View metadata, citation and similar papers atQuantum core.ac.uk Space-Time as a Quantum Causal Set brought to you by CORE provided by CERN Document Server Ioannis Raptis Theoretical Physics Group, Blackett Laboratory, Imperial College of Science, Technology and Medicine, Prince Consort Road, South Kensington, London SW7 2BZ, UK An algebraic representation of the causal set model of the small-scale structure of space-time of Sorkin et al. is proposed. The algebraic model suggested, called quantum causal set, is physically interpreted as a locally finite, causal and quantal version of the kinematical structure of general relativity: the 4-dimensional Lorentzian space-time manifold and its continuous local orthochronous Lorentz symmetries. We also discuss some possible dynamical scenarios for quantum causal sets mainly by using sheaf-theoretic ideas, while at the end we entertain the possibility of constructing an inherently finite and genuinely smooth space-time background free quantum theory of gravity. PACS numbers: 04.60.+n The causal set approach to quantum gravity was ini- tum causal set and bears a sound quantum physical inter- tially proposed by Sorkin and coworkers almost a decade pretation. As with causal sets, some possible dynamical and a half ago [1]. At the heart of this theoretical sce- scenarios for these finite dimensional algebraic analogues nario lies the proposal that the structure of space-time at of the locally finite posets modelling causal sets are briefly quantum scales should be modelled after a locally finite anticipated at the end mainly along sheaf-theoretic lines. partially ordered set (poset) of elements whose partial or- For expository fluency, we have decided to present the der is the small-scale correspondent of the relation that basic tenets of quantum causal set theory by means of a defines past and future distinctions between events in the brief history of the central mathematical and conceptual space-time continuum of macroscopic physics. It is re- developments that led to the quantum causal set idea. markable indeed that from so simple an assumption one The topological ancestors of causal sets are the so- can recover the basic kinematical features of the classi- called finitary substitutes of continuous (ie, 0)mani- C cal space-time manifold of general relativity, namely, its folds, which are T0-topological spaces associated with lo- 0 topological (ie, -continuous), its differential (ie, 1- cally finite open covers of a bounded region X in a topo- smooth) anqd itsC conformal pseudo-Riemannian metricC logical manifold space-time M, and which have the struc- (ie,ametricgab of Lorentzian signature modulo its de- ture of posets [5]. Such posets can be also viewed as sim- terminant, which space-time volume measure anyway is plicial complexes in the homological sense, due to Alexan- accounted for by the local finiteness of the causal sets) drov and Cech,ˇ of nerves associated with open coverings structures, as well as, in a statistical and scale depen- of manifolds [6], and the substitutions of X by them are dent sense, its dimensionality. That alone should suffice regarded as cellular approximations of a 0-manifold— for taking the causal set scenario seriously. they are locally finite discretizations of theC locally Eu- However, by abiding to a Wheeler-type of principle clidean topology of X M. Indeed, that they qualify holding that no theory can qualify as a physical theory as approximations proper⊂ of the topological space-time proper unless it is a dynamical theory, one could main- manifold rests on the fact that an inverse or projective tain that causal set theory would certainly be able to system of those posets ‘converges’, at the projective limit qualify as a physically sound theoretical scheme for quan- of infinite resolution or localization of X by ‘infinitely tum gravity if it somehow offered a plausible dynamics small’ open sets about its points, to a space that is topo- for causal sets. Thus, ‘how can one vary a locally fi- logically equivalent (ie, homeomorphic) to X itself [5]. nite poset?’ has become the central question underlying Subsequently, the aforesaid finitary substitutes of con- the quest for a dynamical theory of causal sets [2]. To tinuous space-time were represented by finite dimen- be sure, a classical stochastic sequential growth dynam- sional, complex, associative incidence Rota algebras and ics for causal sets, supported and guided by a discrete the resulting algebraic structures were interpreted quan- analogue of the principle of general covariance of general tum mechanically [6]. The standard representation of a relativity, has been proposed recently [3] and it has been poset P =(S; ), where S is a set of elements and ‘ ’an regarded as the stepping stone to a deeper quantum dy- irreflexive and→ transitive binary relation between→ them, namics which, in turn, has already been anticipated to by an incidence algebra Ω is given by involve an as yet unknown ‘sum-over-causal set-histories’ argument [1,4]. In the present letter we offer an algebraic alternative P = p q : p; q S Ω(P ):=spanC p q { → ∈ }−→p s;if q = r{ → } to the causal set model of the kinematical structure of (p q) (r s)= → space-time in the quantum deep, which is coined quan- → · → 0 ; otherwise 1 C which depicts the defining C-linear and (associative) mul- of X about its points the commutative algebra 1(X) tiplication structure of Ω(P ). of smooth complex-valued coordinates labelling theC X’s C The interpretation in [6] of the incidence algebras asso- point events, as well as the Z+-graded 1(X)-bimodule ciated with the finitary posets as discrete quantum topo- of smooth complex exterior differentialC forms cotangent logical spaces rests essentially on the following four struc- to every point event in the differential manifold space- tural issues: time [6,8]. At the same time, the aforementioned recovery of the In the Rota algebraic environment, the partial or- classical smooth space-time manifold, in the ideal limit of • der arrow-connections ‘ ’ between the elements of → infinite localization, from the quantal incidence algebraic the posets, that actually define the aforementioned substrata was interpreted physically as Bohr’s correspon- T0-topologies on them, can superpose coherently dence principle, or equivalently, as a classical limit,and with each other (ie,theycaninterfere quantum me- it represented the emergence of the classical local differ- chanically). This possibility for coherent superpo- ential space-time macrostructure from the ‘decoherence’ C sitions, which is encoded in the -linear structure of (ie, the breaking of coherent quantum superpositions of the incidence algebras, is manifestly absent from in) an ensemble of these characteristically alocal quantum the corresponding posets which are merely multi- space-time structures [6,8]. Hence, the finite dimensional plicatively associative structures (ie, arrow semi- incidence algebras associated with Sorkin’s finitary poset groups or monoids, or even small poset categories). substitutes of the space-time continuum encode informa- The incidence algebras are noncommutative. tion about both the topological ( 0) and the differential C • ( 1) structure of the manifold. Arguably, this is an al- The points extracted from these non-abelian alge- gebraicC representation of Sorkin et al.’s thesis mentioned • bras are, in a technical sense, quantum and so are earlier in connection with causal sets that a partial order the (topological) spaces that they constitute. On effectively determines both the topological and the differ- the one hand, the qualification of points as being ential structure of the space-time continuum. quantum comes from their being identified with the On the other hand, in quite a dramatic change of phys- kernels of (equivalence classes of) irreducible (fi- ical interpretation and philosophy nicely recollected in nite dimensional) Hilbert space representations of [9], Sorkin stopped thinking of the locally finite posets the noncommutative incidence algebras, which ker- above as representing finitary discretizations or simpli- nels are in turn primitive ideals in these algebras. cial decompositions (with a strong operational flavor) of On the other hand, the (Rota) topology defined on the space-time continuum and, as noted in the opening these points can be thought of as being quantum, paragraph, he regarded them as causal sets. Then, he and because in the very definition of its generating rela- coworkers posited that the deep structure of space-time is, tion the noncommutativity of the algebras’ product in reality, a causal set [1]. Thus, partial orders stand now structure is crucially involved [6]. Quantum points for causal relations between events in the quantum deep and topological spaces in the sense above have been and not for topological relations proper, as it were, the studied from the more general and abstract per- original ‘spatial’ conception of a poset gave way to a more spective of mathematical structures representing ‘temporal’ one. Of course, the ideas to model causality ‘quantal topological spaces’ known as quantales [7]. after a partial order and that both at the classical and at The incidence Rota algebras and their finite dimen- the quantum level of description of relativistic space-time • sional Hilbert space representations have been seen structure partial order as causality is a more physical no- to be sound finitary-algebraic models of space-time tion than partial order as topology have a long and noble foam [8]. The latter refers to the conception of the ancestry [10]. So, as a consequence of Sorkin’s ‘semantic topology of space-time as a quantum observable:a switch’, the macroscopic space-time manifold should be quantally fluctuating, dynamically variable and in thought of as a coarse approximation of the fundamental principle measurable structure. causal set substrata in contrast to the finitary topologi- cal posets which, as noted earlier, were regarded as be- At this point it must be stressed that the incidence ing coarse approximations of the space-time continuum.