. Then, it
is natural to postulate that – under this identification – for all y∈ O the
difference between ω(Tab (y)) and ω′(Tab(y)) should be given by the point-
splitting formula in terms of the difference between the symmetrized two-
point functions of ωand ω′whenever this formula makes sense; furthermore,
when this formula does not make sense, the difference between ω(Tab(y)) and
ω′(Tab(y)) is ill defined or singular. (This postulate may be viewed as a gen-
eralization to non-globally-hyperbolic spacetimes of the main content of the
stress-energy axioms (1) and (2) of [56].) If so, and if in the globally hyper-
bolic case ω(Tab(y)) is given by the point-splitting prescription as described
9I.e., for which the above italicized definition holds when one replaces ∆Uby ∆U+F
for some (antisymmetric) F∈C∞(U × U).
26
above, then the locally weakly Hadamard condition on the symmetric part
of the two-point function – or slight weakenings thereof (see Footnote 7) –
must be satisfied in order to have an everywhere defined, nonsingular ω(Tab).
If ω(Tab) were singular for every state ω, the theory clearly would be un-
acceptable on physical grounds, since singular ‘back reaction’ effects would
necessarily occur for all states, thereby invalidating the original background
spacetime upon which the quantum field theory was based.
5 Theorems
We are now ready to state and prove our main theorems, which establish
that neither of the two Candidate Conditions of the previous section can be
satisfied by a Klein-Gordon field on a spacetime, (M, gab), with compactly
generated Cauchy horizon. These theorems are direct consequences of the
Propagation of Singularities Theorem of Sect. 3 (applied to the relevant dis-
tributional bisolutions) in combination with the geometrical property of base
points expressed in Proposition 2 of Sect. 2.
Theorem 1. There is no extension to (M, gab)of the usual field algebra on
the initial globally hyperbolic region D(S)which satisfies F-locality at any
base point (see Sect. 2) of the Cauchy horizon.
Proof. Let x∈ B and let Ube any globally hyperbolic neighbourhood of
x. To prove the claimed violation of F-locality, it clearly suffices to prove
that the restrictions of △D(S)and △Uto U ∩ D(S) cannot coincide, where
△D(S)denotes the advanced minus retarded fundamental solution for the
initial globally hyperbolic region D(S), and △Udenotes the advanced minus
retarded fundamental solution for U. However, this follows immediately from
the fact that these two quantities cannot take the same values at the pair
of points (y, z) of Proposition 2 of Sect. 2. Indeed, since (y, z) are spacelike
related in the intrinsic geometry of U,△Umust clearly vanish at this pair
of points. On the other hand, it follows from the explicit description of
the wave-front set of the advanced minus retarded fundamental solution on
any globally hyperbolic spacetime, as given in Sect. 3, that △D(S)must be
singular at the pair (y, z) because they are null related in the spacetime D(S).
2
By repeating this argument with △D(S)(F1, F2) replaced by the commu-
tator ω(φ(F1)φ(F2)−φ(F2)φ(F1)) and using the Propagation of Singularities
27
Theorem, the following closely related theorem also can be readily proven:
Theorem 1′.Under the mild extra technical condition that the algebra ad-
mits at least one state ωfor which the smeared two point function ω(φ(f1)φ(f2))
is distributional, then there is no field algebra whatsoever which satisfies F-
locality (i..e., in the language of [30], spacetimes with compactly generated
Cauchy horizons are non-F-quantum compatible.)
We remark that it is easy to see that Theorems 1 and 1′will continue to
hold if one replaces the notion of F-locality by the weaker notion (see Sect. 4)
of F-locality modulo C∞. (See [11] for further discussion.)
We note that, in the very special case of the massless two-dimensional
Klein-Gordon equation on two-dimensional Misner space, Theorems 1 and 1′
had been obtained previously by relying on the explicitly known propagation
of the two-dimensional wave equation. (See [30].)
The following theorem establishes that Candidate Condition 2 cannot
hold:
Theorem 2. Let Gbe a distributional bisolution on (M, gab )which is every-
where locally weakly Hadamard on the initial globally hyperbolic region D(S).
Then Gfails to be locally weakly Hadamard at any x∈ B in the following se-
vere sense: The difference between Gand any locally constructed Hadamard
parametrix H(n)(see Eq. (25) above) will fail to be given by a locally L2func-
tion on N×N, where Nis any convex normal neighbourhood of x(so that
H(n)is well defined on N×N). Thus, for any n,G−H(n)cannot be given by
a continuous, nor even by a bounded function on N×N, and quantities such
as the renormalized expectation value of φ2or the renormalized stress-energy
tensor must be singular or ill defined at any base point of the Cauchy horizon.
Proof. Let Ube a globally hyperbolic subset of Ncontaining x. Let (y, z)
be as in Proposition 2 of Sect. 2, so that yand zare spacelike-separated in
(U, gab) but are joined by a null geodesic, α, in M. Let y′∈ U lie along α
sufficiently near to yto be contained within the convex normal neighbour-
hood of yappearing in the locally weakly Hadamard property of G. Then
we know by the discussion at the end of Sect. 3 that Gmust fail to belong to
the space L2
loc(y, pa;y′, p′
b) for some pair paat yand p′
bat y′of null tangents
to this null geodesic, which moreover cannot both be zero. Assuming that pa
does not vanish (otherwise, in what follows replace (y, pa) by (y′, p′
a)) we may
conclude by Part (ii) of the Propagation of Singularities Theorem of Sect. 3
that Gcannot belong to the space L2
loc(y, pa;z, qb) for some null covector qb
28
at z. It follows that Gcannot belong to L2
loc(y, z). On the other hand, H(n)
is non-singular at the pair (y, z), since yand zare spacelike-separated in U.
We conclude that G−H(n)cannot arise from a locally L2function on N×N.
2
It is clear from the proof of Theorem 2 that the failure to be locally L2
must already occur if one restricts attention to the part of the neighbourhood
Nwhich lies in the initial globally hyperbolic region D(S). In fact we have:
Theorem 2′(slightly stronger than Theorem 2). Let Gbe a distribu-
tional bisolution on (M, gab)which is everywhere locally weakly Hadamard on
the initial globally hyperbolic region D(S)and let Nbe any convex normal
neighbourhood of any x∈ B. Then G−H(n)fails to be given by a locally L2
function on N∩D(S).
This is now a statement entirely about the behaviour of the quantum
theory on the initial globally hyperbolic region D(S) as one approaches a
base point. Thus it seems fair to conclude from this theorem that something
must go seriously wrong with the quantum field theory on the Cauchy horizon
almost independently of any assumptions (cf. e.g., ‘Candidate Condition 2’
in the previous section) about what would constitute an extension of the
quantum field theory beyond the initial globally hyperbolic region.
It is also worth remarking that nowhere in the proof of Theorem 2 have
we made any use of the positivity conditions (see e.g., [29, 20, 30]) required
of a symmetric distributional bisolution (on a globally hyperbolic spacetime)
in order for it to arise as (twice) the symmetrized two-point function of a
quantum state on the field algebra of Sect. 4. Also, it should be noted that
Theorem 2 would continue to hold if one were to weaken the notion of ‘locally
weakly Hadamard’ along the lines indicated in Footnote 7.
Finally we recall (see end of Sect. 2) that the theorems of this section
will also hold for any spacetime (such as four dimensional Misner space)
which arises as the product with a Euclidean 4 −dspace of a spacetime with
compactly generated Cauchy horizon of lower dimension d.
6 Discussion
In this section, we make some remarks concerning the significance of our
theorems, and also discuss a number of directions in which these theorems
may be further generalized.
29
First, we attempt to clarify the significance of our theorems by contrast-
ing the situation for quantum field theory on spacetimes with compactly
generated Cauchy horizons with that on spacetimes with Killing horizons.
Theorems 1 and 2 here are concerned with a particular instance of a situation
where one asks about the extension of a quantum field theory (in our case,
the covariant Klein-Gordon field) from some given globally hyperbolic space-
time (N, hab) to a larger spacetime (M, gab ). In the situation addressed by
Theorems 1 and 2, (M, gab) is a spacetime with compactly generated Cauchy
horizon, and (N, hab) its initial globally hyperbolic region D(S). However,
another, familiar (see e.g., [18, 50, 2, 32, 56]) instance of such a situation is the
case where (M, gab ) is Minkowski spacetime and (N, hab) the Rindler wedge.
(Equally, we could take the case where (M, gab) is the Kruskal spacetime and
(N, hab ) the exterior Schwarzschild spacetime, etc.) It is interesting to con-
trast these two situations. In the Minkowski-Rindler wedge case, of course,
both spacetimes are globally hyperbolic and it is clear that the field algebra
one would construct for the Rindler wedge – regarding it as a globally hyper-
bolic spacetime in its own right – is naturally identified with the subalgebra
of the Minkowski spacetime field algebra associated with the Rindler wedge
region. Thus the field algebra on Minkowski spacetime (which is, of course,
F-local) certainly constitutes an F-local extension of the field algebra for the
Rindler wedge. Thus there is no analogue of Theorem 1 in this case. Let us
next turn to the question of whether there are any Hadamard states on the
Rindler wedge algebra whose symmetrized two point distributions extend to
everywhere locally weakly Hadamard (symmetric) distributional bisolutions
on the whole of Minkowski space. (We shall refer to this, from now on, as
the question whether ‘Hadamard states have Hadamard extensions’). It is
certainly the case that most Hadamard states on the Rindler wedge have
no Hadamard extension to the whole of Minkowski space. For example,
the Fulling vacuum (i.e., the ground state for the one-parameter group of
wedge-preserving Lorentz boosts), as well as the KMS states with respect
to the group of wedge-preserving Lorentz boosts at all ‘temperatures’ except
T= 1/2π(which corresponds to the Minkowski vacuum state) fail to have
Hadamard extensions.10 In fact, it is well known that, for all these states,
10 In fact, as proven in [32, 31], the Minkowski vacuum state is the only globally
Hadamard state (satisfying a ‘no zero mode’ condition) on Minkowski space which is
globally invariant under the same group of Lorentz boosts and an analogous result holds
for the analogous Kruskal-Schwarzschild situation and a wide range of other analogous
situations involving spacetimes with bifurcate Killing horizons.
30
the expectation value of the stress-energy tensor diverges as one approaches
the horizon. Nevertheless, there do of course exist Hadamard states on the
Rindler wedge which have Hadamard extensions to the whole of Minkowski
spacetime and for which the stress-energy tensor is bounded: namely the
restrictions to the Rindler wedge algebra of Hadamard states on Minkowski
space! Prior to the results in the present paper, it was unclear to what
extent the situation was analogous for Hadamard extensions of Hadamard
states on the initial globally hyperbolic region D(S) of a spacetime (M, gab)
with a compactly generated Cauchy horizon. The work of Kim and Thorne
[33], Hawking [23], Visser [52], and others strongly suggested that (just as
in the Rindler-Minkowski situation) most Hadamard states on D(S) have a
stress-energy tensor which diverges on the Cauchy horizon. However, there
were also examples – albeit in the context of two-dimensional models [37],
or for special models involving automorphic fields [46, 47] (see Sect. 1) –
of states on D(S) for which the stress-energy tensor vanished. Thus one
might have thought that (as in the Rindler-Minkowski situation) there could
still be some/many Hadamard states on D(S) with Hadamard extensions to
(M, gab). Theorem 2 proves that this is not the case. Furthermore, while it
does not rule out the possible existence of Hadamard states on D(S) for which
the stress energy tensor is bounded, it still implies that any such state must
have a stress-energy tensor which is singular at the base points of the Cauchy
horizon. In this important sense, the situation for spacetimes with compactly
generated Cauchy horizons is thus quite distinct from the Minkowski-Rindler
situation.
Finally, we point out that theorems similar to Theorems 1 and 2 will
clearly hold in any spacetime (not necessarily with a compactly generated
Cauchy horizon) which contains an almost closed (but not closed) null geodesic
or a self-intersecting (but not closed) null geodesic (since, clearly, the same
conclusions that hold for the base points of Proposition 2 of Sect. 2 will hold
for any accumulation point, respectively for any intersection point). Thus, in
particular, analogues of Theorems 1 and 2 will hold for points on the ‘polar-
ized hypersurfaces’ in the time-machine models discussed by Kim and Thorne
[33], Gott [21], Grant [22], and others since these contain self-intersecting null
geodesics. For further discussion, see [7] where it is pointed out that, because
of the accumulation of polarized hypersurfaces at the chronology horizons of
these models, analogues of Theorem 2 – but not Theorem 2′– will hold for
points on the chronology horizons of Gott and Grant space. As remarked in
[7], this result holds notwithstanding the existence of the states exhibited by
31
Boulware [3] (for sufficiently massive fields on the initial globally hyperbolic
region of Gott space) and Tanaka and Hiscock [48] (for sufficiently massive
fields on the initial globally hyperbolic region of Grant space) for which the
stress-energy tensor is bounded (i.e., on the initial globally hyperbolic re-
gion). Moreover analogues of Theorems 1 and 2 are expected to hold in most
spacetimes which contain a closed null geodesic since generically, the same
conclusions as hold for the base points in Proposition 2 of Sect. 2 will hold
for each point on such a geodesic. However, we remark that there do ex-
ist very special cases of spacetimes with closed null geodesics for which (for
suitable field theories) F-local field algebras do exist and everywhere locally
Hadamard (symmetric) distributional bisolutions do exist. One such special
spacetime is the double covering of compactified Minkowski space. (We are
grateful to Roger Penrose for pointing this example out to us.) The special
feature of this spacetime which makes it possible to evade the conclusions
of Proposition 2 of Sect. 2 and hence to evade the arguments of Theorems
1 and 2 is that the entire light cone through any point, when globally ex-
tended, refocusses back onto that point. It is not difficult to see that, on
this spacetime, the field algebra obtained by conformally mapping the field
algebra for the massless Klein Gordon equation in Minkowski space extends
to an F-local field algebra (i.e., for the conformally coupled massless Klein-
Gordon equation). Equally, (twice) the conformally mapped symmetrized
two-point function for the massless Klein-Gordon equation on Minkowski
space extends, on this spacetime to an everywhere locally Hadamard distri-
butional bisolution (again of the conformally coupled massless Klein-Gordon
equation). A two-dimensional example with similar behaviour is provided by
the two-dimensional massless Klein-Gordon equation on the two-dimensional
‘null strip’ – i.e., the region between two parallel null lines in two-dimensional
Minkowski space with opposite edges identified (by identifying points inter-
sected by the same null lines).
7 Acknowledgements
We wish to thank Piotr Chrusciel for bringing Theorem 26.1.4 of [28] to
our attention. We also thank Roger Penrose for pointing out the example
mentioned in Sect. 6. B.S.K. thanks Stephen Hawking for conversations in
which he raised some of the issues addressed in this work. M.J.R. wishes
to thank the University of Toronto for hospitality at a late stage of this
32
work. This work was supported by NSF grants PHY-92-20644 and PHY-95-
14726 to the University of Chicago and by EPSRC grant GR/K 29937 to the
University of York.
References
[1] Allen, B.: Vacuum states in de Sitter space. Phys. Rev. D32, 3136–3149
(1985)
[2] Birrell, N.D., Davies, P.C.W.: Quantum Fields in Curved Space. Cam-
bridge: Cambridge University Press, 1982
[3] Boulware, D.G.: Quantum field theory in spaces with closed timelike
curves. Phys. Rev. D46, 4421–4441 (1992)
[4] Brunetti, R., Fredenhagen, K., K¨ohler, M.: The microlocal spectrum
condition and Wick polynomials of free fields on curved spacetimes.
Commun. Math. Phys. 180, 633–652 (1996)
[5] Choquet-Bruhat, Y.: Hyperbolic differential equations on a manifold.
In: De Witt, C., Wheeler, J. (eds.): Battelle Rencontres. New York:
Benjamin, 1977
[6] Chrusciel, P.T., Isenberg, J.: On the dynamics of generators of Cauchy
horizons. In: Hobill, D., Burd, A., Coley, A. (eds.): Deterministic Chaos
in General Relativity. New York: Plenum Press, 1995
[7] Cramer, C.R., Kay, B.S.: Stress-energy must be singular on the Misner
space horizon even for automorphic fields. Class. Quantum Grav. 13,
L143–L149 (1996)
[8] DeWitt, B.S., Brehme, R.W.: Radiation damping in a gravitational
field. Ann. Phys. (N.Y.) 9, 220–259 (1960)
[9] Duistermaat, J.J., H¨ormander, L.: Fourier integral operators II. Acta
Mathematica 128, 183–269 (1972)
[10] Fewster, C.J., Higuchi, A.: Quantum field theory on certain non-globally
hyperbolic spacetimes. Class. Quantum Grav. 13, 51–61 (1996)
33
[11] Fewster, C.J., Higuchi, A., Kay, B.S.: How generic is F-locality? Exam-
ples and counterexamples. (Paper in preparation)
[12] Flanagan, E.E., Wald, R.M.: Does backreaction enforce the averaged
null energy condition in semiclassical gravity? Phys. Rev. D54, 6233–
6283 (1996)
[13] Ford, L.H., Roman, T.A.: Quantum field theory constrains traversable
wormhole geometries. Phys. Rev. D53, 5496–5507 (1996)
[14] Friedman, J.L., Morris, M.S.: Existence and uniqueness theorems for
massless fields on a class of spacetimes with closed timelike curves. To
appear in Commun. Math. Phys.
[15] Friedman, J.L., Papastamatiou, N.J., Simon, J.Z.: Failure of unitarity
for interacting fields on spacetimes with closed timelike curves. Phys.
Rev. D46, 4456–4469 (1992)
[16] Friedman, J.L., Schleich, K., Witt, D.: Topological Censorship. Phys.
Rev. Lett. 71, 1486–1489 (1993)
[17] Frolov, V.P., Novikov, I.D.: Physical effects in wormholes and time
machines. Phys. Rev. D42, 1057–1065 (1990)
[18] Fulling, S.A.: Non-uniqueness of canonical field quantization in Rieman-
nian space-time. Phys. Rev. D7, 2850–2862 (1973)
[19] Garabedian, P.R.: Partial Differential Equations. New York, N.Y.: Wi-
ley, 1964
[20] Gonnella, G., Kay, B.S.: Can locally Hadamard quantum states have
non-local singularities? Class. Quantum Grav. 6, 1445–1454 (1989)
[21] Gott, J.R.: Closed timelike curves produced by pairs of moving cosmic
strings: Exact solutions. Phys. Rev. Lett. 66, 1126–1129 (1991)
[22] Grant, J.D.E.: Cosmic Strings and Chronology Protection. Phys. Rev.
D47, 2388–2394 (1993)
[23] Hawking, S.W.: The chronology protection conjecture. Phys. Rev. D46,
603–611 (1992)
34
[24] Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time.
Cambridge: Cambridge University Press, 1973
[25] Hiscock, W.A., Konkowski, D.A.: Quantum vacuum energy in Taub-
NUT (Newman-Unti-Tamburino)-type cosmologies. Phys. Rev. D26,
1225–1230 (1982)
[26] H¨ormander, L.: The Analysis of Linear Partial Differential Operators I,
(Second Edition). Berlin, Heidelberg, New York: Springer, 1990
[27] H¨ormander, L.: The Analysis of Linear Partial Differential Operators
III. Berlin, Heidelberg, New York: Springer, 1985
[28] H¨ormander, L.: The Analysis of Linear Partial Differential Operators
IV. Berlin, Heidelberg, New York: Springer, 1985
[29] Kay, B.S.: Quantum field theory in curved spacetime. In: Bleuler,
K., Werner, M. (eds.): Differential Geometrical Methods in Theoreti-
cal Physics. Dordrecht: Reidel, 1988, pp. 373–393
[30] Kay, B.S.: The principle of locality and quantum field theory on (non
globally hyperbolic) curved spacetimes. Rev. Math. Phys., Special Issue,
167–195 (1992)
[31] Kay, B.S.: Sufficient conditions for quasi-free states and an improved
uniqueness theorem for quantum field theory on spacetimes with hori-
zons. J. Math. Phys. 34, 4519–4539 (1993)
[32] Kay, B.S., Wald, R.M.: Theorems on the uniqueness and thermal prop-
erties of stationary, nonsingular, quasifree states on spacetimes with a
bifurcate Killing horizon. Phys. Rep. 207(2), 49–136 (1991)
[33] Kim, S-W., Thorne, K.S.: Do vacuum fluctuations prevent the creation
of closed timelike curves? Phys. Rev. D43, 3929–3947 (1991)
[34] K¨ohler, M.: New examples for Wightman fields on a manifold. Class.
Quantum Grav. 12, 1413–1427 (1995)
[35] K¨ohler, M.: The stress energy tensor of a locally supersymmetric quan-
tum field on a curved spacetime. Doctoral dissertation, University of
Hamburg, 1995
35
[36] Junker, W.: Adiabatic vacua and Hadamard states for scalar quantum
fields on curved spacetime. Doctoral dissertation, University of Ham-
burg, 1995. See also Junker, W.: Hadamard states, adiabatic vacua and
the construction of physical states for scalar quantum fields on curved
spacetime. Rev. Math. Phys. 8, 1091–1159 (1996)
[37] Krasnikov, S.V.: On the quantum stability of the time-machine. Phys.
Rev. D54, 7322–7327 (1996)
[38] Leray, J.: Hyperbolic differential equations. Lecture notes, Princeton,
1953 (unpublished)
[39] Lichnerowicz, A.: Propagateurs et commutateurs en relativit´e g´en´erale.
Publ. Math. IHES, no. 10, (1961)
[40] Morris, M.S., Thorne, K.S., Yurtsever, U.: Wormholes, time-machines,
and the weak energy condition. Phys. Rev. Lett. 61, 1446–1449 (1988)
[41] Radzikowski, M.J.: The Hadamard condition and Kay’s conjecture in
(axiomatic) quantum field theory on curved space-time. Ph.D. disser-
tation, Princeton University, 1992. Available through University Mi-
crofilms International, 300 N. Zeeb Road, Ann Arbor, Michigan 48106
U.S.A.
[42] Radzikowski, M.J.: A local-to-global singularity theorem for quantum
field theory on curved space-time. Commun. Math. Phys. 180, 1–22
(1996). See also the Appendix by R. Verch.
[43] Radzikowski, M.J.: Micro-local approach to the Hadamard condition in
quantum field theory on curved space-time. Commun. Math. Phys. 179,
529–553 (1996)
[44] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, I:
Functional Analysis. New York, NY: Academic Press, 1980
[45] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, II:
Fourier Analysis, Self-Adjointness. New York, NY: Academic Press, 1975
[46] Sushkov, S.V.: Quantum complex scalar field in two-dimensional space-
time with closed timelike curves and a time-machine problem. Class.
Quantum Grav. 12, 1685–1697 (1995)
36
[47] Sushkov, S.V.: Chronology protection and quantized fields: Complex
automorphic scalar field in Misner space. gr-qc/9509056
[48] Tanaka, T., Hiscock, W.A.: Massive scalar field in multiply connected
flat spacetimes. Phys. Rev. D52, 4503–4511 (1995)
[49] Thorne, K.S.: Closed Timelike Curves. In: Gleiser, R.J., Kozameh,
C.N., Moreschi, O.M. (eds.): Proceedings of the 13th International
Conference on General Relativity and Gravitation. Cordoba, Argentina
1992. Bristol: Institute of Physics Publishing, 1993, pp. 295–315
[50] Unruh, W.G.: Notes on black hole evaporation. Phys. Rev. D14, 870–
892 (1976)
[51] Verch, R.: Local definiteness, primarity and quasiequivalence of
quasifree Hadamard states in curved spacetime. Commun. Math. Phys.
160, 507–536 (1994)
[52] Visser, M.: Lorentzian Wormholes. New York: AIP Press, 1995
[53] Wald, R.M.: The back reaction effect in particle creation in curved
spacetime. Commun. Math. Phys. 54, 1–19 (1977)
[54] Wald, R.M.: On the trace anomaly of a conformally invariant quantum
field on curved spacetime. Phys. Rev. D17, 1477–1484 (1978)
[55] Wald, R.M.: General Relativity. Chicago, IL: The University of Chicago
Press, 1984
[56] Wald, R.M.: Quantum Field Theory in Curved Spacetime and Black
Hole Thermodynamics. Chicago and London: University of Chicago
Press, 1994
Communicated by A. Connes
37
