Modal realism after string theory METAPHYSICS WITHIN STRING … · Tiziana Vistarini (CU Boulder) Modal realism after string theory October,27, 2017 23 / 67. Modal realism after string

Modalrealism afterstring theory

TizianaVistariniMETAPHYSICS WITHIN STRING PHYSICS

Modal realism after String Theory

Tiziana Vistarini

University of Colorado at Boulder

"Quantum Gravity: Physics and Philosophy" ERCProject Philosophy of Canonical Quantum Gravity

24-27 October 2017

Tiziana Vistarini (CU Boulder) Modal realism after string theory October,27, 2017 2 / 67


TizianaVistarini

This talk is a multilayer presentation. It unravels alandscape of interconnected ideas. The rationalebehind this choice has been somehow imposed bythe multidisciplinary nature of my work during thelast few years.The presentation is based on some parts of myforthcoming book (“Emergent Spacetime in StringTheory" - Routledge, 2017).It is also a work in progress for an invitedcontribution to a miscellaneous volume - “BeyondSpacetime: The Philosophical Foundations ofQuantum Gravity" - submitted to CambridgeUniversity Press.



TizianaVistariniThis presentation mainly divides in two parts.

(1) Background independence of string theoryexplored through the theory’s moduli space - anargument formulated in the mathematicallanguage of deformation theory.(2) The formal articulation of the argument turnsout to be useful to my attempt of naturalizing themetaphysics of possible world.



TizianaVistarini

The hope is the following: on the one sidepromoting a form of inductive metaphysics, namelya metaphysics sensitive to the “empirical sources"imported from physics; on the other side lettingthis type of metaphysical insights add someimportant dimensions to the technical debatewithin the physics circles.



TizianaVistarini

What do I mean here with “empirical sources"?Specifically, empirical sources here amount to bethe fundamental physical scenario (maybe not themost fundamental one)) delivered by string theory.That is, the fundamental physical ontology ofstring theory.Why does the physical ontology of stringtheory count as “empirical sources"?

› Risk of empirical incoherence (Maudlin 2009)› A way of rescuing the theory from the charge ofbeing empirical incoherent (Huggett, Wütrich,2013), along with the claim that string theory isalso empirically adequate, (Huggett, Vistarini,2015), (Dawid, 2013), and (Vistarini, 2017).



TizianaVistarini

As I have been working on string theorybackground independence I discovered somephilosophically interesting features of the theory:if one digs into the mathematical and physicalaspects of that background independence, one mayfind out that the theory’s account of spacetimeemergence is a quite philosophicallyconservative one: it smoothly revises or extendsthe traditional notion of mechanistic explanation.Not a straightforward achievement, if we think tothe historical use of the notion of mechanisticexplanation.



TizianaVistarini

Why smoothly? There is a meaningful sense inwhich we can say that already in any classical,pre-relativistic hamiltonian we can trace an embryoof spacetime emergence - see (Harvey, 2005);(Albert 2016); (Vistarini,2017).And both general relativistic spacetime emergenceand emergence of the extra dimensions (throughdualities arguments) in string theory develop thatnotion in a more radical physical scenario -(Vistarini 2017).



TizianaVistarini

Suppose you have a pre-relativistic, classicalscenario populated by, say, N particles in athree-dimensional Euclidean space. Its hamiltonianlooks like:X

i=1;::N

[:::]+X

K=1;::;N`1Vkj((xk`xj)2+:::+(zk`zj)2):

k 6= j The x,y and z are cartesian coordinates inthe three dimensional space, here appearing to beequipped with an Euclidean metric.



TizianaVistarini

Further simplification: let’s pretend that all weneed to account for ordinary macroscopic systems(clocks, rods, cats and so on) is the set ofinteractions described by the classical potential V -that is, let’s pretend we are inhabitants of a worldwhich is fundamentally classical and in which thereis no such a thing like quantum behaviours.



TizianaVistariniHow would a world like this appear to us? Here

the expression “world’s appearances" does notcontain any reference to our subjective experience.Rather, it simply denotes what the world looks likethrough the outcomes of experiments and ofmeasurements.A physical world like this would appear, withoutany doubt, to be three-dimensional and Euclidean.



TizianaVistarini

Why is that? The answer is given by the structureof the potential V. The latter is an explicitfunction of the three dimensional Euclideandistances. And any potential dictates how anymaterial interaction occurs.One may say that these interactions make theEuclidean distances manifest.



TizianaVistariniThat is, the potential produces a manifest

geometry of the world.In general and not only in this toy case, our accessto any presumed actual geometry can only beachieved by reading into the Hamiltonian.But what if things are not so simple? What if themanifest geometry and the presumed actual one donot coincide?



TizianaVistariniThere is an old story about this possible

geometrical mismatch. It is the parable byPoincairé about the difficulty of establishingbeyond any doubt the status of geometricalknowledge. According to him the issue cannot besettled by means of empirical evidence.Poincairé was using this parable to debate aboutepistemology of geometry. Here I draw a lesson atright angle with all that.



TizianaVistarini

The Poincairé’s story is about an imaginary twodimensional world equipped with Euclideangeometry. It is a disk, the one that contrives bymeans of effects of spatial variation of thetemperature on the lengths of measuring rods.



TizianaVistarini

The inhabitants, unaware of the hidden dynamicsaffecting rods’ lengths, get out of measurementsand of dynamical generalization a manifestgeometrical image of the world that is not thesame as the actual one.



TizianaVistarini

So they conclude to be living in a infiniteLobachevskian plan.One may say in this case the dynamics areproducing a manifest geometry of the world thatdoes not coincide with the actual one.



TizianaVistariniWhat this parable suggests to me is that we may

generalize the classical scenario just described to acase of arbitrarily curved background. In this casethe Hamiltonian would depend on generalizedcoordinates x,y and z. In a world like this there isno general and unique way of defining the lineelement ds2, that is, there is no uniform way ofdefining the distance between particles.



TizianaVistarini

Nonetheless, in a hamiltonian like this, thepotential can appear (by means of localcoordinates transformations) to be function of(‹s)2 = (‹x)2 + (‹y)2 + (‹z)2:

X

i=1;::N

[:::] +X

K=1;::;N`1Vkj((‹x)

2 + (‹y)2 + (‹z)2):

k 6= j.That is, interactions between particles can be seenas manifesting an Euclidean flat space geometry.And so this world now in spite of the fact that it isactually curved, it appears to to be flat andglobally Euclidean.



TizianaVistarini

So, we might wonder, what does it means to saythat space is actually curved in this case?Indeed, the actual curved geometry of thebackground appears to have no role in theproduction of the flat geometrical appearances. Itis the dynamics that do all the work.So, one may take a further step and conclude thatthe hamiltonian produces a manifest geometry ofthe world independently of whether or not thereis an actual background geometry at all.



TizianaVistariniFrom this reformulation of the Poincairé’s

under-determination problem we gain a lesson:still remaining in the classical scenario, there is alegitimate way of reading the physics according towhich space and time are on the side of what ismechanically explained, rather than being on theside of those physical features somehow prior tophysical processes.



TizianaVistarini

As long as we remain confined to classical physics,reading dynamics in one way or the other is amatter of choice. But as we move to the quantumscenario, the reading of emergence turns out to bemore truthful.



TizianaVistarini

My argument in favour of string theory backgroundindependence uses a line of reasoning mimickingthe same logic, but with an important distinction.As soon as we generalize the story to a quantumhamiltonian the manifest image of the world stopsbeing a competitor of the presumed actual one(like in the original Poincairé formulation) sincethey now belong to two distinct levels ofdescription of reality governed by different physicalparameters.



TizianaVistariniThe main claim is that string theory does not posit

any fundamental spacetime geometry. Spacetime isa mechanical byproduct of underlying dynamics.This seems to hold true for both formulations ofthe theory: perturbative and non-perturbative ones- the former gained by perturbating the classicalaction and by reimposing conformal invariance forthe quantum string action, the latter delivered bythe AdS/CFT duality.



TizianaVistarini

The perturbative formulation - at least in myreading - admits general relativistic spacetime inthe sense mentioned above. The theory does notposit any fundamental geometry and generalrelativistic spacetime is a mechanical byproduct ofunderlying strings dynamics.Some important mathematical/physicalresults:(Polchinski,2001);( Witten, 1997);On the philosophical interpretation of thederivation of General Relativity from string theory:(Huggett, Vistarini, 2015);(Vistarini, 2017,forthcoming).



TizianaVistarini

Also, in the non-perturbative case, and in all thosecases in which arguments via duality are involved(T-duality, mirror symmetries) my reinterpretationof Poincairé problem points to the fact that anyposit regarding fundamental geometry in the theory(in this case also involving geometry of the extradimensions) does not play any explanatory power.

S ˆ K#lS



TizianaVistariniModuli spaces are abstract spaces used to

parameterize families of objects. Here our familiesof objects are families of spacetimes (equippedwith metric tensors).What does it mean “spaces of parameters"?A moduli space M is a space of parameters forsome family K of physical spacetimes only if thecorrespondence between ”points" of M andspacetimes in K is well-defined.



TizianaVistarini

Well-defined in some basic mathematical sensemeans something like:

K

#ffiM

each object in K is mapped on a unique point of M,but this is not required in the other way around,that is Ker(ffi) is not required to be equal to ;.Now, this requirement by itself would only producesome humble structure. Better if we require somemore constraints.



TizianaVistarini

In order to avoid some humble structure a secondrequirement may be that whenever two objects inK are somehow “similar" or “close" with respect tosome property, the corresponding “points" on Mmust be close as well with respect to some M’stopology - which in principle is not the same asthat in K.This constraint induces a refinement on theprevious one about having a well-defined map:things in K that get mapped onto the same pointin M, must be at least quite “similar".



TizianaVistariniAll this is a toy version of how you might get a

“fine" moduli space - the one encoding as muchrich information as possible.This moduli space should possibly encodeintra-relations among parts of a single object, ifany, and also trans-objects (or inter-objects)relations among objects in the family.In this context objects are spacetimes.



TizianaVistarini

It turnes out that the “local" topological structureof this simplified version of the theory modulispace, along with some fiber bundle structure(conveying dynamical information) on top orevealthe genuine metaphysical commitment of stringtheory to the non fundamentality of spacetime.But it also reveal that the fundamental ontologyof string theory contains a philosophical notion ofpossibilia that might be seen as a naturalizedversion of the purely metaphysical notion deliveredby Lewis modal realism.



TizianaVistarini

K

#ffiB „ M

whereK =

[

–2BK–; (1)

8 – 2 B, ffi`1(–) = K– -(Kodaira, 2005).B for simplicity is here a one-dimensional set ofparameters. Moreover each K– is an individualspacetime structure. Let’s just focus on thecompact part of the higher dimensional space.Here by heavily relying on (Kodaira, 2005), eachK– is a compact, complex manifold. B is acomplex domain.Tiziana Vistarini (CU Boulder) Modal realism after string theory October,27, 2017 32 / 67


TizianaVistarini

(Kodaira 2005) Suppose given a domain B in anda set fK–j– 2 Bg of complex manifolds K–depending on –. We can say that K– will have aC1 dependence on – and that fK–j– 2 Bg is adifferentiable family of compact complexmanifolds, if there are a differentiable manifold Kas above and a C1 map ffi of K onto B satisfyingthe following conditions:(1)The differential of ffi, ffi?: TpK !̀ Tffi(p)B issurjective at every point p 2 K, where TpK andTffi(p)B are respectively the tangent space to K atthe point p and the tangent space to B at thepoint ffi(p).



TizianaVistarini

(2) K is a non empty complex compact manifoldand for each – 2 B, ffi`1(–) = K– is a compactdifferentiable submanifold of K.



TizianaVistarini(3) There are locally finite open covering

fVjjj = 1; 2; :::g of K and complex-valued C1functions z1j (p); :::; z

nj (p); j = 1; 2; :::; defined on Vj

such that for each – the following coordinatizationform a system of local complex coordinates of K–:

fp! (z1j (p); :::; znj (p))jVj

\

ffi`1(–) 6= ;g;



TizianaVistarini

We are now confined to a case in which deformingthe gluing functions fjk(zk; –) of a compactcomplex manifold means deforming its complexstructure. Although all the K–s of the family,along with K0, share the same topologicalstructure (or stronger, same differentiablestructure), each fiber of the family has its owndistinct complex structure. Generally a complexstructure over a manifold (whether or notcompact) comes along with a hermitian metriccompatible with it, which in principle is not unique.



TizianaVistarini

There is no a priori fixed relation between metricand complex structure, rather many differentcompatibility conditions, namely, many differentmathematical correspondences involvingholomorphic functions in the manifold’s atlas ofcharts. I won’t give here detail on suchmathematical relations. It suffices to say thatgiven a compatibility condition (Kodaira, 2005)between complex structure and an induced metric,any deformation of the complex structure is also adeformation of the metrical one.



TizianaVistarini

Therefore, we have a family of geometricallyinequivalent, topologically equivalent backgrounds.Now, the differentiable structure of any individualfiber in the family is not the same as the“differentiable" structure over the moduli space.The latter is a topological structure technicallyarising from the C1 act of deformations of onefiber into another. Within this type of family ofdeformations, we now pick a specific subtype,namely, first-order infinitesimal deformations.



TizianaVistarini

Deforming the gluing functions by means offirst-order derivation basically means:

@fik

@–=@fij

@fjk

@fjk

@–: (2)

The derivative of the gluing functions gives some“rate of geometrical change" with respect to theparameter – of the family. The derivative givesinformation on all the directions along which theoriginal metric structure may bend when deformed.



TizianaVistarini

The rationale here is still finding a concise, stillgeneral way to express that if we suitably changethe geometrical structure of some provisionallyposited background physical observables of thesystem do not change their expecation values.



TizianaVistarini

Now, by skipping a huge amount of detail, if wetake the derivative of any gluing function over aninfinitesimal disk, we produce, morally speaking, aholomorphic field at – = 0, namely,

„ik(0) =@fik(zk; –)

@–j–=0: (3)



TizianaVistarini

It turns out that „jk(0) is a 1-cocycle of the sheafTK0, namely, the sheaf of holomorphic vector fieldsover K0. Its cohomology class „(0) is an elementof the cohomology group H1(K0; TK0):



TizianaVistariniBut what does it mean? Informally speaking, the

cohomology group H1(K0; TK0) is simply a vectorspace, whose vectors (cohomology classes) can beread in more familiar terms as closed andnon-exact differential 1-form.(In differential geometry, a one-form on adifferentiable manifold is a smooth section of thecotangent bundle).



TizianaVistarini

The cohomology group H1(K0; TK0), generated asvector space by a basis of non isomorphic „s,represents all the infinitesimal deformations of K0.It appears in the Kodaira-Spencer map:

ȷ : TB;0 !̀ H1(K0; TK0):

Locality on the moduli space is not locality in theordinary sense (things are close in spacetime).Rather locality has to do with the degree ofsimilarity among different spacetimes.



TizianaVistariniStill on a conceptual note, I want to make two

points. First, the simplified version of moduli spaceis a mathematical structure depicting a space ofmathematical possibilities that here is confined tothe formal language of smooth deformation theoryapplied to compact complex manifolds. Thingscan be more complex than this if the action ofdeformation is not a smooth one.



TizianaVistarini

Also,by taking in consideration any smoothdeformation of some geometry, it is not always thecase that one can find some physical geometryadmitted in the theory. Physical possibilities are asubset of the mathematical ones.In the part concerning the revision of modalrealism, mathematical possibilities of this sortreplace purely logical possibilities - the lattertraditionally inspired by ordinary intuitions abouthow things might be different from what they are.



TizianaVistariniSecond, the existence of a Kodaira-Spencer map at

– = 0 means that each tangent vector at – = 0 tothe moduli space identifies a possible deformationof the geometry K0.If the Kodaira-Spencer map is surjective, then everyfirst order deformation of a geometry K0 isrepresented by a tangent vector to the modulispace.



TizianaVistariniThe condition under which the Kodaira-Spencer

map is surjective are beyond the goal of this talk.However, if the map is surjective, the family of firstorder deformations K is complete, namely, itcontains all the possible first-order deformations ofK0.So, under the condition of suriectivity of theKodaira-Spencer map we gain a an exhaustivelandscape of mathematical possibilities.



TizianaVistarini

H

#pM

The fiber bundle H is defined as‘

–2M(H– ˆ C).Each disjunct H– ˆ C is a fiber over a point – ofthe moduli space M. Each – parameterizes somespacetime structure. H– is the Hilbert space of thestates of the system Q whose dynamics unfoldalong the spacetime parameterized by – and thevector space of complex numbers C represent allthe numerical values assumed over – by thetransition functions f¸1¸2:::¸n of the system.



TizianaVistarini

A claim of background independence with respectto spacetime geometry may be the following:any local family of different spacetimegeometries, still topologically equivalent,canbe taken as the data for constructing a stringtheory without any prejudice to choice of aparticular member..That is, for any family of topologically equivalentfibers given a set of observables and an arbitrarilypicked member –0 of the family, the valuef¸1:::¸n(–0) =< O¸1O¸2:::O¸n > remains constantover the family, i.e. 8 – 2 M

f¸1:::¸n(–0) = f¸1:::¸n(–):

T-duality of the bosonic string is a toy example ofthis strings physics blindness to geometry.Tiziana Vistarini (CU Boulder) Modal realism after string theory October,27, 2017 50 / 67


TizianaVistariniA claim of background independence with respect

to the topological structure may be the following:any local family of topologically inequivalentspacetimes can be taken as the data forconstructing a string theory without anyprejudice to choice of a particular member.Mirror symmetries not preserving topologicalinvariance.



TizianaVistarini

PART 2



TizianaVistarini

It is held into the philosophy of quantumgravity circles that endorsing Lewis ontology ofmodal realism is incompatible with endorsingthe fundamental physical ontology of anyquantum gravity theory.I argue that this apparent incompatibility canbe bypassed as long as modal realism is revisedin "naturalistic terms". It might turn out thatthis revision, if made in light of string theoryformal articulation and physical ontology, canproduce a metaphysics of possible worldscompatible with the non-fundamentality ofspace and time.



TizianaVistarini

A first step toward this "naturalization" of thesystem may be that of replacing a notion of logicalpossibilities mainly guided by ordinary languageand ordinary intuitions with a notion ofmathematical possibilities imported from theformal articulation of string theory.



TizianaVistarini

FEW INTRODUCTORY REMARKS:(1) Lewis modal realism thesis claims that thisworld is just one among other more or less similarto it.A proposition p is possibly true if and only if p istrue in one of this worlds. A proposition isnecessary true if and only if it is true in everypossible world.In relation to that, Lewis thesis of modal realismheld that what it takes for a proposition like "youare sad" to be true in another world is not for youto be sad in that world since you are not there.Rather it is for your counterpart to be sad in thatworld.



TizianaVistarini

(2) "I believe, and so do you, that things couldhave been different in countless ways. But whatdoes this mean? Ordinary language permits theparaphrase: there are many ways things could havebeen besides the way they actually are. I believethat things could have been different in countlessways; [...] I therefore believe in the existence ofentities that might be called ways things couldhave been. I prefer to call them possible worlds."(1973a: 84)



TizianaVistarini

(3) Lewis believed that modality cannot makesense without the metaphysical assumption ofmodal realism. He thought that we cannotdetermine that " F is possible" without an idea ofwhat a real world where F holds would look like.In deciding whether it is possible for "squirrels tobe inside atoms" we do not simply determinewhether the proposition is grammatically coherent,we actually think about whether a real worldwould accommodate such a state of affairs. Thuswe require modal realism if we want to usemodality at all.



TizianaVistarini

(4) This metaphysical setting turns out to beuseful in the analysis of counterfactuals, which inturn are central to an analysis of causation. Atrivial example of how causation andcounterfactual can be related to each other:as I touch a certain key on my computer, a slidessequence appears in front of your eyes. Should Itouch a different key, a different sequence wouldappear in front of you. That is how the actualorder of the slides you see causally depends on thekeystroke (Lewis, 1986)



TizianaVistarini

Two properties of modal realism here revised:Closeness: Lewis’s theory of counterfactuals relieson comparisons between possible worlds. It relieson comparisons between this world and otherworlds.Properties: The modal realist takes properties tobe sets of possibilia.



TizianaVistarini

Lewis (1987)What does it make a collection of things - the onewhich is the least inclusive - a Lewis possibleworld: mainly the fact that the parts of thecollections are unified by spatio-temporal relations:“whenever two possible individuals arespatiotemporally related they are worldmates".The fact that the most satisfying identikit of aLewis possible world must show its internalspatiotemporal unifying structure suggests thatspace and time are fundamental.



TizianaVistarini

A first comment: “fundamental" in the Lewis senseof world-identification is not the same as“fundamental’ in the sense of quantum gravity.The latter is grounded (at least in the case ofstring theory) on a notion of physical length scale:fundamental physics seems to unfold below acertain threshold where ordinary notion of spaceand time break down.The former instead does not explicitly refer to anidea of fundamentality in terms of quantumphysical scale.



TizianaVistarini

Lewis (1987)As far as I understand, the fundamental role ofspace and time in Lewis has more to do withworld-identification purposes.And within this framework there aren’t any explicitconstraints ruling out that these worlds may beemergent.Lewis idea of fundamentality of space and timedoes not seem to claim something more than theirbeing real. This is not in tension with any quantumgravity scenario. No serious physical theory woulddeny that space and time are real.The Lewis-pluriverse might well be an emergentone.



TizianaVistarini

The line of reasoning about the different meaningsof "fundamental" is an attempt to bypass apowerful objection against Lewis’s approach madeby Christian Wütrich (2015):“despite its modalembarras de richesses, Lewis pluriverse does notcontain our world" (page 9).



TizianaVistarini

A second comment is: Lewis (1986): page 70“Trans-world comparisons, yes; trans-worldspatiotemporal relations, no"This statement concisely exemplify the notion ofisolation. The only trans-world relation allowed byLewis is that defined by a similarity order.“So, things that are parts of the two worlds may be“simultaneous" or not, they may be in the same ordifferent towns, they may be near or far from oneanother, in very natural counterpart theoreticsense. But these are not genuine spatiotemporalrelation across worlds".



TizianaVistarini

The “action" of deforming spatiotemporalstructures produces a notion of locality over thestring theory moduli space that is very differentfrom the ordinary notion of locality in spacetime."Points" are close there in virtue of the degree oftopological similarity of the worlds they bothparameterize. In this sense the mathematical act ofdeformation formally encodes the qualitativeordinary notion of isolation. It exemplifies a nonspatiotemporal relation among worlds related bysimilarity order. In this sense it might beconsidered the rigourous mathematical counterpartof the Lewisian notion of trans-world isolation.



TizianaVistariniA third comment is about a big challenge against

my attempt of rescuing the Lewis approach fromrejection (in light of quantum gravity theoreticalachievements): Humean supervenience.Humean supervenience: there is nothing to realityexcept the spatio-temporal distribution of localnatural properties. Laws supervene on thesedistribution.



TizianaVistarini

String dualities might be considered a violation ofHumean supervenience. String dualities showsstings dynamics insensitivity to differentspatio-temporal structures, along with insensitivityto topologically differences.


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Modal realism after string theory METAPHYSICS WITHIN STRING … · Tiziana Vistarini (CU Boulder) Modal realism after string theory October,27, 2017 23 / 67. Modal realism after string