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An analytic regularisation scheme on curved
spacetimes with applications to cosmological
spacetimes
Antoine Géré
1,a
, ThomasPaul Hack
1,a
, Nicola Pinamonti
1,2,c
1
Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, I16146 Genova, Italy.
2
Istituto Nazionale di Fisica Nucleare, Sezione di Genova, Via Dodecaneso, 33 I16146 Genova, Italy.
EMail:
a
gere@dima.unige.it,
b
hack@dima.unige.it,
c
pinamont@dima.unige.it
November 13, 2015
Abstract.
We develop a renormalisation scheme for time–ordered products in interacting ﬁeld
theories on curved spacetimes which consists of an analytic regularisation of Feynman amplitudes and
a minimal subtraction of the resulting pole parts. This scheme is directly applicable to spacetimes
with Lorentzian signature, manifestly generally covariant, invariant under any spacetime isometries
present and constructed to all orders in perturbation theory. Moreover, the scheme captures correctly
the non–geometric state–dependent contribution of Feynman amplitudes and it is well–suited for
practical computations. To illustrate this last point, we compute explicit examples on a generic curved
spacetime, and demonstrate how momentum space computations in cosmological spacetimes can be
performed in our scheme. In this work, we discuss only scalar ﬁelds in four spacetime dimensions, but
we argue that the renormalisation scheme can be directly generalised to other spacetime dimensions
and ﬁeld theories with higher spin, as well as to theories with local gauge invariance.
Contents
1 Introduction 2
2 Introduction to pAQFT 3
2.1 Basic deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Relation to the standard formulation of perturbative QFT . . . . . . . . . . . . . . . . 5
3 Analytic regularisation and minimal subtraction on curved spacetimes 9
3.1 Analytic regularistion of time–ordered products and the minimal subtraction scheme . 10
3.2 Analytic regularisation of the Feynman propagator H
F
on curved spacetimes . . . . . 14
3.3 Generalised Euler operators and principal parts of homogeneous expansions . . . . . . 18
3.3.1
The diﬀerential form of generalised Euler operators and homogeneous expansions
of Feynman amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Properties of the minimal subtraction scheme . . . . . . . . . . . . . . . . . . . . . . . 23
3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5.1 Computation of the renormalised ﬁsh and sunset graphs in our scheme . . . . . 25
3.5.2 Alternative computation of the renormalised ﬁsh and sunset graphs . . . . . . 27
3.5.3 A more complicated graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Explicit computations in cosmological spacetimes 30
4.1 Propagators in Fourier space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 The renormalised ﬁsh and sunset graphs in Fourier space . . . . . . . . . . . . . . . . 32
4.3 Example: the twopoint function for a quartic potential up to second order . . . . . . 34
4.4 More complicated graphs on cosmological spacetimes . . . . . . . . . . . . . . . . . . . 35
5 Summary and outlook 35
1
arXiv:1505.00286v3 [mathph] 12 Nov 2015
A Conventions and computational details 36
A.1 Propagators of the free Klein–Gordon ﬁeld and their relations . . . . . . . . . . . . . . 36
A.2 Fourier transform of the logarithmic term on Minkowski spacetime . . . . . . . . . . . 37
1. Introduction
In the perturbative construction of models in quantum ﬁeld theory on curved spacetimes one encounters
time–ordered products of ﬁeld polynomials which are a priori ill–deﬁned due to the appearance of UV
divergences. Several renormalisation schemes which deal with these divergences in the presence of non–
trivial spacetime curvature have been discussed in the literature, such as for example local momentum
space methods [
Bu81
], dimensional regularisation in combination with heat kernel techniques [
Lü82
,
To82
], diﬀerential renormalisation [
CHL95
,
Pr97
], zeta–function renormalisation [
BF13
], generic
Epstein–Glaser renormalisation [
BF00
,
HW01
,
HW04
], and, on cosmological spacetimes, Mellin–Barnes
techniques [
Ho10
] and dimensional regularisation with respect to the comoving spatial coordinates
[BCK10].
Some of these schemes, such as heat kernel approaches, zeta–function techniques and local momen
tum space methods are based on constructions which are initially only well–deﬁned for spacetimes with
Euclidean signature. These constructions can be partly transported to general Lorentzian spacetimes
by local Wick–rotation techniques developed in [
Mo99
]. However, whereas the Feynman propagator
is essentially unique on Euclidean spacetimes, this is not the case on Lorentzian spacetimes where
this propagator has a non–unique contribution depending on the quantum state of the ﬁeld model.
Consequently, the Euclidean renormalisation techniques, and the numerous practical computations
already performed by means of these methods – see for example the monographs [
BD82
,
BOS92
,
PT09
]
– are able to capture the correct divergent and geometric parts of Feynman amplitudes, but a priori
not their non–geometric and state–dependent contributions.
A renormalisation scheme which is directly applicable to curved spacetimes with Lorentzian
signature has been developed in [
BF00
,
HW01
,
HW04
] in the framework of algebraic quantum ﬁeld
theory. This scheme implements ideas of [
EG73
] and [
St71
] and is based on microlocal techniques which
replace the momentum space methods available in Minkowski spacetime and have been introduced
to quantum ﬁeld theory in curved spacetime by the seminal work [
Ra96
]. However, although the
generalised Epstein–Glaser scheme developed in [
BF00
,
HW01
,
HW04
] is conceptually clear and
mathematically rigorous, it is not easily applicable in practical computations. On the other hand,
Lorentzian schemes which are better suited for this purpose have not been developed to all orders in
perturbation theory [
CHL95
,
Pr97
], are tailored to speciﬁc spacetimes [
Ho10
] or are not manifestly
covariant [BCK10].
Motivated by this, we develop a renormalisation scheme for time–ordered products in interacting
ﬁeld theories on curved spacetimes which is directly applicable to spacetimes with Lorentzian signature,
manifestly generally covariant, invariant under any spacetime isometries present and constructed
to all orders in perturbation theory. Moreover, the scheme captures correctly the non–geometric
state–dependent contribution of Feynman amplitudes and it is well–suited for practical computations.
In this work, we discuss only scalar ﬁelds in four spacetime dimensions, but we shall argue that the
renormalisation scheme can be directly generalised to other spacetime dimensions and ﬁeld theories
with higher spin, as well as to theories with local gauge invariance. Our analysis will take place in the
framework of perturbative algebraic quantum ﬁeld theory (pAQFT) [
BF00
,
HW01
,
HW04
,
BDF09
,
FR12
,
FR14
] which is a conceptually clear framework in which fundamental physical properties of
perturbative interacting models on curved spacetimes can be discussed. However, we will make an eﬀort
to review how the formulation of pAQFT is related to the more standard formulation of perturbative
QFT.
The renormalisation scheme we propose is inspired by the works [
Ke10
,
DFKR14
] which deal
with perturbative QFT in Minkowski spacetime. In these works, the authors introduce an analytic
regularisation of the position–space Feynman propagator in Minkowski spacetime which is similar to
the one discussed in [
BG72
]. Based on this, time–ordered products are constructed recursively by an
Epstein–Glaser type procedure and it is shown that this recursion can be resolved by a position–space
2
forest formula similar to the one of Zimmermann used in BPHZ renormalisation in momentum space.
In order to extend the scheme proposed in [
DFKR14
] to curved spacetimes, and motivated by
[
BG72
] and by the form of Feynman propagators on curved spacetimes, we introduce an analytic
regularisation ∆
(α)
F
of a Feynman propagator ∆
F
by
∆
(α)
F
:= lim
→0
+
1
8π
2
u
(σ + i)
1+α
+
v
α
1 −
1
(σ + i)
α
+ w ,
where
u
,
v
and
w
are the so–called Hadamard coeﬃcients and
σ
is 1
/
2 times the squared geodesic
distance. This analytic regularisation, namely the construction of certain distributions by means of
powers of the geodesic distance, is reminiscent of the use of Riesz distributions to deﬁne advanced and
retarded Greens functions on Minkowski spacetime. A careful discussion of Riesz distributions and
their extension to the curved case is presented in [
BGP07
]. The regularisation we use is loosely related
to dimensional regularisation because the leading singularity of a Feynman propagator in
N
spacetime
dimensions is proportional to (
σ
+
i
)
1−N/2
, see e.g. [
Mo03
, Appendix A]. A regularisation of the
Feynman propagator similar to the one above has recently been discussed in [
Da15
]. In this work, we
shall combine the analytic regularisation of the Feynman propagator with the minimal subtraction
scheme encoded in a forest formula of the kind discussed in [
Ho10
,
Ke10
,
DFKR14
] in order to obtain
a time–ordered product which satisﬁes the causal factorisation property, i.e. a product which is indeed
“time–ordered”. In order to prove that the analytically regularised amplitudes constructed out of ∆
(α)
F
have the meromorphic structure necessary for the application of the forest formula and in order to show
how the corresponding Laurent series can be computed explicitly, we shall make use of generalised
Euler operators. The practical feasibility of the renormalisation scheme shall be demonstrated by
computing a few examples.
A large part of our analysis is devoted to demonstrating that the scheme we propose is consistent
and welldeﬁned on general curved spacetimes. Readers interested in directly applying our scheme
may use Proposition 3.9 with
(3.22)
and
(3.23)
in order to compute the Laurentseries of the Feynman
amplitudes
(3.11)
regularised by means of
(3.12)
. The correct order of pole subtractions is encoded in
the forest formula
(3.9)
which is explained prior to its display. A number of examples is discussed in
Section 3.5.
In quantum ﬁeld theory on cosmological spacetimes, i.e. Friedmann–Lemaître–Robertson–Walker
(FLRW) spacetimes, one usually exploits the high symmetry of these spacetimes in order to evaluate
analytical expressions in spatial Fourier space. However, the renormalisation scheme discussed in
this work operates on quantities such as the geodesic distance and the Hadamard coeﬃcients, whose
explicit position space and momentum space forms are not even explicitly known in FLRW spacetimes.
Notwithstanding, we shall devote a large part of this work in order to develop simple methods to
evaluate quantities renormalised in our scheme on FLRW spacetimes in momentum space, and we
shall illustrate these methods by explicit examples.
The paper is organised as follows. In the next section we present a brief introduction to pAQFT
and its connection with the more standard formulation of perturbative QFT. Afterwards we introduce
the renormalisation scheme, demonstrate that it is well–deﬁned and analyse its properties in Section 3,
where we also illustrate the scheme by computing examples. In the fourth section we demonstrate
the applicability of the renormalisation scheme to momentum space computations on cosmological
spacetimes. Finally, a few conclusions are drawn in the last section of this paper. Conventions regarding
the various propagators of a scalar ﬁeld theory and a few technical computations are collected in the
appendix.
2. Introduction to pAQFT
2.1. Basic deﬁnitions
Throughout this work, we shall consider fourdimensional globally hyperbolic spactimes (
M, g
), where
g
is a Lorentzian metric whose signature is (
−,
+
,
+
,
+) and we use the sign conventions of [
Wa84
]
regarding the deﬁnitions of curvature tensors.
3
We recall the perturbative construction of an interacting quantum ﬁeld theory on a generic curved
spacetime in the framework of
perturbative algebraic quantum ﬁeld theory (pAQFT)
recently
developed in [
BDF09
,
FR12
,
FR14
] based on earlier work. In this construction, the basic object
of the theory is an algebra of observables which is realised as a suitable set of functionals on ﬁeld
conﬁgurations equipped with a suitable product. In order to implement the perturbative constructions
following the ideas of Bogoliubov and others, the
ﬁeld conﬁgurations φ
are assumed to be oﬀ–shell.
Namely,
φ ∈ E
(
M
) =
C
∞
(
M
) is a smooth function on the globally hyperbolic spacetime (
M, g
) and
observables are modelled by functionals
F
:
E
(
M
)
→ C
satisfying further properties. In particular all
the functional derivatives exist as distributions of compact support, where we recall that the functional
derivative of a functional F is deﬁned for all ψ
1
, . . . , ψ
n
∈ D(M) = C
∞
0
(M) as
F
(n)
(φ)(ψ
1
⊗ ··· ⊗ ψ
n
) :=
d
n
dλ
1
. . . dλ
n
F (φ + λ
1
ψ
1
+ . . . λ
n
ψ
n
)
λ
1
=···=λ
n
=0
∈ E
0
(M
n
).
The set of these functionals is indicated by
F
. Further regularity properties are assumed for the
construction of an algebraic product. In particular, the set of
local functionals F
loc
⊂ F
is formed
by the functionals whose
n
–th order functional derivatives are supported on the total diagonal
d
n
=
{
(
x, . . . , x
)
, x ∈ M} ⊂ M
n
. Furthermore, their singular directions are required to be orthogonal
to
d
n
, namely
WF
(
F
(n)
)
⊂ {
(
x, k
)
∈ T
∗
M
n
, x ∈ d
n
, k ⊥ T d
n
}
where
WF
denotes the wave front
set. A generic local functional is a polynomial
P
(
φ
)(
x
) in
φ
and its derivatives integrated against a
smooth and compactly supported tensor. The functionals whose functional derivatives are compactly
supported smooth functions are instead called regular functionals and indicated by F
reg
.
The quantum theory is speciﬁed once a product among elements of
F
loc
and a
∗−
operation (an
involution on
F
) are given. For the case of free (linear) theories the product can be explicitly given by
a –product
F
H
G =
X
n
~
n
n!
D
F
(n)
, H
⊗n
+
G
(n)
E
, (2.1)
where
H
+
is a Hadamard distribution of the linear theory we are going to quantize, namely a
distribution whose antisymmetric part is proportional to the commutator function ∆ = ∆
R
−
∆
A
and
whose wave front set satisﬁes the Hadamard condition, see e.g. [
Ra96
,
BFK96
] for further details and
Section A.1 for our propagator conventions. Owing to the properties of
H
+
, iterated
H
–products of
local functionals F
1
H
···
H
F
n
are well deﬁned and
H
is associative.
In a normal neighbourhood of (M, g), a Hadamard distribution H
+
is of the form
H
+
(x, y) =
1
8π
2
u(x, y)
σ
+
(x, y)
+ v(x, y) log(M
2
σ
+
(x, y))
+ w(x, y), (2.2)
where
σ
+
(
x, y
) =
σ
(
x, y
) +
i
(
t
(
x
)
− t
(
y
)) +
2
/
2 with
t
a timefunction, i.e. a global timecoordinate,
2
σ
(
x, y
) is the squared geodesic distance between
x
and
y
and
M
is an arbitrary mass scale. The
Hadamard coeﬃcients
u
and
v
are purely geometric and thus state–independent, whereas
w
is smooth
and state–dependent if H
+
(x, y) is the two–point function of a quantum state.
For the perturbative construction of interacting models we further need a
time–ordered product
·
T
H
on local functionals. This product is characterised by
symmetry
and the
causal factorisation
property, which requires that
F ·
T
H
G = F
H
G if F & G , (2.3)
where
F & G
indicates that
F
is later than
G
, i.e. there exists a Cauchy surface Σ of (
M, g
) such that
supp
(
F
)
⊂ J
+
(Σ) and
supp
(
G
)
⊂ J
−
(Σ). However, the causal factorisation ﬁxes uniquely only the
time–ordered products among regular functionals, in which case
F ·
T
H
G =
X
n
~
n
n!
D
F
(n)
, H
⊗n
F
G
(n)
E
, (2.4)
where
H
F
is the time–ordered (Feynman) version of
H
+
, i.e.
H
F
=
H
+
+
i
∆
A
with ∆
A
the advanced
propagator of the free theory, cf. Section A.1. For local functionals,
(2.4)
is only correct up to the
4
need to employ a non–unique renormalisation procedure, cf. Section 3.1. This renormalisation can
be performed in such a way that iterated
·
T
H
–products of local functionals
F
1
·
T
H
··· ·
T
H
F
n
are well
deﬁned with
·
T
H
being associative. Moreover,
H
–products of such time–ordered products of local
functionals are well–deﬁned as well, cf. [
HW02
,
BDF09
,
FR12
,
FR14
]. Consequently, we may consider
the algebra
A
0
H
–generated by iterated
·
T
H
–products of local functionals. This algebra contains all
observables of the free theory which are relevant for perturbation theory.
In the perturbative construction of interacting models, namely when the free action is perturbed by
a non–linear local functional
V
, the observables associated with the interacting theory are represented
on the free algebra
A
0
by means of the
Bogoliubov formula
. This is given in terms of the local
S–matrix, i.e., the time–ordered exponential
S(V ) =
∞
X
n=0
i
n
n!~
n
V ·
T
H
··· ·
T
H
V
 {z }
n times
, (2.5)
where
V
is the interacting Lagrangean. In particular, for every interacting observable
F
the corre
sponding representation on the free algebra A
0
is given by
R
V
(F ) = S
−1
(V )
H
(S(V ) ·
T
H
F ) , (2.6)
where
S
−1
(
V
) is the inverse of
S
(
V
) with respect to the
H
–product. The problem in using
R
V
(
F
)
as generators of the algebra of interacting observables lies in the construction of the time–ordered
product which a priori is an ill–deﬁned operation.
This problem can be solved using ideas which go back to Epstein and Glaser, see e.g. [
BF00
], by
means of which the time–ordered product among local functionals is constructed recursively. The
time–ordered products can be expanded in terms of distributions smeared with compactly supported
smooth functions which play the role of coupling constants (multiplied by a spacetime–cutoﬀ). At
each recursion step the causal factorisation property
(2.3)
permits to construct the distributions
deﬁning the time–ordered product up to the total diagonal. The extension to the total diagonal can be
performed extending the distributions previously obtained without altering the scaling degree towards
the diagonal. In this procedure there is the freedom of adding ﬁnite local contributions supported
on the total diagonal. This freedom is the well known renormalisation freedom. In addition to the
properties already discussed, the renormalised time–ordered product is required to satisfy further
physically reasonable conditions. We refer to [
HW02
,
HW04
] for details on these properties and the
proof that they can be implemented in the recursive Epstein–Glaser construction.
In spite of the theoretical clarity of this construction, the Epstein–Glaser renormalisation is quite
diﬃcult to implement in practise. The aim of this paper is to discuss a renormalisation scheme which
is suitable for practical computations.
2.2. Relation to the standard formulation of perturbative QFT
In this subsection we outline the relation of the pAQFT framework to the standard formulation of
perturbative QFT. As an example, we demonstrate how the twopoint (Wightman) function of the
interacting ﬁeld in
φ
4
theory on a four–dimensional curved spacetime is computed, where we assume
that the quantum state of the interacting ﬁeld is just the state of free ﬁeld modiﬁed by the interacting
dynamics. We further assume that the free ﬁeld is in a pure and Gaussian Hadamard state.
Let us recall the relevant formulae in perturbative algebraic quantum ﬁeld theory where we shall
always try to write expressions both in the pAQFT and in the more standard notation, indicating the
latter by a
.
=
. Given a local action
V
, such as
V
=
R
M
d
4
x
√
−g
λ
4
φ
(
x
)
4
in
φ
4
–theory, the corresponding
S
matrix, which is loosely speaking the “
S
matrix in the interaction picture”, is deﬁned by
(2.5)
and
corresponds to S(V )
.
= T e
i
~
V
.
The interacting ﬁeld, i.e. “the ﬁeld in the interaction picture”
φ
I
(
x
), is deﬁned by the Bogoliubov
formula
φ
I
(x) = R
V
(φ(x)) = S(V )
−1
H
(S(V ) ·
T
H
φ(x))
.
= T (e
i
~
V
)
−1
T (e
i
~
V
φ(x)) (2.7)
similarly to
(2.6)
, where by unitarity
S
(
V
)
−1
=
S
(
V
)
∗
. Interacting versions of more complicated
expressions in the ﬁeld, e.g. polynomials at diﬀerent and coinciding points, are deﬁned analogously. A
5
thorough discussion of the relation between the Bogoliubov formula and the more common formulation
of observables in the interaction picture may be found e.g. in [
Li13
, Section 3.1]. We only remark that,
in the Minkowski vacuum state Ω
0
, the expectation value of the Bogoliubov formula can be shown to
read (also for more general expressions in the ﬁeld)
hφ
I
(x)i
Ω
0
.
=
D
T (e
i
~
V
)
−1
T (e
i
~
V
φ(x))
E
Ω
0
=
D
T (e
i
~
V
φ(x))
E
Ω
0
D
T (e
i
~
V
)
E
Ω
0
,
which is the theorem of Gell–Mann and Low, see [Du96, DF00] for details.
In the algebraic formulation one usually cuts oﬀ the interaction in order to avoid infrared problems
by replacing
λ → λf
(
x
) with a compactly supported smooth function
f
and considers the adiabatic
limit
f →
1 in the end when computing expectation values. As our aim is to compute expectation
values in this section, we shall write the results in the adiabatic limit keeping in mind that proving the
absence of infrared problems, i.e. the convergence of the spacetime integrals, is non–trivial and may
depend on the state of the free ﬁeld chosen. Note that the socalled “in–in–formalism” often used in
perturbative QFT on cosmological spacetimes corresponds to considering a cutoﬀ function
f
of the
form
f
(
t, x
) = Θ(
t −t
0
), i.e.
f
is a step function in time and the parameter
t
0
corresponds to the time
where the interaction is switched on.
Our choice for the quantum state Ω of the interacting ﬁeld implies that e.g. the interacting
twopoint function
hφ
I
(x)
H
φ
I
(y)i
Ω
.
= hφ
I
(x)φ
I
(y)i
Ω
is computed by writing
φ
I
in terms of the free ﬁeld
φ
and computing the expectation value of the
resulting observable of the free ﬁeld in the chosen pure, Gaussian, Hadamard state of the free ﬁeld which
we may thus denote by the same symbol Ω. The interacting vacuum state in Minkowski spacetime is
of this form, whereas interacting thermal states in ﬂat spacetime do not belong to this class, as they
roughly speaking require to take into account both the change of dynamics and the change of spectral
properties induced by V [FL13].
The functionals in the functional picture of pAQFT correspond to Wick–ordered quantities of
the free ﬁeld in the sense we shall explain now. To this avail we recall the form of the (quantum)
H
–product and (time–ordered)
·
T
H
–product in
(2.1)
and
(2.4)
which are deﬁned by means of a
Hadamard distribution
H
+
and its Feynman–version
H
F
=
H
+
+
i
∆
A
. Up to renormalisation of the
time–ordered product, these products computed for the special case of the functional φ
2
(x) give
φ(x)
2
H
φ(y)
2
= φ(x)
2
φ(y)
2
+ 4~φ(x)φ(y)H
+
(x, y) + 2~
2
H
2
+
(x, y) ,
φ(x)
2
·
T
H
φ(y)
2
= φ(x)
2
φ(y)
2
+ 4~φ(x)φ(y)H
F
(x, y) + 2~
2
H
2
F
(x, y) .
This example shows that the
H
–product (
·
T
H
–product) implements the Wick theorem for normal–
ordered (time–ordered) ﬁelds, and thus the previous formulae can be interpreted in more standard
notation as
:φ(x)
2
:
H
:φ(y)
2
:
H
=:φ(x)
2
φ(y)
2
:
H
+4~ : φ(x)φ(y):
H
H
+
(x, y) + 2~
2
H
2
+
(x, y) ,
T
:φ(x)
2
:
H
:φ(y)
2
:
H
=:φ(x)
2
φ(y)
2
:
H
+4~ : φ(x)φ(y):
H
H
F
(x, y) + 2~
2
H
2
F
(x, y) ,
where
:A:
H
:= α
−H
+
(A) := e
−~
H
S
(x,y),
δ
δφ(x)
⊗
δ
δφ(y)
A , (2.8)
H
S
(x, y) :=
1
2
(H
+
(x, y) + H
+
(y, x)) ,
e.g.
:φ(x)
2
:
H
= lim
x→y
(φ(x)φ(y) − H
+
(x, y)) .
(2.8)
is a convenient way to encode the combinatorics of normal ordering whereby the exponential
series terminates for polynomial functionals such as A = φ(x)
2
.
6
The Wick theorem relates (time–ordered) products of Wick–ordered quantities to sums of Wick–
ordered versions of contracted products, where the deﬁnition of “Wick–ordering” and “contraction”
are directly related, they both depend on the Hadamard distribution
H
+
chosen. Thus, if we choose a
particular
H
+
to deﬁne
H
and
·
T
H
in pAQFT, we immediately ﬁx the interpretation of all functionals
in terms of expressions Wick–ordered with respect to H
+
.
For the algebraic formulation the choice of
H
+
is not important, indeed choosing a diﬀerent
H
0
+
with the same properties, one has that
w
:=
H
0
+
− H
+
=
H
0
F
− H
F
, because the advanced propagator
∆
A
is unique and thus universal. Moreover, w is real, smooth and symmetric and
A
H
0
B = α
w
(α
−w
(A)
H
α
−w
(B)) , A ·
T
H
0
B = α
w
(α
−w
(A) ·
T
H
α
−w
(B)) ,
with
α
deﬁned as in
(2.8)
and thus the algebras associated to
H
,
·
T
H
and
H
0
,
·
T
H
0
are isomorphic via
α
w
: A
0
→ A
0
0
,
where we recall that A
0
is algebra
H
–generated by ·
T
H
–products of local functionals.
Hence, one may choose a suitable
H
+
according to ones needs. However, since
α
d
(
A
)
6
=
A
for
functionals containing multiple ﬁeld powers, statements like “the potential is
φ
4
” are ambiguous in
pAQFT, and in fact also in the standard treatment of QFT. They become non–ambiguous only if one
says “the potential is :
φ
4
:
H
, i.e.
φ
4
Wick–ordered with respect to
H
+
”. In pAQFT the corresponding
non–ambiguous statement would be “the potential is the functional
φ
4
in the algebra
A
0
constructed
by means of
H
+
”. If one then passes to the algebra
A
0
0
constructed by means of
H
0
+
, the potential picks
up quadratic and c–number terms as we shall compute explicitly below. Alternatively, this ambiguity
may be seen to correspond to the renormalisation ambiguity of tadpoles in Feynman diagrams.
Given a Gaussian and Hadamard free ﬁeld state Ω, a convenient choice or representation of the
algebra is to take
H
+
= ∆
+
, where ∆
+
(
x, y
) =
hφ
(
x
)
∆
φ
(
y
)
i
Ω
.
= hφ
(
x
)
φ
(
y
)
i
Ω
is the twopoint function
of the free ﬁeld in the state Ω. This corresponds to standard normal–ordering and consequently in this
representation the expectation values of all expressions which contain nontrivial powers of the ﬁeld
vanish, i.e.
hAi
Ω
= A
φ=0
.
= h:A:
∆
i
Ω
. (2.9)
Keeping the state Ω ﬁxed, but passing on to a representation of the algebra with arbitrary
H
+
, the
expectation value is computed as
hAi
Ω
= α
w
(A)
φ=0
.
= h:A:
H
i
Ω
, w = ∆
+
− H
+
,
for instance
hφ
2
(x)i
Ω
= α
w
(φ
2
(x))
φ=0
=
φ
2
(x) + w(x, x)

φ=0
= w(x, x)
.
= h:φ
2
(x):
H
i
Ω
,
which in more standard terms would be computed as
h:φ
2
(x):
H
i
Ω
= lim
x→y
hφ(x)φ(y) − H
+
(x, y)i
Ω
= lim
x→y
(∆
+
(x, y) − H
+
(x, y)) = w(x, x) .
In QFT in curved spacetimes normal–ordering is in principle problematic, because (pointlike)
observables should be deﬁned in a local and generally covariant way, i.e. they should only depend on
the spacetime in an arbitrarily small neighbourhood of the observable localisation [
BFV01
,
HW01
].
This is not satisﬁed for e.g. ﬁeld polynomials Wick–ordered with ∆
+
(
x, y
), because this distribution
satisﬁes the KleinGordon equation and thus it encodes non–local information on the curved spacetime
[
HW01
]. It is still possible to compute in the convenient normal–ordered representation in the following
way. In the example of
φ
4
–theory, one deﬁnes the potential
λ
4
φ
(
x
)
4
as a local and covariant observable
by identifying it with the corresponding monomial in a representation of the algebra furnished by a
purely geometric H
+
, i.e. a H
+
of the form (2.2) with w = 0.
In other words, we set once and for all in the H
+
–representation
V
H
=
Z
M
d
4
x
√
−g
λ
4
φ(x)
4
.
=
Z
M
d
4
x
√
−g
λ
4
:φ(x)
4
:
H
.
7
This does not ﬁx
V
uniquely, because
H
depends on the scale
M
inside of the logarithm, but the
freedom in deﬁning
V
H
, and analogously the free/quadratic part of Klein–Gordon action, as above
corresponds to the usual freedom in choosing the “bare mass”
m
, “bare coupling to the scalar curvature”
ξ
, “bare cosmological constant” Λ, “bare Newton constant”
G
, as well as the “bare coeﬃcients”
β
1
,
β
2
of higher–derivative gravitational terms in the extended Einstein–Hilbert–Klein–Gordon action
S(φ, g
ab
) =
Z
M
d
4
x
√
−g
R − 2Λ
16πG
+ β
1
R
2
+ β
2
R
ab
R
ab
−
(∇φ
2
)
2
−
(m
2
+ ξR)φ
2
2
−
λ
4
φ
4
.
In order to switch to the normalordered representation, we use the map
α
w
deﬁned in
(2.8)
where
w
= ∆
+
−H
+
is the state–dependent part of the Hadamard distribution ∆
+
whose dependence on the
choice of
M
in
H
+
corresponds to the above–mentioned freedom in the deﬁnition of the Wick–ordered
Klein–Gordon action. That is, we have in the normal–ordered representation in the state Ω
V := V
∆
= α
w
(V
H
) =
Z
M
d
4
x
√
−g
λ
4
φ(x)
4
+
3λ
2
w(x, x)φ(x)
2
+
3λ
4
w(x, x)
2
(2.10)
.
=
Z
M
d
4
x
√
−g
λ
4
:φ(x)
4
:
∆
+
3λ
2
w(x, x) :φ(x)
2
:
∆
+
3λ
4
w(x, x)
2
We observe that the combination of the requirements that the interaction potential is a local and
covariant observable and that, in order to compute expectation values in the state Ω, one would like to
compute in the convenient normal–ordered representation with respect to Ω, leads to the introduction
of an eﬀective spacetime–dependent and state–dependent (squared) mass term
µ
(
x
) = 3
λw
(
x, x
) in the
interaction potential which of course leads to additional Feynman graphs in perturbation theory, cf.
Figures 1 and 2. The ﬁeld–independent term
3λ
4
w
(
x, x
)
2
plays no role for computations of quantities
which do not involve functional derivatives of the extended Einstein–Hilbert–Klein–Gordon action
with respect to the metric (an example where it does play a role is the stress–energy tensor), just as
the modiﬁcation of the free action by the change of representation plays no role for the computation
of such quantities. A similar phenomenon as in
(2.10)
occurs in thermal quantum ﬁeld theory on
Minkowski spacetime, where the eﬀective mass generated by changing from the normal–ordered picture
with respect to the free vacuum state to the normal–ordered picture with respect to the free thermal
state is termed “thermal mass”, cf. [Li13, Section 2.3.2.] for details.
After these general considerations, we can proceed to compute as an example the twopoint function
of the interacting ﬁeld
φ
I
in
φ
4
up to second order in
λ
, whereby
φ
I
is assumed to be in a state induced
by a Gaussian Hadamard state of the free ﬁeld. To this avail, we shall exclusively compute in the
associated normal–ordered representation and thus omit the subscripts on the star product, and the
timeordered product, :=
∆
, ·
T
:= ·
T
∆
.
We start from the Bogoliubov formula (2.7) and compute (from now on ~ = 1)
S(V ) = 1 + iV −
1
2
V ·
T
V + O(λ
3
)
S(V )
?−1
= 1 − iV +
1
2
V ·
T
V − V V + O(λ
3
)
φ
I
= φ − iV φ + iV ·
T
φ +
1
2
(V ·
T
V ) φ − V V φ −
1
2
V ·
T
V ·
T
φ + V (V ·
T
φ) + O(λ
3
) .
It remains to compute the
product of
φ
I
(
x
) and
φ
I
(
y
) and to set
φ
= 0 in the remaining expression
in order to obtain the expectation value in the state Ω. The result can as always be conveniently
expressed in terms of Feynman diagrams, where we use the Feynman rules depicted in Figure 1.
In the computation of
hφ
I
(
x
)
φ
I
(
y
)
i
Ω
, many expressions can be shortened considerably by using
the relation ∆
F
−
∆
+
=
i
∆
A
, in particular this holds for the external legs of the appearing Feynman
diagrams. The resulting Feynman diagrams are depicted in Figure 2.
8
Figure 1. The various propagators and vertices in φ
4
–theory, where µ(x) = 3λw(x, x).
Figure 2.
The up–to–second–order contributions to the two–point (Wightman) function
hφ
I
(
x
)
φ
I
(
y
)
i
Ω
of the interacting ﬁeld with potential
λ
4
φ
(
x
)
4
+
µ(x)
2
φ
(
x
)
2
. We omit the labels of the external vertices
after the ﬁrst line using the convention that the left external vertex is always the xvertex.
3. Analytic regularisation and minimal subtraction on curved spacetimes
As discussed above, the main problem in using the Bogoliubov formula (2.6)
R
V
(F ) =
~
i
d
dλ
S(V )
−1
H
S(V + λF )
λ=0
for constructing interacting ﬁelds perturbatively is that it is given in terms of the
S
–matrix, which is
the time–ordered exponential
(2.5)
. Unfortunately, the time–ordered product deﬁned in terms of a
“deformation”
(2.4)
written by means of a Feynman propagator
H
F
is well deﬁned only on regular
functionals because the singularities present in
H
F
forbid their application to more general functionals.
In order to proceed there is the need of employing a renormalisation procedure to construct the
time–ordered products. In this work we discuss the use of certain analytic methods to solve this
problem. The procedure we shall pursue is the following. We deform the Feynman propagator by
means of complex parameter
α
with values in the neighbourhood of the origin obtaining a function
with distributional values
α 7→ H
(α)
F
. The deformation we are looking for needs to be such that in the
limit
α →
0 we recover the ordinary Feynman propagator. Furthermore, when
α
is non–vanishing, but
suﬃciently small, pointwise powers of
H
(α)
F
and integral kernels of more complicated loop diagrams
should be well–deﬁned. If this is the case, since the corresponding distributions obtained in the limit
α →
0 are well deﬁned outside of the total diagonal, the poles of
α 7→ H
(α)
F
and more complicated loop
expressions are supported on the total diagonal. The idea, similar to what happens in dimensional
regularisation, is that it is possible to renormalise these distributions by simply removing the poles.
9
3.1. Analytic regularistion of time–ordered products and the minimal subtraction scheme
In order to discuss the analytic regularisation of time–ordered products, we employ the notation used
e.g. in [
DFKR14
] which eﬃciently encodes the full combinatorics of Feynman diagrams in a compact
form. Namely, the time–ordered product of
n
local functionals
V
1
, . . . , V
n
can be formally deﬁned in
the following way
1
V
1
·
T
H
··· ·
T
H
V
n
:= T
n
(V
1
⊗ ··· ⊗ V
n
) := m ◦ T
n
(V
1
⊗ ··· ⊗ V
n
) , (3.1)
where
m
denotes the pointwise product
m
(
F
1
⊗ ··· ⊗ F
n
)(
φ
) =
F
1
(
φ
)
. . . F
n
(
φ
) and the operator
T
n
is
written in terms of an exponential
T
n
= exp
X
1≤i<j≤n
∆
ij
=
Y
1≤i<j≤n
∞
X
l
ij
≥0
∆
l
ij
ij
l
ij
!
(3.2)
with
∆
ij
:=
H
F
,
δ
2
δφ
i
δφ
j
. (3.3)
Here the functional derivative
δ
δφ
i
acts on the
i−
th element of the tensor product
V
1
⊗ ··· ⊗ V
n
and
H
F
=
H
+
+
i
∆
A
is the time–ordered version of the Hadamard distribution
H
+
entering the
construction of the free algebra
A
0
via
H
. The exponential
(3.2)
admits the usual representation in
terms of Feynman graphs. More precisely, it can be written as a sum over all graphs Γ in
G
n
, the set of
all graphs with vertices
V
(Γ) =
{
1
, . . . , n}
and
l
ij
edges
e ∈ E
(Γ) joining the vertices
i, j
. Furthermore,
in this construction, there are no tadpoles
l
ii
= 0 (cf. Section 2.2 for details on why these are absent)
and the edges are not oriented l
ij
= l
ji
. With this in mind
T
n
=
X
Γ∈G
n
1
N(Γ)
*
τ
Γ
,
δ
2E(Γ)
Q
i∈V (Γ)
Q
E(Γ)3e⊃i
δφ
i
(x
i
)
+
, (3.4)
where
N
(Γ) =
Q
i<j
l
ij
! is a numerical factor counting the possible permutations among the lines
joining the same two vertices, the second product
Q
e⊃i
is over the edges having
i
as a vertex and
x
i
is a point in
M
corresponding to the vertex
i
. Moreover,
τ
Γ
is a distribution which is well–deﬁned
outside of all partial diagonals, namely on M
n
\ D
n
, where
D
n
:= {x
1
, . . . , x
n
x
i
= x
j
for at least one pair (i, j), i 6= j} (3.5)
and τ
Γ
has the form
τ
Γ
=
Y
e=(i,j)∈E(Γ)
H
F
(x
i
, x
j
) =
Y
1≤i<j≤n
H
F
(x
i
, x
j
)
l
ij
. (3.6)
The a priori restricted domain of
τ
Γ
is the reason why
T
n
deﬁned as above is not a well–deﬁned
operation on F
⊗n
loc
. In this context we recall that the total diagonal d
n
⊂ D
n
is deﬁned as
d
n
:= {(x, . . . , x), x ∈ M} ⊂ M
n
. (3.7)
In order to complete the construction we need to extend the obtained distributions to the diagonals
D
n
. This is not a straightforward limit because the singular structure of the Feynman propagator
H
F
contains the one of the
δ
–distribution and because pointwise products of the latter distribution are
ill–deﬁned. Consequently, a renormalisation procedure needs to be implemented in order to extend
τ
Γ
to the full M
n
. This extension is in general not unique, but subject to renormalisation freedom.
1
In fact, in view of locality and covariance a better deﬁnition of the time–ordered product is
T
1
(
V
1
)
·
T
H
· · ··
T
H
T
1
(
V
n
) :=
T
n
(
V
1
⊗· · ·⊗V
n
) where
T
1
:
F
loc
→ F
loc
⊂ A
0
plays the role of identifying local and covariant (smeared) Wick polynomials
as particular elements of the free algebra
A
0
, cf. [
HW04
]. As we shall not touch upon this point in our renormalisation
scheme, we choose to omit T
1
in our formulas for simplicity.
10
Here we shall discuss a procedure to extend the distributions
τ
Γ
to
D
n
called
minimal subtraction
(MS)
, which makes use of an analytic regularisation ∆
α
ij
ij
of ∆
ij
given in terms of a family of
deformations
H
α
ij
F
of the Feynman propagator
H
F
parametrised by complex parameters
α
ij
contained
in some neighbourhood of 0
∈ C
. To this end, we follow [
DFKR14
] and call
t
(α)
an analytic
regularisation of a distribution
t
deﬁned outside of a point
x
0
∈ M
if for all
f ∈ D
(
M
)
ht
(α)
, fi
is a
meromorphic function in
α
for
α
in some neighbourhood of 0 which is analytic for
α 6
= 0. Moreover
t
(α)
may be extended to x
0
for α 6= 0 whereas lim
α→0
t
(α)
= t on M \ {x
0
}.
We shall introduce an analytic regularisation of the Feynman propagator
H
F
in the following
section, but the basic idea of the MS–scheme is independent of the details of the analytic regularisation.
Namely, given any analytic regularisation
H
(α)
F
of
H
F
, we repeat the formal construction of
T
n
presented above by replacing H
F
by H
(α)
F
in (3.3) and ∆
ij
by the induced ∆
α
ij
ij
in (3.2). Proceeding
in this way we deﬁne
T
(α)
n
:= e
P
i<j
∆
α
ij
ij
with α := {α
ij
}
i<j
,
and the corresponding integral kernels
τ
(α)
Γ
of Feynman graphs Γ in analogy to
(3.4)
. We expect that
the distributions
τ
(α)
Γ
are multivariate meromorphic functions which have poles at the origin for some
of the
α
ij
. Hence, in order to obtain well–deﬁned distributions in the limit
α
ij
to 0 and consequently
a renormalised time–ordered product ·
T
H
, all these poles need to be subtracted.
The properties of the analytically regularised Feynman propagator imply that
τ
(α)
Γ
is well–deﬁned
on
M
n
\ D
n
(3.5)
even if all
α
ij
are vanishing. Since
τ
(α)
Γ
is a multivariate meromorphic function
in
α
which is analytic if restricted to
M
n
\ D
n
, we may deduce that the principal part of
τ
(α)
Γ
for
some
α
ij
must be supported on a partial diagonal of
M
n
. In fact, in order for the time–ordered
products to fulﬁl the factorisation property
(2.3)
, the subtraction of the principal parts of
τ
(α)
Γ
needs
to be done in such a way that at each step only local terms are subtracted. However, the previous
discussion only implies that the support of the principal parts is contained in
D
n
, i.e. the union of
all the partial diagonals in
M
n
. In order to satisfy the causal factorisation property, the principal
parts need to be removed in a recursive way starting from the partial diagonals corresponding to two
vertices and proceeding with the partial diagonals corresponding to an increasing number
m ≤ n
of
vertices d
I
:= {(x
1
, . . . , x
n
) ∈ M
n
, x
i
= x
j
, i, j ∈ I ⊂ {1, . . . , n}, I = m}.
The correct recursion procedure is implemented by the so called Epstein–Glaser forest formula,
which is a position–space analogue of the Zimmermann forest formula, see [
Ho10
,
Ke10
,
DFKR14
] for
a careful analysis of the subject. We shall here follow the treatment discussed in [
DFKR14
]. To this
end, we consider the set of indices n := {1, . . . , n} and deﬁne a forest F as
F = {I
1
, . . . , I
k
}, I
j
⊂ n and I
j
 ≥ 2 ,
where for every pair I
i
, I
j
∈ F
I
i
∩ I
j
= ∅ or I
i
⊂ I
j
or I
j
⊂ I
j
.
The set of all forests of n indices together with the empty forest {} is indicated by F
n
.
For every subset
I ⊂ n
we indicate by
R
I
the operator which extracts the principal part with
respect to
α
I
of a multivariate meromorphic function
f
(
{α
ij
}
i<j
), where for every
i, j ∈ I
,
α
ij
=
α
I
,
and multiplies it with −1:
R
I
f := −pp lim
α
ij
→α
I
∀i,j∈I
f({α
ij
}
i<j
). (3.8)
We complement this deﬁnition by setting R
{}
to be the identity.
Given all these data, we deﬁne the renormalised time–ordered product in the MS–scheme as in e.g.
[DFKR14, Theorem 3.1] by
T
n
= (T
n
)
ms
:= lim
α→0
m ◦
X
F ∈F
n
Y
I∈F
R
I
◦ T
(α)
n
, (3.9)
11
where, in the product over
I ∈ F
,
R
I
appears before
R
J
if
I ⊂ J
. Furthermore, for each graph Γ, the
limit
α
=
{α
ij
}
i<j
→
0 is computed by setting
α
ij
=
α
Γ
for every
i < j
before taking the sum over
the forests and ﬁnally considering the limit α
Γ
to 0. In this context we recall that, for every element
of the sum over F
n
, part of the limit α
ij
→ α
Γ
is already taken by applying R
I
, see (3.8).
Given the renormalised
T
n
in the MS–scheme, the corresponding local
S
–matrix may be constructed
as
S(V ) =
∞
X
n=0
i
n
~
n
n!
T
n
(V ⊗ ··· ⊗ V )
for any local interaction Lagrangean V .
In order to implement the minimal subtraction scheme as outlined above we ﬁrst need to specify an
analytic regularisation
H
(α)
F
of the Feynman propagator
H
F
on generic curved spacetimes. Afterwards
we have to demonstrate that for all graphs Γ ∈ G
n
the analytically regularised integral kernels
τ
(α)
Γ
=
Y
e=(i,j)∈Γ
H
α
ij
F
(x
i
, x
j
) =
Y
1≤i<j≤n
H
α
ij
F
(x
i
, x
j
)
l
ij
. (3.10)
appearing in
T
(α)
n
=
X
Γ∈G
n
1
N(Γ)
*
τ
(α)
Γ
,
δ
2E(Γ)
Q
i∈V (Γ)
Q
e⊃i
δφ
i
(x
i
)
+
(3.11)
satisfy the properties necessary for the implementation of the MS–scheme. In particular we need to
demonstrate that the distribution
τ
(α)
Γ
, which is a priori deﬁned only on
M
n
\ D
n
, can be uniquely
extended to the full
M
n
without renormalisation, where the uniqueness of this extension is important
in order to obtain a deﬁnite renormalisation scheme. Moreover, we need to show that this distribution
τ
(α)
Γ
∈ D
0
(
M
n
) is weakly meromorphic in
α
in a neighbourhood of 0, where in view of the forest
formula it is only necessary to show that, setting
α
ij
=
α
I
for all
i, j ∈ I
,
τ
(α)
Γ
is weakly meromorphic
in
α
I
. Additionally, we need to prove that, if
τ
Γ
prior to regularisation is well–deﬁned outside of
the partial diagonal
d
I
, then the pole of
τ
(α)
Γ
with
α
ij
=
α
I
for all
i, j ∈ I
in
α
I
is supported on
d
I
and thus local. Finally, we need to prove that our MS–scheme satisﬁes all properties given in
[
HW02
,
HW04
] which a physically meaningful renormalisation scheme on curved spacetimes should
satisfy, and we need to provide means to explicitly compute the minimal subtraction, which after all is
the main motivation for this work.
Our plan to construct the mentioned quantities and to prove their required properties is as follows.
a)
In Section 3.2 we construct an analytic regularisation
H
(α)
F
of the Feynman propagator based
on the observation that locally
H
F
is of the form
(2.2)
up to considering instead of
σ
+
the half
squared geodesic with the Feynman
–prescription
σ
F
:=
σ
+
i
. Motivated by the fact that the
singular structure of H
F
originates from the form in which σ
F
appears, we set locally
H
(α)
F
:= lim
→0
+
1
8π
2
u
M
2α
σ
1+α
F
+
v
α
1 −
1
M
2α
σ
α
F
+ w, (3.12)
where we use the (arbitrary but ﬁxed) mass scale
M
present in
(2.2)
also for preserving the mass
dimension of H
F
in the regularisation.
b) In Proposition 3.7 we then prove that the relevant distributions
t
(α)
Γ
:=
Y
1≤i<j≤n
1
σ
l
ij
(1+α
ij
)
F
∈ D
0
(M
n
\ D
n
) (3.13)
are multivariate analytic functions. The distribution
(3.13)
only displays the most singular
contribution of
τ
(α)
Γ
(3.10)
, but the subleading contributions are clearly of the same form up to
replacing some of the factors (1 + α
ij
) in the exponents by α
ij
or 0.
12
c)
In order to show that
t
(α)
Γ
can be uniquely extended from
M
n
\D
n
to
M
n
in a weakly meromorphic
fashion, i.e. that the singularities relevant for the forest formula are poles of ﬁnite order, we follow
a strategy similar to the one used in [
HW02
] and consider a scaling expansion with respect to a
suitable scaling transformation. We ﬁrst argue in Proposition 3.8 that an analytically regularised
distribution
t
(α)
∈ D
0
(
M
n
\ d
n
), which can be written as a sum of homogeneous terms with
respect to this scaling transformation plus a suﬃciently regular remainder, can be extended to
M
n
in a weakly meromorphic way, were the uniqueness of the extension follows from its weak
meromorphicity. In Proposition 3.10, we give a suﬃcient condition for the existence of such
a homogeneous expansion and we demonstrate in Proposition 3.12 that the distributions
t
(α)
Γ
satisfy this condition.
d)
The above–mentioned results are proved by means of generalised Euler operators (see [
Da13
]
for a related concept) which can be written abstractly in terms of a scaling transformation, but
also in terms of covariant diﬀerential operators whose explicit form can be straightforwardly
computed as we argue in Section 3.3.1. In Proposition 3.9 we use these operators in order to
demonstrate how the full relevant pole structure of
t
(α)
Γ
can be computed, thus showing the
practical feasibility of the MS–scheme. We ﬁnd that our renormalisation scheme corresponds in
fact to a particular form of diﬀerential renormalisation and expand on this by computing a few
examples in Section 3.5.
e)
Finally, in Proposition 3.14 we prove that the MS–scheme satisﬁes the axioms of [
HW02
,
HW04
]
for time–ordered products and in addition preserves invariance under any spacetime isometries
present.
Remark 3.1.
The local Hadamard expansion
(2.2)
of
H
F
and correspondingly the analytically
continued
H
(α)
F
deﬁned in
(3.12)
are only meaningful on normal neighbourhoods
N
of (
M, g
). In order
to deﬁne
H
(α)
F
and the induced distributions
τ
(α)
Γ
(3.10)
globally, we may employ suitable partitions of
unity. Rather than providing general and cumbersome formulas, we prefer to illustrate the idea at the
example of the triangular graph
τ
Γ
= H
F,13
H
F,23
H
2
F,12
. := H
F
(x
1
, x
3
)H
F
(x
2
, x
3
)H
F
(x
1
, x
2
)
2
the renormalisation of which is discussed in detail in Section 3.5.3.
We recall that a corollary of Lemma 10 of Chapter 5 in [
O’N83
] guarantees that there exists a
covering
C
of
M
consisting of open geodesically convex sets such that
N
i
∩ N
j
is geodesically convex
for every N
i
, N
j
∈ C
2
. With this C at our disposal, we deﬁne the sets
N
12
:=
[
N∈C
N × N ⊂ M
2
, N
123
:=
[
N∈C
N × N × N ⊂ M
3
.
We call sets of the form
N
12
and
N
123
a
normal neighbourhood of the total diagonal
. This
deﬁnition is essentially motivated by the fact that for every
x ∈ M
we can ﬁnd a normal neighborhood
N
x
∈ C
of
x
in
M
. The squared geodesic distance
σ
is then well deﬁned on
N
12
, whereas the same is
in general not true if we replace
C
in the previous formula with a covering of
M
formed by all open
geodesically convex sets.
Setting
σ
ij
:=
σ
(
x
i
, x
j
), we observe that
σ
12
is well–deﬁned on
N
12
, and that
σ
12
,
σ
13
and
σ
23
are well–deﬁned on
N
123
. We now consider smooth and compactly supported functions
χ
12
∈ D
(
N
12
),
χ
123
∈ D
(
N
123
) which are such that
χ
12
= 1 on
d
2
⊂ N
12
and
χ
123
= 1 on
d
3
⊂ N
123
. Note that
by construction
χ
12
and
χ
123
vanish outside of
N
12
and
N
123
respectively. We may now deﬁne the
analytically regularised distribution τ
(α)
Γ
by setting
τ
(α)
Γ
:= H
(α
13
)
F,13
H
(α
23
)
F,23
H
(α
12
)
F,12
2
χ
12
χ
123
+ H
F,13
H
F,23
H
2
F,12
(1 − χ
12
)
+ H
F,13
H
F,23
H
(α
12
)
F,12
2
χ
12
(1 − χ
123
) ,
2
We would like to thank Valter Moretti for pointing this result out to us.
13
where the Feynman propagators are regularised as in
(3.12)
. By construction,
τ
(α)
Γ
is globally well–
deﬁned and the analysis outlined above and performed in the following sections implies that it can be
uniquely extended to a weakly meromorphic distribution on the full
M
3
. Moreover, the local pole
contributions corresponding to
α
12
=
α
I
with
I
=
{
1
,
2
}
and
α
12
=
α
13
=
α
23
=
α
J
with
J
=
{
1
,
2
,
3
}
are clearly independent of the choice of
χ
12
,
χ
123
and
N
12
,
N
123
such that the MS–regularised amplitude
(
τ
Γ
)
ms
is both globally well–deﬁned and independent of the quantities entering the global deﬁnition of
the analytic regularisation.
Keeping this approach to deﬁne global analytically regularised quantities in mind, we shall for
simplicity work only with local quantities in the following.
3.2. Analytic regularisation of the Feynman propagator H
F
on curved spacetimes
Following the plan outlined in Section 3.1, we would like to deﬁne an analytic regularisation
H
(α)
F
of
H
F
by
(3.12)
. To this end, we start our analysis by constructing the distribution 1
/σ
1+α
F
in
M
2
for
α ∈ C \ N
. As anticipated in Section 3.1 we shall make use of scaling properties of 1
/σ
1+α
F
and the
induced quantities t
(α)
Γ
(3.13) with respect to a particular geometric scaling transformation.
For every pair of points
x
1
, x
i
in a normal neighbourhood
N ⊂
(
M, g
) there exists a unique
geodesic
γ
connecting
x
1
and
x
i
. We shall assume that
γ
:
λ 7→ x
i
(
λ
) is aﬃnely parametrised and
that
x
i
(0) =
x
1
whereas
x
i
(1) =
x
i
. For all
λ ≥
0 and all
f ∈ D
(
N
n
) with
N
n
⊂ M
n
a normal
neighbourhood of the total diagonal
d
n
(cf. Remark 3.1), the geometric