Lorentzian 2d CFT from the pAQFT perspective

We provide a detailed construction of the quantum theory of the massless scalar field on 2-dimensional, globally-hyperbolic (in particular, Lorentzian) manifolds using the framework of perturbative algebraic quantum field theory. From this we subalgebras of observables isomorphic to the Heisenberg and Virasoro algebras on the Einstein cylinder. We also show how the conformal version of general covariance, as first introduced by Pinamonti as an extension of the construction due to Brunetti, Fredenhagen and Verch, may be applied to the concept of natural Lagrangians in order to obtain a simple condition for the conformal covariance of classical dynamics, which is then shown to quantise in the case of a quadratic Lagrangian. We then compare the covariance condition for the stress-energy tensor in the classical and quantum theory in Minkowksi space, obtaining a transformation law dependent on the Schwarzian derivative of the transformed coordinate, in accordance with a well-known result in the Euclidean literature.


Introduction
One of the most important problems faced by mathematical physicists nowadays is the search for mathematically rigorous formulation of quantum field theory (QFT). Over the span of six decades, several axiomatic frameworks have been developed (including algebraic quantum field theory [Haa96,HK64]), but none of them can yet claim to include an interacting QFT model in 4 spacetime dimensions. On the other hand, a lot is known about lower-dimensional cases (prominently 2-dimensional) and in the presence of symmetries, e.g. the conformal symmetry. The huge success of conformal field theory (CFT) and its ubiquity in theoretical physics is evidenced by a vast trove of literature and impressive number of results obtained throughout the history of the subject [BPZ84,Gin88,FMS97,Sch08]. In two dimensions, CFT plays a central role in the world-sheet description of string theory. More generally, they describe continuous phase transitions in condensed matter systems, critical points of renormalisation group flows in quantum field theories and provide duals to gravitational theories in anti-de Sitter spacetimes via the AdS/CFT correspondence. From a mathematical point of view, the rigorous formulation of two-dimensional Euclidean chiral CFT has led to the important development of vertex operator algebras (VOA), see e.g. [Kac98,LL04,FB04,BD04], which has been instrumental in various areas of pure mathematics, including the proof of the monstrous moonshine conjecture [FLM89,Bor86,Bor92] and in the study of the geometric Langlands correspondence [FF92,BD96,Fre05,Fre07].
CFT has also provided a rich class of models that satisfy algebraic quantum field theory (AQFT) axioms, as demonstrated for example in [GF93,KL06,KL02,KL04,Bis17,BKL15,LR04,LW11]. The main principles of AQFT can also be applied to describe perturbative QFT. This led to development of perturbative algebraic quantum field theory (pAQFT), which started in the 90s [BDF09, BF00, BF97, DF01, DF03, DF04] (see also [Rej16] and [Düt19] for a review). The advantage of pAQFT is that it combines the ideas of AQFT with the powerful methods of perturbation theory and renormalization and allows one to construct physically interesting models in 4-dimensions, also on curved spacetimes. However, the ultimate goal of pAQFT is to understand how non-perturbative results could be obtained. To this end, it is useful to construct some known non-perturbative models using pAQFT methods and see how convergence and non-perturbative effects arise. An example of such a model has been investigated in [BR18]. This paper is the first step in the research programme aimed at understanding how CFT fits into the framework of pAQFT. The advantages of such a combination are twofold: • Many of the CFT results are proven only in the Euclidean signature. With the aid of pAQFT, we want to show how to obtain them in Lorentizian signature as well.
• Some powerful techniques used in CFT can be applied in pAQFT to obtain nonperturbative results.
In the present paper we concentrate on setting up the general framework, with particular focus on local conformal covariance. We improve on existing results of [Pin09] and apply our methods to define normally-ordered covariant quantities, with Virasoro generators on a cylinder among them. We show that covariant normal-ordering allows one to reproduce the correct Virasoro algebra relations on the cylinder and we demonstrate how the usual "Zeta regularisation" trick can be rigorously understood as the change in the choice of normal ordering.

Mathematical Preliminaries
In this section, we provide an account of the constructions of pAQFT relevant to our discussion. For a more thorough exposition, the reader is directed towards [Rej16]. In particular, whilst we may, from time to time, discuss the possibility of interactions in the classical theory, all of our quantum constructions shall be specific to the free scalar field.
We begin with the kinematics (i.e. states and observables) of our classical theory. Due to our use of deformation quantisation, this will also establish the observables of the quantum theory. Next, we address in Section 2.2 the matter of imposing suitable dynamics on the system, using the generalised Lagrangian formalism. For an appropriately chosen Lagrangian, we are then able to endow our space of observables with a Poisson structure. At this point, the algebra is decidedly "off-shell", as the field configurations we consider include those which do not satisfy the equations of motion. Therefore, in Section 2.3, we make a detour to examine how, in the case of the free scalar field, our construction does indeed recover the canonical (i.e. "equal-time") Poisson bracket on-shell. Here we also briefly explore the 'dg' perspective of QFT at the heart of the Costello-Gwilliam formalism [CG16] and descriptions of 'higher' QFT as outlined in, for example, [BPSW21].
Satisfied with our choice of Poisson structure, we then use it in Section 2.4 to deform the pointwise product of functionals into an associative product ⋆, which is analogous to the operator composition of canonical quantisation. Once the quantum algebra has been established, we discuss the comparison between classical and quantum observables. The difficulty in 'quantising' classical observables is traditionally known as the ordering problem. In an attempt to find the most natural solution to this problem, we then introduce in Section 2.5 the concept of local covariance, where we require our theory to be defined in a coherent manner across multiple spacetimes. This is so that we may be sure our ordering prescription is not dependent on the global geometry of any particular spacetime (which local algebras should in principle be unaware of).

Classical Kinematics
Let M be a smooth manifold (we shall specify dimension and topological constraints later). For the theory of a real scalar field, we take our configuration space, E(M), to be the space of smooth real-valued functions on M. More generally, we might consider the space of smooth sections of some vector bundle E π → M, to which the following constructions can be readily generalised. Note that this space is "off-shell" in the sense that it includes field configurations which may not satisfy any equations of motions later imposed by the dynamics.
Classically, observables are maps F : E(M) → C. Typically, we also assume them to be smooth, with respect to an appropriate notion of smoothness which we shall introduce shortly. The derivative of a functional at a point ϕ ∈ E(M) and in a direction h ∈ E(M) is defined in the obvious way as ǫ , (2.1) whenever this limit exists. If it exists for all ϕ, h ∈ E(M), and the map is continuous with respect to the product topology on E(M) 2 then we say F is C 1 .
Various pieces of notation are commonly used when discussing functional derivatives. For clarity, we collect some of them here. Firstly, note that for a C 1 functional F , the above definition implies that F (1) [ϕ] ∈ E ′ (M), using Schwartz's notation for compactly supported distributions. Hence the bracket ·, · in (2.1) can be seen as denoting the canonical pairing V ′ × V → C, where V is a topological vector space over C and V ′ is its continuous dual space. If M is equipped with a preferred volume form * F (1) [ϕ] may be * As we are only interested in Lorentzian manifolds, we always have the metric volume form. Our definitions of various classes of functionals assume a preferred volume form, other authors opt instead to define δF/δϕ as a distribution density [Hör15,p.145].
for f ∈ D(M).
Whilst it is possible to perform our classical and quantum operations on local functionals, the result is typically not itself local. As such, we need a space of functionals which is algebraically convenient, like F reg (M), but which also contains the physically important subspace F loc (M). The space of mircocausal functionals accomplishes this. Unlike the previous classes of functionals however, its definition requires more than a smooth structure on M. Instead we require the structure of a spacetime, which we define in accordance with [FV12,§2.1] as follows: Definition 2.1 (Spacetime). A spacetime is a tuple M = (M, g, o, t) such that (M, g) is an orientable Lorentzian manifold of some fixed dimension d, o ⊂ Ω d (M) is an equivalence class of nowhere-vanishing volume forms, defining an orientation, and t ⊂ X(M) is an equivalence class of timelike vector fields, where t ∼ t ′ ⇔ g x (t x , t ′ x ) > 0 ∀ x ∈ M. We will typically write F(M), F reg (M), and F loc (M) to refer to the respective spaces of functionals associated to the underlying manifold of M.
For any point x in a spacetime M, we can define the closed past/future lightcone of the cotangent space V ± (x) ⊂ T * x M as comprising covectors k for whichĝ x (k, k) ≥ 0 and ±k(t x ) ≥ 0, for any t ∈ t, whereĝ x is the metric induced on T * x M by g. We can then define the fibre bundles V ± ⊂ T * M such that their fibres at x are V ± (x) respectively.
A functional F ∈ F(M) is microcausal if it satisfies the wavefront set spectral condition (2.5) For detailed definitions and properties of the wavefront set of a distribution, see for example [Hör15,§8], as well as [BDH14]. Briefly put, the wavefront set is a way of characterising the singularity structure of a distribution T ∈ D ′ (M), i.e. the precise manner in which T fails to be a smooth function. It consists of the set of non-zero covectors (x, k) ∈ T * M such that there exists no neighbourhood of x to which the restriction of T is smooth, and the Fourier transform -defined in an arbitrary chart, which turns out to be irrelevant -of T fails to decay rapidly in the direction k. The space of microcausal functionals is denoted F µc (M), and contains both the local and regular spaces of functionals.
The characteristic features of these spaces, as well as the relations between them, are summarised in the following diagram.

Classical Dynamics
There are many ways to specify the dynamics of a classical field theory. In the present formalism it is achieved through a rigorous implementation of the principle of critical action. The foundational idea of this approach, due to Peierls [Pei52], is the formulation of a Poisson structure in terms of the advanced and retarded responses of a field to perturbation. A construction of the classical algebra of observables using the Peierls bracket was set forth in [DF03], and developed in detail in [BFR19]. More recent overviews may be found in, e.g. [Rej16,§4] or [FR15,§5.1].
This approach has the advantage of being manifestly Poincaré covariant, as will be explored further in section 2.5, whilst still endowing our space of observables with a Poisson structure, contrary to a common notion that a choice of Poisson structure requires one to split a spacetime into 'space' and 'time'.
The issue with naïvely written actions for common field theories, such as the Klein-Gordon or Yang-Mills functionals, is that their region of integration must be restricted to a compact subset of spacetime in order to guarantee a finite value is returned. A convenient way to achieve this is to define a map L : D(M) → F loc (M), where the functional L(f ) is interpreted as the action functional with an introduced cutoff function f . Not every such map is suitable however, the necessary criteria are outlined in the following definition (after [Rej16, §4.1]).
is called a generalised Lagrangian if it satisfies the following conditions: 3. If β is an isometry of (M, g) which preserves orientation and time-orientation, then for f ∈ D(M) and ϕ ∈ E(M), This definition refers to the spacetime support, which we denote supp F for a functional F . This is the closure of the set of points x ∈ M such that, for all ϕ ∈ E(M), there exists some perturbation localised to a neighbourhood of x, say ψ ∈ D(U) for some A primary example is the generalised Lagrangian for the Klein-Gordon field on a spacetime M, which is given by where ∇ is the gradient operator associated to the metric g of M and dVol g is its associated volume form.
Heuristically, one may think of the limit of L(f ) as f tends to a Dirac delta δ x as describing the Lagrangian density at x and, if f instead tends to the constant function 1, then L(f ) becomes the action functional S. However one must bear in mind that, in general, these limits may not (and typically will not) yield well-defined local functionals.
Given a generalised Lagrangian L, we define the Euler-Lagrange derivative at a point ϕ ∈ E(M) as the distribution S ′ [ϕ] such that (2.7) where, h ∈ D(M) and f ∈ D(M) is chosen such that f −1 {1} contains a neighbourhood of supp h * . One can use the additivity and support properties to verify that vanishes as a distribution.
Different choices of a generalised Lagrangian may yield the same Euler-Lagrange derivative. If a generalised Lagrangian L 0 satisfies supp L 0 (f ) ⊆ supp df , then clearly its Euler-Lagrange derivative vanishes for all ϕ ∈ E(M). In such a case, we describe L 0 as null. One may add a null Lagrangian to an arbitrary generalised Lagrangian without changing its Euler-Lagrange derivative. Given this, we say that two generalised Lagrangians, L and L ′ define the same action if their difference is null, we denote this fact by [L] = [L ′ ] =: S.
In the case where S is a quadratic action, (i.e. it may be represented by a Lagrangian L such that L(f ) is a quadratic functional for all f ) the map ϕ → S ′ [ϕ], h is linear in ϕ. We assume that this functional can be expressed in the form ϕ → P ϕ, h , where P is a normally hyperbolic differential operator, i.e. P is a second order differential operator of the form ∇ a ∇ a + lower order terms. A more precise definition of normally hyperbolic differential operators can be found in, e.g. [BGP07, §1.5]. As an example, given the free field Lagrangian (2.6), P is simply the Klein-Gordon operator −( + m 2 ).
For interacting theories, one must take a further functional derivative, defining where f is chosen as before. For a broad class of physically interesting actions, there exists a self-adjoint, Green hyperbolic differential operator ([Bae15, Definition 3.2]) P [ϕ] such that (2.9) We refer to P [ϕ]g = 0 as the linearised equations of motion at the configuration ϕ and, if such an operator exists for every ϕ ∈ E(M), we say that the action satisfies the linearisation hypothesis. If ϕ is an on-shell configuration, then Ker P [ϕ] can be thought of as the tangent space at ϕ to the manifold of on-shell configurations. Note that for a free action, P coincides with P [ϕ] for every ϕ ∈ E(M).
Throughout this paper we assume all spacetimes to be globally hyperbolic. A Lorentzian manifold M = (M, g) is globally hyperbolic if there exists a diffeomorphism ρ : M ∼ → Σ × R, such that, for every t ∈ R, ρ −1 (Σ × {t}) is a Riemannian submanifold (referred to as a Cauchy surface) of M.
The key feature of such spacetimes is the existence of Green hyperbolic differential operators P , characterised by the property that the Cauchy problem P ϕ = 0 admits fundamental solutions E R/A : D(M) → E(M) uniquely distinguished by the fact that, for any f ∈ D(M) (2.11) Here J ± (K) denotes the causal future/past of K, i.e. the set of all points connected to some point x ∈ K by a causal future/past directed curve respectively. We call these maps the retarded/advanced propagator respectively. For detailed explanation and proof of the relevant existence and uniqueness theorems, we refer the reader again to [BGP07].
Each propagator is formally adjoint to the other in the sense that, for all f, g ∈ D(M) Their difference E = E R − E A , known as the Pauli-Jordan function, defines a map from D(M) to the space of solutions of P ϕ = 0, and is vital to our construction of a covariant Poisson structure on phase space.
Note that here and in the following we are considering a free theory, governed by the single linear equation P ϕ = 0. However, to generalise to the interacting case, one need only replace P with the linearised operator P [ϕ] defined by (2.9), and note that the fundamental solutions are then defined relative to this linearised operator.
Recall that the phase space of a free field theory is simply the space Ker P of solutions to the equations of motion. Traditionally, we identify this with the space of Cauchy data on some fixed surface, i.e. the strength and momentum-density of a field at some fixed time. [BGP07,Proposition 3.4.7] states that all solutions with spacelike-compact support may expressed as Ef for some f ∈ D(M) and also that the kernel of this map is precisely P (D(M)). In other words, we can identify our phase space with the quotient D(M)/P (D(M)). One could then define the algebra of observables on M to be the space of smooth maps from this space to C, which can be equipped with a nondegenerate Poisson bracket using E as a bivector. This is not, however, the approach that we shall take, which we outline below.
Given two regular functionals F , G ∈ F reg (M), we can use E to define a new functional ( To obtain a closed algebra, we extend the domain of the Pauli-Jordan function to include a suitable class of distributions. As shown in Appendix B, the pairing f, Eg is well defined if f and g are compactly-supported distributions satisfying the (n = 1) wavefront set spectral condition (2.5). In particular, this means (2.13) is well defined for F , G ∈ F µc (M), and one can show (see Appendix B) that the result is again a microcausal functional. Once it is established that {·, ·} is also a derivation over the pointwise product of functionals, we may conclude that (F µc (M), ·, {·, ·}) is a Poisson algebra [BFR19, Theorem 4.1.4], which we shall denote P(M). In the next section, it is precisely this Poisson structure we shall deform in order to arrive at the quantum algebra.

Going On Shell
We shall now explain how this formalism distinguishes between on-shell and off-shell observables. Recall for the free theory we claimed that on-shell observables could be defined as maps from D(M)/P (D(M)) or equivalently the space of on-shell configurations, to C. Broadly speaking, the strategy is to identify this space of maps as a quotient of F µc (M) by a suitable ideal.
A well-known result states that, given a manifold X with some closed submanifold Y ⊆ X, there is an isomorphism (2.14) where I(Y ) ⊆ C ∞ (X) is the ideal of functions vanishing on Y . The construction of the Poisson algebra of on-shell observables may be regarded as an infinite-dimenional analogue of this isomorphism, where C ∞ (X) is replaced with F µc (M). We define the ideal I S ⊆ F µc (M) to be the set of functionals which vanish for all on-shell configurations, i.e.
Crucially, I S is an ideal with respect not only to the pointwise product ·, but also with respect to the Peierls bracket {·, ·}. This can be proved from (2.13) because, if ϕ is a solution, F ∈ I S , and G ∈ F µc (M) then ϕ + ǫEG (1) [ϕ] is also a solution for any ǫ > 0, hence Defining the on-shell algebra as a quotient of two functional spaces, emphasises the algebraic viewpoint on geometry, where a space of maps on an algebraic variety or a topological vector space is used to describe the space itself. The advantage of this viewpoint will become even more apparent after we present a convenient way of characterising I S .
We have already seen variations of the form S ′ [·], h , noting that an on-shell configuration ϕ is precisely one for which the above functional vanishes, for any h ∈ D ′ (M). We can identify h with a constant section of the tangent bundle T E(M) ≃ E(M) × D(M), which we denote X h . Allowing such sections to act on functionals via derivation (in the obvious way), we can rewrite the above functional as X h · L(f ) for any f ∈ D(M) which is suitable in the manner specified after (2.7). To discuss more general variations, we must first discuss a suitable notion of a vector field.
A complete definition of the space of microcausal vector fields requires a few subtleties, and may be found in [Rej16,§4.4]. There it is also noted how such vector fields are related to the space of microcausal observables on the shifted cotangent bundle, T * [1]E(M). Let V µc (M) denote the space of microcausal vector fields. To every functional F ∈ F µc (M) we can associate a one-form dF , i.e. a smooth map V µc (M) → C by dF (X) = X · F . An important characteristic of any X ∈ V µc (M) is that there exists a compact subset K ⊂ M such that X[ϕ] ∈ D(K) for all ϕ ∈ E(M). This means we can define a one-form δ S (X) = dL(f )(X), where f ≡ 1 on a neighbourhood K. We call δ S (X) the variation of the action with respect to X.
The principle of critical action for ϕ ∈ E(M) can be expressed as the condition that, We can begin to see some of the higher structure of this formalism by extending the differential δ S : V µc (M) → F µc (M) to the exterior algebra of V µc (M) (graded such that the degree of k V µc (M) is −k). This yields the cochain complex where δ S is defined in lower degrees via the graded Leibniz rule: for example, a homoge- We call this the Koszul complex associated to δ S , denoted K(δ S ).
One can show that the Peierls bracket also extends to a degree zero Poisson bracket across the entire complex, and that δ S is a derivation over this bracket (i.e. the pair (K(δ S ), {·, ·}) is a dg Poisson algebra). In particular, for a vector field X ∈ V µc (M) and a functional F ∈ F µc (M), this means that δ S {X, F } = {δ S X, F } (as δ S F = 0 for any functional F ). In turn, this establishes that δ S (V µc (M)) is an ideal of the Peierls bracket, and hence that the cohomology of this complex in degree 0 naturally inherits a Poisson structure. Given the fact that δ S (V µc (M)) = I S , this cohomology is H 0 (K(δ S )) = F µc (M)/I S , which we thus call the on-shell algebra of observables.
It is, at this point, natural to ask whether or not there exists a physical interpretation of H −1 (K(δ S )), or the cohomology in yet lower degrees. To answer the first, note that for a vector field X, δ S (X) = 0 implies that the infinitesimal transformation ϕ → ϕ + ǫX[ϕ] leaves the action invariant to first order in ǫ. As such, the kernel of δ S in degree −1 comprises infinitesimal generators of gauge symmetries. The image of δ S in degree −1 contains vector fields of the form δ S (X ∧Y ) = δ S (X)Y −δ S (Y )X. In the physics literature these are referred to as trivial gauge symmetries. They are, in a sense, less insightful because they are defined the same way regardless of the action in question, and also because they act trivially on shell. As such, we can regard H −1 (K(δ S )) as the space of non-trivial gauge symmetries * .
The above discussion motivates us to consider the space • V µc as the primary kinematical object of a physical theory, with δ S representing the choice of dynamics. This perspective is advantageous both in describing conformally covariant field theories (where the generalised Lagrangian formalism proves inconvenient) as well as in the formulation of chiral sectors of a 2-dimensional CFT, where one may require choices of δ S which cannot arise from a generalised Lagrangian.
Finally, as an aside now that we have constructed our on-shell algebra, it is informative to make a comparison to the 'equal-time' (a.k.a. canonical) bracket defined relative to some choice of Cauchy surface Σ.
Definition 2.3 (Canonical Poisson Algebra). Let Σ ⊂ M be a Cauchy surface, we define the associated canonical Poisson algebra as follows: The underlying vector space F can (Σ) consists of functionals F : C ∞ c (Σ)×C ∞ c (Σ) → C which are Bastiani smooth, the arguments of this functional represent the initial field strength and momentum on Σ of some on-shell field configuration. Given a pair F, G of such functionals, their canonical bracket is then defined as It is not immediately obvious why the Peierls bracket should be related to this canonical bracket, other than because E parametrises the space of on-shell field configurations. Especially as the canonical bracket requires a particular Cauchy surface to be specified, a manifestly Lorentz non-covariant choice. However, by sending the initial data (φ, π) ∈ E(Σ)×E(Σ), to their corresponding solution, one can construct a map F µc (M) → F can (Σ) which in turn yields a Poisson algebra homomorphism from the on-shell Peierls bracket to the canonical [FR15, §3.2].

Deformation Quantisation
Having established our Poisson structure, the next step is to deform it to construct our quantum algebra of observables. Here we take an approach that is analogous to Moyal-Weyl quantisation, though in QFT this is made somewhat harder than in the quantummechanical case, due to the infinite degrees of freedom in the configuration space. In particular, as is common in perturbative QFT, our deformation shall be formal, meaning that quantised products will be formal power series in , allowing us to ignore the issue of proving convergence of our formulae.
For regular functionals F , G ∈ F reg (M) we can define the star product of F and G directly as We may write this formula more concisely as where m is the pointwise multiplication map A general result [HR19, Proposition 4.5] states that this exponential form guarantees ⋆ is associative. As mentioned, this deformation is formal, meaning we have actually defined . We can then define the ⋆ product on F reg (M)[[ ]] by linearity to obtain a closed algebra.
Writing the first few terms explicitly, we see F ⋆ G = F · G + i 2 {F , G} + O( 2 ). Thus the classical term of ⋆ (i.e. the coefficient of 0 ) is simply the pointwise product and the Dirac quantisation rule also holds modulo terms of order 2 , hence ⋆ is a deformation of the classical product in the sense of [Rej16, §5.1]. However, if we wished to apply (2.20) to other local functionals, divergences would begin to appear. Consider for example the family of quadratic functionals, for f ∈ D(M) (2.21) A naïve computation of the star product for two such functionals would yield In general, the O( 2 ) term of this product is ill-defined if suppf ∩ suppg = ∅. This is because E is a distribution, as opposed to a smooth function, and the product of two distributions cannot be defined in general.
The solution is to make use of a Hadamard distribution. Physically, a Hadamard distribution is the 2-point correlator function for some 'vacuum-like' state, i.e. W (x 1 , x 2 ) = Φ(x 1 ) ⋆ Φ(x 2 ) . More precisely, a complex-valued distribution W ∈ D ′ (M 2 ; C) is Hadamard if it satisfies the following properties [Rej16] H0 The wavefront set of W satisfies WF(W ) = (x, y; ξ, η) ∈ WF(E) | (x; ξ) ∈ V + (2.23) where H is a symmetric, real distribution. H2 W is a weak solution to P .
H3 W is positive semi-definite in the sense that, ∀ f ∈ D(M; C) W,f ⊗ f ≥ 0.
, for any F , G ∈ F reg (M) and the inverse of this map is simply α −H . Where these two products differ, however, is that ⋆ H can also be extended to a well defined product on F µc (M).
On a generic globally hyperbolic spacetime, it is well-known [FNW81] that there exist infinitely many Hadamard distributions, thus we need never fear that Had(M) is empty. However, there is usually no natural way of selecting which H ∈ Had(M) to use. Thus, whilst we can always construct a well defined algebra for an arbitrary globally hyperbolic spacetime M, it would be unnatural to define the quantum algebra by making such an arbitrary choice. Fortunately, the algebraic structure is defined just as in (2.25). As one might expect, the inverse of this map is α H−H ′ , hence all of our candidate algebras are in fact isomorphic to one another. One way in which we can define the quantum algebra without any undue preference to a particular Hadamard distribution is as follows.
Definition 2.4. The quantum algebra of the free field theory, denoted A(M), is a unital, associative * -algebra whose elements are the indexed sets F = (F H ) H∈Had(M) , subject to the compatibility criterion with a product defined by It is important to bear in mind that, whilst we have deformed the classical algebra F µc (M) into a quantum algebra A(M), we have not yet specified a quantisation map, embedding classical observables into the quantum algebra. We will need to establish such a map before computing commutation relations for the quantum stress energy tensor in section 3.3. However, before considering what a suitable choice of map may be, it is instructive to study how the construction we have just outlined varies as we change the underlying spacetime M.

Local Covariance and Normal Ordering
We have deliberately said little about potential spacetime symmetries in the construction above. The reason being that we take the perspective that covariance under any symmetries a particular spacetime may enjoy is just a special case of a broader property we wish to implement: namely local covariance. The concept of local covariance, introduced in [HW01] and [BFV03], unites the representation of spacetime symmetries as automorphisms of the algebra of observables with the principle that an observable localised to a region O ⊂ M of a spacetime should be 'unaware' of the structure of the spacetime beyond this region.
The foundational idea is that, if there exists a suitable embedding of a spacetime M into a spacetime N , then there should be a corresponding embedding (more precisely, a homomorphism) of observables A(M) → A(N ). A spacetime symmetry is just a suitable embedding of M into itself which also admits an inverse. If the corresponding algebra homomorphism is similarly invertible, then we would have, in particular, an action of the isometry group of M on A(M) as desired.
To formulate local covariance more precisely, it is convenient to invoke the language of category theory. To begin with, by specifying the suitable embeddings of spacetimes, we endow the collection of globally hyperbolic spacetimes with the structure of a category, which is denoted Loc and defined as follows: • An object of Loc is a spacetime M, as specified in definition 2.1, of a fixed dimension d.
• For a pair of spacetimes M = (M, g, o, t) and N = (N, Given an admissible embedding χ : We show later in Section 4.1.2 that even if χ preserves the metric only up to a scale, then F • χ * is still microcausal whenever F is, hence in particular χ * (F µc (M)) ⊂ F µc (N ) for all Loc morphisms χ : M → N . In fact, all of the different spaces of functionals specified in Section 2.1 are each preserved under the map χ * , and thus may be considered functors from Loc to some category of observables.
Next, we need to find a way to specify dynamics in a coherent way across all spacetimes. This involves extending the generalised Lagrangian framework to the concept of a natural Lagrangian. In categorical language, we can define a natural Lagrangian as a natural transformation L : D ⇒ F loc , such that for each M ∈ Loc, L M is a generalised Lagrangian as per Definition 2.2. Here, D is the functor assigning each spacetime its space of compactly-supported test functions, and to each morphism χ : M → N the map χ * : D(M) → D(N ) defined by Spelling this out, the naturality condition reduces to the condition that, for every morphism of spacetimes χ : M → N , f ∈ D(M) and ϕ ∈ E(N ) which is essentially a generalisation of the covariance condition appearing in Definition 2.2 and can be shown to be satisfied by the Klein-Gordon Lagrangian (2.6).
From the naturality condition, one can then show that if χ : M → N , then the Euler-Lagrange derivatives of L M and L N are related by the equation, ∀ ϕ ∈ E(N ) and, in the case of the free scalar field, the causal propagators arising from L M and L N are related by E N (χ * f, χ * g) = E M (f, g). From here, it can be deduced that χ * : is a Poisson algebra homomorphism where each space is equipped with its respective Peierls bracket, hence the assignment P(M) outlined in the above section is covariant (i.e. it defines a functor from Loc to the category of Poisson algebras).
We shall use the generic designation Obs to denote the category our observables (either classical or quantum) belong to. Choices of Obs relevant to our discussion include • Vec, whose objects are vector spaces over C, and whose morphisms are linear maps. This is the most generic space generally considered, and is appropriate when one wishes to treat classical and quantum theories on an equal footing.
• Poi the category of Poisson algebras and Poisson algebra homomorphisms. This is the primary category of observables for classical theories.
• * -Alg, the space of topological * -algebras. We choose this as the target category of quantum theories, as the perturbative nature of our construction requires us to consider unbounded operators, else we would use instead the category of C *algebras.
• In each of the above cases, we may add a dg-structure, i.e. if Obs is any of the above categories, Ch(Obs) comprises cochain complexes which in each degree take values in Obs. Such categories are at the heart of the BV formalism in both the classical and quantum case [GR19], [CG16].
A locally covariant field theory (classical or quantum) is then defined simply as a functor from Loc → Obs. Already this captures a lot of important features, such as the representation of spacetime symmetries as automorphisms of the algebra of observables. Whilst one can go further by imposing additional axioms for such a functor to satisfy, this general definition will suffice for our purposes.
The BV formalism outlined in the previous section can also be made locally covariant. Just like F µc , we can easily promote V µc to a functor Loc → Vec. A choice of natural Lagrangian then yields a natural transformation between the two, δ S : V µc ⇒ F µc . From this it follows that the construction of the Koszul complex K(δ S ) itself defines a functor Loc → Ch(Poi).
We have already sketched an explanation of how our construction of the classical theory may be made locally covariant. If H 0 ∈ Had(N ), then one can show that χ * H 0 ∈ Had(M), thus we can define a map A (χ * H 0 ) (M) → A H 0 (N ) as just the canonical extension of the pushforward χ * : F µc (M) → F µc (N ) to formal power series in . This map satisfies thus it defines a * -algebra homomorphism. The map Aχ : A(M) → A(N ) is then given by which can be shown to satisfy the criteron (2.28), making the map well-defined. With these morphisms, we can then declare A : Loc → Obs to be a locally covariant quantum field theory.
Next, we turn to the topic of normal ordering. On a fixed spacetime M, normal ordering is the process of mapping (some subset of) classical observables into the space of quantum observables. In our case, we seek a map : − : M : F loc (M) → A(M), such that the 0 coefficient of :F : M is F . Given our somewhat indirect definition of A(M), it is helpful to outline here the general strategy for defining a normal ordering prescription, before we turn our attention to any particular maps.
It is easiest to define a normal ordering prescription by a family of map F loc (M) → A H (M) for every choice of H ∈ Had(M). Suppose we denote each such map as F → (: F :) H , they collectively define a map F loc (M) → A(M) if they satisfy, for every (2.35) By choosing a fixed Hadamard state H 0 ∈ Had(M), we can define a quantisation map which has the physical interpretation of normal ordering "with respect to" that state. As indicated above, we first define a map This clearly satisfies the criterion (2.35) above, and hence is a valid normal ordering prescription. We may also characterise this prescription as the only consistent choice such that the map Similar to our definition of a natural Lagrangian, a locally covariant ordering prescrition is defined to be a natural transformation from F loc to A. (Note that we must assume that the target category of each functor is Vec, as normal ordering is linear, but not a homomorphism.) Explicitly, this naturality condition is realised by the equation, for every admissible embedding χ : M → N , (2.37) It is tempting to believe that a covariant prescription across all spacetimes can be found by selecting a suitable Hadamard state for each spacetime. However, it is now a well-established fact that such a choice cannot be made consistently across all spacetimes. (See the remarks following definition 3.2 of [HW01] for a discussion relevant to the scalar field, and [FV12, §6.3] for a more general result.) The solution is to instead define an ordering prescription dependant upon the Hadamard parametrix of the spacetime in question. Before the characterisation via wavefront sets used in (2.23), Hadamard states were defined by the ability to express them locally (i.e. in some neighbourhood of the thin diagonal ∆ ⊂ M 2 ) in what is known as local Hadamard form. In the case of a 2-dimensional spacetime, the local Hadamard form of a state W is where σ(x; y) is the world function, defined as half the geodesic between x and y, t is some choice of a time function (i.e. level sets of t are Cauchy surfaces), σ ǫ is defined by We can then verify that, for H, H ′ ∈ Had(M)

The Massless Scalar Field on a Cylinder
Now that we have constructed both a classical and quantum algebra of observables, and introduced several ordering maps between them, we may study their finer details in an explicit example. As our ultimate goal is to understand conformal field theory from the perspective of pAQFT, the massless scalar field is the obvious place to begin. Moreover, owing to its flat geometry and compact Cauchy surfaces, the Einstein cylinder M cyl -defined as the image of 2D Minkowski space, M 2 , under the identification (t, x) ∼ (t, x + 2π) -provides a natural and convenient setting in which to explore the chiral aspects of the massless scalar field within the pAQFT framework. * This is a direct consequence of the fact that χ * : Had(N ) → Had(M), and that the difference of any pair of elements in Had(M) is smooth.
In this section, we shall see how the quantum algebra of observables for the massless scalar field contains a pair of Heisenberg algebras and a pair of Virasoro algebras, one each for the left and right null-derivatives of the field. In the construction of the Virasoro algebra, we shall also see that the principle of local covariance outlined in Section 2.5 is necessary to recover the 'radially-ordered' form of the Virasoro algebra. The argument involved in this re-ordering constitutes a mathematically rigourous form of the known trick of identifying 1 + 2 + 3 + · · · = ζ(−1).

Minkowski Space
We begin by finding the causal propagator for the massless scalar field in Minkowski space. From this we shall later obtain the propagator for the cylinder, and hence the Poisson algebra P(M cyl ). Moreover we shall begin to see how the classical Poisson algebra of the massless scalar field naturally contains two chiral subalgebras.
The equation of motion for the massless scalar field on Minkowski space is simply By inspection one can then deduce that the distributions both satisfy (3.2) and have the desired supports. Taking their difference we find the Pauli-Jordan function to be We can rewrite this propagator in the form . In other words, we can decouple the u-dependent terms from the v-dependent, defining the summands such that E ℓ does not depend on v and vice-versa.
This split is significant for functionals which depend on the field configuration ϕ only through its left/right null derivative. If we indicate the action of the differential operator ∂ u on a functional F by (∂ * u F )[ϕ] := F [∂ u ϕ], then the functional derivative of ∂ * u F is given by Consequently, the Peierls bracket of two such functionals is This equality motivates the construction of a new Poisson algebra, outlined in the following proposition: Proposition 3.1. The space F µc (M 2 ), equipped with the pointwise product ·, and the bracket is a Poisson algebra, which we denote P ℓ (M 2 ). Furthermore, the map ∂ * u : Next, we must show that {·, ·} ℓ satisfies the Jacobi identity. This we can achieve using (3.8) alongside the observation that ∂ * u is injective (which follows from the fact that ∂ u is surjective). Let F , G, and H all be microcausal functionals. Consider where · · · includes both remaining even permutations of F , G, and H. The right-hand side of this vanishes as the Peierls bracket satisfies the Jacobi identity hence, by injectivity, we see that {F , {G, H} ℓ } ℓ + · · · also vanishes.
Note that, (∂ u ⊗ ∂ u )E˚r ffl = 0, hence the integral kernel of the differentiated propagator is This form of the commutator can be seen as an example of the mutual locality of chiral fields, [Kac98, Definition 2.3], a concept central to many theorems in the VOA framework We shall henceforth refer to {·, ·} ℓ as the chiral bracket, and the analogously defined {·, ·}r ffl as the anti-chiral bracket.
It turns out that the chiral and anti-chiral brackets can be defined on a space of functionals larger than F µc (M 2 ). In an upcoming paper, we shall explore what a suitable enlargement is, and how this relates to the concept that chiral fields are defined over a single light-ray.

The Heisenberg Algebra
We shall now find the advanced and retarded propagators for the Einstein cylinder M cyl . If (u, v) denotes the null coordinates of a point in M 2 , then we define an equivalence relation on M 2 by (u, v) ∼ (u + 2π, v − 2π). The Einstein cylinder is then defined as the quotient space M cyl = M 2 / ∼, with the unique metric such that the covering map π : M 2 → M cyl is a local isometry. We will write points in M cyl as equivalence classes The causal propagator for the cylinder may be obtained from the advanced and retarded propagators of Minkowski spacetime using the method of images. Firstly, note there is an Going from M cyl to M 2 , this map is simply the corestriction of π * to the space of Z invariants. If we denote the inverse of this isomorphism by π * , then we claim the retarded and advanced propagators on the cylinder are given by (3.11) For this map to be well defined, amongst other details, we must show that the domain of E R/A can be extended to the image π * D(M cyl ) , and that the output of E R/A π * contains only Z invariants. Proof of which can be found in Appendix A.
That these maps are then the desired propagators follows from the relationship between the equations of motion on the cylinder and Minkowski. Let U ⊆ M 2 be a sub-spacetime of M 2 and let ι U : U ֒→ M 2 be its inclusion into M 2 . If U is small enough that π • ι U : U → M cyl is an embedding, then we can show from (2.32) that (3.12) Furthermore, ι U is itself an isometric embedding, hence Combining these equations, we find (3.14) One can then show that M 2 is covered by open sets U for which (3.14) holds, and thence that π * P M cyl = P M 2 π * . By acting on the left-hand side of (3.11) with π * P M cyl and the right-hand side with P M 2 π * , we are then able to see why these maps are fundamental solutions to P M cyl .
Throughout this section we shall use the following coordinates for M cyl . Let U = (0, 2π) × R ⊂ R 2 , then (3.15) And, by a standard abuse of notation, for ϕ ∈ E(M cyl ), we shall write (ϕ • ρ)(u, v) as simply ϕ(u, v). As the (u, v) coordinates parametrise M cyl up to a set of measure zero, they are sufficient to define integration on M cyl . In turn, this allows us to define an integral kernel for E cyl by which we may then write in terms of the integral kernel of E as Once again, we see the characteristic splitting of the u-dependent and v-dependent terms of E cyl , which we write E cyl = E ℓ cyl + E˚r ffl cyl , just as before. Just as with Proposition 3.1, we can define a chiral bracket {·, ·} ℓ on F µc (M cyl ) using (∂ u ⊗ ∂ u ) E cyl instead of E cyl , yielding the chiral Poisson algebra P ℓ (M cyl ). The proof that P ℓ (M cyl ) is a Poisson algebra and that ∂ * u : F µc (M cyl ) → F µc (M cyl ) is a Poisson algebra homomorphism carries over essentially unchanged from M 2 . For our choice of chart, we always have that −2π < u − u ′ < 2π, thus the integral kernel for the chiral bracket can be written We shall perform our next set of calculations using {·, ·} ℓ . In an effort to avoid confusion, when we are working in P ℓ (M cyl ), we shall denote the field configuration input to the functional by ψ. We think of ψ as ∂ u ϕ which is realised when we apply the algebra homomorphism . We first define the family of functionals {A n } n∈Z ⊂ F(M cyl ) by Their derivatives are given by  (1) n [ψ] is the conormal bundle to Σ 0 . However, we shall see that they still posess a well-defined chiral bracket, and generate a closed algebra with respect to it.
A direct computation of the chiral bracket yields where we suppress the constant functional for convenience.
This demonstrates that the Lie algebra generated by the A n with the Lie bracket {·, ·} ℓ is isomorphic to the Heisenberg algebra. Moreover, as ∂ * u is a Poisson algebra homomorphism, we see that the algebra generated by A n := ∂ * u A n with the Peierls bracket is also isomorphic to the Heisenberg algebra.
Quantising this family of functionals is relatively simple. Let H ∈ Had(M cyl ) be some Hadamard distribution. As the functionals A n are linear, the definition of the ⋆ H product implies the familiar Dirac quantisation rule is valid: Of course, there is nothing particularly special about the choice of Σ 0 as the Cauchy surface. From the covariance of the Peierls bracket we already know that, for any isometry χ ∈ Aut(M cyl ), the family of functionals {χ * A n } n∈N has the same commutation relations as {A n } n∈N . Moreover, we can see in these functionals the beginnings of conformal covariance, which will be explored further in Section 4. In null coordinates, we can define a conformal transformation of the cylinder as χ[u, v] = [µ(u), ν(v)] where the pair of functions µ, ν ∈ Diff + (R) satisfy µ(u + 2π) = µ(u) + 2π and ν(v + 2π) = ν(v) + 2π. One can then show that the family {χ * A n } n∈N still has the same commutation relations as before in the case that χ is conformal.
We can define a family of functionals akin to A n : , and one can show that, if γ = χ• γ 0 for some conformal transformation χ, then χ * A n = A γ n . In fact, for any other Cauchy surface Σ of M cyl , it is possible to find a conformal transformation χ such that γ = χ • γ 0 is a parametrisation of Σ, hence A γ n is a copy of the Heisenberg algebra associated with the surface Σ. As a sketch: χ is obtained by taking a right-moving null ray passing through a point [u, −u] ∈ Σ 0 , and finding the unique point [u, v] ∈ Σ lying on the same ray. This defines the map ν such that ν(−u) = v, which one can show is an element of Diff + (S 1 ), then any choice of µ ∈ Diff + (S 1 ) completes the definition of χ, for example just the identity function.
These A γ n will not be needed in this paper. However, functionals of this form prove vital for defining truly chiral (i.e. 1-dimensional) algebras as emerging from locally covariant field theory. We shall explore this further in a future paper.

The Virasoro Algebra
As the Virasoro algebra arises from quadratic functionals, the ordering ambiguities we could previously disregard become relevant, and we cannot so easily carry computations from Minkowski space over to the cylinder. To start, the classical functionals are defined analogously to the A n functionals. Again, we begin by defining a family {B n } n∈Z ⊂ F(M cyl ), by B n [ψ] := 2π u=0 e inu ψ 2 (u, −u) du.
As before, we shall compute the chiral bracket of B n with B m in order to obtain the Peierls bracket for the functionals B n := ∂ * u B n . For future reference, the functional derivatives of B n are Here again, the wavefront set of B (1) n [ψ] is contained within the conormal bundle of Σ 0 and hence B n is not microcausal. Moreover, we see that, like A n , these functionals are additive, which means that the support of B (2) n , and hence that of B (2) n , is contained within the thin diagonal ∆ 2 ⊂ M 2 cyl . This will be vital when we later apply the locally covariant Wick ordering prescription outlined in Section 2.5 to these functionals.
The chiral bracket of B n with B m is given by As explained in Section 2.4, it is inconvenient to work directly with A(M cyl ). Instead, we perform our computations in A H (M cyl ) for some suitable choice of Hadamard distribution H. The simplest choice is to take H = W cyl − i 2 E cyl , where W cyl is the ultrastatic vacuum for the cylinder, uniquely distinguished by the fact that it is invariant under time-translations. The integral kernel of W cyl may be written Unlike for the massive scalar field, time-translation is not enough to fix the kernel of W cyl uniquely, owing to the presence of zero mode solutions to the massless Klein-Gordon equation. However, this is no issue in the algebraic approach to QFT, as the construction of our algebra of observables is independent of any choice of ground state and, hence, of any way in which we may choose to handle the problem of zero modes.
Moreover, we are concerned with the ⋆ products of functionals which depend on the field configuration ϕ only through one of its null derivatives. In effect, this means we only depend on W cyl to define the 2-point function for the derivative field (3.28) Taking this derivative annihilates any zero-modes, thus there is no ambiguity in defining the integral kernel of (∂ u ⊗ ∂ u )W cyl .
If we consider the ⋆ H product of two functionals of the form ∂ * u F , we find Analogously to Proposition 3.1, we can hence define a chiral subalgebra of ⋆ H via the following: , equipped with the associative product ⋆ H,ℓ defined by we may now compute the product B n ⋆ H cyl ,ℓ B m . In the abstract algebra, this amounts to computing ⦂B n ⦂ H cyl ⋆ ⦂ B m ⦂ H cyl . Later, we shall compare this to the product of the covariantly ordered B n .
As the B n functionals are quadratic, the power series for their star product truncates at O( 2 ). Thus, it may be written in full as (3.31) First, let us consider the O( ) term (3.32) We can simplify this slightly by reintroducing the A n functionals. Upon doing so, we find (3.33) (Note that for any function ψ the above series is absolutely convergent as the smoothness of ψ guarantees |A n [ψ]| decays rapidly in n.) For the commutator, we need only the anti-symmetric part of (3.33), which is markedly simpler. For now, however, we proceed to compute the O( 2 ) term. To do this, we need the following form of the squared propagator: This can be obtained naïvely by just squaring (3.27) and applying the Cauchy product formula. For a proof that this indeed converges to the correct distribution, see Appendix C. We then find  (3.37) The first series is anti-symmetric under an interchange of n and m, whereas the latter is symmetric and can thus be disregarded. Next, we take two copies of the anti-symmetric series, for the first copy we make the change of variables k → (n − k), and for the second we choose k → (k − m). Recombining these two copies we find Using the * -algebra homomorphism ∂ * u from Proposition 3.2, we can then conclude that Finally, applying α H−H cyl and using the identity (2.27) we obtain the commutation relation It is curious that at this stage we have commutators recognisable as what one might call the 'planar' Virasoro relations (for example [Kac98, (2.6.6)]) for a central charge c = 1, despite the fact that all the functionals in question belong on the cylinder. We will now compute the correction to these relations which occurs when adopting the locally covariant Wick ordering prescription. In doing so, we shall see the result is the 'radially ordered' Virasoro relations.
Recall from section 2.5 that, heuristically, locally covariant Wick ordering is normal ordering with respect to the Hadamard parametrix. In the case of the Minkowski cylinder, the Hadamard parametrix (2.38) is particularly simple. Locally the cylinder is isometric to Minkowski space, hence the parametrix of the cylinder coincides with that of Minkowski. For an arbitrary choice of length scale λ, the singular part of a Hadamard distribution for the undifferentiated field ϕ is Here it is clear that the parametrix exists only locally, as W sing is not spacelike periodic. Passing over to the differentiated field ψ, the singular term becomes (3.43) For the cylindrical vacuum, we have We can think of the above series formally as the derivative of a geometric series. Replacing u − u ′ with z ǫ = u − u ′ − iǫ makes this series absolutely convergent for ǫ > 0, thus we can write the 2-point function as (3.45) Performing an asymptotic expansion of this function near the coincidence limit u−u ′ = 0, we find Which provides an explicit verification that the vacuum state differs from the parametrix only by the addition of a smooth, symmetric function. Moreover, this allows us to calculate :B n : M cyl . As we are working in A H cyl (M cyl ), we need only compute the functional :B n : M cyl H cyl , which is given by Recall that ⦂ − ⦂ H cyl can be interpreted as normal ordering with respect to the vacuum H cyl . Moreover, we established the Hadamard parametrix H sing of the cylinder is effectively the 2-point function of the Minkowski vacuum, embedded into some suitable neighbourhood of ∆ ⊂ M 2 cyl . Accordingly, (3.39) computes the commutation relations for Fourier modes of the stress-energy tensor normally ordered with respect to H cyl , and (3.50) the same but ordered with respect to the Minkowski vacuum.
In the standard approach to CFT in two dimensions, one typically imposes (3.39) as the standard commutation relations for Laurent modes of the stress energy tensor, here understood as a field over the complex plane in a particular sense. Then, mapping the plane to the 'cylinder' via the map z → e iz , one may obtain the radially ordered commutation relations, concordant with (3.50). However, in our framework, it does not make much sense to speak of a Virasoro algebra for the plane, as there is no suitable notion of mode expansion for the stress-energy tensor. In fact, arguably the most significant differences between our approach and the VOA framework is that the latter relies on mode decomposition in order to analyse the singularity structure of quantum fields, whereas we instead use tools from microlocal analysis.

Connection to Zeta Regularisation
There is a well known trick in the physicists' literature to explain (3.49). Firstly, recall that we can write a given B n functional as an infinite series over A m functionals (which is point-wise convergent) as: (3.51) The ⋆ H cyl product of two such functionals is In particular, for n = 0 this means that A k · A n−k = A k ⋆ H cyl A n−k . Hence, we can define a family of observables {(L n )} n∈Z * ⊂ A(M cyl ) by replacing the classical pointwise product · in (3.51) with ⋆. This family would then coincide with {⦂B n ⦂ H cyl } n∈Z * . For n = 0, we may still replace the pointwise product with ⋆ H cyl , but the ordering of the functionals is now significant. Naïvely replacing the classical pointwise product · in (3.51) for n = 0 by the ⋆ product yields the quantum observable (3.53) Casting rigour aside, we could then 'reorder' B 0 by moving every A k in the second series to the left hand side of the product, which would produce the infamous divergent series (3.54) The rigourous and covariant way of reordering B 0 , as we saw in the previous section, is to apply the map α H cyl −H sing . If we define w cyl (u) := (∂ u ⊗ ∂ u ) H cyl (u; 0) − H sing (u; 0) , where we exploit translation invariance to write w cyl as a function of a single variable, then we can write the normally ordered form of B 0 as (3.55) By approximating both H cyl and H sing by smooth functions, we can write where here B k denotes the k th Bernoulli number. This explains the appearance of ζ(−1) in the normal ordering of B 0 without any recourse to intermediate divergent series.
To close out this section, we make a brief remark about how our notion of normal ordering corresponds to the procedure of shuffling creation operators past annihilators, or similarly the normally ordered products of chiral fields [Kac98, (2.3.5)].
Consider the classical product of a collection of A m i , the functional derivative of this may be written where we make use of the fact that we can canonically identify linear classical observables with their quantum counterparts. Applying this procedure iteratively, if we assume that the sequence i → m i is monotonically decreasing, then we can write Given that [A m , A n ] = 0 whenever m and n are either both negative or both positive, we have recovered the familiar result that normal ordering moves A m "to the right" if m ≤ 0 and "to the left" if m > 0.

Conformal Covariance
So far, our classical and quantum algebras of observables are insensitive to any conformal symmetries a given theory may possess. This is because the morphisms in Loc are isometric embeddings, required to preserve the metric exactly. To study the contidions for and consequences of conformal covariance, we must relax this condition to allow conformally admissible embeddings. The category CLoc -first introduced by Pinamonti in [Pin09] -is the natural setting for the study of conformal field theories. It comprises the same objects as Loc, but enlarges the collection of morphisms to conformally admissible embeddings. As one might expect, we upgrade the concept of locally covariant field theory to locally conformally covariant field theory simply by replacing the category Loc with CLoc. In the next section, we show explicitly how this may be done for a large class of classical theories, and for the conformally coupled scalar field in the quantum case.

Conformal Lagrangians
In this section we shall outline the language necessary to identify a particular Lagrangian (more precisely, its corresponding action) as being conformally covariant. In order to do so we must first introduce some notation.
Definition 4.2 (Weighted Pushforward/Pullback). Let χ : M ֒→ N be a conformally admissible embedding with conformal factor Ω 2 . Given ∆ ∈ R, the weighted pushforward with respect to ∆ is defined by where χ * denotes the standard pushforward of test functions (2.30). Similarly, we define the weighted pullback with respect to ∆ by In the following proposition, we collect some useful properties of these maps.
Proof. The first of these results is easiest to see as a consequence of the other two, thus we defer its proof until the end.
Result 2 can be obtained by a direct computation. Firstly, note that if χ * g N = Ω 2 χ g M , and ρ * g O = Ω 2 ρ g N , then the conformal factor for ρ • χ is given by (ρ • χ) * g O = (Ω χ · χ * Ω ρ ) 2 g M . If we select some arbitrary ϕ ∈ E(O), then To prove 3, first note that, because supp χ (∆) * f ⊆ χ(M), we may restrict the first integral to χ(M), where we may consider χ to be a diffeomorphism. Next, recall that a standard result for conformal transformations states χ * (dVol N ) = Ω d dVol M . From this we find Finally, to prove 1, let f ∈ D(M) and take some arbitrary test function h ∈ D(O).
Using the two results we have just established, we see that Thus, as this holds for every choice of h ∈ D(O), we can conclude that ρ Using these definitions, we can then state the condition required for the theory arising from a natural Lagrangian L to be conformally covariant.
where S ′ M is the Euler-Lagrange derivative of L M as defined in (2.7). In this case, we call L a conformal natural Lagrangian.
We can state this condition more elegantly by once again taking the BV perspective where, instead of focussing on the natural Lagrangian L, we use its associated differential δ S : V µc ⇒ F µc .
Firstly, we can use the weighted pullback to define a modification of the functor assigning a spacetime its classical observables, F µc . For ∆ ∈ R, let F (∆) µc be a functor CLoc → Vec which assigns to each spacetime M its microcausal observables F µc (M) as usual, but assigns to χ ∈ Hom CLoc (M; N ) the morphism (4.4) Proposition 4.1 assures us these morphisms compose as they should. Moreover, by using we can see that the wavefront sets of functional derivatives are independent of the choice of ∆. Then, by noting that the joint future/past lightcones V n ± are preserved under pullback by χ are, and both preserved under pushforward by a conformal embedding, the wavefront set spectral condition (2.5) is also preserved. Hence F (∆) Similarly to F µc , for any choice of weight ∆, we can define an extention V (∆) is again the weighted pushforward of test functions. Recall that we defined local covariance in the BV formalism as the condition that δ S is a natural transformation V µc ⇒ F µc , where each is a functor Loc → Vec. Similarly, (4.3) simply states that such a theory is conformally covariant if the same collection of maps comprising δ S also define a natural transformation δ S : V µc , where each is now a functor CLoc → Vec.

Conformally Covariant Classical Field Theory
We can now see how the criterion for conformal covariance that has just been outlined gives rise to classical dynamical structures which vary as one would expect under conformal transformations. The first result compares the linearised equations of motion on two spacetimes related by a conformally admissible embedding.
Proposition 4.2. Let L be a conformal natural Lagrangian which satisfies the linearisation hypothesis (2.9). If χ ∈ Hom CLoc (M; N ) and ϕ ∈ E(N ), then where each differential operator has been implicitly restricted to the space of test functions of the appropriate spacetime.
Proof. The proof is effectively a direct computation. Let g ∈ D(M) and h ∈ D(N ).
Recall from the definition of P N that This then allows us to employ (4.3) as Note the first equality is not immediately obvious: rather, it follows from the locality of L N . In the following line we use (4.3) and, for the final equality, we note that χ (d−∆) * is the adjoint of χ * (∆) . As the choice of h is arbitrary, we may then conclude that the two operators coincide.
Remark. As P N [ϕ] and P M [χ * (∆) ϕ] are both self-adjoint, we can write an equivalent form of (4.6) for linear maps E(N ), namely When a pair of normally-hyperbolic differential operators are related in the above manner, we can similarly relate their fundamental solutions. The following proposition, which reduces to [Pin09, Lemma 2.2] in the particular case of the conformally coupled Klein-Gordon field in 4D, establishes the conformal covariance Pauli-Jordan function arising from a suitable conformal natural Lagrangian. To simplify notation, we shall refer only to a single differential operator on each spacetime, i.e. we suppress the dependence on an initial field configuration ϕ or χ * (∆) ϕ, though this does not mean that the scope of the result is limited to free theories. (4.11) Proof. Recall that the advanced and retarded propagators of P M are uniquely determined by their composition with P M and their support properties. As such, we simply need to establish that the operator on the right-hand side of (4.11) satisfies the relevant criteria.
Firstly, if we act on this operator with P M we see (4.12) If we denote by P c M the restriction of P M to D(M), and likewise P c N , by the symmetry of these operators, we have that Thus, acting on P c M with our candidate propagator yields which is again simply 1 D(M) .
Finally, we must determine the supports of these functions. Let f ∈ D(M). Note that supp (χ Pulling this back to M, we have Conformally admissible embeddings preserve causal structure. In particular, if γ : [0, 1] → M is a causal, future/past-directed curve, then χ • γ is also causal and future/pastdirected. This means that χ J ± M (supp f ) = J ± N (χ(supp f )) . Hence our candidate propagators also meet the desired support criteria, and must genuinely be the advanced and retarded propagators for P M as required.
One can show that conformal invariance as defined in appendix D of [Wal10] implies (4.10), so long as it is also assumed that P M and P N are symmetric in the sense that f, Similar to the case of (isometric) local covariance, the consequence of proposition 4.3 is that we can define a symplectomorphism from the solution space of P M to that of P N . Recall that we can identify the space of solutions to P M with D(M)/P M (D(M)). If f, g ∈ D(M), then (4.13) Moreover, from (4.6), it follows that χ As was the case in Section 2.5, this symplectomorphism of solution spaces in turn gives rise to a Poisson algebra homomorphism relating the Peierls brackets for each spacetime. A quick calculation shows that the map F (∆) µc χ defined in (4.4) is a Poisson algebra homomorphism: for F , G ∈ F µc (M), ϕ ∈ E(N ) we have that We may summarise the above results as ensuring that the following is well-defined: Example 4.1 (The Conformally Coupled Scalar Field). The simplest example of a conformal natural Lagrangian is that of the conformally coupled scalar field. For spacetimes of dimension d, this is given by, for M ∈ CLoc, f ∈ D(M), ϕ ∈ E(M) where R M is the scalar curvature function for the spacetime M and ξ d = d−2 4(d−1) is the conformal coupling constant.
In this case, we can see that the Euler-Lagrange derivative satisfies the desired covariance property with ∆ = (d−2) 2 . Even in this example we see the necessity of phrasing (4.3) in terms of variations of the action. Naïvely, we may have assumed conformal covariance to be given by . However, the presence of the test function f in the above Lagrangian prevents the integration by parts necessary for this equation to hold.

Conformally Covariant Quantum Field Theory
In order to discuss quantisation, we must return our attention to free field theories. In doing so we can once again refer unambiguously to a single operator P M producing the equations of motion on M.
We saw in section 2.4 that quantisation of a free field theory is achieved through the use of arbitrarily selected Hadamard distributions for each P M . The covariance of the quantum algebras was thus dependent on the fact that, given an admissible embedding χ : M → N , the pullback of a Hadamard distribution on N by χ is again a Hadamard distrbution on N . We have already seen that the weighted pullback of the causal propagator on N is the causal propagator on M. The following proof, again adapted from [Pin09], gives the corresponding result for Hadamard distributions.
Proposition 4.4. Let χ ∈ Hom CLoc (M; N ) be a conformally admissible embedding with conformal factor Ω, and let P M , P N be a pair of normally hyperbolic differential operators satisfying is a Hadamard distribution for P N , then Thus, all that remains to be shown is that W M has the appropriate wavefront set: As a distribution in D ′ (M 2 ), as opposed to a continuous map (4.16) This differs from the usual pullback χ * W N only in the multiplication by the smooth function At this point it is convenient to regard χ(M) as a spacetime in its own right, with all the relevant data being that inherited from N by restriction. We then observe that χ factorises as ι • ξ, where the inclusion ι : χ(M) ֒→ N is an isometric embedding, and ξ : M → χ(M) is a conformal diffeomorphism. With this, we write χ * W N = ξ * (ι * W N ). As ξ is a diffeomorphism, we know that WF (ξ * (ι * W N )) = ξ * WF(ι * W N ), and, since ι is an isometric admissible embedding WF(ι * W N ) = Γ χ(M) , where Γ M = WF(W ) for any (and hence every) Hadamard distribution W on M.
If we, by a slight abuse of notation, write W M = χ * (∆) W N , then the above proposition can be expressed as χ * µc χ defined in the previous section, creates the algebra homomorphism required to make the quantum theory conformally covariant.
Firstly we observe that, if H M is the symmetric part of W M etc, then a quick computation confirms that In other words, for a Hadamard distribution H N ∈ Had(N ), F µc χ defines a * -algebra homomorphism A H M (M) → A H N (N ), using the notation introduced in (2.26).
To see that these maps define a homomorphism A(M) → A(N ) note that, if H ′ N ∈ Had(N ) and H ′ M := χ * (∆) H ′ N then, using (4.5), one can show that hence our homomorphisms are compatible with the isomorphisms between different concrete realisations of A(N ) as required.
Thus we have shown that the following definition makes sense.
Definition 4.5 (The Quantum Massless Scalar Field). Let L : D ⇒ F loc be the conformal natural Lagrangian of the massless scalar field in spacetime dimension d. The locally conformally covariant quantum field theory associated to L is a functor A : CLoc → * -Alg, which assigns • To every spacetime M ∈ CLoc, the algebra A(M) defined in Section 2.4.

Primary and Quasi-Primary Fields
Now that we have constructed the quantum theory of the massless scalar field, we can begin comparing our formalism to the standard CFT literature. In formulations of CFT descended from the Osterwalder-Schrader axioms, one defines a field ϕ(z,z), to be primary with conformal weights (h, h) ∈ R 2 if, for a holomorphic function z → w(z) In order to reach an analogous definition of a primary field within the AQFT framework, we must equip our spacetimes with frames. As a motivating example, Minkowski space is naturally equipped with the frame (in null coordinates) (du, dv). The Minkowski metric is then simply ds 2 = du⊙dv, where ⊙ denotes the symmetrised tensor product. A general conformal automorphism, χ, of Minkowski space can be written in the form where either µ, ν ∈ Diff + (R) or Diff − (R). This is readily shown to be conformal as, for (4.20) Hence the conformal factor is the product Ω 2 (u, v) = µ ′ (u)ν ′ (v). To generalise this concept to arbitrary globally-hyperbolic spacetimes, we make the following definition. sends objects in CFLoc to objects in Loc.
A morphism χ : (M, (e ℓ , e˚r ffl )) → (N, (ẽ ℓ ,ẽ˚r ffl ))) is a smooth embedding χ : M ֒→ N such that if M and N are the spacetimes obtained in the above manner from (M, (e ℓ , e˚r ffl )) and (N, (ẽ ℓ ,ẽ˚r ffl ))) respectively, then χ ∈ Hom CLoc (M; N ) . In other words, χ is a conformally admissible embedding of M into N with respect to the metrics and orientations induced by their coframes.
As every 2D globally hyperbolic spacetime is parallelisable, each may be expressed as the spacetime induced by some object of CFLoc, i.e. the map (4.21) is surjective. Furthermore, from the definition of the morphisms in CFLoc, it is evident that this map extends to a fully faithful functor p : CFLoc → CLoc, hence we have an equivalence between the two in the sense of category theory.
Rather than relying solely on this equivalence, however, the following proposition provides a test of whether an embedding χ : M ֒→ N is conformally admissible with respect to the spacetime structure induced by the frames (e ℓ , e˚r ffl ) and (ẽ ℓ ,ẽ˚r ffl ). Proof. Suppose first that the embedding χ satisfies (4.22), then it is clearly conformal, as χ * (ẽ ℓ ⊙ẽ˚r ffl ) = Ω 2 (e ℓ ⊙ e˚r ffl ), (4.23) where the conformal factor is Ω 2 = ω ℓ ωr ffl . To show it is admissible, consider first where the final equality comes from the fact that the product ω ℓ ωr ffl is everywhere positive. Hence, ω ℓ ωr ffl (e ℓ ∧ e˚r ffl ) defines same orientation as e ℓ ∧ e˚r ffl , establishing that χ is orientation preserving.
(4.27) Thus (4.22) is a sufficient condition for χ to be a conformally admissible embedding.
Conversely, let us now assume that χ is conformally admissible. Letẽ ℓ/rffl | χ(M ) denote the restriction ofẽ ℓ/rffl to the image of M under χ. As χ is conformal, the pull-back of each of these 1-forms must be a null 1-form on M with respect to the induced metric. At every point p ∈ M, this tells us that χ * ẽℓ | χ(M ) (p) must be colinear with either e ℓ (p) or e˚r ffl (p). That it must be colinear with e ℓ (p) in particular is due to the fact that χ preserves orientation; a similar argument can then be made forẽ˚r ffl . Thus we have two functions ω ℓ/rffl ∈ E >0 (M) such that χ * ẽℓ | χ(M ) (p) = ω ℓ/rffl e ℓ/rffl . Their product is the conformal factor of χ and hence must be positive. Finally, for χ to preserve time orientation, ω ℓ and ωr ffl must satisfy (4.27), thus each function must be everywhere-positive.
Using these frames, we can define a modified pushforward, similar to (4.1), except now with a pair of weights (λ,λ) ∈ R 2 specified. The weighted pushforward of a test function f ∈ D(M) under a morphism χ : M → N with left/right conformal factors ω ℓ/rffl is given by With this functor, we can finally define a primary field of weight (h, h) to be a natural transformation Φ : D (h, h) ⇒ A, where A : CFLoc → Vec is a locally covariant QFT, which may or may not be the 'pullback' A • p of some theory A : CLoc → Vec. Explicitly, this means that, if M is the spacetime constructed from M ∈ CFLoc according to (4.21), and likewise N arises from N ∈ CFLoc, then we have a pair of linear maps Φ M/N such that, for any χ ∈ Hom CFLoc (M; N), the following diagram commutes Heuristically, we can see how this definition relates to (4.18) by taking the 'limit' of Φ M (f ) as f → δ x , the Dirac delta distribution localised at x ∈ M. Whilst there is no guarantee that Φ M (f ) converges in this limit, (4.29) does converge in the weak- * topology to ω ℓ (x) h ωr ffl (x) h δ χ(x) . If we imagine for a moment that Φ M (x) := lim f →δx Φ M (f ) is welldefined, the statement that Φ is primary with weights (h, h) implies (4.31) Recalling that, if χ : M 2 → M 2 is expressed in null coordinates as χ(u, v) = (µ(u), ν(v)), then ω ℓ = dµ/du and ωr ffl = dν/dv, we see that we have recovered a Lorentzian signature analogue of (4.18) as deisred.
We can also recover the physical interpretations of the sum and difference of h and h, referred to as the scaling dimension ∆ and spin s of the field respectively. For the scalar field, we have already encountered the scaling dimension as the number ∆ appearing in, for example, Definition 4.4. If we consider a field with spin s = 0, the action of the corresponding D functor is (4.32) The right hand side of which is precisely the action of the functor D (∆) as defined in [Pin09]. Hence, any primary field à la Pinamonti's definition Φ : D (∆) ⇒ A defines a primary field of spin 0 in our description: Conversely, a choice of spin 0 primary field Φ : D (∆/2,∆/2) ⇒ A • p unambiguously defines a natural transformation Φ : D (∆) ⇒ A. To see this, note that if M and M represent different frames for the same spacetime M = p(M) = p( M), then the identity morphism of the underlying manifold constitutes a CFLoc morphism M → M, hence we can deduce from (4.30) that Φ M ≡ Φ M . In other words, the spin of a primary field measures how it behaves under a change of frame on a fixed spacetime. Thus, if the spin vanishes, the primary field does not depend on the frame, and can be defined in the same way as in [Pin09].
where e ℓ is the vector field dual to e˚r ffl . To see that this is a primary field consider the upper-right path through the diagram (4.30): To compare this with the lower-left path, we first observe that the algebra isomorphisms α χ * H ′ −H all act by identity on linear functionals, thus if F is linear, Hence the observable we obtain in this way is By fixing (h, h) such that the diagram commutes, we can therefore conclude that ∂Φ is a primary field of weight (1, 0). Similarly, if we consider the field∂Φ, obtained by acting with er ffl instead of e ℓ , we would obtain a primary field of weight (0, 1).
The introduction of frames also allows us to implement rigid transformations. We define the boost and dilation morphisms, for α ∈ R \ {0} as b α : (M, (e ℓ , e˚r ffl )) → (M, ( 1 α e ℓ , αe˚r ffl )), d α : (M, (e ℓ , e˚r ffl )) → (M, (αe ℓ , αe˚r ffl )), where in each case, the smooth embedding inducing the morphism is simply Id M . If we denote the subcategory generated by these morphisms CFLoc 0 , and the restrictions of D (h, h) and A to this subcategory D (h, h) 0 and A 0 respectively, then a quasi-primary field may be defined as a natural transformation D (h, h) 0 ⇒ A 0 , for some pair of weights (h, h) ∈ R 2 . In other words, a field is quasi-primary if it responds to boosts and dilations in the same way a primary field would.
For the massless scalar field, we identify several notable examples of primary and quasiprimary fields below: 1. As demonstrated in the above example, the derivative fields ∂Φ and∂Φ are both We can compare this to :e iaΦ N :, where we have, for g ∈ D(N ) As e iaΦ is a classical primary field of scaling dimension 0, we have e iaΦ as required.

The Stress-Energy Tensor of the Massless Scalar Field
A well known feature of chiral CFTs is the transformation law for the stress-energy tensor, constrained by the famous Lüscher-Mack theorem [LM75]. Here we shall show explicitly that, for the free scalar field in 2D Minkowski space, the stress-energy tensor satisfies precisely this transformation law. And, moreover, that there exist analogous transformation laws on arbitrary globally-hyperbolic spacetimes.
The Note that we can replace the test function f with a compactly supported distribution, so long as its singularity structure is compatible with the constraint that T M (f ) is a microcausal distribution. In particular, the generators of the Virasoro algebra B n from section 3.3 can be expressed as T M cyl (f n ), where the integral kernel of f n is e inu δ(u + v) in the null-coordinates for the cylinder.
Clasically, T is a primary field with conformal weight (2, 0), i.e. T : D (2,0) ⇒ P • p, where P is the classical theory for the massless scalar field, as given in definition 4.4. We shall now study how its quantisation :T : fails to be even quasi-primary.
In order to make our analysis more concrete, we restrict our attention to the subcategory of CFLoc containing the single object M 2 . Here, the locally covariant normal ordering prescription : − : We already know that this vanishes in the classical limit → 0, hence we only need to compute the O( ) term. Recall that in null coordinates we can express a CFLoc morphism M 2 → M 2 using a pair of functions µ, ν ∈ Diff + (R) by χ(u, v) = (µ(u), ν(v)). Upon doing so we see where we have integrated out v and v ′ and defined f (u) := R f (u, v) dv. It only remains to determine (4.39) By Taylor expanding µ(u ′ ) around u, one eventually finds that the limit exists and is equal to Thus we recover the well-known result that, on Minkowski spacetime, the quantum stress-energy tensor transforms almost as a primary of weight (2, 0), but is obstructed by an O( ) correction proportional to the Schwarzian derivative of the transformation. We can now use our framework to generalise this result to any globally hyperbolic spacetime. The failure for (4.30) to commute for χ ∈ Hom CFLoc (M; N) is (4.42) Whilst the right hand side of this equation requires an arbitrary choice of H ′ ∈ Had(N ) and ϕ ∈ E(N ), S is actually independent of both of these choices. As in Minkowski space, the classical term cancels and we are left to compute , where the choice of configuration ϕ has been suppressed as no remaining terms depend on it. If we define h ′ = H ′ − H sing N , then one can show that h ′ , T N D (2,0) χ(f ) (2) , which cancels with the smooth part of χ * H ′ , and hence where we are again using the embedding ι ∆ : x → (x, x) ∈ M 2 . If we take χ : M 2 → M 2 to be as above, we then see that S(χ) = S(µ), hence the original Schwarzian derivative is recovered.
Note that the right-hand side of (4.42) can be defined for any confomally covariant QFT. A Lüscher-Mack theorem for pAQFT would then imply that, as a distribution, this is equal to (4.43) up to multiplication by some constant, which we could then interpret as the central charge of the theory. We stress that such a result has not yet been found, however we intend to return to this issue in future work.

Conclusion and Outlook
In this paper we have shown how CFT fits into the framework of pAQFT. As an example application, we have proposed a fully Lorentzian treatment of 1+1 massless scalar field on the Minkowski cylinder and we have shown how the covariant choice of normal ordering of observables leads to correct commutation relations for Virasoro generators. We have also shown that a change of normal ordering leads to the appearance of an extra term ζ(−1), which is usually explained using the zeta regularisation trick. Here we derive this result completely rigorously, using the pAQFT framework.
In our future work we aim to study further how chiral algebras emerge naturally in our framework and how our approach relates to the standard AQFT treatment (local conformal nets) and the factorisation algebras approach [CG16]. We also plan to study OPEs and interacting theories.

Acknowledgements
We would like to thank Sebastiano Carpi, Chris Fewster and Robin Hillier for very inspiring discussions.

A Method of Images
It is well-known that if a space Y can be expressed as the quotient of some other space X under the action of some group (satisfying certain properties), then we can use this relation in order to build Green's functions on Y out of Green's functions. Here we give a coordinate-free account of some of the necessary results, then explain how this method may be used to construct the retarded/advanced propagators of the cylinder from those of Minkowski space. Proof. We claim that the local definitions where ρ α ∈ D(M) such that ρ α ≡ 1 on supp ϕ ∩ supp Uα G can be glued together to form the desired map. Suppose α, β ∈ A such that U αβ = U α ∩ U β = ∅. One can quickly verify that supp U αβ G ⊆ supp Uα G ∩ supp U β G, hence ρ α ϕ| U αβ = ρ β ϕ| U αβ . In particular this means that supp ((ρ α − ρ β )ϕ) ⊂ M \ supp U αβ G and hence G(ρ α ϕ)| U αβ = G(ρ β ϕ)| U αβ , thus Gϕ is a well-defined function.
Theorem A.1 (The Method of Images). Let π : M → M be a regular covering of M by M. Further, let P and P be a pair of differential operators for M and M respectively, such that π * P = P π * . Further, let G be a fundamental solution to P such that 1. There exists a covering α∈A U α = M such that, ∀ K ⊂ M compact, π −1 (K) ∩ supp Uα G is compact, 2. ∀ ρ ∈ Aut(π), ρ * G = Gρ * .
Then there exists a fundamental solution G for P such that π * G = Gπ * Proof. Because supp π * f = π −1 (supp f ), condition 1 tells us that Gπ * f is well defined and satisfies P Gπ * f = G P π * f = π * f Next, 2 ensures that for any ρ ∈ Aut(π) i.e. Gπ * f is a Aut(π) invariant, and hence can be expressed as π * F for some F ∈ E(M).
As our choice of f was arbitrary, this defines a map f → F , which is clearly linear. As such we denote it G : To show that G is then a fundamental solution for P is a fairly mechanical process: From the injectivity of π * , we may then conclude P Gf = f . Next, using the same trick [ϕ] is some differential operator. If M, M ∈ Loc and π : M → M is such that for every x ∈ M, there exists a subspacetime * N ∋ x such that π| N is an admissible embedding, then Proof. Recall that the naturality of L implies that, for every admissible embedding χ : Applying this to and the composed map π| N = π • ι and then to the inclusion ι : N ֒→ M, we have, for ϕ ∈ E(M) and g ∈ D(M) Given that M is covered by N ⊆ M for which this holds, we may conclude π * (P M [ϕ] g) = P M [π * ϕ]π * g as desired.
Given that the equations of motion are related in this way, we can now show that the propagators are as well: For any f ∈ D(M cyl ), supp π * f is clearly timelike compact, i.e. there exists a pair of Cauchy surfaces Σ ± ∈ M 2 such that supp π * f ⊆ J + (Σ − ) ∩ J − (Σ + ). From this it follows that supp π * f is both past-compact and future-compact. The support properties (2.11) of E R/A imply that supp U E R/A = J ∓ (U), where U is the closure of U.
Next, as the symmetries of the covering map (x, t) → (x + 2πn, t) are translations, and E R/A are both equivariant under translations, we have also satisfied condition 2. Applying Theorem A.1, we thus have a pair of propagators E It is straightforward to verify that these satisfy the support criteria (2.11), hence they are the retarded/advanced propagators for the cylinder. Proof. We shall only prove this fact for M ⊆ R d , but it is possible to 'patch together' the results over an atlas for a more general M. We begin by rephrasing Theorem 8.2.13 of [Hör15]: Suppose that X ⊆ R n , and Y ⊆ R m . Let K ∈ D ′ (X × Y ) and u ∈ E ′ (Y ). Theorem 8.2.13 allows us to define a new distribution K • u, with integral kernel Moreover, whenever K • u does exist, we have WF(K • u) ⊆ {(x, ξ) ∈ T * X | ∃ (y, η) ∈ WF(u) ∪ 0 Y , (x, y; ξ, η) ∈ WF(K)} (B.2) Let F , G ∈ F µc (M), the m th functional derivative of their Peierls bracket can be written, ommitting the dependence on a field configuration ϕ ∈ E(M), as follows: where the sum runs over partitions J 1 ⊔ J 2 = {1, . . . , m}, • is the operation described above, and s J 1 ,J 2 : D(M m ) → D(M m ) is an operation permuting the variables of a given test function according to a permutation σ J 1 ,J 2 ∈ S m such that i ∈ J 1 ⇒ σ J 1 ,J 2 (i) ≤ |J 1 |.
(As F (m) is permutation invariant as a distribution, this is a sufficient characterisation of σ J 1 ,J 2 .) In fact, as we are only testing for microcausality, the only property we need of these distributions is that, for 0 ≤ k ≤ m, the wavefront set of F (k+1) ⊗ G (m−k+1) • E is disjoint from the cones V where the relation (x, ξ) ∼ (y, η) means there exists a null geodesic γ : [0, 1] → M connecting x to y and such that the parallel transport of ξ along γ is η. However, for our purposes, we can use the much simpler estimate i.e. if (x, y; ξ, η) ∈ WF(E) then either (x, ξ) ∈ V + and (y, η) ∈ V − , or (x, ξ) ∈ V − and (y, η) ∈ V + .
In order to apply theorem 8.2.13 to F (k+n) ⊗ G (m−k+n) • (χW ) ⊗n for 0 ≤ k ≤ m, we must show that where Y = M 2n comprises the y i variables in the above integral. The justification of this proceeds similarly to before. Firstly, we note the following estimate, obtained by repeated application of 8.2.9 from [Hör15] WF(W ⊗n ) ⊆ (WF(W ) ∪ 0 M 2 ) n \ 0 M 2n .
To carry out the analogous argument for V m − , one instead starts with the observation that

C Squaring the Propagator
In this section, we explain in detail why the expression (3.34) for [(∂ u ⊗ ∂ u )W cyl ] 2 is valid.
To simplify notation, we shall write (∂ u ⊗ ∂ u )W cyl =: w, and denote by w N the truncation of the series defining w to the first N terms.
Theorem 8.2.4 of [Hör15] gives the necessary conditions for the square of a distribution to exist. However, it does not provide a convenient integral kernel with which to evaluate such products on test functions. A good starting point to this end may be found on page 526 of [CP82], where it is stated that for any pair of cones Γ a , Γ b ⊆Ṫ * M such that Γ a ∩ −Γ b = ∅, the multiplication of distributions, considered as a map D ′ Γa (M) × D ′ Γ b (M) → D ′ (M) is continuous in each of its arguments. In other words, if we take some fixed u ∈ D ′ Γa (M), and a sequence v n converging to v in the sense of D ′ Γ b (M), then u · v n weakly converges to u · v, and vice versa for a sequence in D ′ Γa (M). * Note that here we required the tighter restriction on WF(W ) relative to E: if we had covectors (y i ; η i ) ∈ V + and (y j ; η j ) ∈ V − , for i, j ∈ {1, . . . , n}, then it might be possible to find (x F , y F ; 0, −η F ) ∈ WF(F (k+n) ), hence the above intersection would in general be non-empty, preventing us from proceeding any further.
Let Γ ⊆Ṫ * M 2 cyl be a cone which both contains WF(w) and satisfies Γ ∩ −Γ = ∅. We can show that the smooth distributions w N obtained by truncating the sum appearing in (3.27) converge to w in D ′ Γ . Firstly, we shall pick an open subset U ⊂ M 2 cyl which can be identified with an open subset of R 4 . We shall only prove convergence for the restriction of w N to U, though the full result follows from this with little trouble. Following [Hör15, Definition 8.2.2] for sequential convergence, we must show that, for all χ ∈ D(U) and conic V ⊆ R 4 such that supp χ × V ∩ Γ = ∅, sup ξ∈V (1 + |ξ|) k χw(ξ) − χw N (ξ) → 0 as N → ∞. (1 + n) 1−k .
For k ≥ 3, this establishes the convergence desired. For k = 1, 2, we simply pick a stronger bound for T χ .
Thus we can write, for f ∈ D(M 2 cyl ) w 2 , f = lim N →∞ w N · w, f , which allows us to bring all summation outside of the integrals arising from the duality pairing. Noting that w N is a smooth function for all finite N, we can hence evaluate this pairing directly as where, a priori, the sum over m must be performed first.
As f is smooth, the integral is rapidly decaying as a function of n + m, hence the sum is absolutely convergent. Rearranging the double sum accordingly, it is then clear that the sequence of partial sums converges to w 2 in the weak topology of D ′ (M 2 cyl ).