PoS(FFP14)222 http://pos.sissa.it/ † ∗ [email protected] Speaker. FV is supported by the Netherlands Organisation for Scientific Research (NWO) with a Veni Fellowship. Research in quantum gravity has grownand jauntily in philosophical the issues recent that years, intersecting haveon with a which conceptual long Loop history. Quantum Gravity Inphilosophy has of this grown, science paper the and I way the analyze itis main the deals based. methodological conceptual with and basis some philosophical In classical assumptions particular,relationalism problems on have I of had which in emphasize it the the construction of importance the that theory. atomism (in the broadest sense) and ∗ † Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

Frontiers of Fundamental Physics 14 -15-18 FFP14, July 2014 Aix Marseille University (AMU) Saint-Charles Campus, Marseille Francesca Vidotto Radboud University Nijmegen, Institute for Mathematics,Mailbox Astrophysics 79, and P.O. Particle Box Physics, 9010, 6500E-mail: GL Nijmegen, The Netherlands Atomism and Relationalism as guiding principles for Quantum Gravity PoS(FFP14)222 ? The action at bodies Francesca Vidotto , whose combination yields all the beautiful . Plato did not like the atomistic/materialistic atoms 2 abhors a vacuum ) and its enormous success as funding stone of the new developing hypotheses non fingo position, defended by Descartes and Leibniz, denying the existence of space but as a . Only with the introduction of the notion of field this worry was removed: forces are Scientists are guided in their investigation of Nature by ideas brought to them by philosophers. I think this is particularly true when the theory we are searching for requires a deep rethinking What does exist? Does space exists? or it emerges from the relations between Faraday and Maxwell described the electromagnetic phenomena as a manifestation of a field. General Relativity identifies spacetime and the gravitational field. This field, like all fields, Nonetheless, Newton hesitated in proposing such an idea of space. He called this a working The XVII-century debate on the nature of space should be framed in a culture dominated Not always scientists are aware ofthey this, but address, still their have way emerged ofof working in and these a the debates, kind humus of that of questions shapeddirections that philosophical for our our debates present science. investigations, of can long help history. us to Being recognizeof aware basic paths concepts and such as new space anda time. theory In that order hopes to to analyze the providepossible recent a reconstruction findings new in of fundamental quantum the view gravity, thread on from the the nature ancient of time space to and our time, modern l debate. start1. by a The historical framework: from Democritus to Einstein The physics of the XXexist century is took a the manifestation of notion some of field.a field particle So to is it an just is ontological every the particle, excitation extreme:is as of everything discovered space, a by that as field, Quantum discovered a Mechanics: by manifestation Einstein: of space the quantum and nature time of are every the field.should expression of So exhibit it the quantum gravitational field. propertiesproperties at as some well. This scale, is therefore the beauty space and and the difficulty time of must quantum gravity: have it quantum obliges us to a com- Atomism and Relationalism as guiding principles for Quantum Gravity idea that space canbut exist a as revolution a brought to separatepostulates humanity entity the by existence with Leucippus of respect elementary and units, to Democritus the in the the bodies V is century not BCE. a They primitive idea, science made the later scientists forgethis the doubts law about of it. gravitation, What concerneddistance acting Newton on more was an that emptyfields, stage, that were permeate space. leading to the possibility of an hypothesis ( rainbow of different things that we observe inis space. the world. Space is The here atoms associated randomly with movesgives the in notion an a of stage: vacuum. ontological this This status is aschool: to contradictory Aristotle notion, the thought because that non-being. Nature That’s why this position was attacked by Athens’ net of relations. Onwhich the Democritean other bodies side there moves was ondeterministic the an law substantivalist infinite of position immutable the proposedposition fixed new by to empty Mechanics. Newton, became space, in the mainstream The according one empirical to nowadays. the success of Mechanics has brought this by the aristotelian/platonic thinking.relational The mainstream side of the debate was there taken by the views either, at the point never toera, mention only Democritus. Atheniensis wisdom Trough survived Greek to decadence the and centuries. the Christian PoS(FFP14)222 ] and ]. The 3 1 . If this is done Francesca Vidotto L covariant fields - ]. This is the beginning and the core 5 , 4 covariant quantum fields - Fields Spacetime Fields - 3 : Particles Fields Space Time Space Fields : Particles : Particles General : Particles : General : Particles Fields Spacetime Fields : Particles l : Quantum prevent a component of a field to be measured with arbitrary precision ] in which the two proved that in a quantum field theory the Heisenberg , its location should be determined with some precision x 2 : Matter Time : Matter Maxwe - : Particles Space Time Space : Particles do not Quantum gravity Quantum Newton Faraday relativity Special theory Quantum relativity General Descartes In modern terms, Bronstein’s argument would be the following. In order to measure some As it often happens in science, the contemporary questioning of the nature of space started with he found that Landau’s intuition in this case was correct [ uncertainty relations field value at a location intuition was that an arbitrarily sharpwith spatiotemporal the localization Heisenberg would uncertainty have relations. been in Nielssuggestion contradiction Bohr was guessed wrong. immediately, To and prove correctly, it thatlead wrong, this he to embarked a in classic a research paper program [ with Rosenfeld, that of quantum gravity. at a spacetime point. The Bohr-Rosenfeldif analysis repeated was with done the using gravitational the electromagnetic field? field: This what question engaged Landau’s friend Bronstein [ a mistake. The problem was toand extend Heisenberg’s Peierls uncertainty relations suggested to that fields.component In once of 1931 a applied Landau field at to a given fields, spacetime point the could be uncertainty measured with relation arbitrary precision would [ imply that no Atomism and Relationalism as guiding principles for Quantum Gravity plete rethinking of what wedescription of mean quantum by fields space in andity). order time. to We have make to In it learn covariant orderof (i.e. a to standard new compatible do field language with this, for theory General describing we on Relativ- presents the need the flat world. discreteness to spacetime, typical A sharp nor of language every that our quantumspace which of system. and is Riemannian We what neither have is continuous to that quantum understand geometry time. what as is quantum space 2. The end of space and time, the beginning of quantum gravity PoS(FFP14)222 , p in ∆ ¯ h (2.2) (2.1) / is the ¯ h . In the G > E x and distorts ∆ 2 c / , where Francesca Vidotto 2 E c / ∼ M GM ∼ R . 3 . But in this case the region of size , L ¯ hG Lc cm = 33 3 − . Sharp localization requires large energy. c pG 10 cp = ∼ ∼ 4 acts as a gravitational mass 4 E c 3 EG E c ¯ hG . , and since Heisenberg uncertainty gives = L x r ∆ = 2 becomes larger than c ∼ R MG R o l = is concentrated in a sphere of radius L . This is a very well known consequence of Heisenberg uncertainty: M L / ]: “Without a deep revision of classical notions it seems hardly possible ¯ h 4 arbitrarily small in order to get a sharper localization, the concentrated > , the quantum nature of the particle implies that there is an uncertainties , that is when | x L L p | . The average absolute value of the momentum cannot be smaller than its L , we find that it is not possible to localize anything with a precision better than / L ¯ h . Quantum Mechanics can be interpreted as the discovery of a minimal action c > can be decreased only up to a minimum value, which clearly is reached when the associated to its position and its momentum. To have the location determined with , this should be greater than p L L p ∆ ∆ In General Relativity any form of energy In Bronstein’s words [ This simple derivation, using only semiclassical physics, characterizes the physics of quantum Combining the relations above, we obtain that the minimal size where we can localize a quan- and that we wanted to mark will be hidden beyond a black hole horizon, and we loose localization. x sharp location requires large momentum. In turn, large momentum implies large energy relativistic limit, where rest mass is negligible, spacetime around itself. The distortionblack hole increases forms when when energy a is mass concentrated, to the point that a all physical interactions, or equivalentlya the finite fact number that of a distinguishable finiteamount region (orthogonal) of of quantum information phase in states, space the andminimal contains length. state therefore only of there a is system. a minimal Quantum gravity is theto discovery extend that the there quantum isforces theory a us of to gravity take also seriouslyRosenfeld to the argument, showing connection [the that between quantum short-distance] fields gravity domain." can and befails Bronstein’s defined geometry. in in result the arbitrary It presence small shows regions of of that gravity.as space, the Therefore a Bohr- the quantum quantum field gravitational in fieldneeded cannot space. to be The define treated smooth simply standard metric quantumgenuine geometry field, quantum of is theory physical itself of space, geometry. affected which by is quantum the theory. ground What we need is a fluctuation, therefore spacetime. The existence ofgives a it minimal a length universal character, scaleRelativity analogous is can to be the seen Special main as Relativity featurespeed the and of of discovery Quantum light quantum of Mechanics. gravity the and existence Special of a maximal local physical velocity, the we have which is called the Planck scale.Below Above this this length scale, scale, it we makes can no treat sense spacetime as to a talk smooth about space. distance or extension. horizon radius reaches precision Therefore tum particle without having it hidden by its own horizon, is the length Atomism and Relationalism as guiding principles for Quantum Gravity by having a particle at Newton constant. For energy will grow to theL point where Solving this for ∆ PoS(FFP14)222 ) ) 3 2 ( ( SU SU Francesca Vidotto for weak interactions and ) , that is compact yielding a discrete ) 2 ] is a state defined by the invariance ( 2 9 ( , SU 8 SU 5 for electromagnetic interactions, ) 1 ( U ], describing the geometry at some fixed time. In the quantum theory, the tetrad turns out General Relativity is traditionally formulated using the metric, but it admits an equivalent for- So consider now the tetrad formulation. The invariance under diffeomorphisms implies the As in every Yang-Mills theory, the gauge invariance of the triads brings in a gauge field. The variables used in Loop Quantum Gravity are group variables, as the variables used to de- Background independence is at the core of General Relativity: spacetime is not a stage where 7 , 6 spectrum for the corresponding observables.observables In such particular, as the areas, geometry volumes can anding be angles, to described constructed the through starting triads. from the operator correspond- mulation where the fundamental object areexpressed in reference terms fields: of the the tetrads, tetrads. thereforeto The it the metric is always can tetrad possible always one. to be passthe from On gravitational the the metric field. formulation other A hand,includes description only fermions, of so in General reality the Relativity cannot in tetradform. be tetrad formulation variables complete assumes fermions in somehow can a a more be framework fundamental that coupled does to not independence by coordinate transformations. Thisassociated means a that tetrad for that each is pointof locally of the Lorentz spacetime Lorentz invariant. there transformation, We since is can time reducethis is pure ourselves gauge. gauge to It in the is General rotational this Relativity part andstarting gauge we invariance, point can that for always naturally fix the arises quantum in[ the theory. classical The gravitational quantum theory, the states have to be thought as boundary states This is an object incurvature. the In Lorentz the quantum algebra, theory thatexponential the codes of gauge the the invariant information gauge observables are field. aboutthis defined intrinsic the by In name and the the Loop extrinsic path-ordered for languageholonomy, the of namely quantum particle the theory). physics, parallel In this transport the is of language a of the Wilson differential gauge loops geometry, field (from it seen is as called a connection over the scribe the other fundamental interactions. The group is for strong interactions. Gravitationalsymmetry, as interactions the can others. also be characterized in terms of a gauge under the rotations of theto triads, its the physical excitations size) of and each the spacetime adjacency relation quanta between (its them spin, (coded corresponding in an abstract graph). principal bundle. The canonical analysisthe of triad. the theory shows that this is the conjugate variable to to be the generator ofimplies SU(2) that transformations, spacetime satisfying is the quantizedmore algebra with a a of tetrad discrete angular for spectrum. momentum. eachthe spacetime Notice This orientations point, that of but now a the we tetrad reference don’tmatter. for fields have each is any In spacetime not Loop quanta. relevant, Quantum only For Gravity the each a of relations spinnetwork these, between state adjacent [ quanta fields interact, but itis is that an they interacting areare field associated respectively to as some the gauge others. symmetry. The In modern the understanding Standard of Model, fields these symmetries Atomism and Relationalism as guiding principles for Quantum Gravity 3. Quanta of spacetime PoS(FFP14)222 Σ W W , as it Σ . How do we will depend on Σ . The quantum Francesca Vidotto are the transition W . For instance, on Σ will not be a function W processes boundaries . Obviously, M ∂ ]). Thus, consider a finite bounded 7 = , Σ 6 already specifies the shape of can be expressed as the Feynman path Σ W ]. What happens at the boundary of a process are functions of the values of the field on the 10 ? Quantum gravity teaches us that we have not Σ 6 W . Formally, M of the system with another completely generic physical system. ), which contains the entire relevant geometrical information on the Σ portion of the trajectories of a system, where “finite" means finite in time , with fixed values on the boundary M interaction finite of a spacetime region. Formally, this will be given by the Feynman path integral in , which determine probabilities of alternative sets of outcomes. Σ W ]. It computes the probabilities for the different possible effects of such interactions. A of spacetime. For a field theory on a given fixed spacetime, this simply means to consider 10 M In fact, the gravitational field on the boundary of Let us now apply this perspective to field theory and in particular to gravity. The central idea This happens as a consequence of a general property of parametrized systems: the temporal We want now to discuss the dynamical properties of spacetime. In classical mechanics, a dy- Quantum Mechanics describes how physical systems affect one another in the course of inter- Now let us take this same idea to gravity. Then what we want is the transition amplitude to do it. is now to consider a but also in space (this is called the “boundary formalism" [ information is stored and mixedseparated among from the other dynamical variables variables,tivistic as instead context, as in this being unparametrized holds signed Newtonian for out mechanics. temporal and as In well the as general for rela- spacial locations: if simply a physical Notice that the structureconsistent of with the arbitrary displacements theory of isprocesses. what largely we determined decide to by consider the as fact the that boundaries between this description is now specify the shape, namely the geometry, of includes all relevant metric informationswill that be can a function be of gathered the on boundary fields the and surface nothing itself. else. Therefore common language for describing suchBut processes this is anthropomorphic in language terms is of misleading “preparation" [ and “measurement". namical system can be defined bydefining the the relations allowed between trajectories. initial In and Quantumnot final Mechanics, coordinates deducible the and from momenta, trajectories between the interactions interaction are outcomes, and the theory describes actions [ Atomism and Relationalism as guiding principles for Quantum Gravity 4. Quantum relations analogs of the relations betweenamplitudes values of physical variables at the region the evolution of the field inTherefore a spacial the box, transition with given amplitudes boundarysurface, values will at the be the final boundaries functions spacelike of of the surface,box. box. the but field In also values other on on the words,entire “sides": the the boundary transition initial of the amplitudes the spacelike timelike spacetime surfaceintegral of region that the bounds field in the the time lapsed between the initial and final surfaces. that depend on the valuethe of boundary the gravitational field (as well as any other field which is present) on of space and time(up variables, to but diffeomorphisms simply of a function of the gravitational field on the boundary the internal region, at fixed boundary values of the gravitational (and other) fields these values of the fields as well as on the (spacetime) shape and geometry of PoS(FFP14)222 ]. ]. the 16 10 , is 15 , 14 is the “sum , W 13 , Francesca Vidotto are only localized 12 , 11 bodies . Intuitively, Σ : this is called a spinfoam. The explicit form Σ 7 → a process: because it is actually like a Feynman sum of includes all dynamical objects, included the gravitational is Locality a spacetime region. A state is not somewhere in space: it ← bodies is between systems, where the relation is instantiated by a physical bulk region bounded by finite relations State Boundary, space region Process Spacetime region Quantum Mechanics General Relativity is one of the most important result in quantum gravity of the last years [ This structural identification is in fact much deeper. As noticed, the most remarkable aspect The formal structure of Quantum Mechanics is relational, because Quantum Mechanics gives The structure of General Relativity is also relational, because the localization of dynamical Therefore in quantum gravity the quantum dynamics will be captured by a transition amplitude A fundamental ingredient in XX century physics is locality: interaction are local, namely they which is a function of the (quantum) state of the field on a surface W of quantum theory isamplitudes are that not the affected by boundary displacing thesystem". between boundary between processes “observed The system" can and same be “observing physical is moved theory true is at for therefore wish. a spacetime:Because description of Final of boundaries locality how total are and arbitrary because arbitrarily partitionsand of partitions of drawn gravity, of nature in these spacetime. affect partitions A spacetime. one are spacetime another. is at The a the process, same a time state is subsystems what split happens at its boundary. probability amplitudes for processes, and a process is what happens between interactions [ objects is not given with respect to a fixed background structure; instead, 5. The relational structure of quantum gravity Therefore Quantum Mechanics describes theother. universe States in are terms descriptions of of the mannersis a way therefore system systems based can affect affect on one another an- interaction. system. Quantum Mechanics require spacetime contiguity. But the contrary isare also true: contiguous the only is way to by ascertainstructural that analogy means two between objects of the relations having on whichspacetime them Quantum is Mechanics interact. is based. based and Quantum those Therefore gravity onin makes which locality a this spacetime reveals connection region: completely a a explicit. fundamental process A process is not Atomism and Relationalism as guiding principles for Quantum Gravity boundary. over geometries" on a of with respect to one another, where field. The relevant relation thatbeing “next builds to the one spacetime another" structure inpatchwork is spacetime. of of We spacetime can course regions view adjacent contiguity: a to general the one relativistic another fact theory at of as their a boundaries. dynamical description of the way twoanother. processes Vice versa, interact, a or spacetime two region everything that spacetime can regions happen pass between information its boundaries: to one W PoS(FFP14)222 , Phys. Nucl. , ? Quan- Zh. Eksp. bodies Francesca Vidotto Phys.Rev.D , 1936:140–157, 9. , D52:5743–5759, Class. Quant. Grav. Mathematiks-fysike Phys. Rev. Phys. Z. Sowjetunion 8 Matvei Petrovich Bronstein and Soviet Theoretical Physics in , 69:56–69, 1931. Covariant Loop Quantum Gravity. An introduction to Quantum Cam, 2014. , 12:319–352, 2008, hep-th/0509122. , B799:136–149, 2008, 0711.0146. Zeitschrift fur Physik , 99:161301, 2007, 0705.2388. , 12:65, 1933. Nucl. Phys. . Birkhäuser Verlag, 1994. , 6:195, 1936. , B798:251–290, 2008, 0708.1236. , B575:318–324, 2003, hep-th/0306025. Tear. Fiz. Lett. Adv. Theor. Math. Phys. quantentheorie. Phys. Rev. Lett. Meddeleser the Thirties 1995, gr-qc/9505006. Gravity and Spinfoam Theory. Phys. parameter. 25:125018, 2008, 0708.1595. Quantum Gravity. 2009, 0909.0939. 76:084028, 2007, 0705.0674. What does exist? Does space exists? or it emerges from the relations between [5] M.P. Bronstein. Kvantovanie gravitatsionnykh voln (quantization of gravitational waves). [1] L. Landau and R. Peierls. Erweiterung des unbestimmheitsprinzips fur die relativistische [2] N Bohr and L Rosenfeld. Det kongelige danske videnskabernes selskabs. [4] M.P. Bronstein. Quantentheorie schwacher gravitationsfelder. [3] Gennady Gorelik and Victor Frenkel. [6] Robert Oeckl. A ’general boundary’ formulation for quantum mechanics and quantum[7] gravity. Robert Oeckl. General boundary quantum field theory: Foundations and probability interpretation. [8] Carlo Rovelli and Lee Smolin. Spin networks and quantum gravity. [9] Francesca Vidotto Carlo Rovelli. Atomism and Relationalism as guiding principles for Quantum Gravity 6. Conclusion tum gravity is the questwhile for doing a this, synthesis quantum gravity between would Quantum achievetionalism: a Mechanics synthesis space and also is General between a substantivalism Relativity. field, and that But rela- by come atoms to of existence space, only through defined its through interactions. their Space relations. is constituted References [14] Laurent Freidel and Kirill Krasnov. A New Spin Foam Model for 4d Gravity. [12] Jonathan Engle, Roberto Pereira, and Carlo Rovelli. The loop-quantum-gravity vertex-amplitude. [13] Jonathan Engle, Etera Livine, Roberto Pereira, and Carlo Rovelli. LQG vertex with finite Immirzi [15] Wojciech Kaminski, Marcin Kisielowski, and Jerzy Lewandowski. Spin-Foams for All Loop [16] Etera R. Livine and Simone Speziale. A new spinfoam vertex for quantum gravity. [10] Carlo Rovelli. Relational Quantum Mechanics.[11] 1995, quant-ph/9609002. Jonathan Engle, Roberto Pereira, and Carlo Rovelli. Flipped spinfoam vertex and loop gravity.