This is a repository copy of Algebraic Classical and Quantum Field Theory on Causal Sets. White Rose Research Online URL for this paper: https://eprints.whiterose.ac.uk/157643/ Version: Accepted Version 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 Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [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.