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Gravity as a four dimensional algebraic quantum
field theory
G´abor Etesi
Department of Geometry, Mathematical Institute, Faculty of Science,
Budapest University of Technology and Economics,
Egry J. u. 1, H e´ p., H-1111 Budapest, Hungary ∗
October 27, 2014
Abstract
Based on an indefinite unitary representation of the diffeomorphism group of an oriented 4manifold an algebraic quantum field theory formulation of gravity is exhibited. More precisely the
representation space is a Krein space therefore as a vector space it admits a family of direct sum
decompositions into orthogonal pairs of maximal definite Hilbert subspaces coming from the Krein
space structure. It is observed that the C∗ -algebra of bounded linear operators associated to this
representation space contains algebraic curvature tensors. Classical vacuum gravitational fields i.e.,
Einstein manifolds correspond to quantum observables obeying at least one of the above decompositions of the space. In this way classical general relativity exactly in 4 dimensions naturally embeds
into an algebraic quantum field theory whose net of local C∗ -algebras is generated by local algebraic
curvature tensors and vector fields. This theory is constructed out of the structures provided by an
oriented 4-manifold only hence possesses a diffeomorphism group symmetry.
Motivated by the Gelfand–Naimark–Segal construction and the Dougan–Mason construction
of quasi-local energy-momentum we construct certain representations of the limiting global C∗ algebra what we call the “positive mass representations”. Finally we observe that the bunch of
these representations give rise to a 2 dimensional conformal field theory in the sense of G. Segal.
AMS Classification: Primary: 83C45, 81T05, Secondary: 57N13
Keywords: General relativity; Algebraic quantum field theory; Four dimensions
1
Introduction
The outstanding problem of modern theoretical physics is how to unify the obviously successful and
mathematically consistent theory of general relativity with the obviously successful but yet mathematically problematic relativistic quantum field theory. It has been generally believed that these two
fundamental pillars of modern theoretical physics conflict each other not only in the mathematical tools
they use but even at a deep foundational level [7]: classical concepts of general relativity such as the
space-time event, the light cone or the event horizon of a black hole are too “sharp” objects from a
∗ e-mail:
etesi@math.bme.hu
G. Etesi: Gravity as algebraic quantum field theory
2
quantum theoretic viewpoint meanwhile relativistic quantum field theory is not background independent from the aspect of general relativity. We do not attempt here to survey the vast physical and
even mathematical and philosophical literature created by the unification problem; we just mention that
nowadays the two leading candidates expected to be capable for a sort of unification are string theory
and loop quantum gravity but surely there is still a long way ahead.
Nevertheless we have the conviction that one day the language of classical general relativity will
sound familiar to quantum theorists and vice versa i.e., conceptual bridges must exist connecting the
two theories. In this note an effort has been made to embed classical general relativity into a quantum
framework. This quantum framework is algebraic quantum field theory formulated by Haag–Kastler
and others during the past decades, cf. [6]. Recently this language also appears to be suitable for
formulating quantum field theory on curved space-time [8] or even quantum gravity [2].
In more detail we will do something very simple here. Namely using structures given by an oriented
smooth 4-manifold M only, our overall guiding principle will be seeking unitary representations of
the corresponding orientation-preserving diffeomorphism group Diff+ (M). There is a unique such
representation on the space of sections of ∧2 M ⊗R C via pullback. However the natural scalar product
on this space—namely the one given by integration of the wedge product of two 2-forms—is indefinite
hence cannot be used to complete the space of smooth 2-forms into a Hilbert space. Rather in struggling
with the completion problem one comes up with a complete topological vector space K (M) with a
non-degenerate indefinite Hermitian scalar product h · , · iL2 (M) i.e., a Krein space. Therefore the bare
space—i.e., not considered as a Diff+ (M)-module—admits non-canonical decompositions K (M) =
H + (M) ⊕ H − (M) into maximal definite orthogonal Hilbert subspaces H ± (M) with respect to the
indefinite scalar product.
Given this Krein space carrying a unitary representation of the diffeomorphism group the next obvious task is to explore the C∗ -algebra of its bounded linear operators. So far our construction is merely
representation theory. However it comes as a surprise (at least to the author) that precisely in 4 dimensions among these operators one can discover curvature tensors! This is because of the well-known fact
that the curvature tensor Rg of a pseudo-Riemannian 4-manifold (M, g) can be regarded as a section
of End(∧2 M ⊗R C) i.e., gives rise to a linear operator acting on K (M). This permits to construct a
diffeomorphism-invariant algebraic quantum field theory {O 7→ U(O)}OjM whose C∗ -algebras U(O)
of local quantum observables are generated by bundle endomorphisms and Lie derivatives commuting
outside O with the O-preserving subgroup of the diffeomorphism group. The local algebras obtained
this way are natural generalizations of the CCR algebra. This construction satisfies the Haag–Kastler
axioms [6, pp. 105-107] of algebraic quantum field theory. As a result classical general relativity
naturally embeds into a quantum framework if one interprets classical curvature tensors as quantum
observables. The appearence of the curvature tensor as a local quantum observable is reasonable even
from the physical viewpoint: in local gravitational physics the metric tensor has no direct physical
meaning only its curvature can cause local physical effects such as tidal forces. Moreover if one wishes
at least in principle the metric i.e., the geometry locally can be reconstructed from its curvature [5].
A meaningful quantum field theory must exhibit stability i.e., unitary representations of its local
observables which are the “positive mass” representations in the sense of Wigner. In our case this
directly leads to the long-standing problem of gravitational mass [11]. It is quite interesting that the
Gelfand–Naimark–Segal construction in the theory of C∗ -algebras and the Dougan–Mason construction
of quasi-local energy-momentum [3] in general relativity naturally meet up here and provide us with
unitary representations in which the expectation values of quasi-local infinitesimal translations along
surfaces in a state corresponding to a gravitational field give Dougan–Mason-like quasi-local quantities.
These representations and the corresponding quasi-local energy-momenta are constructed from im-
G. Etesi: Gravity as algebraic quantum field theory
3
mersed surfaces in M with a choice of a complex structure on them. However the whole construction
is expected to be independent of this choice leading to the by-now classical observation of Witten [15]
that in fact one has to deal with a conformal field theory on these surfaces. We construct this theory: its
conformal blocks are the Clifford algebras generated by finite energy meromorphic sections of certain
unitary holomorphic vector bundles on punctured Riemannian surfaces.
Our algebraic quantum field theory of gravity possesses a giant symmetry group leading to some unexpected consequences. The most fundamental of these is that already the desire to distinguish between
pure gravity and matter requires breaking the diffeomorphism symmetry in some extent. We already
mentioned that after breaking the diffeomorphism symmetry K (M) has an extra structure: a family
of orthogonal splittings into positive and negative definite subspaces. Consequently it is natural to ask
which field operators Q obey at least one of these splittings i.e., have the form Q(H ± (M)) j H ± (M)
with respect to a decomposition K (M) = H + (M) ⊕ H − (M)? In the classical situation this holds for
the curvatures of solutions of the vacuum Einstein’s equation (possibly with a cosmological constant)
with respect to the canonical splitting into (anti)self-dual 2-forms induced by the metric. Indeed, Einstein’s equation says that term mixing H + (M) with H − (M) is proportional to the traceless part of
the energy-momentum tensor of matter. Therefore generally a local quantum observable corresponding
to gravity coupled to matter must be an operator which mixes H + (M) and H − (M) in some splitting
and this mixing term is identified with the “material content” of the operator relative to this splitting.
From this we draw the second conclusion that our diffeomorphism-invariant quantum field theory
lacks any causal structure. (This is in accordance with recent expectations of Lorentz symmetry violations in high energy cosmic processes.) Indeed, the operational description of causality requires a
distinction between pure gravity and matter. The causal future J + (p) ⊂ M of an event p ∈ M in spacetime is by definition the hull of all future-inextendible worldlines of particles departing from p and
moving forward in time locally not faster than light. The causal past J − (p) ⊂ M is defined similarly.
The collection of these causal futures and pasts furnish the space-time with a special topology, the
causal structure. The Lorentzian metric is a mathematical fusion of the geometry captured by a Riemannian metric and the causal structure captured by a “causal topology”. But from this it is clear that
the construction of a causal structure refers to not only gravity but other entities of physical reality as
well which are moreover classical: pointlike particles, electromagnetic waves, time, etc. However they
cannot exist for instance in a vacuum space-time considered in the strict sense. Very strictly speaking
even the interpretation of a space-time point as a “physical event” fails in an empty space-time. Henceforth we are convinced that causality cannot be a fundamental feature of a fully quantum description
of gravity if it is a diffeomorphism-invariant quantum field theory. As a technical consequence we
will prefer to use Riemannian metrics in this note. To summarize: from our standpoint causality is
an emergent statistical phenomenon created by the highly complex interaction of gravity and matter.
Consequently in order to understand it first we should should be able to dinstinguish pure gravity from
matter hence break the diffeomorphism symmetry.
This note is organized as follows. In Sect. 2 we construct a natural indefinite unitary representation
of the orientation-preserving diffeomorphisms of an oriented 4-manifold. Then we take a look of the
C∗ -algebra of bounded linear operators of this representation space. We identify its “classical part”
with local Einstein spaces. In Sect. 3 we introduce an algebraic quantum field theory and in Theorem
3.1 we construct certain representations of its algebras of local observables what we call “positive mass
representations”. In Sect. 4 we bunch these representations together into a conformal field theory.
´
Acknowledgement. The author is grateful to A.G.
Horv´ath, L.B. Szabados and P. Vrana for the stimulating discussions and to the Alfr´ed R´enyi Institute of Mathematics for their hospitality. This work was
supported by OTKA grant No. NK81203 (Hungary).
G. Etesi: Gravity as algebraic quantum field theory
2
4
A representation of the diffeomorphism group
Let M be a connected orientable smooth 4-manifold, possibly non-closed (i.e., it can be non-compact
and-or with non-empty boundary). Fix an orientation on M. Given only these data at our disposal it
is already meaningful to talk about the group of its orientation-preserving diffeomorphisms Diff+ (M).
Our guiding principle simply will be a search for unitary representations of Diff+ (M). A bunch of
representations arise in a geometric way as follows. Consider T (r,s) M ⊗R C, the bundle of complexified (r, s)-type tensors with the associated vector spaces Cc∞ (M; T (r,s) M ⊗R C) of their compactly supported smooth complexified sections. Then the group Diff+ (M) acts from the left via pushforward on
Cc∞ (M; T (r,0) M ⊗R C) for all r ∈ N while from the right via pullback on Cc∞ (M; T (0,s) M ⊗R C) for all
s ∈ N. However these representations are typically not unitary because the underlying vector spaces do
not carry extra structures in a natural way.
The only exception is the 2nd exterior power ∧2 M ⊂ T (0,2) M of the cotangent bundle with the
corresponding space of sections Cc∞ (M; ∧2 M ⊗R C) =: Ω2c (M; C), the space of complexified smooth
2-forms with compact support. Indeed, this vector space has a natural non-degenerate Hermite scalar
product h · , · iL2 (M) : Ω2c (M; C) × Ω2c (M; C) → C given by integration on oriented smooth manifolds;
more precisely for α, β ∈ Ω2c (M; C) put
hα, β iL2 (M) :=
Z
α ∧β
(1)
M
(complex conjugate-linear in its first variable). Note however that this scalar product is indefinite: an
unavoidable fact which plays a key role in our considerations ahead. Consequently this scalar product
cannot be used to complete Ω2c (M; C) into a Hilbert space. Instead with respect to (1) there is a nonunique direct sum decomposition
−
Ω2c (M; C) = Ω+
c (M; C) ⊕ Ωc (M; C)
with the property that they are maximal definite orthogonal subspaces i.e., ±h · , · iL2 (M) |Ω±c (M;C) :
±
+
−
Ω±
c (M; C) × Ωc (M; C) → C are both positive definite moreover Ωc (M; C) ⊥L2 (M) Ωc (M; C). There2
±
fore these restricted scalar products can be used to complete Ω±
c (M; C) into L Hilbert spaces H (M)
respectively.
Lemma 2.1. Given a connected orientable smooth 4-manifold M with a fixed orientation, up to unitary
transformations the Hilbert spaces H ± (M) are independent of the particular choice of the decomposition used to construct them.
Proof. Pick two particular decompositions
0
−
0
Ω2c (M; C) = Ω+
c (M; C) ⊕ (Ωc (M; C)
and
00
−
00
Ω2c (M; C) = Ω+
c (M; C) ⊕ Ωc (M; C)
into maximal definite pairwise orthogonal subspaces with corresponding Hilbert spaces H ± (M)0 and
H ± (M)00 respectively. Let U : Ω2c (M; C) → Ω2c (M; C) be a unitary transformation with respect to (1)
which rotates the first decomposition into the second one. Then obviously the restricted-extended maps
U ± : H ± (M)0 → H ± (M)00 are unitary equivalences hence give rise to Hilbert space isomorphisms
H ± (M)0 ∼
= H ± (M)00 as claimed. 3
G. Etesi: Gravity as algebraic quantum field theory
5
−
Given some decomposition Ω2c (M; C) = Ω+
c (M; C) ⊕ Ωc (M; C) consider the partially completed space
+
−
K (M) := H (M) ⊕ H (M) in the above sense. One can complete K (M) to an L2 Hilbert space
with respect to the definite scalar product
(α, β )L2 (M) := hα + , β + iL2 (M) − hα − , β − iL2 (M)
(2)
where K (M) 3 ω = (ω + , ω − ) ∈ H + (M)⊕H − (M). Although this Hilbert space structure on K (M)
depends on the decomposition the induced norm is always the same hence we obtain a well-defined
topology on K (M), independent of the particular decomposition K (M) = H + (M) ⊕ H − (M) we
began with.
Summing up, starting with an M we can complete Ω2c (M; C) into a pair (K (M), h · , · iL2 (M) ) consisting of a complete topological vector space K (M) and a non-degenerate indefinite Hermitian scalar
product
h · , · iL2 (M) : K (M) × K (M) −→ C
(3)
induced by integration (1) with the following properties. The space admits a family of direct sum
Hilbert space structures K (M) = H + (M) ⊕ H − (M) such that the summands are orthogonal and
maximal definite closed subspaces for the scalar product:
H + (M) ⊥L2 (M) H − (M),
h · , · iL2 (M) |H ± (M) : H ± (M) × H ± (M) −→ C are positive or negative definite, respectively.
Definition 2.1. Let M be a connected oriented smooth 4-manifold. The pair (K (M), h · , · iL2 (M) )
consisting of a complete topological vector space and a non-degenerate indefinite Hermitian scalar
product on it is called the Krein space of M.
Moreover (K (M), h · , · iL2 (M) ) carries a representation of Diff+ (M) from the right given by the unique
extension of the pullback of 2-forms: ω 7→ f ∗ ω for ω ∈ Ω2c (M; C) and f ∈ Diff+ (M). It is easy to
check that these operators are unitary with respect to (3) and if the corresponding diffeomorphism is
the identity outside a compact subset then they are also bounded for the unique norm induced by (2) .
This representation has the following immmediate properties:
Lemma 2.2. Consider the indefinite unitary reprsentation of Diff+ (M) from the right on the Krein
space (K (M), h · , · iL2 (M) ) constructed above.
(i) A vector ω ∈ K (M) satisfies f ∗ ω = ω for all f ∈ Diff+ (M) if and only if ω = 0 (“no vacuum”);
(ii) The closed subspaces B(M) j Z (M) ⊂ K (M) generated by exact or closed 2-forms respectively are invariant under the action of Diff+ (M).
Proof. (i) Assume that there exists an element 0 6= ω ∈ K (M) stabilized by the whole Diff+ (M).
Consider a 1-parameter subgroup { ft }t∈(−1,+1) ∈ Diff+ (M) such that f0 = IdM and let X be the vector
field on M generating this subgroup. Differentiating the equation ft∗ ω = ω with respect to t ∈ (−1, +1)
at t = 0 we obtain LX ω = 0 where LX is the Lie derivative by X. Since an arbitrary compactly supported
vector field generates a 1-parameter subgroup of Diff+ (M) we obtain that in fact ω = 0 which is a
contradiction.
(ii) The statement readily follows by naturality of exterior differentiation i.e., d( f ∗ ϕ) = f ∗ dϕ for
all f ∈ Diff+ (M) and ϕ ∈ Ωkc (M; C). 3
Remark. 1. We succeeded to construct a faithful, reducible, unitary representation of the diffeomorphism group out of the structures provided only by an orientable smooth 4-manifold.1 But the scalar
1 In
fact our construction so far works in any 4k (k = 1, 2, . . . ) dimensions if the diffeomorphism group acts on 2k-forms.
G. Etesi: Gravity as algebraic quantum field theory
6
product h · , · iL2 (M) on K (M) is indefinite therefore there is an extra structure on K (M) namely
a family of decompositions K (M) = H + (M) ⊕ H − (M) into orthogonal pairs of maximal definite
Hilbert subspaces. Note that such decompositions cannot hold for K (M) as a Diff+ (M)-module
or in other words such decompositions break the diffeomorphism symmetry. Their relevance is that
the classical vacuum Einstein equation can be viewed as saying that there is a canonical splitting
K (M) = H + (M) ⊕ H − (M) in which the curvature is blockdiagonal.
2. We also remark that making use of the structures given by an orientable differentiable manifold
only one cannot construct canonically a Hilbert space structure on K (M) i.e., as a Hilbert space its
splittings like K (M) = H + (M) ⊕ H − (M) are unavoidable without using some extra piece of data.
3. From the mathematical viewpoint in many important cases we do not loose any information on
the topology of the original manifold if we replace M by the representation space (K (M), h · , · iL2 (M) ).
Indeed, restricting Ω2c (M; C) to closed forms then dividing by the exact ones we can pass to compactly
supported cohomology Hc2 (M; C); then by the Poincar´e duality Hc2 (M; C) ∼
= (H 2 (M; C))∗ we can switch
to the ordinary de Rham cohomology. If we assume that M is compact and simply connected then the
singular cohomology H 2 (M; Z) maps injectively into H 2 (M; C) hence finally the scalar product (3)
descends to the topological intersection form
qM : H 2 (M; Z) × H 2 (M; Z) −→ H 4 (M; Z) ∼
=Z
of the underlying topological 4-manifold. However taking into account that by assumption M has a
smooth structure we can refer to Freedman’s fundamental result [4] that qM uniquely determines the
topology of M.
Taking into account that all of our constructions so far are local we can consider all the objects on open
subsets. Take the Krein space (K (M), h · , · iL2 (M) ) with ∗ being the adjoint operation for the indefinite
scalar product (3). Moreover let k·kL2 (M) be the unique norm on K (M) induced by a particular definite
scalar product (2). Given a linear map A : K (M) → K (M) take its usual operator norm
kAk :=
kAωkL2 (M)
sup
06=ω∈K (M)
kωkL2 (M)
and consider the Banach ∗-algebra B(M) := {A ∈ EndK (M) | kAk < +∞} of all bounded linear operators on (K (M), h · , · iL2 (M) ). One can check that B(M) is in fact a C∗ -algebra. Pick a relatively
compact open subset O j M and define the local C∗ -algebra (with unit)
B(O) := B ∈ B(M) B|K (M\O) , Diff+
(M)
=
0
O
i.e., B(O) ⊂ B(M) consists of operators which commute on the closed subspace K (M \ O) ⊂ K (M)
+
with the subgroup Diff+
O (M) ⊂ Diff (M) consisting of diffeomorphisms preserving O j M.
Consider the assignment {O 7→ B(O)}OjM for all relatively compact open subsets. This assignment has the following properties. Take two relatively compact open subsets O j V ; then the embedding eVO : U → V induces a unit-preserving injective homomorphism EVO : B(O) → B(V ) of local
C∗ -algebras such that the corresponding diagram
O
eVO
V
/ B(O)
/
EVO
B(V )
G. Etesi: Gravity as algebraic quantum field theory
7
is commutative. This allows us to define the algebra B(O) for any open O j M and B(M) arises
as the C∗ -algebra direct (inductive) limit of these local algebras. Note that by construction for all
O j M we simply have unital sub-C∗ -algebras B(O) j B(M) acting on the common Krein space
(K (M), h · , · iL2 (M) ). However observe that if we consider the dual process namely the restriction
then the elements of these local algebras do not behave well because they lack the presheaf property in
general.
As a consequence of the geometric origin of the global C∗ -algebra B(M), it has an interesting
subalgebra C (M). Indeed, consider the sheaf CM over M whose spaces of local sections C (O) over
open subsets are algebras of local smooth bundle (i.e., fiberwise) morphisms
C∞ (O ; End(∧2 O ⊗R C)) for all open O j M.
In contrast to general elements of B(O), sections of C (O) behave well under restriction due to their
presheaf property. Given two open subsets O j V j M the restriction map rVO : V → O induces a
unit-preserving injective homomorphism RVO : C (V ) → C (O) of algebras such that the corresponding
diagram
/ C (O)
OO O
RVO
rVO
V
/
C (V )
is commutative. If M happens to be compact then C (M) can be completed to a unital C∗ -algebra and
there is an obvious embedding of unital C∗ -algebras C (M) $ B(M).
Remark. It is possible to demonstrate that for compact M the C∗ -algebra of global sections C (M) of
the sheaf CM is strongly Morita equivalent to the unital commutative C∗ -algebra C0 (M; C) consisting
of bounded functions. By the classical result of Gelfand we know that M can be recovered from
C0 (M; C). Consequently we can think of the whole C∗ -algebra B(M) as a sort of enhancement of M
to a “noncommutative space” in the spirit of noncommutative geometry.
Examples. The time has come to take a closer look of the these C∗ -algebras emerging through a unitary
representation of the diffeomorphism group of an oriented 4-manifold M.
1. Let (M, g) be a 4-dimensional Riemannian Einstein manifold i.e., assume that g is a Riemannian
metric on M with Ricci tensor rg satisfying the vacuum Einstein equation rg = ΛM g with a cosmological
constant ΛM ∈ R. In this special situation the vast symmetry group of the original theory reduces to
the stabilizer subgroup Iso+ (M, g) $ Diff+ (M) leaving the geometry (M, g) unaffected. In this realm
the Riemannian metric together with the orientation gives a Hodge operator ∗ : ∧2 M → ∧2 M with
∗2 = Id∧2 M . This induces a usual real splitting
∧2 M = ∧+ M ⊕ ∧− M.
(4)
It is a well-known fact [10] but from our viewpoint is an interesting coincidence that in exactly 4
dimensions the full Riemannian curvature tensor can be regarded as a real linear map Rg : ∧2 M → ∧2 M
which decomposes as
+ sg
Bg
Wg + 12
Rg =
s
B∗g
Wg− + 12g
with respect to the splitting (4). Here the traceless symmetric maps Wg± : ∧± M → ∧± M are the
(anti)self-dual parts of the Weyl tensor, the diagonal sg : ∧2 M → ∧2 M is the scalar curvature while
G. Etesi: Gravity as algebraic quantum field theory
8
Bg : ∧+ M → ∧− M is the traceless Ricci tensor together with its adjoint B∗g : ∧− M → ∧+ M. Observe that the Einstein equation exactly says that Bg = 8πT0 where T0 is the traceless part of the
energy-momentum tensor and the vacuum is equivalently characterized by the condition Bg = 0 i.e.,
Rg ∈ C∞ (M; End(∧2 M)) obeys (4). The pointwise splitting above in addition yields a canonical decomposition
−
Ω2c (M; C) = Ω+
c (M; C) ⊕ Ωc (M; C)
of the space of 2-forms into (anti)self-dual forms which is the same as decomposing this space into
mutually orthogonal maximal definite subspaces with respect to the scalar product (1). Therefore in
the presence of a metric—which is a way to break the original symmetry group Diff+ (M) down to
a smaller one—there is a distinguished splitting K (M) = H + (M) ⊕ H − (M) obeyed by the curvature Rg . Switching to our notation we conclude that Rg ∈ C (M) such that Rg (H ± (M)) j H ± (M).
Moreover by the usual symmetries of the curvature tensor Rg is self-adjoint for (3). Note that all the
conclusions hold for the complexified curvature tensor of a Lorentzian manifold.
Therefore we come up with a natural embedding of classical Riemannian vacuum general relativity
into a quantum framework (although note that formally all of our conclusions work for the Lorentzian
case, too):
C. The real Riemannian curvature tensor Rg of an orientable Riemannian Einstein 4-manifold (M, g)
can be interpreted as an operator Rg ∈ C (M). The existence of a metric breaks the original symmetry
group Diff+ (M) down to the finite dimensional group Iso+ (M, g). This operator as well as the reduced
symmetry group obeys the canonical splitting K (M) = H + (M) ⊕ H − (M) induced by the metric.
Moreover Rg is a real self-adjoint operator with respect to the scalar product (3) on K (M).
2. Next we consider a generic element RM ∈ C (M). In this case the previous picture can almost
be reversed in the sense that at least locally such operators correspond to Einstein metrics i.e., classical
structures and these local operators obey the canonical splittings of the local Krein spaces induced by
these local metrics.
More precisely we claim that
Lemma 2.3. Let M be a connected oriented smooth 4-manifold. Take any RM ∈ C (M). Assume that
(i) In every point x ∈ M it is real and is a smooth algebraic curvature tensor i.e., the restricted map
Rx : ∧2x M → ∧2x M is real moreover is symmetric and satisfies Bianchi’s first identity;
(ii) For every point x ∈ M there exists a non-singular Riemannian metric gx on some neighbourhood
of x such that its Riemann tensor at x satisfies Rgx = Rx and its Ricci tensor at x satisfies rgx = Λx gx
with a constant Λx ∈ R.
Then about every point x ∈ M one can find a neighbourhood x ∈ O and a local Riemannian metric gO
on O such that
(i) Its Riemannian curvature coincides with RM on O more precisely RgO = RM |O = RO ∈ C (O);
(ii) The metric also satisfies the vacuum Einstein equation on U with the local cosmological constant
Λx ∈ R i.e., rgO = Λx gO on O.
Note that these local Riemannian Einstein metrics may not fit together into a global one over M.
Proof. By referring to a local solvability result of Gasqui [5] for a pseudo-Riemannian algebraic curvature tensor the conditions in the lemma guarantee the existence of a neighbourhood x ∈ V and a
Riemannian metric gV on V such that
G. Etesi: Gravity as algebraic quantum field theory
9
(i) gV = gx at least in x ∈ V ;
(ii) RgV = Rx at least in x ∈ V ;
(iii) rgV = Λx gV along V with the local cosmological constant Λx ∈ R.
We would like to extend part (ii) of this statement from the point x to a whole but probably smaller
neighbourhood x ∈ O j V . Since by the Definition CM is a sheaf recall [14] that there exists an isomorphism of sheaves τ : CM −→ C M where C M is the e´ tal´e space of CM defined by C M = ∪x∈M Cx with
Cx being the stalk of CM at x ∈ M. This latter space arises as a direct (inductive) limit
Cx = lim C (W )
−→
W 3x
with well-defined restriction maps rW
x : C (W ) → Cx for all x ∈ W j M open. But note that for any
RM ∈ C (M) ∩C∞ (M ; End(∧2 M ⊗R C)) its value Rx at x ∈ M satisfies
rxM (RM ) = Rx
consequently Rx ∈ Cx . In addition the local Riemannian curvature RgV obviously satisfies RgV ∈ C (V )
and part (ii) of Gasqui’s result shows that
rVx (RgV ) = Rx .
However these last two equations imply by the aid of the definition of a stalk that there exists a neighbourhood x ∈ O j V j M such that in fact
M
rO
(RM ) = rVO (RgV ).
Therefore we put the neighbourhood around x to be this U and the metric on it to be the restriction
gO := rVO (gV ).
Moreover part (iii) of Gasqui’s theorem obviously provides us that gO also satisfies the local Einstein
equation with cosmological constant Λx ∈ R as desired. 3
We conclude that generic elements of C (M) can be interpreted as a certain structure on the manifold
which at least locally resembles to classical Riemannian general relativity (we note again that all of
these conclusions are valid in the Lorentzian case, too):
SC. Given a connected oriented smooth 4-manifold M a generic element RM ∈ C (M) satisfying the
technical conditions of Lemma 2.3 gives rise to local Riemannian Einstein metrics on open subsets O
around every points of M. The corresponding restricted operators RO ∈ C (O) and reduced symmetry
groups obey the local canonical splittings K (O) = H + (O) ⊕ H − (O) induced by the local metrics.
3. Finally we take a further departure from classical general relativity and explore the deep quantum
regime. Of course the trouble is how to describe a generic bounded linear operator Q ∈ B(M) in terms
of a geometric linear operator R ∈ C (M). Our quantum instinct tells us that a truely quantum operator
should be constructed by somehow smearing a semiclassical operator over regions in M. This instinct
will be justified by the famous Schwartz kernel theorem applied below.
Fix a semiclassical operator R ∈ C (M) ∩C∞ (M; End(∧2 M ⊗R C)) and a point x ∈ M. Then on any
2-form ω ∈ Ω2c (M; C) its action can be expressed in a fully local form (Rω)x = Rx ωx . We can generalize
this as follows. Pick finitely many distinct further points y1 , . . . , yn(x) ∈ M where n(x) ∈ N may depend
G. Etesi: Gravity as algebraic quantum field theory
10
on x ∈ M. Consider diffeomorphisms fi ∈ Diff+ (M) such that f0 = IdU hence f0 (x) = x moreover
fi (x) = yi for i = 1, . . . , n(x). An operator Q ∈ B(M) from R ∈ C (M) ∩C∞ (M; End(∧2 M ⊗R C)) and
f0 , . . . , fn(x) ∈ Diff+ (M) is constructed such that on vectors ω ∈ Ω2c (M; C) ⊂ K (M) forming a dense
subset in any point x ∈ M it has the shape
n(x)
(Qω)x :=
∑
n(x)
fi∗ (Rω) = Rx ωx +
i=0
∑ fi∗(Rω).
(5)
i=1
Note that this linear operator is not local in the sense that its effect on ωx depends not only on Rx and
ωx but on the value of R and ω in further distant points y1 , . . . , yn(x) ∈ M as well. The question arises
how to generalize this construction for infinite sums or even integrals. For all points y ∈ M pick up
unique diffeomorphisms fy ∈ Diff+ (M) such that fy (x) = y and fx = IdM . Then for all ω ∈ K (M) the
assignment y 7→ fy∗ (Rω) gives a function from M into ∧2x M ⊗R C. Suppose moreover that there is a
complex measure µx on M constructed as follows. Recall that a double 2-form K is a section of the
bundle (∧2 M ⊗R C) × (∧2 M ⊗R C) over M × M. Then the kernel K is a right tool how to integrate this
function against this measure i.e., for all x ∈ M and ω ∈ K (M) we put
Z
y∈M
fy∗ (Rω)dµx (y) :=
Z
y∈M
Kx,y ∧ (Rω)y ∈ ∧2x M ⊗R C.
Consequently the appropriate way to generalize the discrete formula (5) is to set
Z
(Qω)x :=
Kx,y ∧ (Rω)y .
y∈M
Of course this integral to make sense we have to specialize the precise class of these “kernel functions”.
We do not do it here but note that the more singular the kernel is, the more general the resulting bounded
linear operator is. The general situation is covered by the Schwartz kernel theorem: there exists a
distributional double 2-form K ∈ D 0 (M × M ; (∧2 M ⊗R C) × (∧2 M ⊗R C)) such that any bounded
linear operator can be constructed in the form
hα, Qβ iL2 (M) = (K, α ⊗ (Rβ ))M×M
where this latter bracket is the pairing between dual spaces (cf. e.g. [12, Vol. I. Section 4.6]).
We summarize again our findings as follows:
Q. Over a connected oriented smooth 4-manifold M a generic element Q ∈ B(M) always can be
constructed from a geometric one R ∈ C (M) by a smearing procedure provided by the Schwartz kernel
theorem. In this general situation no pointwisely given geometric object has a meaning because the
original symmetry group Diff+ (M) is unbroken. This is in accord with the physical expectations.
We have completed the exploration of the elements of C (M) and B(M).
3
Gravity as an algebraic quantum field theory
Before proceeding further let us summarize the situation we have reached in Sect. 2. To any given
smooth oriented 4-manifold M one can associate a Krein space (K (M), h · , · iL2 (M) ) such that this
space carries an indefinite unitary representation of the group Diff+ (M) from the right. Moreover
G. Etesi: Gravity as algebraic quantum field theory
11
the associated C∗ -algebra B(M) of bounded linear operators acting on K (M) from the left contains
algebraic curvature tensors. Classical solutions of the vacuum Einstein equations i.e., Riemannian
Einstein manifolds (M, g) can be characterized by the fact that they obey a canonical splitting of the
Krein space induced by the metric moreover they are real and self-adjoint operators. (In the case of
Lorentzian metrics the same holds except that the corresponding operators are not real.) Therefore
one is tempted to look at the curvature operators as local quantum observables in a quantum field
theory possessing a huge symmetry group coming from diffeomorphisms. We would like to make
these observations more formal by constructing something which resembles an algebraic quantum field
theory in the sense of [6] with local quantum observables containing gravitational curvature tensors.
For this aim we have to associate a “net of local algebras” to M i.e., we need an assignment O 7→ U(O)
of certain algebras for all open O j M such that the basic axioms of this theory having still meaning
in our more general context should be satisfied. The simplest and most natural way of doing this is as
follows.
Recall that the space of local smooth complexified (0, 4)-type algebraic curvature tensors over M is
C∞ (M; (S2 ∧2 M ∩ Ker b) ⊗R C) where b : C∞ (M; (∧1 M)⊗4 ) → C∞ (M; (∧1 M)⊗4 ) is the usual algebraic
Bianchi map. Making use of a metric i.e., pseudo-Euclidean structures on the fibers, the corresponding
(2, 2)-type algebraic curvature tensors fulfill a subspace of C∞ (M; End(∧2 M ⊗R C)). However now
we lack any preferred metric hence only the whole endomorphism space is at our disposal. Consider
the Krein space (K (M), h · , · iL2 (M) ) with ∗ being the correspondiong adjoint operation and the norm
k · kL2 (M) on K (M) given by any (2) as usual. This gives the C∗ -algebra B(M) of all bounded linear
operators on K (M). For an R ∈ Cc∞ (M; End(∧2 M ⊗R C)) and a vector field X ∈ Cc∞ (M; T M ⊗R C) with
1
∗
the associated Lie derivative LX write B(R) := e 2 (R+R ) as well as B(LX ) := eLX (note that LX is already
self-adjoint because it generates diffeomorphisms which are unitary). Fix a relatively compact open
subset O j M and let U(O) ⊂ B(M) be the C∗ -algebra generated by operators B(R), B(LX ) ∈ B(M)
+
which commute on the closed subspace K (M \ O) ⊂ K (M) with the subgroup Diff+
O (M) ⊂ Diff (M)
consisting of O-preserving diffeomorphisms:
+
U(O) := B(R), B(LX ) ∈ B(M) B(R)|K (M\O) , Diff+
(M)
=
0,
B(L
)|
,
Diff
(M)
=
0
.
X
K
(M\O)
O
O
By construction U(O) j B(M) therefore as usual, all these local algebras act on K (M) and the global
algebra U(M) is constructed as the C∗ -algebra direct (inductive) limit of these local algebras.
Definition 3.1. For a relatively compact open O j M the algebra U(O) is called the local generalized
CCR algebra of local quantum observables while U(M) is the global generalized CCR algebra of M.
Remark. 1. By construction U(O) contains curvature tensors hence solutions of classical general relativity via their curvatures can be incorporated into U(M).
2. U(O) also contains a usual CCR algebra at least when O ⊂ M is a coordinate ball. Consider
the maximal subspace of those pairs of local endomorphisms and local vector fields which either commute: [R1 + R∗1 , R2 + R∗2 ] = 0, [LX , LY ] = 0, [R + R∗ , LX ] = 0 or are canonically conjugate to each other
i.e., [R + R∗ , LX ] = cIdK (O) with some c ∈ C which means that for all ω ∈ Ω2c (O; C) the identity
[R + R∗ , LX ]ω = cω holds and extends continuously over K (O). Then the sub-C∗ -algebra in U(O)
generated by B(R), B(LX ) from these restricted classes of algebraic curvature tensors and vector fields
respectively, yields a usual CCR algebra; R + R∗ and LX play the role of the position operator Q and
its canonically conjugate momentum operator P, respectively. This standard CCR algebra within U(O)
describes the “free graviton part” while the rest of U(O) the “self-interacting part” of this quantum field
theory.
G. Etesi: Gravity as algebraic quantum field theory
12
Putting things together then let us consider the algebraic quantum field theory defined by the assignment
O 7−→ U(O), U j M is relatively compact open.
Moreover U(M) is taken to be the C∗ -algebra direct (inductive) limit of the U(O)’s as usual. Note
that the formulation of this theory rests only on the smooth structure on M hence does not refer to
any metric on M for instance. The Krein space (K (M), h · , · iL2 (M) ) carries an action of all U(O)’s
from the left and a unitary one of Diff+ (M) from the right. Elements of the algebra U(O) are the local
quantum observables and those of the group Diff+ (M) are the symmetry transformations. The states
are continuous normalized positive linear functionals on U(O) and the expectation value of B ∈ U(O)
in the state Φ is Φ(B) ∈ C. Now we introduce the concept of a “quantum gravitational field”.
Definition 3.2. Let M be a connected oriented smooth 4-manifold and (K (M), h · , · iL2 (M) ) its Krein
space. Given a local generalized CCR algebra U(O) generated by B(R)’s and B(LX )’s as above a local
observable Q of the infinitesimal form
d m1
n1
mk
nk
Q := (B (tR1 )B (tLX1 ) · · · B (tRk )B (tLXk ))
dt
t=0
is called a local quantum gravitational field on O j M.
Take any splitting K (M) = H + (M) ⊕ H − (M) into maximal definite orthogonal subspaces (note
that this breaks the diffeomorphism symmetry). The off-blockdiagonal part of Q is the material content
of the local quantum gravitational field relative to the splitting. In particular a Q is called a local
quantum vacuum gravitational field relative to the splitting if its material content relative to the splitting
vanishes i.e., it respects the splitting i.e., Q(H ± (M)) j H ± (M).
In a quantum field theory the algebra of quantum observables must possess positive mass and energy
representations in the spirit of E. Wigner. Let us therefore proceed further and construct some representations of our C∗ -algebra U(M) what we will call positive mass representations. When doing this
we touch upon the problem of gravitational mass and energy which is probably the most painful part of
current general relativity [11]. Therefore our considerations here are tentative.
Theorem 3.1. Take a compact oriented surface-without-boundary Σ. Let (Σ, p1 , . . . , pn ) denote a
generic smooth immersion Σ ⊂ M where the points p1 , . . . , pn ∈ Σ are the preimages of the multiple
points of this immersion. Moreover let ω ∈ Ω2c (M; C). Assume that
(i) ω|Σ satisfies
1 R
2πi Σ ω
= 1 and
1 R
∗
2πi Σ (B B)ω
= 0 for all B ∈ End(Ω2c (M; C)) ∩ U(M);
(ii) For any complex structure C = C(Σ) on Σ there exists a positive definite unitary holomorphic
vector bundle structure induced by ω|C on the rank 4 vector bundle E := T M ⊗R C|C over C ⊂ M
satisfying dimC H 0 (C; O(E)) = 4.
Then (Σ, p1 , . . . , pn , ω) gives rise to a unitary representation πΣ,ω of U(M) on a Hilbert space HΣ,ω
as follows. For every B ∈ U(M) there exists a complex 4-vector PC,ω,B ∈ H 0 (C; O(E)) defined by (6)
below, called the quasi-local energy-momentum of the corresponding state [B] ∈ HΣ,ω . Its length
mC,ω,B := kPC,ω,B kL2 (C) = 0
with respect to a natural Hermitian scalar product ( · , · )L2 (C) on H 0 (C; O(E)) is called the mass of the
state. HΣ,ω also carries a unitary representation of Diff+ (M) such that there exists a unique vector
ΩΣ,ω ∈ HΣ,ω of unit length left invariant by the diffeomorphism group and having zero mass.
The unitary representation πΣ,ω on HΣ,ω is called a positive mass representation of U(M) with
vacuum vector ΩΣ,ω ∈ HΣ,ω .
G. Etesi: Gravity as algebraic quantum field theory
13
Proof. The unique continuous extension of the map ΦΣ,ω : U(M) → C given by
1
B 7−→ ΦΣ,ω (B) :=
2πi
Z
Bω ∈ C
Σ
on elements B ∈ End(Ω2c (M; C)) ∩ U(M) is a continuous normalized positive functional on U(M) by the
assumptions (i) of the theorem. Hence the GNS construction can be used to construct a corresponding
unitary representation of the C∗ -algebra U(M). Recall that this goes as follows. One has the induced
left-multiplicative Gelfand ideal IΣ,ω := {B ∈ U(M) | ΦΣ,ω (B∗ B) = 0} ⊂ U(M). The functional provides us with a well-defined positive definite scalar product ([A], [B])Σ,ω := ΦΣ,ω (A∗ B) on U(M)/IΣ,ω
with A ∈ [A], B ∈ [B] where [A] := A + IΣ,ω , etc. Making use of this scalar product one completes
U(M)/IΣ,ω to a Hilbert space HΣ,ω and then lets U(M) act from the left by the continuous extension of
πΣ,ω (A)[B] := [AB] from U(M)/IΣ,ω to HΣ,ω . Two representations πΣ1 ,ω1 and πΣ2 ,ω2 are unitary equivalent if and only if there is a positive real number a ∈ R+ such that ΦΣ2 ,ω2 = aΦΣ1 ,ω1 i.e., there exists
an element f ∈ Diff+ (M) satisfying Σ2 = f (Σ1 ) and ω2 = a f ∗ ω1 . In fact the case of a = 1 with any f
is nothing else than a representation of the conjugate algebra f ∗ U(M)( f −1 )∗ on HΣ1 ,ω1 since Diff+ (M)
acts on U(M) induced by its unitary representation on K (M) as in Sect. 2. In particular the identically
zero curvature R = 0 gives B(0) = 1U(M) and its class
ΩΣ,ω := [1U(M) ] ∈ HΣ,ω
is the unique vector which is left invariant by this adjoint action of Diff+ (M).
Consider the smooth complex vector bundle E := T M ⊗R C|Σ satisfying degE = 0 and rkE = 4.
The 2-form ω can also be used to construct a Hermitian
metric on it. Indeed, such a Hermitian metric
1
is defined by h(X,Y ) := 2 ω(X, iY ) − ω(iX,Y ) for all vector fields X,Y ∈ C∞ (Σ; E). By assumption
this makes E into a smooth positive definite unitary vector bundle (E, h) over Σ (hence in particular
ω|Σ must be non-degenerate). Take a smooth connetion ∇E : C∞ (Σ; E) → C∞ (Σ; E ⊗C (∧1 Σ ⊗R C))
satisfying ∇E h = 0 which means that it is compatible with the unitary structure. Picking any complex
structure on Σ we can identify it with a compact Riemann surface C = C(Σ). The (0, 1)-part ∂ E of
the connection endows (E, h) with the structure of a unitary holomorphic vector bundle over C. The
finite dimensional subspace of holomorphic sections of this bundle is denoted by H 0 (C; O(E)). The
Riemann–Roch–Hirzebruch theorem gives dimC H 0 (C; O(E)) = 4(1 − g(C)) but by assumptions (ii) in
the theorem this vector space is supposed to be precisely 4 dimensional. In addition one also obtains a
non-degenerate definite L2 Hermitian scalar product on C∞ (C; E) by putting
(X,Y )L2 (C) :=
1
2πi
Z
h(X,Y )ω.
C
In usual Poincar´e-invariant quantum field theory a 4 dimensional set of infinitesimal space-time
symmetries is distinguished because they commute hence are regarded as infinitesimal translations; the
corresponding operators are interpreted as energy-momentum operators acting on the Hilbert space of
the theory. Here the space-time symmetry transformations are all the diffeomorphisms hence in our
algebraic quantum field theory the corresponding infinitesimal transformations are the Lie derivatives
with respect to vector fields. However in general one cannot find a distinguished 4 dimensional commuting subspace which could be called as “infinitesimal translations”. To overcome this difficulty we
will follow the Dougan–Mason quasi-local construction [3] (or [11, Chapter 8]) to find a distinguished
4 dimensional subspace of vector fields by holomorphicity:
G. Etesi: Gravity as algebraic quantum field theory
14
Definition 3.3. A vector field X ∈ Cc∞ (M; T M ⊗R C)) is a called an infinitesimal quasi-local translation
of Σ if X|Σ ∈ H 0 (C; O(E)) ⊂ C∞ (Σ; E) where C = C(Σ) is a complex structure on the surface Σ ⊂ M.
Let’s continue the proof. If ω ∈ Ω2c (M; C) and X ∈ Cc∞ (M; T M ⊗R C) then LX ω ∈ Ω2c (M; C) is meaningful; therefore if B ∈ End(Ω2c (M; C)) then we make sense of LX B ∈ B(M) by setting
(LX B)ω := LX (Bω) − B(LX ω) for all ω ∈ Ω2c (M; C)
and define LX B for B ∈ U(M) by continuity. Proceeding further, if [B] ∈ U(M)/IΣ,ω ⊂ HΣ,ω then we
put πΣ,ω (LX )[B] := [LX B]. If { ft | t ∈ R} ⊂ Diff+ (M) is a 1-parameter subgroup generated by X and
B ∈ End(Ω2c (M; C)) then we can see that πΣ,ω (LX )[B] ∈ HΣ,ω satisfies
∗ ∗
f
B
f
−
B
t
−t
= 0.
lim π
(L
)[B]
−
X
Σ,ω
t→0 t
Σ,ω
Finally we define πΣ,ω (LX )ψ by continuity for a generic state ψ ∈ HΣ,ω . The expectation value of LX
in a state 0 6= ψ ∈ HΣ,ω is therefore
(ψ , πΣ,ω (LX )ψ)Σ,ω
∈ C.
kψk2Σ,ω
Take an infinitesimal quasi-local translation LX of Σ, an element B ∈ U(M) and regard the corresponding
expectation value as a component of the quasi-local energy-momentum 4-vector for [B] ∈ HΣ,ω . By
construction
Z
1
∗
(B∗ LX B)ω
([B], πΣ,ω (LX )[B])Σ,ω = ΦΣ,ω (B LX B) =
2πi
C
consequently the expectation value can be expressed as the evaluation on X of a C-linear functional
PC,ω,B on H 0 (C; O(E)) whose shape is
R
(B∗ LY B)ω
C
PC,ω,B (Y ) := R
(B∗ B)ω
for all Y ∈ H 0 (C; O(E)).
(6)
C
This is called the quasi-local energy-momentum 4-vector of the state 0 6= [B] ∈ HΣ,ω . The Hermitian
∗
structure ( · , · )L2 (C) gives H 0 (C; O(E)) ∼
= H 0 (C; O(E)) hence we can regard PC,ω,B as a 4-vector
PC,ω,B ∈ H 0 (C; O(E)) by setting (PC,ω,B , X)L2 (C) := PC,ω,B (X). Moreover mC,ω,B := kPC,ω,B kL2 (C) = 0
is interpreted as the mass of [B] ∈ HΣ,ω . The invariant vector satisfies πΣ,ω (LX )ΩΣ,ω = [LX 1U(M) ] = 0
hence has vanishing mass as claimed. 3
Remark. 1. The formula (6) for the quasi-local energy-momentum formally remains meaningful for
quantum gravitational fields introduced in Definition 3.2. Hence the corresponding quantitites PC,ω,Q
and mC,ω,Q are interpreted as the quasi-lolcal energy-momentum 4-vector and the mass of a quantum
gravitational field Q. Among local quantum gravitational fields one can recognize classical curvature
tensors hence we obtain quasi-local quantities for classical general relativity, too.
2. The usual axioms of algebraic quantum field theory (cf. e.g. [6, pp. 58-60 or pp. 105-107])
typically make no sense in this very general setting. But for clarity we check them one-by-one in order
to see in what extent our algebraic quantum field theory more general is than the usual ones.2
2 We
quote from Haag [6, p. 60]: “On the other hand the word axiom suggests something fixed, unchangeable. This
is certainly not intended here. Indeed, some of the assumptions are rather technical and should be replaced by some more
natural ones as deeper insight is gained. We are concerned with a developing area of physics which is far from closed and
should keep an open mind for modifications of assumptions, additional structural principles as well as information singling
out a specific theory within the general frame.”
G. Etesi: Gravity as algebraic quantum field theory
15
[6, Axiom A on p. 106] can be translated to saying that the Hilbert space HΣ,ω carrying a unitary
representation πΣ,ω of the global generalized CCR algebra U(M) is also acted upon by the orientationpreserving diffeomorphism group Diff+ (M) in a unitary way such that there exists a unique invariant
ray generated by the vector ΩΣ,ω ∈ HΣ,ω . Moreover Dougan–Mason quasi-local translations of Σ ⊂ M
give rise to quasi-local energy-momentum 4-vectors in this representation such that the state ΩΣ,ω has
zero quasi-local energy-momentum hence mass. Consequently this vector (or ray) might be called the
vacuum in this representation. This is interesting because the concepts of mass and energy are quite
problematic in classical general relativity [11] as well as the concept of the vacuum in a general quantum
field theory [8]. Nevertheless this construction of representations of the generalized CCR algebra—
which mixes ideas of quasi-local constructions in classical general relativity [3, 11] and standard GNS
representation theory of C∗ -algebras—contains a technical ambiguity namely a choice of a complex
structure on an embedded surface in M. However one expects the whole machinery to be independent
of this choice. We treat this problem in Sect. 4.
[6, Axioms B and C on p. 107] dealing with the additivity of local algebras and their hermiticity by
construction hold here.
[6, Axiom D on p. 107] can be translated by saying that since the diffeomorphism group is the
symmetry group of the theory, it acts on the net of local algebras like
f ∗ U(O)( f −1 )∗ = U( f (O))
(7)
for all f ∈ Diff+ (M) i.e., symmetry transformations map the local algebra of a region to that one of the
transformed region. This continues to be valid here.
[6, Axiom E on p. 107] holds in a trivial way as an unavoidable consequence of the vast diffeomorphism invariance. It is easy to see that [U(O), U(V )] = 0 if and only if O ∩ V = 0.
/ Indeed, (7) shows
that regardless what U(O) actually is, it must commute with those elements of Diff+ (M) which are the
identity on O; consequently if A ∈ U(O) ⊂ B(M) then A|K (M\O) ∈ Z(B(M \ O)) = C IdK (M\O) . But
K (V ) ⊂ K (M \ O) if O ∩ V = 0/ hence the claim follows. Therefore there is no causality hence no
dynamics present here. Hence the reason we prefer to use Riemannian metrics over Lorentzian ones
throughout the paper. We can also physically say that this theory represents a very elementary level of
physical reality where even no causality exists yet. Causality should emerge through breaking of the
diffeomorphism symmetry.
[6, Axiom F on p. 107] This completeness requirement claims for the validity of Schur’s lemma i.e.,
the only bounded operator which commutes with all quantum gravitational fields should be a multiple
of the identity operator 1 ∈ B(M). This also holds in our case.
[6, Axiom G on p. 107] about “primitive causality” has no meaning in this general setting.
We finished the comparison with the usual axioms of algebraic quantum field theory.
4
Positive mass representations and conformal field theory
In Sect. 3 by the aid of the Krein space K (M) of a 4-manifold M the global generalized CCR-algebra
U(M) has been introduced together with a bunch of its unitary representations πΣ,ω constructed by
standard means from a smooth immersion (Σ, p1 , . . . , pn ) of a surface Σ into M and a regular element
ω ∈ K (M). If a complex structure C = C(Σ) is put onto the surface as well then the quasi-local energy
momentum PC,ω,B ∈ H 0 (C; O(E)) and mass mC,ω,B ∈ R+ ∪ {0} of a non-zero state [B] ∈ HΣ,ω can
be defined enriching πΣ,ω further to a positive mass representation. However on physical grounds we
expect the whole construction to be independent of these technicalities i.e., any choice of these complex
G. Etesi: Gravity as algebraic quantum field theory
16
structures have to result in the same construction. Following Witten [15] this means that a conformal
field theory lurks behind the curtain. We can indeed find this theory which however turns out to be a
very simple topological conformal field theory in the sense that its Hilbert space is finite dimensional
and the correlation functions are insensitive for the insertion of marked points i.e., how the immersion
looks like.
In constructing this topological conformal field theory we will follow G. Segal [9]. That is first
construct a “modular functor extended with an Abelian category possessing a symmetric object” (cf.
in particular [1, Definition 5.1.12]) and [1, Chapters 5 and 6] in general). In other words we have to
construct an assignment
τ : (Σ, p1 , . . . , pn ) 7−→ τ(Σ, p1 , . . . , pn )
(8)
which somehow associates to surfaces with marked points finite dimensional complex vector spaces
satisfying certain axioms. Consider a positive mass representation πΣ,ω of U(M) constructed out of
(Σ, p1 , . . . , pn , ω) as in Theorem 3.1. Recall that the marked points pi ∈ Σ correspond the multiple points
of the immersion Σ ⊂ M (the case (Σ, 0,
/ ω) is an embedding). Then to a positive mass representation
of U(M) a holomorphic vector bundle E of conformal blocks τ(Σ, p1 , . . . , pn ) over the coarse moduli
space Mg,n of complex structures on (Σ, p1 , . . . , pn ) will be assigned in manner that if 0 6= [B] ∈ HΣ,ω
is a state then its quasi-local energy-momentum 4-vector PC,ω,B gives rise to a conformal block in this
conformal field theory i.e., a section PΣ,ω,B of E . This section will be moreover (projectively) flat with
respect to the natural (projectively) flat connection ∇ on E (the Knizhnik–Zamolodchikov connection).
We begin with the following simple observation (an elementary version of Uhlenbeck’s singularity
removal theorem [13]).
Lemma 4.1. Take any compact Riemann surface C = C(Σ) with distinct marked points p1 , . . . , pn ∈ C
and a holomorphic unitary vector bundle F over C \ {p1 , . . . , pn }. Let s ∈ H 0 (C \ {p1 , . . . , pn }; O(F))
2 (C; O(F)) i.e., having locally finite energy over C.
be a holomorphic section with the property s ∈ Lloc
If s is singular in pi ∈ C then one can find a local gauge transformation about this point such that
the gauge transformed section extends holomorphically across it i.e., pointlike singularities of locally
finite energy meromorphic sections over C are removable. More precisely there exists a unique unitary
holomorphic vector bundle F 0 over C satisfying F 0 |C\{p1 ,...,pn } ∼
= F so that for any locally finite energy
0
0
section s ∈ H (C \ {p1 , . . . , pn }; O(F)) there exists a section s ∈ H 0 (C; O(F 0 )) so that s0 |C\{p1 ,...,pn } is
gauge equivalent to s.
Proof. First we prove the existence of the unique extendibility of the unitary bundle (F, h). Consider
a local holomorphic coordinate system (U, z) on C such that z(U) = D(0) ⊂ C some open disc about
the origin and U contains only one marked point pi ∈ U satisfying z(pi ) = 0. Cutting out the open
neighbourhood U ⊂ C of pi we obtain a manifold-with-boundary C \U and ∂ (C \U) ∼
= S1 . Consider the
restriction (F, h)|∂ (C\U) regarded as a smooth U(k)-bundle over S1 . Taking smooth local trivializations
the corresponding smooth local transition functions of (F, h)|∂ (C\U) give rise to a monodromy map
µ : S1 → U(k) where k = rk F. However π0 (U(k)) ∼
= 1 hence this monodromy map together with its
derivatives along S1 extends over pi as the identity consequently (F, h)|U\{pi } can be extended over this
point as a smooth unitary vector bundle (Fi0 , h0i )|U . Consider a smooth trivialization Fi0 |U ∼
= U × Ck and
write in this smooth gauge the restriction of the partial connection defining the holomorphic structure
on F as ∂ F |U\{pi } = ∂ + αU\{pi } . Then the Hermitian scalar product on Fi0 satisfies
∂ F (h0i |U\{pi } ) = ∂ (h0i |U\{pi } ) + αU\{pi } (h0i |U\{pi } ) = 0
0 := −(∂ h0 | )(h0 | )−1 on U defines a
and h0i |U\{pi } extends smoothly over pi as h0i |U . Therefore αi,U
iU
iU
0
smooth extension of αU\{pi } over pi in a manner that ∂ Fi0 |U := ∂ + αi,U is the restriction of a com-
G. Etesi: Gravity as algebraic quantum field theory
17
patible partial connection ∂ Fi0 yielding a compatible holomorphic structure on (Fi0 , h0i ). Performing
this procedure around every marked points we obtain a unique unitary holomorphic vector bundle i.e.,
(F 0 , h0 , ∂ F 0 ) with ∂ F 0 h0 = 0.
Now we come to the extendibility of sections. Compatibility provides us that in a local holomorphic
trivialization F 0 |U ∼
= U × Ck the coefficients of h0 |U are holomorphic functions. Performing a GL(k, C)valued holomorphic gauge transformation if necessary we can pass to a local holomorphic trivialization
in which h0 |U has the standard form. Take any holomorphic section of F or equivalently, a meromorphic
section of F 0 with singularities in the marked points i.e., pick any
s ∈ H 0 (C \ {p1 , . . . , pn }; O(F)) ∼
= H 0 (C \ {p1 , . . . , pn }; O(F 0 ))
with local shape s|U (z) = s1 (z) f1 + · · · + sk (z) fk in this local trivialization. Since s|U is holomorphic
outside 0 ∈ C each component s j : U → C admit Laurent expansions
+∞
s j (z) =
∑
amj zm , amj ∈ C.
m=−N j
Moreover the local L2 -norm of the section in this special gauge looks like
ks|U k2L2 (U)
1
=
2πi
Z 1
2
k
2
|s (z)| + · · · + |s (z)|
ω|U =
U
Z |s1 (z)|2 + · · · + |sk (z)|2 ϕU (z, z)dz ∧ dz
U
where ϕU is a smooth nowhere vanishing function on U. Assume that the section has locally finite
energy. On substituting the above expansions into this integral the finiteness then dictates to conclude
that N j = 0 for all j = 1, . . . , k and i = 1, . . . , n hence in fact s is holomorphic over the whole C as
claimed. 3
Now we turn to the construction of the relevant modular functor. Suppose that Σ ⊂ M is a compact
surface without boundary. Choose any complex structure C = C(Σ) on it and n distinct marked points
p1 , . . . , pn ∈ C. Let E := T M ⊗R C|C\{p1 ,...,pn } be a holomorphic unitary vector bundle over the punctured surface as in Theorem 3.1. Or rather more generally, if C = tiCi is an abstract compact nonpunctured Riemann surface with connected components Ci then let E 0 be a holomorphic unitary vector
bundle over C with rk(E 0 |Ci ) = 4, deg(E 0 |Ci ) = 0 and dimC H 0 (Ci ; O(E 0 |Ci )) = 4. Then in terms of the
restricted bundle E := E 0 |C\{p1 ,...,pn } our choice is as follows:
τ(Σ, p1 , . . . , pn ) :=

n
C
0

X ∈ H (C \ {p1 , . . . , pn }; O(E)) kXkL2
 Cl
loc


C
(C) < +∞
o
if (Σ, p1 , . . . , pn ) 6= 0;
/
(9)
if (Σ, p1 , . . . , pn ) = 0/
that is, this vector space is the underlying vector space of the complexified Clifford algebra of
0
Re(H (C \ {p1 , . . . , pn }; O(E))) , Re( · , · )L2 (C) ∼
= R4Euclid ,
the vector fields of M having locally finite energy and being holomorphic except in the marked points
when restricted to C.
G. Etesi: Gravity as algebraic quantum field theory
18
Lemma 4.2. Let (Σ, p1 , . . . , pn ) be a smooth surface with marked points and take a complex structure
C = C(Σ) rendering it as a Riemann surface with marked points (C, p1 , . . . , pn ). Also take the holomorphic unitary vector bundle E over C \ {p1 , . . . , pn } as before. Attach to every marked point pi ∈ C the
single label
ν := {a holomorphic section of E has a finite energy singularity in pi ∈ C}.
Then the assignment (8) with the choice (9) is a modular functor which is not normalized in the sense
that τ(S2 , 0)
/ = ClC (H 0 (CP1 ; O(E))) instead of τ(S2 , 0)
/ = C.
Moreover the vector spaces τ(Σ, p1 , . . . , pn ) fit together into a trivial holomorphic vector bundle E
over the coarse moduli space Mg,n of genus g Riemann surfaces with n marked points carrying a flat
connection ∇ (the Knizhnik–Zamolodchikov connection). The vector PC,ω,B ∈ H 0 (C; O(E)) is the value
at C ∈ Mg,n of a section PΣ,ω,B of this bundle over Mg,n satisfying ∇PΣ,ω,B = 0.
Proof. We check the three relevant axioms of [1, Definition 5.1.2]. First of all Lemma 4.1 yields that if
(Σ, p1 , . . . , pn ) 6= 0/ then
τ(Σ, p1 . . . , pn ) ∼
= ClC (H 0 (C; O(E 0 )))
consequently the vector spaces are finite dimensional. It also readily follows from (9) that
τ((Σ1 , p1 , . . . , pn ) t (Σ2 , q1 , . . . , qm )) ∼
= τ(Σ1 , p1 , . . . , pn ) ⊗C τ(Σ2 , q1 , . . . , qm )
as in [1, part (iii) of Definition 5.1.2]. The second axiom to check is the glueing axiom [1, part
(iv) of Definition 5.1.2]. Let γ ⊂ (Σ, p1 , . . . , pn ) be a closed curve without self-intersections. Cut
(Σ, p1 , . . . , pn ) along γ. The resulting surface has naturally the structure of a not necessarily connected
˜ p1 , . . . , pn , q1 , q2 ) where the two new marked points q1 , q2 come from the circle
punctured surface (Σ,
˜
γ. Putting E := E|C\({p1 ,...,pn }∪γ) into (9) by the aid of Lemma 4.1 we obtain that locally finite energy
˜ p1 , . . . , pn , q1 , q2 ) correspond to those on (C, p1 , . . . , pn ) consequently
meromorphic sections on (C,
˜ p1 , . . . , pn , q1 , q2 ) ∼
τ(Σ,
= τ(Σ, p1 , . . . , pn )
hence the glueing axiom indeed holds in a trivial way here.
The third axiom to check is the functorial behaviour under diffeomorphisms [1, part (ii) of Definition 5.1.2]. In turn this is equivalent to checking the existence of a Knizhnik–Zamolodchikov
connection. Let Mg,n be the coarse moduli space of connected non-singular Riemann surfaces of
genus g and n marked points. We take a complex vector bundle E over Mg,n whose fibers over
(C, p1 , . . . , pn ) ∈ Mg,n are the individual conformal blocks τ(Σ, p1 , . . . , pn ) constructed from the holomorphic bundle E over C \ {p1 , . . . , pn } or equivalently E 0 over C. Recall that M is acted upon by its
+
diffeomorphism group. Hence the subgroup Diff+
Σ (M) ⊂ Diff (M) consisting of Σ-preserving diffeomorphisms acts on the real smooth punctured surface such that it deforms its complex structure i.e.,
(Σ, p1 , . . . , pn ) and f (Σ, p1 , . . . , pn ) correspond in general to different points in Mg,n . This subgroup
also acts on C∞ (Σ; E 0 ) by pushforward. Consequently it transforms the subspaces τ(Σ, p1 , . . . , pn ) ∼
=
C
C ∞
0
0
0
Cl (H (C; O(E ))) ⊂ Cl (C (Σ; E )) giving rise to linear isomorphisms
τ(Σ, p1 , . . . , pn ) ∼
= τ( f (Σ, p1 , . . . , pn )) for all f ∈ Diff+
Σ (M).
These linear isomorphisms can be interpreted as parallel translations along E by a flat connection ∇
called the Knizhnik–Zamolodchikov connection. Note that since the representation of Diff+
Σ (M) on
C∞ (Σ; E 0 ) is not only projective but in fact a true representation, the resulting connection is not only
G. Etesi: Gravity as algebraic quantum field theory
19
projectively but truely flat on E . In particular the bundle E as a complex vector bundle is trivial over
Mg,n but is equipped with a holomorphic structure. Via Lemma 4.1 the holomorhic section PC,ω,B can
be regarded as a meromorphic one i.e., PC,ω,B ∈ H 0 (C \ {p1 , . . . , pn }; O(E)). Define a section PΣ,ω,B
of E on Mg,n by putting PΣ,ω,B (C) := PC,ω,B . It follows from the invariance of the definition (6) of the
quasi-local energy-momentum 4-vector
2
PC,ω,B ∈ H 0 (C \ {p1 , . . . , pn }; O(E)) ∩ Lloc
(C; O(E)) ∼
= H 0 (C; O(E 0 ))
⊂ ClC H 0 (C; O(E 0 )) = τ(Σ, p1 , . . . , pn )
under diffeomorphisms that as the complex structure varies PΣ,ω,B of E satisfies ∇PΣ,ω,B = 0 i.e., is
parallel for the Knizhnik–Zamolodchikov connection.
We conclude that the assignment (8) with (9) is a C-extended modular functor as in [1, Definition
5.1.2] i.e., a weakly conformal field theory a´ la G. Segal [9]. 3
After having constructed the modular functor, we find the vector space on which it acts hence exhibit the
conformal field theory given by (8) and (9). This step is very simple: the space (Σ, p1 , . . . , pn ) identified
with an oriented smooth cobordism between the disjoint compact oriented 1-manifolds S1p1 t · · · t S1pk
and S1pk+1 t · · · t S1pn . To the oriented 1-manifold S1p1 t · · · t S1pk t (S1pk+1 )∗ t · · · t (S1pn )∗ , regardless what
it actually is, we associate the finite dimensional vector space S ⊗C S∗ where S ∼
= C4 is the unique
irreducible Clifford-module. The resulting conformal field theory is a topological one because its state
space is finite dimensional and its correlation functions are insensitive for the marked points.
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