Quantum Field Theory on Curved Backgrounds. II. Spacetime Symmetries
QUANTUM FIELD THEORY ON CURVED
BACKGROUNDS. II.
SPACETIME SYMMETRIES
ARTHUR JAFFE AND GORDON RITTER
Abstract. We study space-time symmetries in scalar quantum field
theory (including interacting theories) on static space-times. We first
consider Euclidean quantum field theory on a static Riemannian mani-
fold, and show that the isometry group is generated by one-parameter
subgroups which have either self-adjoint or unitary quantizations. We
analytically continue the self-adjoint semigroups to one-parameter uni-
tary groups, and thus construct a unitary representation of the isometry
group of the associated Lorentzian manifold. The method is illustrated
for the example of hyperbolic space, whose Lorentzian continuation is
Anti-de Sitter space.
1. Introduction
The extension of quantum field theory to curved space-times has led to
the discovery of many qualitatively new phenomena which do not occur
in the simpler theory on Minkowski space, such as Hawking radiation; for
background and historical references, see [2, 6, 18].
The reconstruction of quantum field theory on a Lorentz-signature space-
time from the corresponding Euclidean quantum field theory makes use of
Osterwalder-Schrader (OS) positivity [15, 16] and analytic continuation. On
a curved background, there may be no proper definition of time-translation
and no Hamiltonian; thus, the mathematical framework of Euclidean quan-
tum field theory may break down. However, on static space-times there is a
Hamiltonian and it makes sense to define Euclidean QFT. This approach was
recently taken by the authors [11], in which the fundamental properties of
Osterwalder-Schrader quantization and some of the fundamental estimates
of constructive quantum field theory1 were generalized to static space-times.
The previous work [11], however, did not address the analytic continuation
which leads from a Euclidean theory to a real-time theory. In the present
article, we initiate a treatment of the analytic continuation by constructing
unitary operators which form a representation of the isometry group of the
Lorentz-signature space-time associated to a static Riemannian space-time.
Our approach is similar in spirit to that of Fröhlich [4] and of Klein and
Date: February 22, 2007.
1For background on constructive field theory in flat space-times, see [8, 9].
http://arxiv.org/abs/0704.0052v1
2 ARTHUR JAFFE AND GORDON RITTER
Landau [13], who showed how to go from the Euclidean group to the Poincaré
group without using the field operators on flat space-time.
This work also has applications to representation theory, as it provides
a natural (functorial) quantization procedure which constructs nontrivial
unitary representations of those Lie groups which arise as isometry groups
of static, Lorentz-signature space-times. These groups are typically non-
compact. For example, when applied to AdSd+1, our procedure gives a
unitary representation of the identity component of SO(d, 2). Moreover,
our procedure makes use of the Cartan decomposition, a standard tool in
representation theory.
2. Classical Space-Time
2.1. Structure of Static Space-Times.
Definition 2.1. A quantizable static space-time is a complete, con-
nected orientable Riemannian manifold (M,gab) with a globally-defined (smooth)
Killing field ξ which is orthogonal to a codimension-one hypersurface Σ ⊂M ,
such that the orbits of ξ are complete and each orbit intersects Σ exactly
once.
Throughout this paper, we assume that M is a quantizable static space-
time. Definition 2.1 implies that there is a global time function t defined
up to a constant by the requirement that ξ = ∂/∂t. Thus M is foliated by
time-slices Mt, and
M = Ω− ∪ Σ ∪Ω+
where the unions are disjoint, Σ =M0, and Ω± are open sets corresponding
to t > 0 and t < 0 respectively. We infer existence of an isometry θ which
reverses the sign of t,
θ : Ω± → Ω∓ such that θ
2 = 1, θ|Σ = id.
Fix a self-adjoint extension of the Laplacian, and let C = (−∆ +m2)−1
be the resolvent of the Laplacian (also called the free covariance), where
m2 > 0. Then C is a bounded self-adjoint operator on L2(M). For each s ∈
R, the Sobolev space Hs(M) is a real Hilbert space, defined as completion
of C∞c (M) in the norm
(2.1) ‖f‖2s = 〈f,C
−sf〉.
The inclusion Hs →֒ Hs+k for k > 0 is Hilbert-Schmidt. Define S :=⋂
s<0Hs(M) and S
s>0Hs(M). Then
S ⊂ H−1(M) ⊂ S
form a Gelfand triple, and S is a nuclear space.
Recall that S ′ has a natural σ-algebra of measurable sets (see for instance
[7, 8, 17]). There is a unique Gaussian probability measure µ with mean
zero and covariance C defined on the cylinder sets in S ′ (see [7]).
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES3
More generally, one may consider a non-Gaussian, countably-additive
measure µ on S ′ and the space
E := L2(S ′, µ).
We are interested in the case that the monomials of the form A(Φ) =
Φ(f1) . . .Φ(fn) for fi ∈ S are all elements of E , and for which their span
is dense in E . This is of course true if µ is the Gaussian measure with
covariance C.
For an open set Ω ⊂ M , let EΩ denote the closure in E of the set of
monomials A(Φ) =
iΦ(fi) where supp(fi) ⊂ Ω for all i. Of particular
importance for Euclidean quantum field theory is the positive-time subspace
E+ := EΩ+ .
2.2. The Operator Induced by an Isometry. Isometries of the under-
lying space-time manifold act on a Hilbert space of classical fields arising in
the study of a classical field theory. For f ∈ C∞(M) and ψ : M → M an
isometry, define
fψ ≡ (ψ−1)∗f = f ◦ ψ−1.
Since det(dψ) = 1, the operation f → fψ extends to a bounded operator
on H±1(M) or on L
2(M). A treatment of isometries for static space-times
appears in [11].
Definition 2.2. Let ψ be an isometry, and A(Φ) = Φ(f1) . . .Φ(fn) ∈ E a
monomial. Define the induced operator
(2.2) Γ(ψ)A ≡ Φ(f1
ψ) . . .Φ(fn
and extend Γ(ψ) by linearity to the domain of polynomials in the fields,
which is dense in E .
3. Osterwalder-Schrader Quantization
3.1. Quantization of Vectors (The Hilbert Space H of Quantum
Theory). In this section we define the quantization map E+ → H , where
H is the Hilbert space of quantum theory. The existence of the quantization
map relies on a condition known as Osterwalder-Schrader (or reflection)
positivity. A probability measure µ on S ′ is said to be reflection positive if
(3.1)
Γ(θ)F F dµ ≥ 0
for all F in the positive-time subspace E+ ⊂ E . Let Θ = Γ(θ) be the
reflection on E induced by θ. Define the sesquilinear form (A,B) on E+×E+
as (A,B) = 〈ΘA,B〉E , so (3.1) states that (F,F ) ≥ 0.
Assumption 1 (O-S Positivity). Any measure dµ that we consider is re-
flection positive with respect to the time-reflection Θ.
4 ARTHUR JAFFE AND GORDON RITTER
Definition 3.1 (OS-Quantization). Given a reflection-positive measure dµ,
the Hilbert space H of quantum theory is the completion of E+/N with
respect to the inner product given by the sesquilinear form (A,B). Denote
the quantization map Π for vectors E+ → H by Π(A) = Â, and write
(3.2) 〈Â, B̂〉H = (A,B) = 〈ΘA,B〉E for A,B ∈ E+ .
3.2. Quantization of Operators. The basic quantization theorem gives a
sufficient condition to map a (possibly unbounded) linear operator T on E
to its quantization, a linear operator T̂ on H . Consider a densely-defined
operator T on E , the unitary time-reflection Θ, and the adjoint T+ = ΘT ∗Θ.
A preliminary version of the following was also given in [10].
Definition 3.2 (Quantization Condition I). The operator T satisfies QC-I
i. The operator T has a domain D(T ) dense in E .
ii. There is a subdomain D0 ⊂ E+ ∩D(T )∩D(T
+), for which D̂0 ⊂ H
is dense.
iii. The transformations T and T+ both map D0 into E+.
Theorem 3.3 (Quantization I). If T satisfies QC-I, then
i. The operators T ↾D0 and T
+↾D0 have quantizations T̂ and T̂+ with
domain D̂0.
ii. The operators T̂ ∗ =
T̂ ↾D̂0
and T̂+ agree on D̂0.
iii. The operator T̂ ↾D0 has a closure, namely T̂
Proof. We wish to define the quantization T̂ with the putative domain D̂0
(3.3) T̂ Â = T̂A .
For any vector A ∈ D0 and for any B ∈ (D0 ∩ N ), it is the case that
Â = Â+B. The transformation T̂ is defined by (3.3) iff T̂A = ̂T (A+B) =
T̂A+ T̂B. Hence one needs to verify that T : D0 ∩ N → N , which we now
The assumption D0 ⊂ D(T
+), along with the fact that Θ is unitary,
ensures that ΘD0 ⊂ D(T
∗). Therefore for any F ∈ D0,
(3.4)
〈ΘF, TB〉E = 〈T
∗ΘF,B〉E = 〈Θ(ΘT
∗ΘF ) , B〉E = 〈ΘT
+F,B〉E = 〈T̂
+F, B̂〉H .
In the last step we use the fact assumed in part (iii) of QC-I that T+ :
D0 → E+, yielding the inner product of two vectors in H . We infer from
the Schwarz inequality in H that
|〈ΘF, TB〉E | ≤ ‖T̂+F‖H ‖B̂‖H = 0 .
As 〈ΘF, TB〉E = 〈F̂ , T̂B〉H , this means that T̂B ⊥ D̂0. As D̂0 is dense in
H by QC-I.ii, we infer T̂B = 0. In other words, TB ∈ N as required to
define T̂ .
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES5
In order show that D̂0 ⊂ D(T̂
∗), perform a similar calculation to (3.4)
with arbitrary A ∈ D0 replacing B, namely
(3.5)
〈F̂ , T̂ Â〉H = 〈ΘF, TA〉E = 〈Θ(ΘT
∗ΘF ) , A〉E = 〈ΘT
+F,A〉E = 〈T̂+F, Â〉H .
The right side is continuous in Â ∈ H , and therefore F̂ ∈ D(T ∗). Further-
more T ∗F̂ = T̂+F . This identity shows that if F ∈ N , then T̂+F = 0.
Hence T+↾D0 has a quantization T̂
+, and we may write (3.5) as
(3.6) T ∗F̂ = T̂+F̂ , for all F ∈ D0 .
In particular T̂ ∗ is densely defined so T̂ has a closure. This completes the
proof. �
Definition 3.4 (Quantization Condition II). The operator T satisfies QC-II
i. Both the operator T and its adjoint T ∗ have dense domains D(T ),D(T ∗) ⊂
ii. There is a domain D0 ⊂ E+ in the common domain of T , T
+, T+T ,
and TT+.
iii. Each operator T , T+, T+T , and TT+ maps D0 into E+.
Theorem 3.5 (Quantization II). If T satisfies QC-II, then
i. The operators T ↾D0 and T
+↾D0 have quantizations T̂ and T̂+ with
domain D̂0.
ii. If A,B ∈ D0, one has 〈B̂, T̂ Â〉H = 〈T̂+B̂, Â〉H .
Remarks.
i. In Theorem 3.5 we drop the assumption that the domain D̂0 is dense,
obtaining quantizations T̂ and T̂+ whose domains are not necessarily
dense. In order to compensate for this, we assume more properties
concerning the domain and the range of T+ on E .
ii. As D̂0 need not be dense in H , the adjoint of T̂ need not be defined.
Nevertheless, one calls the operator T̂ symmetric in case one has
(3.7) 〈B̂, T̂ Â〉H = 〈T̂ B̂, Â〉H , for all A,B ∈ D0 .
iii. If Ŝ ⊃ T̂ is a densely-defined extension of T̂ , then Ŝ∗ = T̂+ on the
domain D̂0.
Proof. We define the quantization T̂ with the putative domain D̂0. As in
the proof of Theorem 3.3, this quantization T̂ is well-defined iff it is the case
that T : D0 ∩ N → N . For any F ∈ D0 ∩ N , by definition ‖F̂‖H = 0.
〈TF, TF 〉H = (TF, TF ) = 〈ΘTF, TF 〉E = 〈F, T
∗ΘTF 〉
where one uses the fact that D0 ⊂ D(T
+T ). Thus
〈TF, TF 〉H =
ΘF, T+TF
= 〈F, T+TF 〉H .
6 ARTHUR JAFFE AND GORDON RITTER
Here we use the fact that T+T maps D0 to E+. Thus one can use the
Schwarz inequality on H to obtain
〈TF, TF 〉H ≤ ‖F̂‖H ‖T̂
= 0 .
Hence T : D0 ∩ N → N , and T has a quantization T̂ with domain D̂0.
In order verify that T+↾D0 has a quantization, one needs to show that
T+ : D0 ∩N ⊂ N . Repeat the argument above with T
+ replacing T . The
assumption TT+ : D0 → E+ yields for F ∈ D0 ∩ N ,
〈T+F, T+F 〉H = 〈T
∗ΘF, T+F 〉E = 〈ΘF, TT
+F 〉E = 〈F̂ , T̂ T
+F 〉H .
Use the Schwarz inequality in H to obtain the desired result that
〈T+F, T+F 〉H ≤ ‖F̂‖H ‖T̂ T
= 0 .
Hence T+ has a quantization T̂+ with domain D̂0, and for B ∈ D0 one has
T̂+B = T̂+B̂. In order to establish (ii), assume that A,B ∈ D0. Then
〈B̂, T̂ Â〉H = 〈ΘB,TA〉E = 〈Θ(ΘT
∗ΘB) , A〉E = 〈ΘT
+B,A〉E
= 〈T̂+B, Â〉H = 〈T̂+B̂, Â〉H .(3.8)
This completes the proof.
4. Structure and Representation of the Lie Algebra of
Killing Fields
For the remainder of this paper we assume the following, which is clearly
true in the Gaussian case as the Laplacian commutes with the isometry
group G. (A further explanation was given in [11].)
Assumption 2. The isometry groups G that we consider leave the measure
dµ invariant, in the sense that Γ, defined above, is a unitary representation
of G on E .
4.1. The Representation of g on E .
Lemma 4.1. Let Gi be an analytic group with Lie algebra gi (i = 1, 2),
and let λ : g1 → g2 be a homomorphism. There cannot exist more than one
analytic homomorphism π : G1 → G2 for which dπ = λ. If G1 is simply
connected then there is always one such π.
Let D = d/dt denote the canonical unit vector field on R. Let G be a
real Lie group with algebra g, and let X ∈ g. The map tD → tX(t ∈ R)
is a homomorphism of Lie(R) → g, so by the Lemma there is a unique
analytic homomorphism ξX : R → G such that dξX(D) = X. Conversely,
if η is an analytic homomorphism of R → G, and if we let X = dη(D),
it is obvious that η = ξX . Thus X 7→ ξX is a bijection of g onto the set
of analytic homomorphisms R → G. The exponential map is defined by
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES7
exp(X) := ξX(1). For complex Lie groups, the same argument applies,
replacing R with C throughout.
Since g is connected, so is exp(g). Hence exp(g) ⊆ G0, where G0 denotes
the connected component of the identity in G. It need not be the case for
a general Lie group that exp(g) = G0, but for a large class of examples
(the so-called exponential groups) this does hold. For any Lie group, exp(g)
contains an open neighborhood of the identity, so the subgroup generated
by exp(g) always coincides with G0.
We will apply the above results with G = Iso(M), the isometry group of
M , and g = Lie(G) the algebra of global Killing fields. Thus we have a bijec-
tive correspondence between Killing fields and 1-parameter groups of isome-
tries. This correspondence has a geometric realization: the 1-parameter
group of isometries
φs = ξX(s) = exp(sX)
corresponding to X ∈ g is the flow generated by X.
Consider the two different 1-parameter groups of unitary operators:
(1) the unitary group φ∗s on L
2(M), and
(2) the unitary group Γ(φs) on E .
Stone’s theorem applies to both of these unitary groups to yield densely-
defined self-adjoint operators on the respective Hilbert spaces.
In the first case, the relevant self-adjoint operator is simply an extension
of −iX, viewed as a differential operator on C∞c (M). This is because for
f ∈ C∞c (M) and p ∈M , we have:
Xpf = (LXf)(p) =
f(φs(p))|s=0.
Thus −iX is a densely-defined symmetric operator on L2(M), and Stone’s
theorem implies that −iX has self-adjoint extensions.
In the second case, the unitary group Γ(φs) on E also has a self-adjoint
generator Γ(X), which can be calculated explicitly. By definition,
e−isΓ(X)
Φ(fi)
Φ(fi ◦ φ−s).
Now replace s→ −s and calculate d/ds|s=0 applied to both sides of the last
equation to see that
Φ(fi)
Φ(f1) . . .Φ(−iXfj)Φ(fj+1) . . .Φ(fn) .
One may check that Γ is a Lie algebra representation of g, i.e. Γ([X,Y ]) =
[Γ(X),Γ(Y )].
4.2. The Cartan Decomposition of g. For each ξ ∈ g, there exists some
dense domain in E on which Γ(ξ) is self-adjoint, as discussed previously.
8 ARTHUR JAFFE AND GORDON RITTER
However, the quantizations Γ̂(ξ) acting on H may be hermitian, anti-
hermitian, or neither depending on whether there holds a relation of the
(4.1) Γ(ξ)Θ = ±ΘΓ(ξ),
with one of the two possible signs, or whether no such relation holds.
Even if (4.1) holds, to complete the construction of a unitary representa-
tion one must prove that there exists a dense domain in H on which Γ̂(ξ) is
self-adjoint or skew-adjoint. This nontrivial problem will be dealt with in a
later section using Theorems 3.3 and 3.5 and the theory of symmetric local
semigroups [12, 4]. Presently we determine which elements within g satisfy
relations of the form (4.1).
Let ϑ := θ∗ as an operator on C∞(M), and consider a Killing field X ∈ g
also as an operator on C∞(M). Define T : g → g by
(4.2) T (X) := ϑXϑ.
From (4.2) it is not obvious that the range of T is contained in g. To prove
this, we recall some geometric constructions.
Let M,N be manifolds, let ψ : M → N be a diffeomorphism, and X ∈
Vect(M). Then
(4.3) ψ−1∗Xψ∗ = X(· ◦ ψ) ◦ ψ−1.
defines an operator on C∞(N). One may check that this operator is a
derivation, thus (4.3) defines a vector field on N . The vector field (4.3) is
usually denoted
ψ∗X = dψ(Xψ−1(p))
and referred to as the push-forward of X.
We now wish to show that g = g+⊕ g−, where g± are the ±1-eigenspaces
of T . This is proven by introducing an inner product (X,Y )g on g with
respect to which T is self-adjoint.
Theorem 4.2. Consider g as a Hilbert space with inner product (X,Y )g.
The operator T : g → g is self-adjoint with T 2 = I; hence
(4.4) g = g+ ⊕ g−
as an orthogonal direct sum of Hilbert spaces, where g± are the ±1-eigenspaces
of T . Further, ∂t ∈ g− hence dim(g−) ≥ 1. Elements of g− have hermitian
quantizations, while elements of g+ have anti-hermitian quantizations.
Proof. Write (4.2) as
(4.5) T (X) = θ−1∗Xθ∗ = θ∗X .
Thus T is the operator of push-forward by θ. The push-forward of a Killing
field by an isometry is another Killing field, hence the range of T is contained
2It is not the case that g− consists only of ∂t. In particular, dim(g−) = 2 for M = H 2.
It can occur that dim g+ = 0.
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES9
in g. Also, T must have a trivial kernel since T 2 = I, and this implies that
T is surjective. It follows from (4.5) that T is a Hermitian operator on
g. Hence T is diagonalizable and has real eigenvalues which are square
roots of 1. This establishes the decomposition (4.4). That elements of
g− have hermitian quantizations, while elements of g+ have anti-hermitian
quantizations follows from Theorem 3.3. �
A Cartan involution is a Lie algebra homomorphism g → g which squares
to the identity. It follows from (4.2) that T is a Lie algebra homomorphism;
thus, Theorem 4.2 implies that T is a Cartan involution of g. This implies
that the eigenspaces (g+, g−) form a Cartan pair, meaning that
(4.6) [g+, g+] ⊂ g+, [g+, g−] ⊂ g−, and [g−, g−] ⊂ g+ .
Clearly g+ is a subalgebra while g− is not, and any subalgebra contained in
g− is abelian.
5. Reflection-Invariant and Reflected Isometries
Let G = Iso(M) denote the isometry group of M , as above. Then G
has a Z2 subgroup containing {1, θ}. This subgroup acts on G by conjuga-
tion, which is just the action ψ → ψθ := θψθ. Conjugation is an (inner)
automorphism of the group, so
(ψφ)θ = ψθφθ, (ψθ)−1 = (ψ−1)θ.
Definition 5.1. We say that ψ ∈ G is reflection-invariant if
ψθ = ψ,
and that ψ is reflected if
ψθ = ψ−1.
Let GRI denote the subgroup of G consisting of reflection-invariant elements,
and let GR denote the subset of reflected elements.
Note that GRI is the stabilizer of the Z2 action, hence a subgroup. An
alternate proof of this proceeds usingGRI = exp(g+). Although GR is closed
under the taking of inverses and does contain the identity, the product of two
reflected isometries is no longer reflected unless they commute. Generally,
the product of an element of GR with an element ofGRI is neither an element
of GR nor of GRI . The only isometry that is both reflection-invariant and
reflected is θ itself. Thus we have:
GR ∩GRI = {1, θ} ⊂ GR ∪GRI ( G.
Theorem 5.2. Let G0 denote the connected component of the identity in
G. Then G0 is generated by GR ∪ GRI . (This is a form of the Cartan
decomposition for G.)
10 ARTHUR JAFFE AND GORDON RITTER
Proof. Since g = g+ ⊕ g− as a direct sum of vector spaces (though not of
Lie algebras), we have
exp(g)
exp(g+) ∪ exp(g−)
Choose bases {ξ±,i}i=1,...,n± for g± respectively. Then we have:
{exp(sξ+,i) : 1 ≤ i ≤ n+, s ∈ R}∪{exp(sξ−,j) : 1 ≤ j ≤ n−, s ∈ R}
Furthermore, exp(sξ−,i) is reflected, while exp(sξ+,i) is reflection-invariant,
completing the proof. �
Corollary 5.3. The Lie algebra of the subgroup GRI is g+.
To summarize, the isometry group of a static space-time can always be
generated by a collection of n (= dim g) one-parameter subgroups, each of
which consists either of reflected isometries, or reflection-invariant isome-
tries.
6. Construction of Unitary Representations
6.1. Self-adjointness of Semigroups. In this section, we recall several
known results on self-adjointness of semigroups. Roughly speaking, these
results imply that if a one-parameter family Sα of unbounded symmetric
operators satisfies a semigroup condition of the form SαSβ = Sα+β, then
under suitable conditions one may conclude essential self-adjointness.
A theorem of this type appeared in a 1970 paper of Nussbaum [14], who
assumed that the semigroup operators have a common dense domain. The
result was rediscovered independently by Fröhlich, who applied it to quan-
tum field theory in several important papers [5, 3]. For our intended appli-
cation to quantum field theory, it turns out to be very convenient to drop
the assumption that ∃ a such that the Sα all have a common dense domain
for |α| < a, in favor of the weaker assumption that
α>0D(Sα) is dense.
A generalization of Nussbaum’s theorem which allows the domains of the
semigroup operators to vary with the parameter, and which only requires the
union of the domains to be dense, was later formulated and two independent
proofs were given: one by Fröhlich [4], and another by Klein and Landau [12].
The latter also used this theorem in their construction of representations of
the Euclidean group and the corresponding analytic continuation to the
Lorentz group [13].
In order to keep the present article self-contained, we first define symmet-
ric local semigroups and then recall the refined self-adjointness theorem of
Fröhlich, and Klein and Landau.
Definition 6.1. Let H be a Hilbert space, let T > 0 and for each α ∈ [0, T ],
let Sα be a symmetric linear operator on the domain Dα ⊂ H , such that:
(i) Dα ⊃ Dβ if α ≤ β and D :=
0<α≤T Dα is dense in H ,
(ii) α→ Sα is weakly continuous,
(iii) S0 = I, Sβ(Dα) ⊂ Dα−β for 0 ≤ β ≤ α ≤ T , and
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES11
(iv) SαSβ = Sα+β on Dα+β for α, β, α + β ∈ [0, T ].
In this situation, we say that (Sα,Dα, T ) is a symmetric local semigroup.
It is important that Dα is not required to be dense in H for each α; the
only density requirement is (i).
Theorem 6.2 ([12, 4]). For each symmetric local semigroup (Sα,Dα, T ),
there exists a unique self-adjoint operator A such that3
Dα ⊂ D(e
−αA) and Sα = e
−αA|Dα for all α ∈ [0, T ].
Also, A ≥ −c if and only if ‖Sαf‖ ≤ e
cα‖f‖ for all f ∈ Dα and 0 < α < T .
6.2. Reflection-Invariant Isometries.
Lemma 6.3. Let ψ be a reflection-invariant isometry and assume ∃ p ∈ Ω+
such that ψ(p) ∈ Ω+. Then ψ preserves the positive-time subspace, i.e.
ψ(Ω+) ⊆ Ω+.
Proof. We first prove that ψ(Σ) ⊆ Σ. Suppose not; then ∃ p ∈ Σ with ψ(p) 6∈
Σ. Assume ψ(p) ∈ Ω+ (without loss of generality: we could repeat the same
argument with ψ(p) ∈ Ω−). Then Ω+ contains (θψθ)(p) = θψ(p) ∈ Ω−, a
contradiction since Ω−∩Ω+ = ∅. We used the fact that θ|Σ = id so θ(p) = p.
Hence ψ restricts to an isometry of Σ. It follows that the restriction of ψ to
M ′ =M \Σ is also an isometry. However, M ′ = Ω− ⊔Ω+, where ⊔ denotes
the disjoint union. Therefore ψ(Ω+) is wholly contained in either Ω+ or Ω−,
since ψ is a homeomorphism and so ψ(Ω+) is connected. The possibility
that ψ(Ω+) ⊆ Ω− is ruled out by our assumption, so ψ(Ω+) ⊆ Ω+. �
Lemma 6.3 has the immediate consequence that if ξ ∈ g+ then the one-
parameter group associated to ξ is positive-time-invariant. This result plays
a key role in the proof of Theorem 6.4.
6.3. Construction of Unitary Representations. The rest of this section
is devoted to proving that the theory of symmetric local semigroups can be
applied to the quantized operators on H corresponding to each of a set of
1-parameter subgroups of G = Iso(M). The proof relies upon Lemma 6.3,
and Theorems 3.3, 3.5 and 6.2.
Theorem 6.4. Let (M,gab) be a quantizable static space-time. Let ξ be a
Killing field which lies in g+ or g−, with associated one-parameter group of
isometries {φα}α∈R. Then there exists a densely-defined self-adjoint opera-
tor Aξ on H such that
Γ̂(φα) =
e−αAξ , if ξ ∈ g−
eiαAξ if ξ ∈ g+.
3The authors of [4, 12] also showed that
bD :=
0<α≤S
0<β<α
Sβ(Dα)
, where 0 < S ≤ T,
is a core for A, i.e. (A, bD) is essentially self-adjoint.
12 ARTHUR JAFFE AND GORDON RITTER
Proof. First suppose that ξ ∈ g−, which implies that the isometries φα are
reflected, and so Γ(φα)
+ = Γ(φα). Define
Ωξ,α := φ
α (Ω+).
For all α in some neighborhood of zero, Ωξ,α is a nonempty open subset of
Ω+, and moreover, as α → 0
+, Ωξ,α increases to fill Ω+ with Ωξ,0 = Ω+.
These statements follow immediately from the fact that, for each p ∈ Ω+,
φα(p) is continuous with respect to α, and φ0 is the identity map.
Since φα(Ωξ,α) ⊆ Ω+, we infer that Γ(φα)EΩξ,α ⊆ E+. By Theorem 3.5,
Γ(φα) has a quantization which is a symmetric operator on the domain
Dξ,α := Π(EΩξ,α).
Note that Dξ,α is not necessarily dense in H .
4 We now show that Theorem
6.2 can be applied.
Fix some positive constant a with Ωξ,a nonempty. Note that
0<α≤a
Ωξ,α = Ω+ ⇒
0<α≤a
EΩξ,α = E+.
It follows that
Dξ :=
0<α≤a
is dense in H . This establishes condition (i) of Definition 6.1, and the
other conditions are routine verifications. Theorem 6.2 implies existence of
a densely-defined self-adjoint operator Aξ on H , such that
Γ̂(φα) = exp(−αAξ) for all α ∈ [0, a] .
This proves the theorem in case ξ ∈ g−.
Now suppose that ξ ∈ g+, implying that the isometries φα are reflection-
invariant, and
Γ(φα)
+ = Γ(φα)
−1 = Γ(φ−α) on E .
Lemma 6.3 implies that Γ(φα)E+ ⊆ E+. By Theorem 3.3, Γ(φα) has a
quantization Γ̂(φα) which is defined and satisfies
Γ̂(φα)
∗ = Γ̂(φα)
on the domain Π(E+), which is dense in H by definition. In this case we
do not need Theorem 6.2; for each α, Γ̂(φα) extends by continuity to a one-
parameter unitary group defined on all of H (not only for a dense subspace).
By Stone’s theorem,
Γ̂(φα) = exp(iαAξ)
for Aξ self-adjoint and for all α ∈ R. The proof is complete. �
4Density of Dξ,α would be implied by a Reeh-Schlieder theorem, which we do not prove
except in the free case. Theorem 6.2 removes the need for a Reeh-Schlieder theorem in
this argument.
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES13
7. Analytic Continuation
Each Riemannian static space-time (M,gab) has a Lorentzian continuation
Mlor, which we construct as follows. In adapted coordinates, the metric gab
on M takes the form
(7.1) ds2 = F (x)dt2 + Gµν(x)dx
µdxν .
The analytic continuation t → −it of (7.1) is standard and gives a metric
of Lorentz signature, ds2lor = −F dt
2 + G dx2, by which we define the
Lorentzian space-time Mlor. Einstein’s equation Ricg = k g is preserved
by the analytic continuation, but we do not use this fact anywhere in the
present paper.
Let {ξ
i : 1 ≤ i ≤ n±} be bases of g±, respectively. Let A
i = Aξ(±)
the densely-defined self-adjoint operators on H , constructed by Theorem
6.4. Let
(7.2) U
i (α) = exp(iαA
i ) , for 1 ≤ i ≤ n±
be the associated one-parameter unitary groups on H .
We claim that the group generated by the n = n+ + n− one-parameter
unitary groups (7.2) is isomorphic to the identity component of
Glor := Iso(Mlor),
the group of Lorentzian isometries. Since locally, the group structure is
determined by its Lie algebra, it suffices to check that the generators satisfy
the defining relations of glor := Lie(Glor).
Since quantization of operators preserves multiplication, we have
(7.3) X,Y,Z ∈ g, [X,Y ] = Z ⇒ [Γ̂(X), Γ̂(Y )] = Γ̂(Z).
In what follows, we will use the notation ĝ± for {Γ̂(X) : X ∈ g±}.
Quantization converts the elements of g− from skew operators into Her-
mitian operators; i.e. elements of ĝ− are Hermitian on H and hence, ele-
ments of i ĝ− are skew-symmetric on H . Thus ĝ+ ⊕ i ĝ− is a Lie algebra
represented by skew-symmetric operators on H .
Theorem 7.1. We have an isomorphism of Lie algebras:
(7.4) glor ∼= ĝ+ ⊕ i ĝ− .
Proof. LetMC be the manifold obtained by allowing the t coordinate to take
values in C. Define ψ :MC →MC by t 7→ −it. Then glor is generated by
i }1≤i≤n+ ∪ {ηj}1≤j≤n− , where ηj := iψ
It is possible to define a set of real structure constants fijk such that
(7.5) [ξ
fijkξ
14 ARTHUR JAFFE AND GORDON RITTER
Applying ψ∗ to both sides of (7.5), the commutation relations of glor are
seen to be
(7.6) [ηi, ηj ] = −fijkξ
together with the same relations for g+ as before. Now (7.3) implies that
(7.6) are the precisely the commutation relations of ĝ+ ⊕ i ĝ−, completing
the proof of (7.4). �
Corollary 7.2. Let (M,gab) be a quantizable static space-time. The unitary
groups (7.2) determine a unitary representation of G0lor on H .
7.1. Conclusions. We have obtained the following conclusions. There is a
unitary representation of the group G0lor on the physical Hilbert space H
of quantum field theory on the static space-time M . This representation
maps the time-translation subgroup into the unitary group exp(itH), where
the energy H ≥ 0 is a positive, densely-defined self-adjoint operator corre-
sponding to the Hamiltonian of the theory. The Hilbert space H contains a
ground state Ψ0 = 1̂ which is such that HΨ0 = 0 and Ψ0 is invariant under
the action of all spacetime symmetries. We obtain these results via analytic
continuation from the Euclidean path integral, under mild assumptions on
the measure which should include all physically interesting examples. This is
done without introducing the field operators; nonetheless, Theorems 3.3 and
3.5 do suffice to construct field operators. In the special case M = Rd with
G = SO(4), we obtain a unitary representation of the proper orthochronous
Lorentz group, G0lor = L
+ = SO
0(3, 1).
8. Hyperbolic Space and Anti-de Sitter Space
Consider Euclidean quantum field theory on M = H d. The metric is
ds2 = r−2
dx2i ,
where we define r = xd for convenience. The Laplacian is
(8.1) ∆
d = (2− d)r
+ r2∆
The d−1 coordinate vector fields {∂/∂xi : i 6= d} are all static Killing fields,
and any one of the coordinates xi (i 6= d) is a satisfactory representation of
time in this space-time. It is convenient to define t = x1 as before, and to
identify t with time.
The time-zero slice is M0 = H
d−1. From
H d = {v ∈ Rd,1 | 〈v, v〉 = −1, v0 > 0}
it follows that Isom(H d) = O+(d, 1) and the orientation-preserving isometry
group is SO+(d, 1).
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES15
Figure 1. Flow lines of the Killing field ζ = (t2 − r2)∂t +
2tr ∂r on H
For constant curvature spaces, one may solve Killing’s equation LKg = 0
explicitly. Let us illustrate the solutions and their quantizations for d = 2.
The three Killing fields
(8.2) ξ = ∂t, η = t∂t + r∂r, ζ = (t
2 − r2)∂t + 2tr ∂r
are a convenient basis for g. Any d-dimensional manifold satisfies dim g ≤
d(d + 1)/2, manifolds saturating the bound are said to be maximally sym-
metric, and H d is maximally symmetric.
Now, ∂tf(−t) = −f
′(−t) so ∂tΘ = −Θ∂t, hence ∂t ∈ g−. Similar calcula-
tions show [Θ, η] = 0 and Θζ = −ζΘ. Thus η spans g+, while ∂t, ζ span g−.
The commutation relations5 for g are:
[η, ζ] = ζ, [η, ∂t] = −∂t, [ζ, ∂t] = −2η.
These calculations verify that (g+, g−) forms a Cartan pair, as defined in
(4.6).
The flows associated to (8.2) are easily visualized: ξ is a right-translation,
and η flow-lines are radially outward from the Euclidean origin. The flows
of ζ are Euclidean circles, indicated by the darker lines in Figure 1. Hence
the flows of η are defined on all of E+, while the flows of ζ are analogous to
space-time rotations in R2, and hence, must be defined on a wedge of the
Wα = {(t, r) : t, r > 0, tan
−1(r/t) < α}.
The simple geometric idea of Section 6.2 is nicely confirmed in this case: the
flows of η (the generator of g+) preserve the t = 0 plane, and are separately
isometries of Ω+ and Ω−.
Corollary 7.2 implies that the procedure outlined above defines a uni-
tary representation of the identity component of Iso(AdS2) on the physical
Hilbert space H for quantum field theory on this background, including
theories with interactions that preserve the symmetry. Since Iso(AdSd+1) =
5Note that quite generally [g−, g−] ⊆ g+ so it’s automatic that [ζ, ∂t] is proportional
to η.
16 ARTHUR JAFFE AND GORDON RITTER
SO(d, 2), we have a unitary representation of SO0(1, 2). The latter is a non-
compact, semisimple real Lie group, and thus it has no finite-dimensional
unitary representations, but a host of interesting infinite-dimensional ones.
Appendix A. Euclidean Reeh-Schlieder Theorem
We prove the Euclidean Reeh-Schlieder property for free theories on curved
backgrounds. It is reasonable to expect this property to extend to interact-
ing theories on curved backgrounds, but it would have to be established for
each such model since it depends explicitly on the two-point function.
The Reeh-Schlieder theorem guarantees the existence of a dense quanti-
zation domain based on any open subset of Ω+. For this reason, one could
use the Reeh-Schlieder (RS) theorem with Nussbaum’s theorem [14] to con-
struct a second proof of Theorem 6.4 under the additional assumption that
M is real-analytic.
Fortunately, our proof of Theorem 6.4 is completely independent of the
Reeh-Schlieder property. This has two advantages: we do not have to assume
M is a real-analytic manifold and, more importantly, our proof of Theorem
6.4 generalizes immediately and transparently to interacting theories as long
as the Hilbert space H is not modified by the interaction.
We state and prove this using the one-particle space; however, the result
clearly extends to the quantum-field Hilbert space.
Theorem A.1. Let M be a quantizable static space-time endowed with a
real-analytic structure, and assume that gab is real-analytic. Let O ⊂ Ω+
and D = C∞(O) ⊂ L2(Ω+). Then D̂
⊥ = {0}.
Proof. Let f ∈ L2(Ω+) with f̂ ⊥ D . For x ∈ Ω+, define
η(x) := 〈f̂ , δ̂x〉H = 〈Θf,Cδx〉L2 .
Real-analyticity of η(x) follows from the real-analyticity of (the integral
kernel of) C, which in turn follows from the elliptic regularity theorem in the
real-analytic category (see for instance [1, Sec. II.1.3]). Now by assumption,
for any g ∈ C∞c (O), we have
0 = 〈ĝ, f̂〉H = 〈Θf,Cg〉L2(M).
Let g → δx for x ∈ O. Then 0 = 〈Θf,Cδx〉L2 ≡ η(x). Since η|O = 0, by
real-analyticity we infer the vanishing of η on Ω+, completing the proof. �
Acknowledgements. We are grateful to Hanno Gottschalk and Alexander
Strohmaier for helpful discussions, and G.R. is grateful to the Universität
Bonn for their hospitality during February 2007.
References
[1] Lipman Bers, Fritz John, and Martin Schechter. Partial differential
equations. American Mathematical Society, Providence, R.I., 1979. Lec-
tures in Applied Mathematics 3.
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES17
[2] N. D. Birrell and P. C. W. Davies. Quantum fields in curved space, vol-
ume 7 of Cambridge Monographs on Mathematical Physics. Cambridge
University Press, Cambridge, 1982.
[3] W. Driessler and J. Fröhlich. The reconstruction of local observable
algebras from the Euclidean Green’s functions of relativistic quantum
field theory. Ann. Inst. H. Poincaré Sect. A (N.S.), 27(3):221–236,
1977.
[4] J. Fröhlich. Unbounded, symmetric semigroups on a separable Hilbert
space are essentially selfadjoint. Adv. in Appl. Math., 1(3):237–256,
1980.
[5] Jürg Fröhlich. The pure phases, the irreducible quantum fields, and
dynamical symmetry breaking in Symanzik-Nelson positive quantum
field theories. Ann. Physics, 97(1):1–54, 1976.
[6] Stephen A. Fulling. Aspects of quantum field theory in curved space-
time, volume 17 of London Mathematical Society Student Texts. Cam-
bridge University Press, Cambridge, 1989.
[7] I. M. Gel′fand and N. Ya. Vilenkin. Generalized functions. Vol. 4.
Academic Press [Harcourt Brace Jovanovich Publishers], New York,
1964 [1977]. Applications of harmonic analysis, Translated from the
Russian by Amiel Feinstein.
[8] James Glimm and Arthur Jaffe. Quantum physics. Springer-Verlag,
New York, second edition, 1987. A functional integral point of view.
[9] Arthur Jaffe. Constructive quantum field theory. In Mathematical
physics 2000, pages 111–127. Imp. Coll. Press, London, 2000.
[10] Arthur Jaffe. Introduction to Quantum Field Theory. 2005. Lecture
notes from Harvard Physics 289r, available in part online at
http://www.arthurjaffe.com/Assets/pdf/IntroQFT.pdf.
[11] Arthur Jaffe and Gordon Ritter. Quantum field theory on curved
backgrounds. i. the euclidean functional integral. Comm. Math. Phys.,
270(2):545–572, 2007.
[12] Abel Klein and Lawrence J. Landau. Construction of a unique self-
adjoint generator for a symmetric local semigroup. J. Funct. Anal.,
44(2):121–137, 1981.
[13] Abel Klein and Lawrence J. Landau. From the Euclidean group to
the Poincaré group via Osterwalder-Schrader positivity. Comm. Math.
Phys., 87(4):469–484, 1983.
[14] A. E. Nussbaum. Spectral representation of certain one-parametric
families of symmetric operators in Hilbert space. Trans. Amer. Math.
Soc., 152:419–429, 1970.
[15] Konrad Osterwalder and Robert Schrader. Axioms for Euclidean
Green’s functions. Comm. Math. Phys., 31:83–112, 1973.
[16] Konrad Osterwalder and Robert Schrader. Axioms for Euclidean
Green’s functions. II. Comm. Math. Phys., 42:281–305, 1975. With
an appendix by Stephen Summers.
http://www.arthurjaffe.com/Assets/pdf/IntroQFT.pdf
18 ARTHUR JAFFE AND GORDON RITTER
[17] Barry Simon. The P (φ)2 Euclidean (quantum) field theory. Princeton
University Press, Princeton, N.J., 1974. Princeton Series in Physics.
[18] Robert M. Wald. Quantum field theory in curved space-time. In Grav-
itation et quantifications (Les Houches, 1992), pages 63–167. North-
Holland, Amsterdam, 1995.
E-mail address: arthur jaffe@harvard.edu
Harvard University, 17 Oxford St., Cambridge, MA 02138
E-mail address: ritter@post.harvard.edu
Harvard University, 17 Oxford St., Cambridge, MA 02138
	1. Introduction
	2. Classical Space-Time
	2.1. Structure of Static Space-Times
	2.2. The Operator Induced by an Isometry
	3. Osterwalder-Schrader Quantization
	3.1. Quantization of Vectors (The Hilbert Space  H of Quantum Theory)
	3.2. Quantization of Operators
	4. Structure and Representation of the Lie Algebra of Killing Fields
	4.1. The Representation of g on  E
	4.2. The Cartan Decomposition of g
	5. Reflection-Invariant and Reflected Isometries
	6. Construction of Unitary Representations
	6.1. Self-adjointness of Semigroups
	6.2. Reflection-Invariant Isometries
	6.3. Construction of Unitary Representations
	7. Analytic Continuation
	7.1. Conclusions
	8. Hyperbolic Space and Anti-de Sitter Space
	Appendix A. Euclidean Reeh-Schlieder Theorem
	Acknowledgements
	References
