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Article: Dable-Heath, Edmund, Fewster, Chris orcid.org/0000-0001-8915-5321, Rejzner, Kasia orcid.org/0000-0001-7101-5806 et al. (1 more author) (2020) Algebraic Classical and Quantum Field Theory on Causal Sets. Physical Review D. 065013. ISSN 2470-0029 https://doi.org/10.1103/PhysRevD.101.065013

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[email protected] https://eprints.whiterose.ac.uk/ Algebraic Classical and Quantum Field Theory on Causal Sets

Edmund Dable-Heath,∗ Christopher J. Fewster,† Kasia Rejzner,‡ and Nick Woods§

Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom (Dated: February 24, 2020) The framework of perturbative algebraic quantum field theory (pAQFT) is used to construct QFT models on causal sets. We discuss various discretised wave operators, including a new proposal based on the idea of a ‘preferred past’, which we also introduce, and show how they may be used to construct classical free and interacting field theory models on a fixed causal set; additionally, we describe how the sensitivity of observables to changes in the background causal set may be encapsulated in a relative Cauchy evolution. These structures are used as the basis of a deformation quantization, using the methods of pAQFT. The SJ state is defined and discussed as a particular quantum state on the free quantum theory. Finally, using the framework of pAQFT, we construct interacting models for arbitrary interactions that are smooth functions of the field configurations. This is the first construction of such a wide class of models achieved in QFT on causal sets.

CONTENTS 2. S-matrix and interacting fields 21

I. Introduction 1 V. Conclusions and Outlook 22

II. Preliminaries 3 Acknowledgments 22 A. Causal sets 3 B. Causal set Cauchy surfaces 4 A. Interacting star product in terms of graphs 23

III. Classical field theory on a fixed causal set 5 References 24 A. Kinematical structure 5 B. Classical dynamics 7 1. Discretized retarded wave equations 7 I. INTRODUCTION 2. Causal sets with a preferred past structure 8 Presently, our understanding of nature is split into two 3. Retarded Green function for PΛ on the domains: one theory applies to quantum phenomena, and regular diamond lattice 9 is relevant on the small scale; and a very different theory 4. Edge effects at past infinity and the applies to gravity, space, and time, and is important for Cauchy problem 11 large-scale phenomena. Quantum gravity seeks to unify C. Peierls bracket 12 these two into one single description of nature. Whilst 1. Tentative definition 12 many attempts have been made, and various methods 2. Justification of the formula for the bracket 12 suggested, the problem of finding the unified theory of D. Free Dynamics 13 quantum gravity still remains open. One of the funda- E. Interacting theory 14 mental conceptual problems that any such theory has to 1. Interacting and linearized interacting address is the understanding of the nature of space-time equations of motion 14 at small scales and the interplay of geometry and quan- 2. Interacting Poisson bracket 14 tum phenomena. This paper brings together two frameworks that have IV. Quantum Theory 14 been used to develop theories that combine quantum ef- A. Construction of the Quantum Algebra 15 fects and geometry. The first, causal set theory [1–3], 1. Exponential products 15 is based on the idea that the spacetime that we observe 2. Formal power series 17 is not fundamental, but rather emergent from a discrete B. Weyl Algebra 17 underlying structure. It is conjectured that in the small C. States and the GNS representation 18 scale, spacetime is a discrete set of points and the only D. Interacting theory 20 structure on this set is a partial order relation, inter- 1. Motivating the approach 20 preted as the causal structure. The second framework is that of algebraic quantum field theory (AQFT) [4, 5] (see [6] for a recent pedagogi- ∗ [email protected] cal introduction), and its generalization to curved space- † [email protected] times: locally covariant quantum field theory (LCQFT) ‡ [email protected] [7, 8] (see also [9] for review) and perturbative algebraic § [email protected] quantum field theory (pAQFT) [10–13] (see also [14] for 2 review). In LCQFT a model is defined by the assignment drawn to the interesting paper [21] in which discrete of topological -algebras (often C∗-algebras) to globally d’Alembertians are formulated and the corresponding hyperbolic spacetimes∗ and algebra morphisms to causal free theories quantised using the broad methodology of embeddings of spacetimes. This assignment has to satisfy [22]. The approach taken here is complementary in some a number of axioms that generalize the Haag-Kastler ax- respects: ref [21] is concerned with causal sets equipped ioms. In pAQFT, these topological *-algebras are formal with a slicing, which does not appear in our approach, power series in ~ and the coupling constant λ. but is essential to the definition of the symplectic form In this paper we apply LCQFT and pAQFT methods given in [21]. By contrast, we follow the spirit of Peierls to QFT on causal sets. This brings benefit to both causal covariant definition of the Poisson bracket, leading to a set theory and AQFT. In the first instance, the methods quantisation that can be applied to interacting theories. of pAQFT have been successfully applied to construct in- Another interesting contrast is that our use of ‘preferred teracting QFT models in the continuum and now we use past’ structures for one of the discrete d’Alembertians the same framework to construct interacting QFT mod- considered, is much more local in nature than the global els on causal sets. To our best knowledge, this is the first slicing structure of [21]. Nonetheless there are some very instance, where the general framework for introducing close parallels between the resulting discrete equations. interaction in causal set theories has been proposed. The idea of augmenting causal sets with some extra On the AQFT side, studying the discrete models allows structure has a precedent in the works of Cortês and one to avoid many of the technical difficulties related to Smolin [23, 24], where elements of the causal set (events) UV divergences and study in detail the purely algebraic carry momentum and energy, transmitted along causal aspects of pAQFT and how the topology change affects links and conserved at each event. This is local in na- LCQFT. ture, but seems to be very different from our idea of aug- The main advantage of the algebraic framework is that menting the causal set with the ‘preferred past structure’. many of the concepts used in the continuum translate Nevertheless, it would be interesting to look for parallels very straightforwardly to the discrete case. For example, between our approaches. instead of assigning algebras to spacetimes, we assign al- In section 3.3, we discuss relative Cauchy evolution gebras to causal sets. To follow the spirit of pAQFT, (RCE) on causal sets. In [7] the RCE was introduced we start by defining the classical field theory on a causal as the way to characterize the dynamics in LCQFT, see set and then deform it using a simple formal deforma- [9] for further developments, and [25] for an application tion quantization prescription. The problem of defining to the characterization of background independence in classical dynamics on causal sets is, in our opinion, of perturbative quantum gravity. Relative Cauchy evolu- interest on its own, since the usual canonical formal- tion measures the response of the dynamics to a local ism does not apply in this situation. Instead, we use modification of the background spacetime (just as the a variant of the Peierls prescription [15] that allows us stress-energy tensor in a continuum theory is obtained to introduce a Poisson bracket on the space of observ- as a functional derivative of the action with respect to ables. We also show how to introduce interactions in the metric). To define the RCE for causal sets, we first this framework (following [12]) using classical Møller op- identify distinguished regions, which we call past and fu- erators. This is covered in section 3. For our construc- ture infinity, using the notion of layers [1]. Then we tions to work, we need to define, on a given causal set, consider two finite causal sets whose future and past in- the retarded (or advanced) Green function for the dis- finity regions may be identified, so differences between cretized field equation we consider. The retarded Green the sets are localised in between. The RCE measures function is also a starting point in the approach of [16]. the response of the observables (classical or quantum) to We discuss various choices for discretization of the wave that small change of the background causal set. In this equation and for construction of Green functions. These work we study the RCE in the classical theory, but the include one [1, 17–19] which works well for sprinklings (lo- generalization to quantum theory should be straightfor- cally finite subsets of Lorentzian manifolds, constructed ward. We hope that RCE combined with ideas about by randomly selecting points from a given manifold us- dynamical generation of causal sets [26] will allow us to ing a Poisson distribution) [1] and the continuum limit is 1 understand how the evolution of observables on a causal achieved by an averaging procedure. Another choice is set is related to the evolution of the causal set itself. based on an additional ‘preferred past structure’, which In section 4 we quantize the free theory using defor- we introduce in this work. It works well on a regular mation quantization. In particular, we construct the diamond lattice, for example. Weyl algebra from the Poisson algebra of the classical After this paper was completed, our attention was theory and discuss states. We also show how to recover the Hilbert space representation of the Weyl algebra by considering the GNS representation. For the latter, one 1 Although the expectation value of the discretized field equation needs to fix a state and a possible choice in causal set the- converges in mean, the variances diverge unless further nonlo- ory is provided by the Sorkin-Johnston (SJ) state [27, 28]. cal corrections are applied [1]; see [20] for quantisations of such This is a pure state which, as we emphasise, is closely models. connected to a choice of inner product on the space of 3 off-shell linear observables (in a finite causal set this is spacetime event – with a relation satisfying the axioms just RN , N N = 1, 2,... ). The original SJ state of:  is related in∈ this way{ to the} standard Euclidean inner RN x y z = x z, transitivity (1) product on .   ⇒  However, if we want to take the continuum limit, it is x y and y x = x = y, acyclicity (2)   ⇒ better to modify the inner product on the space of linear I(x,y) < , local finiteness (3) observables, so that the state we obtain in the continuum | | ∞ is Hadamard. As shown in [29], the continuum SJ state where fails to be Hadamard on a large class of globally hyper- I(x,y)= z C x z y (4) bolic spacetimes (ultrastatic slabs). It was later proven { ∈ |   } in [30] that modifying the inner product on the space is the set known as the causal interval (or Alexandrov of smooth compactly-supported functions by means of set). We write x y if x y and x = y. The physi- changing the volume form on the underlying space-time cal interpretation≺ of x y is that the6 event x is in the results in the construction of a class of Hadamard states, causal past of y (allowing for equality). Some of the main interpreted as “softened” SJ states. This strategy for ob- building blocks of the theory are defined as follows: taining Hadamard states was first suggested by Sorkin in [16], as an alternative to the construction by Brum Definition II.1. A chain in a causal set (C, ) is a to-  and Fredenhagen [31]. The latter also produces a class tally ordered subset of C. A pair x,y C is a link, de- ∈ of Hadamard states that can be interpreted as “softened” noted x y, if x y and there is no w C such that ≺∗ ≺ ∈ SJ states; an analogous construction for Dirac fields can x w y. In particular, if x y, then I(x,y)= x,y . ≺ ≺ ≺∗ { } be found in [32]. (More discussion appears at the start A path is a chain such that each pair of consecutive ele- of section 4.) ments is a link. The results mentioned above suggest that one should Thus, a finite chain of length n is an ordered set of be able to modify the inner product used for the construc- elements tion of the state in the discrete setting, in such a way that the continuum limit would yield Hadamard state. We x1 x2 xn−1 xn , (5) plan to follow this line of research in our future work. ≺ ≺···≺ ≺ Last but not least, we close section 4 with the construc- while a finite path of length n is an ordered set of elements tion of the quantum interacting algebra AV (C), using the with framework of pAQFT. As mentioned before, this result is x x ... xn xn . (6) of particular interest, since, to our best knowledge, this 1 ≺∗ 2 ≺∗ ≺∗ −1 ≺∗ is the first systematic construction of interacting causal We will denote such path by (x1,...,xn). This is an set quantum field theory models. analogue of a causal curve. In analogy with the continuum, it is convenient to in- troduce the following notation. II. PRELIMINARIES Definition II.2. Given x C, we introduce the causal ∈ A. Causal sets past J −(x)= y C y x A feature common to many quantum gravity theories { ∈ |  } − − is the idea that the fundamental structure of spacetime is of x; for a subset A C we write J (A)= x∈AJ (x). It ⊂ − − ∪ − discrete, and the continuum that we observe is emergent is also useful to define J0 (x)= J (x) x and J0 (A)= − \{ } from this underlying structure. Causal sets originated as x∈AJ0 (x). Analogously, we also introduce the causal ∪ + a suggested space of histories of a “sum-over-histories” ap- future J . proach to quantum theory, analogous to Feynman’s path An interesting class of causal sets are those that can be integral formulation. By discretising spacetime it also formed by taking a subset of points in a Lorentzian man- provides us with a regularization scheme to deal with UV ifold M =(M,g), with a (subset of) the inherited causal divergences in QFT. Causal set theory models spacetime order. These are called embedded causal sets. For exam- as a discrete structure of points, which are linked by a ple a regular diamond lattice can be embedded within causal relation which respects the causal ordering of con- Minkowski spacetime. tinuum spacetimes. Further, the macroscopic volume of The discussion of continuum limits can be facilitated a region of spacetime is proportional to the number of by considering causal sets equipped with a length scale, elements in the causal set contained in the region. Here forming triples (C, ,ℓ). If M =(M,g) is a time-oriented we present an overview of the causal set theory.  D-dimensional Lorentzian manifold, a sequence (Cn, n The mathematical structure of causal sets is that of  ,ℓn) (n N) of embedded causal sets will be said to have a partially ordered set [3, 33]. Thus, the standard con- M as its∈ continuum limit if, for all n, tinuum structure of spacetime is replaced by (C, ), a  discrete set of points C – with each point representing a Cn Cn , p n q = p n q, (7) ⊂ +1  ⇒  +1 4

. C = Cn is dense in M and, for all p,q C, B. Causal set Cauchy surfaces n ∈ D + − S lim ℓn ICn (p,q) = VolM(JM(p) JM(q)). (8) n→∞ | | ∩ Here we consider natural analogues to the notion of a We emphasize that these continuum limits are to be re- Cauchy surface for causal sets. We start with maximal garded as theoretical constructions: a universe that ac- anti-chains [35]. tually is a causal set would be fundamentally discrete with a continuum as an approximation at suitable scales. Definition II.5. An anti-chain is a collection of ele- C Our continuum limits provide one way to control such ments Σ such that x,y Σ neither x y nor y x.A ⊂maximal anti-chain∀ is∈ an anti-chain such≺ that approximations. ≺ In the causal set literature, one often considers ran- any element not in it is related to it, which partitions domly chosen locally finite embedded causal subsets of the causal set as a union of mutually disjoint subsets C =Σ J +(Σ) J −(Σ). a given D-dimensional Lorentzian manifold M =(M,g). ∪ 0 ∪ 0 There is a specific choice of a measure – the Poisson mea- A maximal anti-chain can be regarded as a generali- sure – on these subsets, so that, fixing a length scale ℓ, sation of an instantaneous time hypersurface. For our the probability that a randomly chosen C has n points in purposes it will be more convenient to generalise the idea a volume V is: that a Cauchy surface is a set on which initial data can (ρV )ne−ρV Prob( C V = n)= , (9) be posed for normally hyperbolic operators. For second | ∩ | n! order operators in the continuum, the initial data con- where ρ = ℓ−D is the fundamental density. In particu- sists of the field and its normal derivative; in the discrete lar, the expected number of points in a given spacetime setting the derivative is replaced by a finite difference and volume V obeys it is therefore convenient to replace maximal anti-chains by thickened objects that we will call Cauchy slices. D ℓ E C V = VolM(V ). (10) We start by defining Cauchy slices identified as fu- | ∩ | Causal sets obtained in this way are called sprinklings. To ture/past infinity. In a finite causal set – our main inter- generate the link matrix in a sprinkling, we say that two est – one can always find elements that have no future elements p and q are linked if and only if their Alexandrov or no past. The definition of past and future infinity is neighbourhood does not contain another element of the formulated in terms of layers, as introduced in [1], which sprinkling (see [3] for details). give a meaning to the spacetime separation of two points by using the notion of the causal interval (4) to find a Remark II.3. It is important to note that a generic em- ‘proximity measure’ n between two points: bedded causal set χ : C ֒ M, does not inherit the local → structure of that spacetime, since there could be direct n(x,y)= I(y,x) 1 . (13) links between points x,y C such that χ(x),χ(y) M | |− appear widely separated with∈ respect to the metric∈g. Using this, the i’th layer below x C, Li(x), can be For concrete computations, we typically label the el- defined as: ∈ ements of a causal set by natural numbers. One may − . always choose a natural labelling which respects the or- Li (x) = y C y x, n(x,y)= i . (14) dering such that if xn xm then n < m, n, m N [34]. { ∈ | ≺ } ≺ ∈ We caution the reader that, when we represent a field on One can also define dual layers using the reversed order: a causal set by a column vector φn of its values at xn, + . the elements at the top of the vector correspond to the Li (x) = y C y x, n(y,x)= i , (15) values of φ in the far past. Once this labelling has been { ∈ | ≻ } found, two adjacency matrices can be constructed, both Figure 1 illustrates how the layers are defined for a reg- of which are lower triangular matrices which vanish on ular lattice and a simple sprinkling. the diagonal: The notion of future and past infinity is formalised as Definition II.4. The causal or chain matrix contains all follows. of the relations between any causally related spacetime Definition II.6 (Past and Future Infinity). For n N, elements: − ∈ the n-layer past infinity Cn is defined by 1, if y x Cxy = ≺ (11) − . − 0, otherwise. Cn = x C Li (x)= , i n ( { ∈ | ∅ ∀ ≥ } = x C n(x,y) < n, y x . (16) The link matrix is given by { ∈ | ∀ ≺ } 1, if y x Similarly, the n-layer future infinity is defined by Lxy = ≺∗ (12) 0, otherwise . ( C+ = x C L+(x)= , i n n { ∈ | i ∅ ∀ ≥ } where was introduced in Definition II.1. = x C n(y,x) < n, y x . (17) ≺∗ { ∈ | ∀ ≻ } 5

FIG. 1. An illustration of how layers are defined on a regular diamond lattice (left) and a less symmetric causal set (right).

− As n(x,y) 1 for y x, one notes that C1 consists of Then we have ≥ ≺ + all points having no predecessor; similarly, C1 consists − − − of those with no successor. Further, if y x Cn then Cn Rn (20) ≺ ∈ − ⊂ all z y obey n(y,z)

field φ, remembering that low values of the index corre- A special case is given by linear observables, defined, spond to spacetime events in the ‘far past’. for each f CN ( C = N), by ∈ | | If the causal set is equipped with a length scale, it N . i T Φf : R C; Φf (φ) = f φi f φ, (25) becomes possible to discuss dimensionful fields, saying → ≡ that φ has dimension d to mean dimensions of [length]d. E C RN where φ ( ) ∼= and we have used the Einstein On the basis that a length is a quantity whose numerical summation∈ convention for repeating indices. The space value increases in inverse proportion to a decrease in the of linear observables is denoted by X(C). This space has units of length, a scalar field of dimension d on C should a vector space structure, inherited from F(C), and this transform under a change of length scale ℓ λℓ by 7→ structure is compatible with the addition on the labeling −d space, i.e. Φg + λΦh =Φg λh, for any λ C. φ(p) λ φ(p). (21) + ∈ 7→ Remark III.3. In the last expression of formula (25), we ′ ′ ′ More generally, if (C, ,ℓ) is embedded within (C , ,ℓ ), made implicit use of the Euclidean metric on RN and the  ′ ′  a dimension d scalar field φ on C pulls back to a field induced inner product. This metric allows us to identify on C defined by elements of RN with observables and will be used to raise φ(p)=(ℓ′/ℓ)dφ′(p). (22) and lower indices. To see how this is consistent with the viewpoint on continuum limits and dimensions taken We wish to consider a field theory on a causal set that earlier, consider φ,f that are smooth functions on D- has a claim to be an analogue of the wave equation on dimensional M that have supports intersecting compactly a continuum Lorentzian spacetime. It is therefore neces- (for simplicity) and have dimensions dφ,df . Then one has sary to be able to compare the continuum and discrete −dφ−df situations. One way of doing this is through a suitable ΦfC (φC)= ℓ φ(p)f(p) , (26) continuum limit. First, a dimension d scalar field φ on p∈C a time-oriented Lorentzian spacetime M may be pulled X −d R back to a function φC(p) = ℓ φ(p) on any causal set so if we have a sequence of functions φCn : Cn and C → (C, ,ℓ) embedded in M. This viewpoint allows us to fCn : Cn with continuum limits φ and f respectively, work solely with numerical scalar fields on causal sets. then → Next, consider a situation in which M is the continuum −dφ−df −D D limit of a sequence of causal sets (Cn, n,ℓn), as defined lim ΦfCn (φCn )= ℓ lim ℓ φCn (p)fCn (p)  n→∞ n→∞ in section IIA. We say that a sequence of functions φC : p∈Cn n X C R has a continuum limit as a continuous dimension n −dφ−df −D d field→ φ : M R if = ℓ f(x)φ(x)dµg(x). → ZM d (27) φ(p)= lim ℓnφCn (p) (23) n→∞ Hence the continuum analog of the inner product is the C . C for all p = n n. choice of a volume form ε on spacetime M = (M,g) For example,∈ we will shortly discuss discretised ana- S (take e.g. the invariant volume form dµg induced by logues of the d’Alembertian, which changes dimensions the Lorentzian metric), allowing one to identify f by two powers of length in the continuum. If the contin- C∞(M, C) with the observable ∈ uum limit φ just described is twice continuously differen- c tiable, a family of discrete d’Alembert operators Pn on a sequence of causal sets (Cn, n,ℓn) would therefore be Φf (ϕ)= f(x)ϕ(x)ε(x) . (28) expected to obey  ZM

d−2 The consequences of choosing a different inner product ℓ (PnφC )(p) (φ)(p) (24) n n −→ will be discussed in more detail in section IV A. as n for every p Cn. → ∞ ∈ n Next we introduce the notation for functional deriva- Observables are defined similarly to the continuum tives. The functional derivative of F F(C) at point S case: ϕ E(C) in the direction of ψ E(C) is∈ defined by: ∈ ∈ Definition III.2 (Causal set observables). Observables . 1 on a causal set C are smooth maps from the configuration F (1)(ϕ),ψ = lim (F (ϕ + tψ) F (ϕ)) , (29) space E(C), to C, i.e. they are elements of C∞(E(C), C) t→0 t − ≡ D E F(C). The space of observables is equipped with the nat- where t R. We will also use the notation ural structures of addition ∈ δF F (1)(ϕ) (ϕ) . (30) (F + G)(ϕ)= F (ϕ)+ G(ϕ) ≡ δφ and multiplication E C RN Note that since ( ) ∼= , the functional derivative δF C RN FG(ϕ)= F (ϕ)G(ϕ) . δφ (ϕ) at point ϕ is a linear -valued functional on 7 and therefore can be identified with an element of CN that all points in the same layer below p contribute with and we write its components as δF (ϕ), i = 1,...,N. equal weight to (Pφ) ; we will not impose this and in- δφi p We introduce a product on X(C), induced by the deed will introduce an ‘undemocratic’ example that may component-wise multiplication of the smearing functions be defined on causal sets with a preferred past structure. g CN , or the Hadamard product: Our last general requirement is that each (Pφ)p should ∈ have nontrivial dependence on φ ; in [36] it was assumed . p Φg Φh =Φg∗h , (31) that the coefficient should be independent of p, but one ∗ could certainly envisage prescriptions in which the coef- where (g h)i = gihi, with no summation over the re- ficient was variable and determined by the statistics of ∗ peated indices. the causal order, restricted to the past of p. Another natural product on X(C) is the pointwise In a natural labelling of the causal set, these require- product of observables, inherited from F(C): ments ensure that P is in particular lower triangular and its diagonal entries are all nonvanishing. Consequently, (Φg Φh)(φ)=Φg(φ)Φh(φ) , P is invertible and it may easily be seen that P −1 is also · a retarded operator. Clearly the solution to (33) is then which does not leave X(C) invariant. Let Freg(C) denote φ = E+f, where the subalgebra of F(C) generated by X(C) with respect to . . This is the analog of regular functionals in continuum E+ = P −1K (34) pAQFT.· They form a -algebra, where the operation is just the complex conjugation.∗ ∗ defines the retarded Green operator. Note that the com- posite of retarded operators is retarded. As in [16], we define the advanced Green operator to be . B. Classical dynamics E− =(E+)T , (35) 2 1. Discretized retarded wave equations and the advanced-minus-retarded operator is the anti- symmetric matrix As in continuum QFT, we will construct the interact- E = E− E+ =(E+)T E+. (36) ing theory as a perturbation of a free field equation. The − − − starting-point is therefore a suitable discretization of the By construction, (E f)p is a linear combination (with continuum field equation real coefficients) of fq with p q, and therefore an ad- vanced operator by analogy with previous definitions. We φ = f (32) have followed the existing literature by emphasising the retarded equations and Green operators as the starting- to a causal set. Several possible causal set point. It would be possible, though less physically well- d’Alembertians or ‘box operators’ have been discussed motivated, to base the discussion on advanced operators. previously [1, 18, 19, 36], and we will give a specific ex- As a specific example, we recall the d’Alembertian de- ample below as well as introducing a new type of box 1 fined in [1] (we multiply by a factor of 2 and adapt to operator. We study equations taking the form our sign conventions) Pφ = Kf (33) . (PSφ)p = φp 2 φq 2 φq + φq . neglecting edge effects for the moment – they will be dis- −  − − − −  q∈L (p) q∈L (p) q∈L (p) cussed in Sec. III B 4. Here f,φ E(C) are the source X1 X2 X3  (37) and solution respectively, while P ∈and K are linear maps Sorkin also included a factor of ℓ−2, where ℓ is the funda- on E(C). The map K is newly introduced here, and can mental length scale associated with the sprinkling, which absorb factors (it sometimes turns out to be more conve- is not present here because of the way we treat dimen- nient to discretise 1  rather than ) but also provides 2 sionful fields. In matrix form, additional freedom to determine the way in which a con- tinuum source is discretized. 1, p = q Various requirements on P were set down in [36]. First, (PS)pq = 2, 4, 2, p = q, n(p,q) = 1, 2, 3 respectively in addition to linearity, P is required to be a retarded op- − − 6 0, otherwise, erator, meaning that (Pφ)p is a linear combination of φq  (38) with q p. We also require that K be retarded in this  sense and that both operators are real. As will be seen,  this requirement ensures the causal nature of solutions to (33). Second, the prescription for constructing P and 2 This differs from the convention used e.g. in [12, 14, 37], where K should be independent of the way in which the causal the operator P in the continuum is −, rather than , so that set is labelled – a covariance requirement. In [36] a re- E± in those references are Green functions for − and E ends quirement of ‘neighbourly democracy’ is imposed, namely up with the opposite sign. 8 and is lower-triangular in a natural labelling. In [1] the where continuum limit of the operator (37), averaged over sprin- M −1 klings into two-dimensional Minkowski space 2, was Mean φq = U φq (41) 1  q∈U | | shown to be the continuum d’Alembertian 2 . Gen- q∈U eralizations exist to d-dimensional spacetimes for d > 2, X but involve more layers and different coefficients, to ob- is the arithmetic mean taken over a subset U C. Here it M tain the correct continuum limit for sprinklings into d is necessary to treat points in C− separately because⊂ they [17–19]. 2 do not have preferred pasts. Note that (PΛφ)p involves a sum over points of at most rank 2 below p — to be precise, those in the causal interval between p and its 2. Causal sets with a preferred past structure preferred past Λ(p) — and that the coefficients associated with each contributing point are determined by the rank As an alternative to the principle of neighbourly relative to p and so are independent of the way that the democracy, we propose a new type of discretized causal set is labelled. d’Alembertian for causal sets, which will be investigated It is convenient to present PΛ as a matrix. To this in more detail elsewhere. It is based on a ‘preferred past end, we define a lower triangular matrix Ω with vanishing structure’ defined as follows. diagonal given by Definition III.4. Given a causal set C, a preferred (2- − step) past structure is a map Λ: C C2 C so that, for − \ → 1 Λ(p) q p each p C C2 , the preferred past Λ(p) of p is a point Ωpq = ≺ ≺ (42) of rank∈2 in\ the past of p. The corresponding preferred (0 otherwise, past matrix is a lower triangular matrix with vanishing diagonal entries, given by which encodes information about the causal intervals as- sociated with the preferred past structure, and also a 1 if y = Λ(x), diagonal weight matrix W , Λxy = δΛ(x) y = (39) (0 otherwise. −1 − We will regard the causal interval between Λ(p) and ( q Ωpq) p / C2 Wpp = ∈ − (43) p as an elementary non-atomic volume in the causal set. (0 p C2 . − P ∈ Lemma II.8 shows that every point outside C2 has points of rank 2 in its past. Therefore every causal set in which The second case deals with edge effects to avoid an in- every point has at most finitely many past-directed links finite value. In fact its value will not matter. Then the (and therefore at most finitely many points of rank 2 discretised operator may be written as in its past) admits (at least one) preferred 2-step past structure. P = 1 +Λ 2W Ω. (44) In general, there may exist more than one possible Λ − preferred past structure, in which case a choice must be made. Ideally, there should be some additional rule for One reason for regarding PΛ as a causal set analogue selecting Λ in a given causal set to restrict the choice. of half the d’Alembertian is that it produces a valid dis- 1  For example, one could require that Λ(p) of p is a point cretisation of the continuum operator 2 using regular with maximal layer number (among all the points in the diamond lattices. Consider the lattice (mℓ√2,nℓ√2) : past of p of rank 2). Consider the regular diamond lattice m, n ❩ embedded in M , using ({u,v)-coordinates ∈ } 2 in M2, a portion of which is illustrated in the left-hand related to the standard inertial Minkowski coordinates part of Fig. 1. The points of rank 2 below x are in the by u = t x, v = t + x. The continuum metric and − 2 third row, and there is a unique point with maximal layer d’Alembertian are ds = dudv and  = 4∂u∂v. Each number, i.e., the centre point in that row, belonging to lattice cell therefore has spacetime volume ℓ2 (explaining the third layer below x; therefore the ‘maximal layer rule’ the factor of √2 above), so ℓ is a natural length scale selects a unique preferred past structure in this example. associated with the lattice and indeed one has For general sprinklings, it is not yet clear to us what rule is the most appropriate one. Other possible rules for 2 + − ℓ I(p,q) VolM2 (JM (p) JM (q)) , (45) selecting Λ will be investigated in our future work. | |∼ 2 ∩ 2 Using a preferred past structure, we may introduce a when p and q are widely separated lattice points and J ± new type of discretised retarded d’Alembertian. An ex- M2 ample, developed especially with two-dimensional con- on the right-hand side refer to the causal future/past of tinuum spacetimes in mind, is given as follows: the continuum spacetime. The sequence of such lattices with ℓr = ℓ/r (r N) has M as its continuum limit, ∈ 2 − associating the length scale ℓr with each. We denote the φp 2 Mean φq + φΛ(p) p / C2 (PΛφ)p = − Λ(p)≺q≺p ∈ corresponding causal sets by (Cr, ,ℓr), with ordering    −  φp p C , p q in all cases determined by the causal order of M .  ∈ 2 2 (40)   9

(u,v)

(u,v δ) (u δ, v) − −

(u δ, v δ) − − FIG. 2. Parametrization of points in a segment of a regular diamond lattice, embedded into M2, where δ = ℓ√2.

Suppose, for simplicity, that φ is a smooth dimensionless scalar field on M2, which pulls back to causal set Cr by restriction. Writing δr = ℓ√2/r, we have

(PC , φC )(u,v)= φ(u,v) φ(u δr,v) φ(u,v δr)+ φ(u δr,v δr) . (46) r Λ r − − − − − − Taking Taylor series to second order,

2 δr 2 3 φ(u δr,v)= φ(u,v) δr∂uφ(u,v)+ ∂ φ(u,v)+ O(δ ) − − 2 u r 2 δr 2 3 φ(u,v δr)= φ(u,v) δr∂vφ(u,v)+ ∂ φ(u,v)+ O(δ ) − − 2 v r φ(u δr,v δr)= φ(u,v) δr(∂uφ(u,v)+ ∂vφ(u,v)) − − − δ2 + r (∂2φ(u,v) + 2∂ ∂ φ(u,v)+ ∂2φ(u,v)) + O(δ3) (47) 2 u u v v r with error terms uniform in r. Therefore,

2 3 (PCr ,Λ)(u,v)= δr ∂u∂vφ(u,v)+ O(δr ) , (48) and it follows that

−2 1 ℓ (PC , φC )(u,v) 2∂u∂vφ(u,v)= (φ)(u,v) (49) r r Λ r −→ 2 as r , which is the claimed continuum limit. → ∞

Given this result, a natural choice for K is to set KΛ = preferred past associated with a given causal set. One 1 1 2 . However, this prescription is not the only possibility could remove the element of choice by averaging PΛ over and should be reconsidered near the past boundary of the all such possibilities. causal set if there is one. See further comments below. We remark that Sorkin’s operator PS does not have the 1  continuum limit 2 on the regular lattice; instead, it is 3. Retarded Green function for PΛ on the regular diamond adapted to sprinklings into M2. lattice

The extent to which PΛ, or similar operators, approx- imate the d’Alembertian in higher dimensions on regu- The retarded Green function may be computed exactly lar or sprinkled lattices will be reported elsewhere. Our for PΛ on the regular diamond lattice in M2 for various main purpose in introducing it here is to illustrate the choices of operator K, which may help to illustrate the point that there are discretised d’Alembertians that do additional freedom that it represents. The starting ob- not obey the neighbourly democracy principle, but are servations are that Ω precisely coincides with the link 1 1 still naturally associated with the causal set, augmented matrix L, and that W = 2 . Thus we have by a preferred past structure. Our hope is that some generalization of this ansatz could be applied in arbitrary PΛ = 1 L +Λ. (50) dimensions in such a way that the dimension itself is not − an input (as in the proposals [17–19]), but an emergent Lemma III.5. For the regular diamond lattice, and tak- 1 1 quantity. Typically, there will be more than one possible ing KΛ = 2 , the retarded and advanced Green functions 10 are φp

E+ = 1 (P )−1 = 1 (1 L + Λ)−1 = 1 (1 + C) (51) Λ 2 Λ 2 − 2 − . + T 1 1 T φq1 φq2 EΛ =(EΛ ) = 2 ( + C ) . (52) Proof. Direct calculation gives

− 1 q IM2 (p) φΛ(p) Λpq +[CΛ]pq = ∈ (53) (0 otherwise, FIG. 3. An isolated diamond from a regular diamond lattice. because [CΛ]pq = 1 if and only if q Λ(p). Similarly, ≺ − operator E+ is a valid discretisation of the continuum 2 q IM2 (p) Λ ∈ − retarded Green function. Evidently the same will hold Lpq +[CL]pq = 1 q J˙M (p) p (54)  ∈ 2 \{ } for the advanced Green operator. 0 otherwise, + A different discretisation of EM2 was considered by and one sees immediately that [38], namely + 1 (1 + C)(1 L +Λ)=Λ+ CΛ (L + CL) +1 + C [EDSX ]pq = Cpq , (60) − − 2 −C which takes the value 1 when p q and vanishes other- = 1 , (55) 2 ≺ | {z } wise. This may be reproduced from our operator PΛ by −1 1 − changing KΛ to so (PΛ) = + C, giving (51). The result for EΛ is obvious. + 1 1 KDSX = P E = P C = (1 L + Λ)C Λ DSX 2 Λ 2 − + 1 1 This result shows that [EΛ ]pq takes the value 2 if p q = 2 (L Λ). (61) and zero otherwise.  − Let us now consider the continuum limit of these op- + Yet a further possibility would be to discretise EM2 by erators as the mesh of the diamond lattice tends to zero. Suppose f is a smooth compactly supported function on + 1 1 1 Etrap = + (L + CL) , (62) −2 8 4 M2 of dimension [L] for simplicity, and pull it back to 2 + 1 1 the causal set Cr (as in Sec. III B 2) by (fCr )p = ℓ f(p). with components [Etrap]pq equal to 8 when p = q, 4 Then ˙− 1 − for q JM2 (p) p , 2 for q IM2 (p) and vanishing 2 ∈ \{−} ∈ + ℓr otherwise. Here J˙M (p) is the boundary of the causal (E fC )p = f(q) (56) 2 Cr ,Λ r 2 − + qp past JM2 (p). The discretisation Etrap, which amounts to X discretising (58) using the trapezium rule in u,v coordi- On the other hand, the retarded Green function on M2 nates, corresponds to is given by 1 Ktrap = (1 + L + Λ) . (63) + ′ 1 ′ 8 EM2 (t,x; t ,y)= 2 θ((t t ) x y ) − −| − | + = 1 θ(t t′)θ((t t′)2 x y 2) , (57) These definitions have the same continuum limit, EM, but 2 − − −| − | correspond to different discretisations of the inhomoge- where (t,x) M2 is a point in 2D Minkowski spacetime neous wave equation. Consider the isolated diamond in (with signature∈ (+ )) and θ is the Heaviside step func- Figure 3. Then: − + 1 ′ tion. Thus, for fixed (t,x), EM takes value for (t ,y) 2 2 1 1 inside the closed past lightcone of (t,x) and vanishes oth- setting K = 2 corresponds to sampling the value + • of the test function f on the diamond by taking its erwise. The function EM2 f is dimensionless, and given by value only at the future-most point fp; 1 + + 1 setting KDSX = 2 M samples f by taking fq1 + (EM f)(p)= f(q)dµg(q) . (58) • 2 2 − fq2 f p ; J (p) Λ( ) Z M2 − 1 1 It follows that setting Ktrap = 8 ( + L + Λ) samples f by taking • 1 4 (fp + fq1 + fq2 + fΛ(p)). + + (EC ,ΛfCr )p (EM2 f)(p) (59) r → This illustrates a basic fact that a continuum operator as r , because the spacetime volume of each dia- may have many valid discretisations, and indicates the → ∞ 2 mond [u,u + δr] [v,v + δr] is ℓ . This shows that our flexibility introduced by the operator K. × r 11

− − 4. Edge effects at past infinity and the Cauchy problem Ck , which is just the space E(Ck ). Therefore the solu- tion space is In causal sets with a past boundary, i.e., points with E+ . +E − no predecessors in the causal order, the form (33) of the Sol = E (Ck ) . (67) wave equation given above should be reconsidered near + to that boundary. In fact we have already anticipated The space of such solutions will be denoted ESol. A par- − this in our definition of PΛ, which treats points in C2 ticularly simple situation occurs if the solutions are also differently to those in the bulk. In the same way, one in bijection with data on future infinity, in which case might expect that the operator PS might be modified for − + + + points in C , because these points do not have layer-3 α = S E E − (68) 3 k (Ck ) predecessors. | The simplest possibility to take the edge effects into + − + is an isomorphism α : E(Ck ) E(Ck ) that will be account is to first fix the discretised d’Alembertian op- called the Cauchy evolution. Of→ course, this requires − ± erator P outside Ck (the k-step past infinity), for some among other things that C have equal cardinality. fixed k, using some discretization of the wave equation, k − As a slight digression we note that, in circumstances and then set (Pφ)p = φp for p C . This is already the ∈ k where the Cauchy evolution is defined, we can use it to case with PΛ for k = 2, as defined by (40). As for the compare the dynamics of the theory on two causal sets C right-hand side of the equation (33), the definition of K Λ and C whose k-layer past and future infinity regions are should be modified so that the diagonal elements of K Λ in order-preserving isomorphism with each other, thus in- would be 1 except for entries corresponding to points in 2 ducinge linear isomorphisms ι± : E(C±) E(C±). Writ- C−, where the value would be 1. As P is then triangular k k 2 ing α+ and α+ for the two Cauchy evolutions,→ the relative with nonvanishing diagonal elements, this prescription Cauchy evolution is a linear isomorphism on thee solution ensures that P remains invertible. The values of φ on E+ C − space Sol( e) defined by Ck are then treated as Cauchy data for the solution and we replace the wave equation (33) by rce . + − −1 + −1 + + (φ) = E (ι ) (α ) ι Sk φ, (69) Pφ = Kf + φ− , (64) which is an isomorphism; note thate we also have the iden- − − tity where φ = Sk φ is the projection of φ onto past infinity, (S− was defined in (18)), and it is understood that the k +rce + − −1 + −1 + + − − − Sk (φ)= α (ι ) (α ) ι Sk φ (70) source f should vanish in Ck . Note that Kφ = φ , − so one also has Pφ = K(f + φ ). Thus our prescription + + allows the Cauchy data to be combined with the source in which the comparison of α ande α is apparent. Rel- in a natural way. In these circumstances we will say that ative Cauchy evolution provides a way of discussing the P has a k-layer Cauchy problem. Recalling that Cauchy response to changes in causal set geometrye by reference to data for the scalar field in the continuum consists of both solutions of the wave equation on the unperturbed causal values and normal derivatives on a Cauchy surface, it is set. It was first introduced in [7], where it was formu- natural enough that the Cauchy data on a causal set lated in locally covariant QFT on continuum spacetimes, involves values taken on at least two layers. for perturbations in the background metric, localised be- Given the assumptions made on P and K, the solution tween two Cauchy surfaces. In that situation both back- to (64) is grounds must be globally hyperbolic and share the same Cauchy surface topology (the Cauchy surfaces must be φ = E+f + E+φ−. (65) related by orientation-preserving diffeomorphism). By contrast, the causal set framework would permit a per- Two special cases are of interest. First, the situation in turbed causal set that modelled a change of topology which φ− = 0, in which case φ = E+f, in line with the relative to the unperturbed one, provided that suitable continuum idea that the retarded Green function should identifications can be made in the past and future infinity produce solutions vanishing in the far past. Second, if regions. f = 0, φ = E+φ− may be interpreted as the solution When it is defined, the relative Cauchy evolution can to the source-free equation with Cauchy data φ−, which be pulled back to the map on observables, as follows. + + thus takes the form Consider F FSol(C), where FSol(C) denotes the space ∈ + of functionals on ESol. Define Pφ = φ− (66) . rce(F )(φ) = F (rce(φ)) . (71) and which will be called the homogeneous wave equation in the sequel. This map describes the change to the observable F re- By definition, solutions to the homogeneous wave equa- sulting from the perturbation to the underlying causal tion are in bijection with the Cauchy data specified on set. 12

C. Peierls bracket retarded and advanced responses of the field equation to linear perturbations, supposing that the unperturbed 1. Tentative definition equation has a k-layer Cauchy problem. In the origi- nal work of Peierls [15] the idea is to define a covariant The next step is to define a Poisson structure on the bracket, using the Lagrangian formalism, by studying the space of observables on a fixed causal set. We do this response of a given observable (say F ) to a perturbation using the method of Peierls [15]. of the action by another observable, G. One compares Using the commutator function (36) (in analogy to two situations: C∞ E C C [15]), we define the following bracket on ( ( ), ) F is evaluated at the solution to the perturbed • N N problem, which coincides with the free solution in δF δG F,G = Eij , (72) the far past (retarded response) and { } δφi δφj i=1 j=1 X X F is evaluated at the solution to the perturbed • where we used the Euclidean inner product to raise one problem, which coincides with the free solution in j the far future (advanced response) of the indices in Ei (compare with Remark III.3). To simplify the notation, we will drop the summation sym- δF The bracket of F and G is the linear term (in G) of bols, and use the condensed notation F,i, so the the difference between the retarded and the advanced δφi ≡ formula above becomes response of F to G. We start with analyzing the situation, where we add a ij (1) T (1) F,G = F,iE G,j =(F ) EG , (73) source λg to the theory. Heuristically, this means adding { } a linear functional λΦg to the action, where g is sup- − using the index-free notation in the last expression. For ported outside Ck and both λ and g are real. We will linear observables this reduces to implement this by a direct modification to the field equa- tion. The idea of Peierls is to study the effect of hav- T Fg,Fh = g Eh, (74) ing such a perturbation on the observables. Let φ be { } a solution to the non-perturbed field equation (64) with where Fg,Fh X(C). − − ∈ Cauchy data φ at Ck and let φλ be a solution to the Proposition III.6. The bracket (72) is a Poisson perturbed equation bracket, in particular, it satisfies the Jacobi identity: for − any F,G,H C∞(E(C), C): Pφλ = K(f + λg)+ φ , (78) ∈ F, G,H + G, H,F + H, F,G = 0. (75) with the same Cauchy data. Now consider another linear { { }} { { }} { { }} observable Φh with h real. The retarded response opera- + Proof. The argument is standard but we give it for com- tor DΦg is, in direct analogy to [15], a transformation of pleteness and for comparison with a later result. The observables defined by antisymmetry is obvious from the definition of E. It re- mains to prove the Jacobi identity. Expanding (75) we + 1 DΦ Φh (φ)= lim (Φh(φλ) Φh(φ)) . (79) find: g λ→0 λ −   ij kl ij kl F,i E (G,k E H,l ),j +G,i E (H,k E F,l ),j In this case, Eq. (78) gives +H, Eij(F, EklF, ), = 0 (76) i k l j + 1 T + + DΦ Φh (φ)= lim h (E (f + λg) E (f)) g λ→0 λ − of which the first term equals   = hT E+g, (80) ij kl if kl F,i E G,kj E H,l +F,i E G,k E,j H,l + ij kl which is independent of the solution φ, i.e., D Φh is +F, E G, E H, . (77) Φg i j lj a constant functional. Just as we defined the advanced Due to the antisymmetry of E and because E is indepen- Green function to be the transpose of the retarded ver- dent of the field, all the terms in (76) cancel out, so the sion, we now define the advanced response by reversing Jacobi identity follows. the roles of the perturbation and the observable used to test the response . D− Φ = D+ Φ . (81) 2. Justification of the formula for the bracket Φg h Φh g

We now discuss a sense in which (72) corresponds to With this definition, a discrete version of the Peierls bracket [15], by showing − T + T + T T − that it represents the difference between suitably defined DΦg Φh (φ)= g E h = h (E ) g = h E g, (82)   13 so the Peierls bracket is Linear functionals considered in the previous section

+ − T + − are obviously local. Other examples are local polynomials Φg, Φh P ei(φ)= D Φh D Φh = h (E E )g { } Φg − Φg − which are finite sums of terms with the form T = g Eh, (83) i F (φ)=(φ φ) gi , (88) ∗ · · · ∗ in agreement with our definition (72). Turning to nonlinear perturbations and nonlinear ob- using the Hadamard product (31). servables, let F C∞(E(C), C) (some examples appear at the end of this∈ subsection). Heuristically, perturbing D. Free Dynamics the action by λF has the effect of perturbing the field equation by λ δF (φ). Taking (66) as the starting point, δφp this suggests that the interacting field equation should Let us define the Poisson algebra assigned to a causal C take the form:3 set as . δF − P(C) =(F, .,. ) , (89) Pφ = λK( δφ (φ)) + φ , (84) { } which is the causal set counterpart of the off-shell clas- where F is supported away from the past infinity C−. k sical algebra in continuum pAQFT (equations of motion The matrix K is there for consistency with linear per- are not imposed). Typically, one would now quotient turbations (compare with (78)). This is an artifact of it by the ideal generated by the equations of motion, the way we choose to discretize the interaction term. As to obtain the on-shell algebra. The potential problem for the notion of support of F , we adopt the following that arises in the causal set context is that the retarded definition Green function is the inverse to P but its transpose is supp F not (unless in very special cases). As a result, in gen- . eral, EP = (E− E+)P = 0, which means that the = p C φ E(C), λ C s.t. F (φ + λδ ) = F (φ) . p Peierls bracket would− not be6 well-defined on the quotient { ∈ |∃ ∈ ∈ 6 (85)} algebra. Hence, instead of quotienting by the ideal generated by where δp(q) = δpq. This is a straightforward general- ization of the notion used in continuum. An obvious the equations of motion, we propose to quotient P(C) by consequence of this definition is that if p / supp F , then the ideal generated by functionals F with the property δF ∈ δφ (φ) = 0. Conversely, we can express the support as ij p E F,j 0 , (90) ≡ δF supp F = supp (φ) , (86) denoting this quotient by P(C). Note that on P(C) the δφ φ∈E(C)   Poisson bracket .,. is non-degenerate. [ { } To see how the above quotiente is related to implement-e where supp( δF (φ)) consists of points p, for which δF (φ) δφ δφp ing the dynamics, recall that in continuum we have the is non-zero. exact sequence [39]: Let F,G C∞(E(C), C) be supported away from past ∈ P E P infinity. Then the retarded response for the discretized 0 D(M) D(M) Esc(M) Esc(M) , (91) wave equation (64) is → −→ −→ −→ where M is a globally hyperbolic spacetime, P is a nor- + + ij DF (G)= G,i(E ) F,j , (87) mally hyperbolic operator, D(M) and Esc(M) are space − + of functions with compact and spatially compact sup- and defining DG(F ) = DF (G), as before, the bracket port, respectively. We also know that the space of linear (72) agrees with the Peierls construction. observables is isomorphic to D(M) by means of Note that in general δF (φ) can be very non-local, i.e. δφp it can depend on values of φ at points other than p. This D(M) g Φg; Φg(φ)= φ(x)g(x)dµ (92) motivates the following definition: ∋ 7→ Z Definition III.7. A functional F C∞(E(C), C) is Hence quotienting the algebra of regular functionals by called local if δF (φ) is a function of p∈and φ(p) (only). δφp the ideal generated by elements of the form ΦPf , f D(M) is the same as quotienting by the ideal generated∈ by linear observables Φg with the property g ker E. This result extends to more singular functionals∈ by 3 minus Note that since Pφ is heuristically the variation of the continuity. Clearly, our condition (90) is the causal set free action, in order to implement the interaction, we have to subtract the variation of F on the left-hand side (or add it on analogue of quotienting by the kernel of E. This con- the right-hand side). This differs from the usual convention used dition is also the natural generalization of the condition in [12, 14, 37], where P = −, rather than P = , as we assume proposed in [16] for linear observables. Sorkin argues that in the present work. the kernel of E in the causal set situation is typically very 14 small. There are many very small eigenvalues of E, but Proof. The proof is analogous to that of Proposition III.6. only a few of them are exactly 0. This leads Sorkin to The only difference is in the proof of the Jacobi identity. conclude that the equations of motion on a causal set can Expanding the first term in (98), we obtain be implemented only in approximate sense [16]. We hope il jk il jk to address this point in future work. F,iEλV G,jlEλV H,k + F,iEλV G,j (EλV ),l H,k il jk +F,iEλV G,jEλV H,kl. (99) E. Interacting theory Due to the antisymmetry of EλV , only the central term of the expansion of each bracket remains as the others can- Next we want to introduce the interaction. We will use cel across all three expanded brackets. The derivatives the framework proposed in [12] and further developed in of the retarded Green function are: [37]. + jk + jm + nk (EλV ),l = λ(EλV ) V,lmn(EλV ) , (100)

1. Interacting and linearized interacting equations of where V is the interaction term. Inserting this into (99) + T + motion and expanding each EλV = (EλV ) EλV , we obtain twelve terms altogether. These cancel− in pairs due to the Let V C∞(E(C), C), where C = N and let λ be antisymmetry of EλV and the fact that V,lmn is sym- the coupling∈ constant. We work| perturbatively,| so the metrical with respect to its indices (see the Appendix B space of observables is now extended to include formal of [40] for more details of the proof). power series in the coupling constant λ, i.e. it becomes F(C)[[λ]]. The interacting field equations are given by We can also introduce the retarded Møller map, which (84), which we can also write as maps solutions to free discretised retarded field equations to solutions to interacting field equations: (84) (1) − Pφ λK(V (φ)) = φ . (93) + (1) − rλV (φ)= φ + λE V (rλV (φ)) , (101) The interacting field equations linearized about φ E(C), are ∈ Its inverse is given by r−1 + (1) Pψ λKV (2)(φ)ψ = ψ− . (94) λV (φ)= φ λE V (φ) . (102) − − The retarded Møller map on configurations induces the where V (2)(φ) is an N N matrix with components × corresponding map on observables, (2) (V (φ))ij = V,ij(φ) . (95) . (rλV F )(φ) = F rλV (φ) , (103) ◦ where F F(C)[[λ]]. Analogously to the continuum case, 2. Interacting Poisson bracket the Peierls∈ bracket (96) satisfies:

−1 The prescription for the Poisson bracket of the inter- F,G λV = r rλV F,rλV G . (104) { } λV { } acting theory with the interaction λV is given by

. ij G,H = G,iEλV (φ) H,j , (96) IV. QUANTUM THEORY { }λV + T + + where EλV (φ)=(E (φ)) E (φ), and E (φ) is So far, we have constructed a Poisson algebra of ob- λV − λV λV the retarded Green function for the interacting linearized servables for the free and interacting classical field theory field equations (94). Starting from the free Green func- on a causal set. We now pass to the quantum theory us- tion E+, we construct the interacting one using the Neu- ing the method of deformation quantization, following mann series: ideas applied in the context of continuum field theory by [11, 13, 41]. The main idea behind this approach is to ∞ n E+ = E+ + λnE+ V (2)E+ . (97) phrase the problem of quantization as the mathemati- λV cal problem of finding a non-commutative ~-dependent n=1 X   product ⋆~, such that, given a Poisson algebra (with a commutative product and Poisson bracket .,. ): Proposition III.8. The bracket (96) is a Poisson · { } bracket, in particular, it satisfies the Jacobi identity: for ∞ ~→0 any F,G,H C (E(C), C), one has F ⋆~ G F G, ∈ −−−→ · 1 ~→0 F, G,H + G, H,F + H, F,G = 0. (98) i~ (F ⋆~ G G ⋆~ F ) F,G . { { }} { { }} { { }} − −−−→{ } 15

In our case, F and G are smooth functions on the con- understand the class of states that can have Hadamard figuration space E(C). Furthermore, the ⋆~ product is continuum limits; and if the causal set has infinitely required to be of the form: many elements then there is still the possibility that dif-

∞ ferent states could yield inequivalent representations of n the CCR algebra. The situation is of course better still F ⋆~ G = ~ B (F,G) , (105) n in the case of finite causal sets, our main focus, where n=0 X the Stone-von Neumann theorem ensures that all suf- where Bn are differential operators on E(C). For simplic- ficiently regular representations are unitarily equivalent ity, we focus on finite causal sets, so that for a causal set up to multiplicity. In this situation any pure state will E C RN of size N, ( ) ∼= and the differential calculus on it lead to the same Hilbert space representation. Therefore is well understood. We can then express the SJ state is a valid starting point for a more refined discussion. As we will show, it is mathematically sim- N N n δ F i1,...,in,j1,...,jn ple to describe, and closely related to our choice of the Bn(F,G)= (Bn) ··· δφi1 ...δφin Euclidean inner product in Sec. IIIA. Alternative inner i1=1 j =1 X Xn products produce states that may be seen as precursors δnG (106) of the softened SJ states of [30–32]. See Remark IV.3 for × δφj1 ...δφjn some further comments. A great advantage of the algebraic viewpoint is that The higher orders in ~ present in (105) distinguish it is much more straightforward to introduce interactions this approach from the Dirac quantization, so one than in constructions based on Fock space. Applying avoids contradiction with the Groenewald-van Hove no- the ideas of [11, 37, 51], we will show how to pass to go theorem[42, 43]. the interacting theory, using further deformation of the Deformation quantization has the advantage that the non-commutative product ⋆~. construction of the algebra of observables can be done completely abstractly, without the need for existence of a distinguished state (e.g., a vacuum) and without in- A. Construction of the Quantum Algebra voking Fock space methods. At a later stage, one can then seek suitable states on the abstract algebra and use these to form Hilbert space representations by the GNS 1. Exponential products construction. The choice of states has both a mathematical and a Deformation quantization of the classical theory starts physical aspect. There is a precise mathematical defini- with the free theory, i.e. a linearized wave equation (66) tion of a state, as a positive normalised linear functional and its retarded Green function E+. From this we obtain on a -algebra (and this can be extended to algebras of E and the Peierls bracket. Let us for the moment restrict formal∗ power series, as will be described below). However ourselves to the subspace of F(C) that consists of smooth not all such linear functionals need qualify as physically functionals F with the property that there exists N N relevant. For QFT in continuum curved spacetimes it is such that F (n)(ϕ) = 0 for all n>N, ϕ E(C). We∈ call known that it is not possible to single out a unique distin- such functionals polynomial and denote∈ the correspond- guished state that is locally and covariantly determined ing vector space by Fpol(C). This space can be equipped by the geometry, assuming certain additional physically with various types of noncommutative product, of which motivated assumptions – see [44] for a formal proof and we will describe two. First, the Moyal–Weyl product is [45] for a review. Nonetheless, there are circumstances . 1 ~ in which a distinguished global state (or distinguished F ⋆G = m e 2 i DE (F G) , (107) ‘in’ and ‘out’ states) with good properties (specifically, ◦ ⊗ the Hadamard condition) may be determined [46–48]. where F,G Fpol(C), m is the multiplication on Fpol(C) One proposal for a global geometrically determined state, induced by∈ pointwise multiplication of functionals in arising from the causal set programme, is the Sorkin- Fpol(C) and for a given N N matrix M, Johnston (SJ) state [27, 49]. In the continuum this is × known to have certain problems; in particular, it gener- . δ δ δ δ DM = Mij M, δφ δφ , (108) ally fails to be Hadamard [29, 50]. Nonetheless, SJ states δφi δφj ≡ ⊗ retain interest as a specific construction of a state where D E ⊗2 ⊗2 particular examples are otherwise sparse; furthermore, maps Fpol(C) Fpol(C) . With the appropriate there are softened versions [30–32] of the SJ construction choice of units, ~ →is just a number and can be set equal . in which the Hadamard property is restored at the price to 1. We obtain a non-commutative algebra A(C) = of losing uniqueness. (Fpol(C), ⋆), which, as in the classical case, is the ana- As the definition of Hadamard states centres on the logue of the continuum off-shell algebra. Second, the UV behaviour of their n-point functions, it may seem Wick product is defined by that these problems are vitiated in discrete spacetimes. . ~DW This is not quite so, because ideally one would like to F ⋆H G = m e (F G) , (109) ◦ ⊗ 16

. where Formally, we can consider Φx = Φ(δx), where δx is the Dirac delta supported at some x M, and we find: i ∈ W = E + H, (110) 2 ~ [Φx, Φy]⋆H = i E(x,y) . is a complex hermitian matrix that has the physical in- terpretation of the two-point function of a quasifree state Now fix an inertial coordinate system and consider the . on A(C). Denote AH (C) =(Fpol(C), ⋆H ). t = 0 Cauchy surface. Let x and y denote spacelike We require W to have the following properties, which vectors on this surface. From properties of E follows then constrain H: that: W1) E = 2Im W , i.e., H = Re H (recall that E is real [Φ(0,x), Φ(0,y)]⋆ = ∆(0, x; 0, y) = 0 by definition). ˙ 0 [Φ(0,x), Φ(0,y)]⋆ = ∂y ∆(0, x; 0, y)= i~δ(x y) , W2) W is a positive definite matrix, meaning that − f †Wf 0, where f † is the hermitian conjugate where dot denotes the time derivative. These are the ≥N of f C . standard equal-time commutation relations, so we have ∈ recovered the usual formulas from the deformation quan- W3) ker W ker E (a proxy for W solving the equations tization approach. of motion)⊂ In order to find a specific choice of W , we will follow This is almost the same as in the continuum, but modify- the ideas of [27, 49] and take W as the Sorkin-Johnston ing the condition that W solves the equations of motion. (SJ) two-point function. We recall, that according to [16], This is related to the general difficulty with going on-shell W is the unique N N matrix satisfying the following discussed before. properties: × Remark IV.1. Note that both ⋆ and ⋆H are of the form (105) with (B )i1,...,in,j1,...,jn = (i/2)nE ...E for SJ1) W W = iE, where bar denotes the complex con- n i1j1 injn − i1,...,in,j1,...,jn jugation, the former and (Bn) = Wi1j1 ...Winjn for the latter. SJ2) W 0, Physically, passing from ⋆ to ⋆H corresponds to ≥ normal-ordering with respect to the quasifree state deter- mined by W . In deformation quantization, the products SJ3) WW = 0 . ⋆ and ⋆ are regarded as equivalent, because they are H It was shown in [16] that the unique W satisfying the related by a gauge transformation α : A(C) A (C), H H axioms above is given by which is given by →

~ 1 . 2 DH 2 αH = e , (111) W = (iE + E ) , (113) 2 − where p where the square root is the unique positive semi-definite 2 . ij δ2F square root of the positive semi-definite matrix (iE) = DH (F ) = H F,ij H, 2 , ≡ δφ (iE)(iE)†. D E More explicitly, we can write Remark IV.3. Note that we have made implicit use of the Euclidean inner product in order to identify the 2- −1 −1 F ⋆H G = αH (αH F ⋆αH G) , (112) point function with a matrix. As already indicated in Remark III.3, this choice is to some extent arbitrary and −1 and we identify αH (F ) :F :H as the normal (Wick) could be changed. Since SJ3) crucially depends on this ≡ −1 ordering operation. Applying αH to a local functional choice, a different auxiliary inner product would produce F Floc(M) in continuum is analogous to normal order- a different 2-point function and hence a different state. ∈ ing using the point-splitting prescription. The significance of this fact becomes acute in the con- Example IV.2 (Minkowski spacetime). Consider the ex- tinuum. Consider the real scalar field on (M,g) with ample of continuum free scalar field theory on Minkowski M = ( τ,τ) Σ being an ultrastatic slab spacetime. − × C∞ R spacetime M. Let Φf , Φg be smeared fields, as defined Choosing the inner product on c (M, ) to be the one by (28). For the Klein-Gordon operator P =  + m2, induced by the volume form dµg implies that the unique there exist the unique retarded and advanced Green func- W solving SJ1)-3) is the 2-point function of the SJ state tions E± and their difference is the commutator function which is known not to be Hadamard in general [29, 50]. − + 1 E = E E . With the star product taken to be ⋆H , However, replacing dµg by ρ dµg produces Wρ, which is for any choice− of the symmetric part H, we have a 2-point function of a Hadamard state, if ρ is an appro- priately chosen smooth function on the interval ( τ,τ), ′ − [Φf , Φf ′ ]⋆ =Φf ⋆H Φf ′ Φf ′ ⋆H Φf = i~ E,f f . tending to zero at both endpoints (see [30] for details). H − h ⊗ i 17

2. Formal power series typically represented in Hilbert spaces by unbounded op- erators. If we want to work with bounded operators in- Going beyond the polynomial observables requires stead, a suitable candidate for the algebra of observables some caution, since the power series defining the star is provided by the Weyl algebra, defined by exponenti- ~ product might not converge. One possibility is to con- ating linear functionals. In this section we treat as a sider analytic functions (e.g. exponentials, as discussed number, rather then a formal parameter. in the next section) or to extend the framework to allow Recall that elements of RN are identified with linear also formal power series. The latter is actually necessary real observables on a causal set C of size N by means of if we want to introduce interactions (see Section IV D). (25). The Poisson bracket .,. on the space of observ- { } N Let F(C)[[~]] denote the vector space consisting of for- ables is given by (72), so for g,h R we have ~ ∈ mal power series in the formal parameter , with coef- . ficients in F(C). Formulas (107) and (109) can be eas- Φg, Φh = g,Eh = σ(g,h) . (114) { } h i ily adapted to this setting, but now we interpret ~ as a formal parameter rather than a number. The result- Definition IV.4. Each linear real observable Φg, g ~ . ing algebras will be denoted by A = (F(C)[[~]], ⋆) and RN , defines a Weyl functional W(g) C∞(E(C), C) by∈ ~ . ~ iΦg ∈ AH =(F(C)[[ ]], ⋆H ), respectively. W(g)= e .

Proposition IV.5. The Weyl functionals satisfy the B. Weyl Algebra Weyl commutation relations

i~ The algebra of observables A(C) introduced in the pre- W(g) ⋆ W(˜g)= e− 2 σ(g,g˜)W(g +g ˜) , (115) vious section can be equipped with a topology that makes it into a topological unital -algebra. Such algebras are with respect to the star product, and W(g)∗ = W( g). ∗ −

Proof. This is a simple computation. The functional derivative of the operators in the direction of an arbitrary field in the configuration space h E(C) is: ∈ N N (1) d (W(g)) (φ),h = exp i gi(φj + λhj) = i gjhj W(g)(φ) (116) dλ     j=1 j=1 D E X λ=0 X     hence: n N (n) ⊗n (W(g)) (φ),h = i gjhj W(g)(φ). (117)   j=1 D E X   Therefore, the following formula is obtained from the star product:

∞ ~n iE ⊗n W(g) ⋆ W(˜g)= (W(g))(n), (W(˜g))(n) n! 2 n=0 *   + X n ∞ ~ n n N i ( 1) ij = − giE g˜j W(g +g ˜) 2 n!   n=0 i,j=1 X   X   ~ N i ij = exp giE g˜j W(g +g ˜) − 2  i,j=1 X i~ i~ − {F ,F˜} − σ(g,g˜) = e 2 g g W(g +g ˜)= e 2 W(g +g ˜) , (118) as required.

We may now introduce the Weyl C∗-algebra for the sider the non-separable Hilbert space H = L2(RN ,dµ) of free scalar field on a causal set C. For details see for square integrable functions with the counting measure µ example [52, section 8.3.5] or [53, section 14.2]. Con- on RN . This can also be identified with l2(RN ), the space 18 of square-summable sequences indexed over RN , because there is no complete proof written down anywhere in the any element of H may be written literature, so for completeness we provide it here as well. Definition IV.9. Let M = (M,g) be a globally hyper- 2 . ∞ C . ∞ n cgeg , with cg < , (119) bolic manifold, E = C (M, R), D = C (M , C). De- | | ∞ n g∈RN g∈RN fine the space of regular polynomials Fpol(M) as the space X X of functionals on E of the form: where eg g∈RN is the orthonormal basis for H given by { } N (eg)h = δgh. The representation of Weyl generators is ⊗k given by F (ϕ)= F0 + ϕ ,fk , (123) k=1 i~ X . − 2 σ(g,h) (π(W(h))a)g = e ag+h . (120) C C where F0 , ϕ E, fk Dk and .,. denotes the for any a l2(RN ). One may easily check that the Weyl usual pairing∈ induced∈ by integrating∈ withh i the invariant ∈ k relations are fulfilled, by explicit computation. Using this volume form dµg over the whole M . representation, we define a C∗-norm . on the generators k k (by taking the corresponding operator norm as operators Proposition IV.10. Let AH (M)=(Fpol(M), ⋆H ) for on H). We are now ready to define the Weyl C∗-algebra. some choice of a Hadamard function (by that we mean a bi-distribution satisfying the continuum version of W1)- Definition IV.6. The Weyl C∗-algebra is generated by i W3), see [14] for the precise definition) W = 2 E + H. W N the operators (g) g∈R and completed with respect Set ~ = 1. The functional given by evaluation at zero to the C∗-norm{ . } k k . ω (F ) = F (0) , F AH (M) , (124) 0 ∈ C. States and the GNS representation is a quasi-free Hadamard state on AH (M) with 2-point function W . Within the framework of algebraic quantum theory, a physical system is described by the algebra of observables Proof. Take F as in (123) and write (it is useful to keep associated with it. The abstract algebra may be linked ~ explicit at this stage) to the standard formulation of quantum theory by means of a Hilbert space representation. If we start with a C∗- ∞ ~n algebra, we can represent it by bounded operators. For ω (F ∗ ⋆ F )= (F (n)(0))∗,W ⊗nF (n)(0) 0 H n! a general topological unital -algebra, we have to work n=0 ∗ X D E with unbounded operators as well. N 2 k Choosing a Hilbert space representation is equivalent = F + ~ k! f ,wkfk , (125) | 0| k to choosing an algebraic state, by virtue of the Gelfand- k=1 Naimark-Segal (GNS) construction. X where w is a distribution in D′ defined by the following Definition IV.7. Let A be a topological unital - k n distributional kernel: algebra, then an algebraic state is a linear functional∗ ω : A C such that: . → wk(x1,...,xk,y1,...,yk) = ∗ 1 ∗ ω(a a) 0 a A, ω( ) = 1. (121) ((Φx1 ,... Φx ) ⋆H (Φy1 ,... Φy ))(0) , (126) ≥ ∀ ∈ k k

Theorem IV.8 (GNS). Let ω be a state on a unital where Φx is the evaluation functional at x M, i.e. for . ∈ -algebra A. Then there exists a representation π of the ϕ E: Φx(ϕ) = ϕ(x). ∗algebra by linear operators on a dense subspace K of some ∈ Hence for the positivity of ω0 it is sufficient to show Hilbert space H and a unit vector Ω K, such that N ¯ ∈ that all wk, k are positive type, i.e., w(F,F ) 0. We proceed by∈ induction. First, note that for k = 1≥we ω(A)=(Ω, π(A)Ω) , (122) have w1 = W , which is by assumption positive type. We and K = π(A)Ω,A A . need to prove the induction step, i.e. assuming wn−1 { ∈ } is positive definite, we want to show that wn is positive For the details of the proof, which is constructive and type. builds the Hilbert space from the algebra, see for example Our proof is similar to the one used in [52] for states [14, Section 2.1.3] or [6, §2.3]. Let us now record some on the Weyl algebra, but is more general. First we recall general facts about states in the pAQFT framework. the Schur product theorem about positive semi-definite Firstly, we establish that evaluation at the zero vector matrices: if A 0 and B 0 (A, B are positive semi- ≥ ≥ is a state on AH (C). The corresponding result in the con- definite), then their Hadamard product A B is also pos- tinuum case is known, but to the best of our knowledge itive semi-definite. ∗ 19

Let x (x ,...,xn), y (y ,...,yn). We express wn as ≡ 1 ≡ 1 n ∗ ∗ wn(x,y)= (Φ ⋆H Φy )(0)((Φx1 ... Φx ... Φx ) ⋆H (Φy1 ... Φy ... Φy ))(0) , (127) xi j · · i · · n · · j · · n i,j=1 X d d where indicates that the given symbol is omitted. Let f = f1 ... fn Dn, where fi D, i = 1,...,n. Using (127) we obtain · · ∈ ∈ b n f,w¯ nf = aijbij , (128) i,j=1 X where

aij f¯i,Wfj , ≡ ∗ bij (f ... fi ... fn) ,wn (f ... fj ... fn) (129) ≡ 1 · · · · −1 1 · · · · D E b b

Define n n matrices A [aij] and B [bij]. These Remark IV.11. We could replace Fpol(M) with a larger are both× positive semi-definite.≡ To see this,≡ we consider space of functionals, e.g. the microcausal functionals [10]. λ Cn and define The proof is then exactly the same, but the test function ∈ spaces D are replaced by appropriate spaces of distri- n n n . . butions satisfying given wavefront set conditions. f˜λ = λifi , fˆλ = λif ... fi ... fn . (130) 1 · · · · i=1 i=1 Let us now state the discrete version of the Proposi- X X b tion IV.10. Again, we set ~ = 1. It follows that Proposition IV.12. The functional given by evaluation † ˜ ˜ λ Aλ = fλ,W fλ 0 , (131) at zero ≥ D E . since W is positive semi-definite, and ω (F ) = F (0) , F AH (C) , (136) 0 ∈ † λ Bλ = fˆλ,wn−1fˆλ 0 , (132) is a quasi-free state on AH (C), with 2-point function W = ≥ i E + H. D E 2 using the assumption in the induction step. By Schur product theorem the Hadamard product A B is also Corollary IV.13. The functional given by positive semi-definite and we note that ∗ . ωH,0(F ) = αH (F )(0) (137) ¯ † f,wnf = λ1(A B)λ1 , (133) C i ∗ is a state on A( ) and if we take W = 2 E + H to be that of the SJ state, then where λ1 = (1,..., 1), so we conclude that

f,w¯ nf 0 , (134) ωH,0 = ωSJ . (138) ≥ which proves the induction step. Proof. The 2-point function of ωH,0 is given by It remains to show that ω0 is a quasifree state. This, i however, is straightforward, since for f1,...,f2k D the ωH,0(Φf ⋆Φg)=(Φf ⋆H Φg)(0) = W =( 2 E +H) . (139) correlation function is given by ∈

ω0(Φf1 ⋆H ⋆H Φf2 )= W (fs(e),ft(e)), (135) ··· k In particular, for Weyl generators, we obtain G G2 e G X∈ k Y∈ ~ − 2 hg,Hgi where fi D, i = 1,..., 2k, G2k is the set of directed ωH,0(W(g)) = e , (140) graphs with∈ vertices labelled 1,..., 2n, such that each vertex is met by exactly one edge and the source and so H is the covariance of the state ωH,0. target of each edge obey s(e)

∞ ~ ∗ or with original functionals F,G within the algebra AH . c = n=0 cn, such that b = c c. This does not make Correlation functions are then computed using the rule: any difference for what follows. P

ωH,0(:F :H ⋆ :G:H )= ω0(F ⋆H G)=(F ⋆H G)(0) . (141) D. Interacting theory

1. Motivating the approach Let us now discuss the generalization to the situation, where Fpol(C) is replaced with the space F(C)[[~]] of for- mal power series. We finish this section with the construction of the in- teracting theory for a given interaction V F(C). We use We need the notion of states on the formal power se- the framework of perturbative AQFT [11∈, 14, 51], where ~ F ~ ries algebra A = ( [[ ]], ⋆). Condition (121) has to be the interacting fields are constructed with the use of understood in the sense of the formal power series. For quantum Møller operators. The motivation comes from ∞ ~n ∞ ~n A = n=0 An and ω = n=0 ωn, the normalization the interaction picture in quantum mechanics. Consider condition is that ω(1)= ω0(1) = 1. We have P P the continuum theory of the scalar field on Minkowski spacetime with the free Hamiltonain H0 and let the inter- ∗ ∗ ~ ∗ ∗ ∗ action Hamiltonian begiven by :LI (0, x): dσ, where ω(A A)= ω0(A0A0)+ (ω0(A1A0 + A0A1)+ ω1(A0A0)) Σ the integration goes over some fixed− Cauchy surface and + ... , (142) R :LI : is the normal-ordered Lagrangian density. Heuristically, we would like to be able to use Dyson’s and, by definition, positivity for a formal power series formula for interacting fields, which reads: means that the lowest non-vanishing term has to be pos- 0 −1 0 ∗ ϕI (x)= U(x ,s) ϕ(x)U(x ,s) itive (see [54]), so if ω0(A0A0) = 0, then we require ∗ 6 −1 0 0 ω0(A0A0) 0, i.e. ω0 is a state in the usual sense. Alter- = U(t,s) U(t,x )ϕ(x)U(x ,s) , (143) natively, one≥ could use the stronger notion of positivity, proposed in [55], where one says that a formal power se- where ϕ(x) is the free field, ϕI (x) is the interacting field ∞ ~ ries b = n=0 bn is non-negative, if there exists a series and the interacting time evolution operator is given by: P

∞ n n i λ 4 4 UI (t,s)=1+ T (:LI (x1): ... :LI (xn):)d x1 ...d xn , (144) n! R3 n n=1 ([s,t]× ) X Z where λ is the coupling constant, T denotes time-ordering the expression we use is now fully covariant, so there and :LI : is an operator-valued function given by is no need for singling out a Cauchy surface. The nor-

0 0 mal ordering is achieved by fixing a Hadamard function L (x)= eiH0x :L (0, x): e−iH0x , i I I W = 2 E +H and using the corresponding ⋆H product in the free theory. Finally, the time-ordered products cor- Unfortunately, there are many problems with the above responding to the above choice of a Hadamard function idea, already in Minkowski spacetime: have to be constructed. In continuum, this is achieved through the Epstein-Glaser renormalization [56], but on Typical Lagrangian densities, e.g. :LI (x): = • :ϕ(x)4: cannot be restricted to a Cauchy surface causal sets we are able to use a more direct method, as as operator-valued distributions. This is the source shown below. The problem with overall convergence can- of the UV problem. On causal sets, the major prob- not be addressed with our methods, so we will work with ~ lem with such quantities is the lack of a good analog formal power series in . As an intermediate step, we will of a Cauchy surface. also use formal power series in the coupling constant λ. In pAQFT time-ordered products are not just formal There is a problem with taking the adiabatic limit, expressions, but they stem from a binary product T , so • as the integral of the Lagrangian density over x that: · does not exist if Σ is non-compact. 4 4 T f1(x)LI (x)d x fn(x)LI (x)d x The overall sum might not converge. M ··· M • Z Z  4 4 In pAQFT, the first two problems are addressed as fol- := f1(x)LI (x)d x T ... T fn(x)LI (x)d x. M · · M lows: quantities like LI (0, x) are replaced by smeared Z Z 4 (145) ones, of the form: V f(x)LI (x)d x, where f ≡ M ∈ D(M) plays the role of the adiabatic cutoff. Note that Constructing T will be one of the main tasks in the fol- R · 21 lowing section. Remark IV.14. On Minkowski spacetime, the operator T formally corresponds to a path integral involving a ‘Gaus- sian measure’ with covariance i~∆F, i.e. 2. S-matrix and interacting fields formal TF (ϕ) = F (ϕ φ)dµ ~ F (φ) . (149) − i ∆ We start with the algebra A(C), constructed in sec- Z tion IV A 2, but we introduce a new formal parame- Therefore, one can think of the pAQFT formalism as a ter λ, which plays the role of the coupling constant. way to make path integrals and other formulas used in ~,λ C F C ~ In this section A ( ) ( ( )[[ ,λ]], ⋆). We fix a perturbative QFT rigorous. We hope that this statement ≡i ~,λ C Hadamard function W = E + H and denote AH ( ) can be made more precise in the context of causal sets, ~ 2 ≡ (F(C)[[ ,λ]], ⋆H ). On this algebra there is a distinguished where the path integral has better chances of being well- state given by evaluation at 0, i.e. defined. . Using the time-ordered product, we introduce the for- ω0(F ) = F (0) . (146) mal S-matrix for the interaction V and coupling constant Different choices of W lead to different, but isomorphic λ. It is given by A~,λ C algebras (all of them isomorphic to ( )), each one ∞ i n n with its own distinguished state, given by evaluation at . ~ λV λ i S(λV ) = e = V T ... T V . (150) 0. In each case, the 2-point function of this state is by T ~nn! · · n=0 definition W . X n ~,λ C ~,λ C Since A ( ) and AH ( ) are related through the iso- Next, we define the interacting fields. For a classical ob- ~,λ | {z } morphism αH , we obtain a family of states on A (C) servable F F(C), the corresponding quantum interact- labeled by H, i.e.: ∈ ing field is given by RλV (F ), where RλV is the retarded quantum Møller operator defined by ωH,0(F )= αH (F )(0) .

. d −1 The 2-point functions of these states are given by: RλV (F ) = i~ S(λV ) ⋆H S(λV + µF ) − dµ µ=0 T i −1 i αH (Φf ⋆ Φg)(0) = (Φf ⋆H Φg)(0) = f Wg. ~ λV ~ λV = e ⋆H e T F . (151) T T · If W satisfies SJ1)-3), then ωH,0 is the SJ state on     A~,λ(C). Note the analogy of this formula to the Dyson series men- Now we are ready to introduce time-ordered products. tioned in section IV D 1 as the motivation for the pAQFT To this end, we will need the Feynman propagator. In approach. our framework, there is a “Feynman-like” propagator for We can also use the Møller operator to deform the free every choice of W , so it is a state-dependent notion. We star product and obtain the interacting one, using the define it as formula: . −1 i F ⋆ int G = R (R (F ) ⋆ R (G)) . (152) ∆F = (E+ + E−)+ H, (147) H, λV λV H λV 2 int . This way we obtain the interacting algebra AH (C) = where H is the symmetric part of the 2-point function (F(C)[[~,λ]], ⋆H,int). Given a state ω on the free algebra, int W . In what follows, we will keep H fixed and refer to we can construct the state ωint on AH (C) using the pull- ∆F simply as the Feynman propagator. back: F Given ∆ , we define the time-ordered product T by · . ~ ωint(F ) = ω RλV (F ) , (153) . D F ◦ F T G = m e 2 ∆ (F G) · ◦ ⊗ where F F(C)[[~,λ]]. The natural choice of a state ∞ ~n ∈ = F (∆F)i1j1 ... (∆F)injn G . in this context is ω0 for the free theory. Next, we want n! ,i1...in ,j1...jn n=0 to choose observables. The first natural candidate is the X (148) smeared interacting field itself, i.e.

int . We can also write it as Φf = RλV (Φf )

−1 −1 F T G = T(T F T G) , The n-point correlation function of smeared interacting · · fields is given by: where the product on the right-hand side is the usual int point-wise product of functionals in F(C)[[~,λ]] and the ωn (g1,...,gn) operator T is given by . int int = ω (Φ ⋆H ⋆H Φ ) 0 g1 ··· gn . ~ D 2 ∆F =(RλV (Φg1 ) ⋆H ⋆H RλV (Φg1 ))(0) . (154) T = e . ··· 22

Note that the product we used is the product of the free and theory, since, following the philosophy of the interaction 1 picture, interacting fields are constructed within the free [F,G] = F,G . (157) ~ ⋆H,int λV field algebra. i ~=0 { }

Alternatively, we can write the above correlation func- With these formulas at hand, one can now implement any tion as interacting theory in numerical simulations, provided the free theory is known. This opens up perspectives for more int ωn (g1,...,gn) examples of interesting causal set field theories, where −1 the influence of adding different interaction terms can be = ω0 RλV R (RλV (Φg1 ) ⋆H ⋆H RλV (Φg )) ◦ ◦ λV ··· n tested and compared with the continuum. = ωint(Φg1 ⋆H,int . . . ⋆H,int Φgn ) . (155)

Here we did not modify the observables, but changed the V. CONCLUSIONS AND OUTLOOK product and changed the state. Other natural observ- ables to consider include all the local polynomials (88). In this paper we have shown how to construct a large Remark IV.15. In the pAQFT setting there are two class of QFT models on causal sets, using methods of equivalent ways of treating the interacting theory. On perturbative algebraic quantum field theory (pAQFT). ~,λ the one hand, one can work with the algebra AH (C) = For the purpose of defining the free classical theory (the (F(C)[[~,λ]], ⋆H ) and identify physical observables with starting point of our construction) we discussed a number elements of this algebra by means of RλV . For example, of discretized d’Alembert operators and their retarded ~,λ Green functions. We have also proposed a new ansatz take Φf , as above. Inside AH (C) the free quantum ob- servable corresponding to this object is just Φf , while the for a class of such discretized wave operators, which uses int the notion of preferred past structure. The latter is an interacting observable is identified as Φf = RλV (Φf ). For computing the correlation functions we use the prod- additional structure augmenting those of a causal set. uct ⋆H and the state ω0 (given by evaluation at φ = 0). However, we hope that in our future research we will On the other hand, we can model interacting fields understand better how to obtain this structure more in- int ~ trinsically. In particular, we want to determine, using using AH (C)=(F(C)[[ ,λ]], ⋆H,int). In this case, the numerical simulations, how much choice there is in typi- interacting observable corresponding to Φf is just Φf , but for computing the correlation functions we use the cal sprinklings in the definition of a preferred past struc- ture. The element of choice can be removed altogether product ⋆H,int and the state ωint. Note that in one approach we work with complicated by defining a discrete d’Alembertian that is the average observables, but a simple product and a simple state, of PΛ over all possible preferred past structures Λ. We while in the other approach we have simple observables, also hope to be able to generalize the ansatz (40) so that but the product and the state become complicated. The the dimension of spacetime itself is not an input (as in crucial difference between the two approaches is how we [17–19]), but an emergent quantity. identify physical objects (e.g. the linear field) with ele- The quantization scheme we have proposed works for ments of F[[~,λ]], which is the underlying vector space a very general class of interactions and can be used to ~ test ideas of both causal set theory and pAQFT in new in both algebras A ,λ(C) and Aint(C). H H ways. In particular, one can study how approximating Remark IV.16. Note that in QFT on causal sets there the continuum works for interacting theories. One can is a natural UV regularization due to the existence of also investigate how our method of introducing interac- fundamental length scale. Since the theory is defined on tions using pAQFT framework relates to the more tra- discrete sets, none of the problems that appear in con- ditional approach using path integrals. On finite causal tinuum, due to singularities of the Feynman propagator, sets both approaches can be studied by numerical as well occur here. Hence there is no need for renormalization. as analytical methods, which is typically not the case in However, one has to be careful when taking the contin- continuum QFT. uum limit, since the UV divergences could again occur, if not taken care of properly. We hope to address this issue in our future work. ACKNOWLEDGMENTS

Alternative formulas for RλV and ⋆int, in terms of Feynman-like diagrams, have been derived in [37]. Even We thank William Cunningham, Fay Dowker, Eli though [37] is formulated for the continuum case, the re- Hawkins, Lisa Glaser, Ian Jubb, Christoph Minz, Rafael sults are algebraic in nature, so apply also to causal sets Sorkin, Sumati Surya and Stav Zalel for very useful dis- (see the Appendix for explicit formulas and more detail). cussions about causal sets! We also thank Ted Jacobson In the same work, it has also been shown that and Lee Smolin for useful comments on the manuscript and for pointing out some relevant references. K.R. would like to acknowledge the financial support of EP- RλV (F ) = rλV (F ) , (156) EP/P021204/1 ~=0 SRC (through the grant ) and would like

23 to thank the Perimeter Institute for hospitality and on- Theorem A.4. going support. The work of E.D.-H. and N.W. was partly ~ e(γ)−v(γ) v(γ) supported by summer studentships from the Department ( i ) ( λ) F ⋆T,int G = − − ~γ(F,G) (A3) of Mathematics, University of York and E.D.-H. also re- Aut(γ) γ∈G3(2) | | ceived support from the 2016 EPSRC DTP funds. X Next, we give the formula for the interacting star prod- uct in the more standard formulation, where classical ob- Appendix A: Interacting star product in terms of servables are identified with quantum ones by means of graphs −1 normal ordering (i.e. by applying the map αH , defined in section IV A) For the convenience of the reader, in this section we G summarize the results of [37] concerning formulas for Definition A.5. 6(n) is the set of (isomorphism classes4) of graphs with directed and undirected edges ⋆H,int and RλV in terms of graphs. and labelled vertices 1,...,n such that: Definition A.1. Let G(n) denote the set of directed Each unlabelled vertex is at least 3-valent, with at • graphs with n vertices labelled 1,...,n (and possibly un- least one ingoing and one outgoing edge; labelled vertices with valency 1) and there exist no directed cycles; ≥ • for 1 j

there are no loops. Definition A.10. G12(1) G11(1) is the subset of • graphs such that no unlabelled⊂ vertex has one incoming With this, for the given interaction V , the interacting edge, one outgoing edge, and no unlabelled edge. observable corresponding to F is Any graph in G11(1) can be obtained by adding vertices ( i)d(γ)−v(γ)( λ)v(γ)~e(γ)−v(γ) along directed edges of a graph in G (1). In this way, R (F )= − − ~γ(F ) 12 λV Aut(γ) the formula for the Møller map can be re-expressed as G γ∈X11(1) | | (A5) v(γ)+d(γ)~e(γ)−v(γ) ։ F ( i) where undirected edges represent ∆ , unlabelled vertices RH,λV (F )= − γ (F ) , Aut(γ) correspond to derivatives of λV and d(γ) is the num- γ∈G12(1) X | | ber of directed edges. As before,− we can also give a non- (A6) − perturbative (in λ) formula, where we sum up the con- where directed edges represent EλV , undirected edges tributions containing E− to obtain an expression that represent ∆F, and unlabelled vertices represent deriva- depends only on the full interacting Green function E− . tives of λV . λV −

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