Quantum Space-Time As a Quantum Causal

Home , Rafael Sorkin

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, ∞- 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 ∞(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 ∞(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 • ( ∞) 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 topological 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. algebras associated with Sorkin’s finitary topological Hence, the local finiteness of the open coverings of the posets were shown to encode combinatorial information bounded region X in the continuous space-time manifold not only about the topology of the space-time manifold, M relative to which the topological posets were defined but also about its differential structure. In fact, they in [5] was replaced by the slightly different local finiteness were seen to be Z+-graded discrete differential manifolds of the causal sets, although the underlying mathematical [6,8]. Thus, they were physically interpreted as the retic- structure, the poset, remained the same [9]. ular substitutes of bounded regions X of the ∞-smooth In the sequel, after having replaced the finitary topo- space-time manifold M of macroscopic physics.C More- logical posets by causal sets as the truly fundamental over, it has been argued that, similarly to the case of the structures underlying the classical space-time manifold of finitary posets, an inverse system or net of incidence alge- macroscopic experience, it was a rather natural step to bras yields at the projective limit of infinite localization associate with the latter incidence Rota algebras and, like

2 in the case of the incidence algebras representing discrete ity in its gauge-theoretic formulation in terms of the quantum topological spaces [6,8], to interpret the result- spin-Lorentzian connection on a principal fiber bun- +A ing structures as quantum causal sets [11]. Hence, in the dle with structure group L over the ∞-smooth space- new non-abelian algebraic realm of quantum causal sets, time manifold M. Furthermore, finitary-algebraicC ver- the causal arrows ‘ ’ defining their causal set correspon- sions, of strong categorical (ie, functorial) character, of dents can superpose→ quantum mechanically with each the principles of locality and general covariance were for- other. This lies at the heart of the conception of quantum mulated sheaf-theoretically mainly due to the fact that causality and qualifies the epithet ‘quantum’ in front of the s were represented by suitable finitary spacetime ‘causal sets’. Quantum causal sets, like their causal set sheafA morphisms. counterparts, stand for locally finite, causal and quan- The following step in the development of quantum tal models of the kinematics of the classical space-time causal set theory was to carry the entire de Rham theory of general relativity which, as mentioned above, can be of differential forms on the classical ∞-smooth space- summarized into the assumption of a (4-dimensional) time manifold, virtually unaltered, to theC reticular realm Lorentzian ∞-smooth manifold. of the curved finitary spacetime sheaves of quantum Having inC hand sound reticular and quantal models of causal sets with the help of some rather ‘universal’ sheaf- the kinematics of gravitational space-time, and by abid- cohomological techniques borrowed from ADG [14]. The ing to the aforesaid Wheeler-type of criterion for a phys- central achievement of that transcription was the sheaf- ical theory, the next stage in the development of quan- cohomological classification of the non-flat reticular spin- tum causal sets was the search for a plausible dynamics Lorentzian connections the quanta of which, originally for them. The basic idea in [12] was that in order to coined causons in [12],A were taken to represent dynami- construct a dynamical theory of quantum causality, and calquantaofcausality—elementary particle-like entities, since Zeeman had shown that causality as a partial or- anticipated to be closely related to gravitons, that dy- der determines the conformal metric structure of the flat namically propagate in a discrete manner in the ‘curved (ie, non-dynamical) Minkowski space of special relativ- quantum space-time vacuum’ represented by the curved ity and its global orthochronous Lorentz symmetries up finitary spacetime sheaves of quantum causal sets. to conformal transformations [10], one should somehow However, it is fair to say that, the aforementioned look for a scenario to curve quantum causality. Then, kinematical developments aside, an explicit dynamics for the main intuition, principally motivated by the fact that quantum causal sets has not been proposed yet. The the curved space-time of general relativity is Minkowski formulation of a finitary, causal and quantal analogue space localized or gauged, was that we could possibly ar- of the classical vacuum Einstein equations for gravity, rive at dynamical variations of quantum causal sets by which in turn can be interpreted as the dynamical equa- localizing (or gauging) them and, as a result, by math- tions for the reticular sort of Ashtekar’s (self-dual) spin- ematically implementing this localization process sheaf- Lorentzian connection variable inhabiting the relevant theoretically. In other words, a possible reply to Sorkin’s curved principal finitary spacetimeA sheaves of quantum question ‘how can one vary a poset?’ mentioned earlier is causal sets [12], is currently under way [15] again along ‘by sheaf-theoretic means’. To this end, we first needed to the general lines of Mallios’ ADG. A ‘covariant’ (ie,La- adapt basic sheaf-theoretic notions and constructions to grangian or action based) path integral over the space a locally finite setting. With that need in mind, finitary of connections , or even more generally, a ‘sum-over- spacetime sheaves of (algebras of) continuous observables quantum causalA set-histories’ scenario appears to be in- on Sorkin’s finitary substitutes were first defined in [13]. tuitively more well suited for such a quantum dynamics Shortly after, and in a way akin to the aforesaid tran- than a canonical (ie, Hamiltonian) approach mainly due sition from finitary substitutes to causal sets and then to the obvious absence in the reticular environment of to quantum causal sets, finitary spacetime sheaves of the the causal or the quantum causal sets of (a generator incidence algebras modelling quantum causal sets were of) infinitesimal time increments [16]. This also seems defined in [12] along the lines of Mallios’ Abstract Differ- to accord with Sorkin et al.’s anticipation noted earlier ential Geometry (ADG). The latter pertains to the geom- of a ‘sum-over-causal set-histories’ dynamical scheme for etry of vector, (differential) module and algebra sheaves causal sets. which generalizes the usual differential calculus on ∞- Another possible quantum dynamics for quantum manifolds to an abstract differential geometry on, in prin-C causal sets, now perhaps one with a more canonical fla- ciple, any (topological) base space [12]. Principal finitary vor, arises when, instead of working with the individual spacetime sheaves of quantum causal sets over the causal incidence algebras representing quantum causal sets per sets of Sorkin et al., having for structure group reticular se, one works with the ‘universal’ incidence algebra of versions of the continuous local orthochronous Lorentz all finitary incidence algebras that are partially ordered + group manifold L = SO(1, 3)↑ of general relativity, to- by inclusion [17]—again, itself a poset or net represent- + gether with non-flat finitary ` = so(1, 3)↑ sl(2, C)- ing ‘quantum causal refinement’ [12]. This poset may be valued spin-Lorentzian connection 1-forms ,' were seen thought of as the incidence algebraic analogue of the lat- to be sound models of a locally finite, causalA and quantal tice of all (finite) topologies studied in [18] in the context version of the kinematics of classical Lorentzian grav- of quantum topology, and it may be interpreted as some

3 kind of kinematical state space for quantum causal sets. smoothness of the space-time manifold M,andthetwo One could then try to define suitable generalized position notorious problems associated with it, namely, the ‘inner and momentum-like observables of quantum causal topol- product problem’ [23] and the ‘problem of time’ [24]. ogyala ` Isham, impose some non-trivial canonical com- I wish to thank Chris Isham, Tasos Mallios, Chris Mul- mutation relations on them, and look for a Hamiltonian- vey, Steve Selesnick, Rafael Sorkin and Roman Zapatrin like operator in terms of them that effects contiguous for numerous exchanges about causal sets, some possible dynamical transitions between quantum causal sets, as dynamical scenarios for them, as well as about their inci- it were, dynamical quantum (causal) topology changes in dence algebraic and sheaf-theoretic representations. This the universal incidence algebra. This could also be done research was supported by an EU Marie Curie Postdoc- ‘by hand’, that is, one could first define creators and toral Fellowship held at Imperial College. annihilators of vertices in causal sets, look for their incidence algebraic correspondents in the associated quantum causal sets and their finite dimensional Hilbert space representations as in [8], then consider their commutation relations, and finally proceed in a canonical Hamiltonian fashion to implement dynamical quantum causal topology changes as briefly alluded to above [19]. [1] L. Bombelli, J. Lee, D. Meyer, and R. D. Sorkin, Phys. In parallel with these canonical scenarios, one could Rev. Lett. 59, 521 (1987); R. D. Sorkin, in Proceedings also try to define up-front the notion of quantum causal of the Third Canadian Conference on General Relativity histories and the finitary sheaves thereof in the univer- and Relativistic Astrophysics, edited by F. Cooperstock sal incidence algebra [20] and proceed in a consistent- and B. Tupper (World Scientific, Singapore 1990), and histories-like fashion to look for a quantum measure for in Relativity and Gravitation: Classical and Quantum, quantum causal histories, relative to which the ‘sum- edited by J. D’Olivo et al. (World Scientific, Singapore 1991). over-quantum causal set-histories’ is going to be weighed, [2] Rafael Sorkin in private communication. in the form of some kind of decoherence functional on [3]D.P.RideoutandR.D.Sorkin,Phys.Rev.D61, 024002 them [21]. Alternatively, one could seek directly for a (2000). discrete quantum causal dynamics and for functorial ex- [4] Two immediate results, of utmost importance to current pressions of discrete causal covariance and quantum en- quantum gravity research, anticipated to come from the tanglement by applying ideas from linear logic and the formulation of a sound quantum dynamics for causal sets theory of polycategories, both commonly used nowadays would be the derivation of the law for the entropy of the to study discrete dynamical systems in theoretical com- horizon of a black hole, as well as an estimate and an puter science, to quantum causal histories [22]. explanation of the (small) value of the cosmological con- Certainly though, and in spite of the prominent ab- stant (R. D. Sorkin, theoretical physics research seminar, sence of an explicit formulation of a quantum dynam- Imperial College 4/12/2001). ics, whether covariant, canonical or other, for either the [5]R.D.Sorkin,inGeneral Reativity and Gravitation,Vol. causal or the quantum causal sets, the finitary-algebraic 1, edited by B. Bertotti et al. (Consiglio Nazionale delle and sheaf-theoretic approach to quantum space-time Ricerche, Roma 1983), and Int. J. Theor. Phys. 30, 923 structure supporting quantum causal set theory has pro- (1991). vided us with strong clues on how to construct an in- [6]I.RaptisandR.R.Zapatrin,Int.J.Theor.Phys.39, trinsically finite quantum gravity that is genuinely ∞- 1 (2000); R. R. Zapatrin, Pure Math. Appl. (2002) (to smooth space-time background independent. ForC one appear), math.CO/0001065. thing, it has shown us that one can carry out the usual [7] C. J. Mulvey, Suppl. Rend. Circ. Mat. Palermo, II (12), differential geometric constructions, that are of vital im- 99 (1986); C. J. Mulvey and J. W. Pelletier, J. Pure Appl. portance for the formulation of general relativity, over Algebra (2002) (to appear). reticular base spaces that may appear to be singular or [8] I. Raptis and R. R. Zapatrin, Class. Quant. Grav. 20, 4187 (2001). pathological, or at best ‘anomalous’ and ‘odd’, from the [9]R.D.Sorkin,inThe Creation of Ideas in Physics,edited perspective of the classical ∞-smooth space-time con- C by J. Leplin (Kluwer Academic Publishers, Dordrecht tinuum. This is one of the benefits we get from ap- 1995). plying ADG to the finitary-algebraic setting, for ADG [10] A. A. Robb, “Geometry of Space and Time”, 1st edi- has time and again proven itself when it comes to evad- tion, Cambridge University Press, Cambridge (1914); ing ∞-smoothness and the singularities that go with it C E. C. Zeeman, J. Math. Phys. 5, 490 (1964); [12,14]. If anything, such an independence of quantum A. D. Alexandroff, Can. J. Math. 19, 1119 (1967); causal set theory (and of a future dynamics for it) from D. Finkelstein, Phys. Rev. 184, 1261 (1969), and Int. the smooth space-time continuum is more than welcome J. Theor. Phys. 27, 473 (1988). from the point of view of modern research in quantum [11] I. Raptis, Int. J. Theor. Phys. 39, 1233 (2000). gravity, since the theory appears to avoid ab initio the in- [12] A. Mallios and I. Raptis, Int. J. Theor. Phys. 40, 1885 finite dimensional diffeomorphism group Diff(M) of gen- (2001). eral relativity, which comes hand in hand with the ∞- [13] I. Raptis, Int. J. Theor. Phys. 39, 1703 (2000). C

4 [14] A. Mallios and I. Raptis, Int. J. Theor. Phys. (2002) (to appear), gr-qc/0110033. [15] I. Raptis, Locally Finite, Causal and Quantal Einstein Gravity (2002) (in preparation). [16] Rafael Sorkin in private communication. [17] Rafael Sorkin in private communication. [18] C. J. Isham, Class. Quant. Grav. 6, 1509 (1989). [19] Chris Isham in private communication. [20] F. Markopoulou, Class. Quant. Grav. 17, 2059 (2000); I. Raptis (2001) (pre-print), quant-ph/0107037. [21] Rafael Sorkin in private communication. [22] R. Blute, E. T. Ivanov, and P. Panangaden (2001) (pre-print), gr-qc/0109053, and (2001) (pre-print), gr- qc/0111020. [23] The problem of fixing the inner product in the Hilbert space of physical states by requiring that it is invariant under Diff(M). [24] The problem of requiring that the dynamics is encoded in the action of Diff(M) on the space of states.