THE PRINCIPLE OF RELATIVE LOCALITY Laurent Freidel ILQG March 2011 G. Amelino-Camelia, J. Kowalski-Glikman, L. Smolin hep-th:1101.0931, ... 11

Monday, March 7, 2011

Look around What do you see? What do you directly experience? Do you see space? spacetime? Do you experience directly being in a spacetime?

Monday, March 7, 2011

We don’t really see spacetime, we see momentum space... As naive observers we do not directly observe spacetime points we do not directly observe events macroscopically displaced from us Our most fundamental measurements are all about energymomentum quanta we absorb and emit We see photons arriving with different momenta and energies at different angles Physics happens in phase space.We do not have to assume that the projection from phase space to space time is trivial. Monday, March 7, 2011

Absolute locality Spacetime is constructed by inference from energy and momenta measurement e.g. Einstein procedure of photon exchange to give coordinates to distant events via momentum space measurements (x,t)=(0,S) (x,t)=(S/2, S/2)

(x,t)=(0,0)

One fundamental hypothesis is that the energy of the probe we use is inessential. This is the absolute locality hypothesis. Why? Is it a low energy approximation? Monday, March 7, 2011

Absolute locality As we are going to see the absolute locality hypothesis is equivalent to the assumption that momentum space is a linear manifold. The notion of locality is related to hypothesis about the geometry of momentum space What if momentum space is a non linear manifold? Do we still all infer the same spacetime? Do we still infer the same spacetime at different energies? Introducing the possibility for momentum space to be non linear allows us to propose a framework in which locality is relaxed in a controlled manner

Monday, March 7, 2011

ges of light sigto acthe symmetry of true theWaterloo,Canada ground state that governs its low lying excitations. This is no less of spacetime itself, moreover sics, 31 Caroline St.is N, ON N2L 2Y5, we take into acforth excitations. Thispresumably is no less true of spacetime itself, moreover in general relativity, and in any description of the back and forth in general relativity, and presumably in any description of the ulting bruary 28, 2011 quantum dynamics of spacetime, the symmetry of the ground nergy, quantum dynamics of spacetime, the symmetry ofclasthe ground sume resulting state is dynamically determined. We also expect that the his we presume state is dynamically determined. We also expect that the clasf light spacetime geometry of general relativity is a semiclassiIn this work we propose a framework in which we can relax in a changes ofsical light sical spacetime geometry of general relativity is a semiclassiming cal approximation to notion a more fundamental quantum geometry. sedcontrolled to assuming manner the of locality cal approximation to a more fundamental quantum geometry. econIn this paper we show how simple physical assumptions about Abstract ch other, reconIn this paper we show how simple physical assumptions about softhey Quantum gravity the geometry of momentum space may control the departure photons they What could be the motivation? the geometry of momentum space may control the departure ry of momentum space in more detail of the spacetime description from thefrom classical one. one. of the spacetime description the classical of the We work a first regime where weattention neglect quantum mechanics Wein will attention to an approximation in which e energy of the We restrict will first restrict to both an approximation in which httial? that h¯ that and GNewton mayboth beare neglected whilewhile their ratio Might and gravity neglected h¯ andboth GNewton may be neglected while their ratio near composition pre- we pre! ye should ! servaom !, ourGobservah¯ h¯ →0 = mp = mp (1) (1) m us? GNewton GNewton Lkwhich (p) ≡postuk⊕p y, postu1 . In this approximation quantum and gravita1 . In is isheld fixed time, equivais held fixed quivais held fixed this approximation quantum and gravita01 ∆θ = T−man∆E is a linear manIn this limit there is a fundamental energy scale which we throw throw ∆S = 0use eequanta we use one expects a deformation of momentum space 1 inand aΓ++ simple We work in unitsc in c = 1. e= We units in which = which 1. ∆E1 and = work N ++in∆E

Quantum gravity limit

−

= N−++ ∆E

Monday, March 7, 2011

−

Geometry of momentum space Operational point of view: a local observer is equipped with a calorimeter and a clock.

From her measurement she concludes that each isolated system possesses conserved Energy momentumspace space2 P from m but there 4may be newquantities: ometry of momentum

or energy given by m p . the dynamics of interacting particles. W She can two type → 0realise we expect no of measurements: choice of calorimeter and other instrume ewton ry to be relevant. server there is a preferred coordinate on One particle measurements:But measurement of the mass the we also assume that and the of dynamics n energy scale, but not kinetic energy variantly in terms of geometry of P and uming that momentum choice of calorimeter’s coordinates. We Multi particle measurements: scattering processes, interactions, time. This is in accord sure the energy and momenta of excitat merging. mentioned in the openstate, hence the origin of momentum sp entum space by asking cally well defined. is measured by a scale Our local observer can make two kin l derive the properties One type of measurement can be done o Monday, March 7, 2011

cle which is a function of the particles four momenta. She can also interact. unction of the four momenta. She can also measure the kinetic energy K of a particle of mass m moving hich is a function of the four momenta. She can also espect to her, but local to her. We postulate that these energy K of a particle of mass m moving es place in phase space and there is no invariant 2 tic energy K of a particle of mass m moving with respect to her, but local her. We postulate that these tion the that gives a description of processes ina to ure kinetic energy K of particle of mass m moving A. The metric geometry of momentum space rements determine the metric geometry of momentum but local to her. We postulate that these From measurements their measurements local observers the can metric geometry of momentum determine er, but local to her. We postulate that these 2 We One postulates that single particle measurements determine the espect to her, but local to her. postulate that these criptions of particles moving and interacting We interpret the mass as the geodesic distance from the First we describe the metric geometry. Our local observer mine the metric geometry of momentum 2 space. We interpret the mass as the geodesic distance from the me, but different observers construct different geometry of ntum space P from the measurements made of etermine the metric geometry of momentum can measure the rest energy or relativistic mass of a partiurements determine metric geometry of momentum which are observer-dependent slices of phase this gives the dispersion relation origin, this gives the dispersion relation he mass as the geodesic distance from the cle which is a each function of the four momenta. She can also nteracting particles. We assume that to

Geometry of momentum space

isinterpret amass lorentzian manifold withthean origin ret the asthe themetric geodesic from the0K offrom PWe from measurements made of mass as thedistance geodesic distance the measure kinetic energy a particle of mass m moving 2 2 with respect dispersion relation to2her, but local to her. We postulate that these eter and other instruments carried by our ob2 2 2 D (p) ≡ D (p, 0) = m . (2) (2) the dispersion relation articles. We assume that to each ,section this gives the dispersion relation D (p) ≡ D (p, 0) = m . measurements determine the metric geometry of momentum we introduce an operational approach to The mass is interpreted as the timelike distance from the origin

referred coordinate on momentum . space. space, We interpretkthe mass as the geodesic distance from the of momentum space, which we build on inour sec- oba er instruments carried by 2 2 2 2that 2on 2 beenergy The measurement of kinetic defines geodesic origin, this gives thethe dispersion relationdise a(p) dynamics ofD particles a curved momentum 2 2 2 ≡ (p, 0) = m . (2) me the dynamics can expressed coDa modified (p)on≡version DD (p, 0)≡=space, menergy . 0) (2) D (p, = m . (2) momentum k . $ erdinate how of(p) spacetime geometry easurement of kinetic defines the geodesic a 2 2 2 distance between a particle p at rest and a particle p of identical D (p) ≡ D (p, 0) = m . (2) the dynamics which is P formulated on mosrom of geometry of and do not depend on the $ $ dynamics can be expressed coetween a particle at rest and a particle p of identical and kinetic energy K, is D(p) = D(p ) = m and e. In mass these sections we consider apgeneral mo-that The measurement of kinetic energy defines the geodesic disf kinetic energy defines the geodesic dist of kinetic energy defines the geodesic dismeasurement of kinetic energy defines the geodesic diskinetic energy thethat geodesic between a meter’s coordinates. We the ka spacelike meaeThe geometry, which do illuminates adefines varietynote of new tance between a particle$ p at distance rest and a particle p$ of identical y of P and not depend on the nd kinetic energy K, that is D(p) = D(p ) = m and $ $ $ $ ) = m and hat might be experimentally probed correspond2 $ mass and kinetic energy K, that is D(p) = D(p between a particle p at rest and a particle p of identical icle p at rest and a particle p of identical particle pandat atnote restexcitations and aparticle particle pthe identical particle rest and amomentum ofof identical mass D (p, p ) = −2mK. (3) nd momenta of ground vature, torsion non-metricity ofthe inates. We that ka above mea$ )D2= $ $ $ ver, one advantage of this approach is that with (p, p ) =and −2mK. (3) and kinetic energy K, that is D(p) = D(p m origin of momentum space, k = 0, is physienergy K, that is D(p) = D(p ) = m and ergy K, that is D(p) = D(p ) = m and 2 $ a that the geometry of momenofphysical excitations the ground ble principlessign theabove geometry of momenThe minus express the fact D (p, p ) = −2mK. (3) The minus sign express the fact that the geometry of momenn be reduced to three choices, depending on the d. mentum space, k = 0, is physitum space is Lorentzian. From these one can tum space measurements is Lorentzian. From these measurements one can 2 $ a 2 $ meter. As we show in section IV, this gives this 2 Dcan $make D (p, p ) = −2mK. (3) 3 (p, p ) = −2mK. (3) 3 2 P from reconstruct a metric on P erver two kinds of measurements. D (p, p ) = −2mK. (3) oth great elegance and experimental specificity. from these measurements we can reconstruct the metric on reconstruct a metric on P d, but there may be new ometry of momentum space inus sign express the fact that the geometry of momenwe make some preliminary observations as to dk2 = hab (k)dka dkb . (4) surement can be done only with a single parake kinds ofmaymeasurements. m oroftwo energy given by m the dynamics of interacting particles. metry momentum space be probed 2 .experab p minus sign express the fact that the geometry of momenace is Lorentzian. From these measurements one can dk = h on (k)dk . (4) xpress the fact that the geometry ofmomena dk bmomener which wewe conclude. ess the fact that the geometry of es, as will see, a metric momentum 3 npace be done only with a single parG → 0 we expect no choice of calorimeter and other instrum Newton is Lorentzian. From these measurements one can B. The algebra of interactions truct a metric on P entzian. From these measurements one can type of measurement involve multi particles zian. From these measurements one can 3 ill see, a metric on momentum metry to be relevant. server there is a preferred coordinate o 3 struct a metric on P ric on3 P A key mathematical we describe the construction of the connection on monnection. underly2 Theab B. algebraidea ofNow interactions Monday, March 7, 2011

Geometry2 ofAffine momentum space connection from add

2 P from measurem e may be new ometry of momentum space When we define operationally momentum space is to show that t The main point of this section y given by m p . the dynamics of interacting particles. We assum of an affine connection on momentum space. 0 we expect of calorimeter and other instruments carri use oneno type ofchoice calorimeter, is left invertible. In order to do so we need relevant. server there is amount a preferred coordinate on moment chosing another calorimeter will to a redefinition the composition rule that transform well unde But we! also assume that the dynamics can be e p → p = φ(p) the addition rule transform as ( y scale, but not variantly in terms of geometry of P and do not d hat momentum choice of calorimeter’s coordinates. We note tha (p ⊕ q) = φ The theory has to be invariant under diffeomorphism µ φ his is in accord sure the energy and momenta of excitations abov on momentum space. ed in the openGivenhence a rulethe of origin adddition we define space, ka = state, of momentum pace by asking calorimeter regauging ! 0 "α cally well defined. ured by a scale τp µ (p) ≡ Our local observer can make two kinds of m the properties One type of measurement can be done only with mentum space. ticle and it defines, as we will see, a metric o just described, space. The other type of measurement involve m and defines a connection. A key mathematical ing our construction is that a connection on a Monday, March 7, 2011

in n = proceeds nin + nout .byThis by the of an al-Thatstructure. Ts This the proceeds construction of construction an al- out structure. is,Thus, causal the ∂ ∂ n = n + n . This proceeds by the construction of an alstruc in out ab which then determines the connection. associativity en gebra, determines the connection. associativity of combi − ((p ⊕ q)c − (p ⊕ q)c )q,p=o = Tc (0) (9) the from part gebra, which∂p then the assoot ∂qbdetermines Associated to each interaction thereconnection. must be a combination turn maps athere each interaction must be a combination turn maps to curvature parallel tr In the multiple particle case we should have a rule to associate a rule for momentum, which will in general be non-linear. We of momentu Associated to each interaction there must be a combination turn tum, which will in general be non-linear. We of momentum space make Similarly the curvature of P is a measure of the lack of assototal momenta to the combination of particles denote this rule for two particles by guishable, a rule for momentum, which will in general be non-linear. We of m for two particles by guishable, and hence mea ciativity of the combination rule tocurvature measure denote this rule for two particles by guish " to measure the We postulate that there exists a composition of momenta " (p, q) → p = (p ⊕ q) (5) a a discuss belo (p, q) → pa = (p ⊕ q)a (5) where τ(p to m discuss below. " ∂ ∂ ∂ abc (p, q) → p = (p ⊕ q) (5) to p.discu It ca a a Hence the momentum space P has the structure of an algebra 2 ((p ⊕ q) ⊕ k − p ⊕ (q ⊕ k)) | = R (0) d d q,p,k=o entum space P has the structure of an algebra ∂p ∂q ∂k c b] product defined [aby the rule ⊕. We assume that more compliroduct rule ⊕. We assume that more compliHence the momentum space P has theofstructure of up an algebra (10) More complicated interaction processes areproduct build by iteration of cated processes are built up by iterations this – but are built up by iterations of this product – but where the bracket denote the anti-symmetrisation. defined by the product rule ⊕. We assume that (p ⊕ q)commutativity ⊕ k more complithis composition e.g to begin with we assume neither linearity, nor with (p ! q) !k e cated assume neither linearity, nor commutativity We note that there is no physical reason to expect a comprocesses are built up by iterations of this product – but nor associativity. We do not rule assume that it is linear or commutative ornonassociative bination for momentum to be associative, once it is y. to We begin we assume neither nor willwith also need an operation thatlinearity, turns outgoing momenta 1 commutativity p ⊕ p = p + p + T (p , p ) + · · · linear. Indeed, the lack of associativity means there is a physneed an operation that turns outgoing momenta 1 2 1 2 1 2 Outgoing momenta can be turned in ingoing momenta: into incoming momenta. This is denoted, p → $p and it satnor associativity. m P ical distinction between the two processes of Figure 1, which 4 momenta. This is denoted, p → $p and it satthere is an operation isfies We will also need an operation that turns outgoing momenta The corre

Geometry of momentum space

ond order into incoming momenta. This is denoted, p → $p and it satpN satisfying (6) 0 ($p) ⊕ p = 0 4 τPN = ∂Q ((pN ⊕ p) ⊕pN Q)|Q=pN ∂q (pN ⊕ q)|q=0 isfies ($p) ⊕ pτp= N ⊕p 0 (6) 4 Andwe Then have the conservation law of⊕energy and ((p p)$ ⊕ (p ⊕ q))|q=0 paNleft Nmomengenerally ($p) ⊕ = (p ⊕ ∂ k)q= k,Nwhere isLeft inverse. We alsomore ask that Loop ($p) ⊕ p momen= 0 of interaction, four (6) e the lawgiving, of energy and tumconservation for any process, for each type = ∂q (pN ⊕ (p ⊕ q))|q=0 n functions onfor PL.F , depending momenta four of interacting particess, giving, each interaction, kikkawa, sabinin, 0type of on = ∂q ((pNlaw ⊕ p) ⊕energy q)|q=0 and momen- p τpN ⊕p Then we have the conservation of cles, which vanish , depending on momenta of interacting partiMonday, March 7, 2011

Abstract

The main point of thi senote I describethe the geometry geometry of of momentum space space in of more detail I describe momentum moreconnectio detail anin affine is left 2invertible. In Abstract The composition rule defines an affine connection on there may be new ometry of momentum space P from meas we restrict to non diffeomorphism that preserve the product identity the composition rule ections and linear composition In this note I describe the geometry of momentum space in nergy given by m . the dynamics of interacting particles. We a ! p ons and non linear composition the ad # One first notes that under a diffeomorphism# p →$p = φ(p) & % β ∂ ! no " choice of calorimeter and other instruments ∂p ∗ P0 we expect α n T→ µ α 0 0 τˆp0 µ1≡ ∂Connections (p ⊕ q) | = (p) τ and non linear compositio µisq=0 φ q φ(p)on mo o The be relevant. server there a preferred coordinate operator(p ⊕ (p ⊕ p)) ν ⊕ q) ν b q) = bp ⊕ ((p ⊕ p) ⊕∂φ

Geometry of momentum space

(p) q)a |q=0 P ⊕ TU∗ P =M × P∂Talso a ≡ q0∗(p But we assume that the dynamics can but not ⊕ p) ⊕ q = p ⊕ ⊕ q)(p of ⊕ Given (pthe ⊕ p))differential ⊕ =p⊕ ⊕ a q)rule of((padddi written(pshorthandly in(pterms ofp)φ⊕

nergy scale, hich ca be variantly in terms of geometry of P and do ng that momentum !, G → 0 (pcoordinates. ⊕ p) ⊕ q = p ⊕ (p ⊕ q)no 0 choice of calorimeter’s We hows that τ is a map from T P to T (P ), where P denote the mo transforms as a map from 0 p p p 0 e. This in accord ∂q (pN ⊕ q)|q=0 τ ⊕energy τpNNis pN Q)|Q=p N ⊕ p) 0 ⊕p PN = ∂Q ((psure N momenta the and of excitations p 0p P its tangent space at the point p. Moreover it is clear that τ p L (p) ≡ k ⊕ 0∂ τ = ∂ ((p ⊕ p) ⊕ Q)| τ k p Q N Q=p ntioned in the openP p ⊕p It can be interpreted asN a⊕ paralell transport operator = ∂ ((p p) ⊕ (p ⊕ q))| 0 q=0of momentum space q p N state, hence the origin N roperty we get that τ is invertible with inverse = ∂q ((pN ⊕ p) ⊕p (pN ⊕ q))|q um space by asking p 01 = ∂q (pNcally ⊕ (p well ⊕ q))defined. = ∂q (pN ⊕ (p ⊕ q)) ∆θ = T ∆E measured by a scale −p0 α 0 Our make |Q=p τpN ⊕p = ∂q ((pN ⊕ p) ⊕local q) τobserver τp≡⊕p∂Q=(pcan ∂q# ((pQ) ⊕µp) ⊕two q). kinds N 0 erive the properties One type of measurement can be done only ∆S = 0 pN +1 = pN ⊕ p, p1 = p One additional hypothesis: mono alternaticity n momentum space. 0 = pNand ⊕ p, p1 a = paralell p as we N +1ticle defines, will see,operato a met his means that we pcan identify τit as transport p imit just described, ++pN +1 ⊕ q = pN ⊕ (p ⊕ q) ++ S2 to S =q = Γ−pThe ∆E =q)type N− from of∆E measurement be1space. 0 to p if invo p− ⊕paralel ⊕ other (p ⊕ µ (p) ∈ Tp P is said N +1 N transported geodesic motion Definition 1. A leftainvertible composition law onmathema a manifold and defines connection. A key αis a C ∞ map ++ A left invertible composition law on a manifold P Pµ (p) =isτ that (p)Paα (0). ∆E ing = ourNconstruction connection o N N

N

N

N

N

Monday, March 7, 2011

N

Geometry of momentum space Momenta combine into interactions: The rule: ! (k, q) → ka = ka ⊕ qa P

ka ⊕ pa

ka

pa pa 0

!pa

Monday, March 7, 2011

can be thought as a rule for combining geodesics on a curved manifold, so it defines a connection or parallel transport. ka ⊕ dpa

= ka + = ka +

b U (k)a dpb dpa + Γbc a kb dpc

+

is studied in detail in [2], but the basics are as follows. The algebra of the combination rule determines a connection on P by rules defines connection on2 P from me but The therecomposition may be new ometryanofaffine momentum space energy given by m the dynamics of interacting particles. We ∂ p . ∂ Abstract ab (p ⊕ q) | = −Γ (0) (8) c q,p=o c → 0 wethe expect choice of calorimeter and other instrumen wton e I describe geometry of momentum space in more detail ∂pno ∂q a b y to be relevant. server there is a preferred coordinate on m transforms as an affine connexion bc The torsion of Γa is a measure of theassume asymmetric partdynamics of the c But we also that the energy scale, but not ons and non linear composition Torsion measures non commutativity combination rule variantly in terms of geometry of P and d ming that momentum choice of calorimeter’s coordinates. We n ∂ ∂ a b ab ab me. This−∂ is− in accord ∂ [(p ⊕ q) − (q ⊕ p) ]| = T (0) c ⊕sure c energy p,q=0 ((p q)c − (p ⊕ q)c )q,p=o = Tc (0) of excitatio (9) p q c momenta the and ∂paopen∂qb mentioned in the state, hence the origin of momentum spa b b What about associativity? ntum space by asking U (p)a ≡ cally ∂ofq (p q) | a q=0 Similarly the curvature P⊕ is a measure of the lack of assowell defined. s measured by a scale ciativity of the combination rule Our local observer can make two kind derive the properties !, G → 0 One type of measurement can be done on in momentum space. Lk (p) ≡ k ⊕ p ticle and it defines, as we will see, a m ∂ ∂ ∂ abc e limit just described, 2 ((p ⊕ q)space. ⊕ k − pThe ⊕ (qother ⊕ k))type |q,p,k=o = R d (0) in of measurement d 01 ∂p[a ∂qb] ∂kc ∆θ = T− ∆E and defines a connection. A key(10) mathem ing our construction is that a connection

Geometry of momentum space

Monday, March 7, 2011

there is a connection on P . The geometry of momentum space from the bc The torsion of Γain is measure of the part oflating the in is studied in detail [2],a but the basics are asymmetric as follows. The algebra of the combination determinesof a connection on P space combination rule The rule geometry momentum The by composition rules defines connection on2 P from meas there may be new ometryanofaffine momentum space

Geometry of momentum space

∂ ∂ a∗ ab Freidel nergy given by the⊕dynamics interacting We a − m p∂. ∂ ((p q)c − (pLaurent ⊕ofq)ab = Tcparticles. (0) (9) c )q,p=o the ident ∂p ∂q (p ⊕ q) | = −Γ (0) (8) a b c q,p=o c → 0 awe expect no choice of calorimeter and other instruments

n

Perimeter Institute ∂pa ∂qfor b Theoretical Physics, 31 Caroline St. N, ON N2L 2Y5,

o be relevant. server a preferred coordinate on mo Similarly the curvature of there P is aismeasure of the lack of assobc The torsion measure of the asymmetric part of the February 28, 2011 transform as of an Γaffine a is aconnexion But we also assume that the dynamics can ciativity of the combination rule combination rule nergy scale, but not Curvature measures non associativity variantly in terms of geometry of P Thus, and do th ng that momentum ∂ ∂ ab coordinates. We no choice of calorimeter’s − ((p ⊕ q) − (p ⊕ q) ) = T (0) (9) c c from par c q,p=o ∂ ∂ ∂ e. This2 is in accord ∂pa ∂qb abc Abstract ((p ⊕ q) ⊕ k− p ⊕ (qand ⊕ k)) =R (0) t sure the energy momenta excitations dparallel d |q,p,k=o of ∂p ∂q ∂k ntioned in the openc I describe [a b] In this momentum space in more detail Similarly the note curvature of Pthe ishence ageometry measure the lack assostate, theofof origin of of momentum space (10) um space by ofasking ciativity the combination rule cally well defined. where the bracket denote the anti-symmetrisation. measured by a scale 1define Connections and non linear composition Our local observer can make two kinds where τ( To the connection away from 0 we “translate” the addition We note that there is no physical reason to expect a comerive the∂ properties ∂ ∂ abc be done to p.only It c One type of measurement can 2 ((p ⊕ q) ⊕ k − p ⊕ (q ⊕ k)) | = R (0) d bination rule for momentum to be associative, it is nond q,p,k=o L (p) ≡ k ⊕ p once using the left translation operator k n momentum space. ∂p[a ∂qb] ∂kc ticleofand it defines,means as wethere willissee, a met linear. Indeed, the lack associativity a physimit just described, (10) 01 ∆θ =type T− ∆E space. The other of measurement invo ical distinction two processes of Figure 1, which where the bracket between denote thethe anti-symmetrisation. with defines reason a connection. key mathema ∆Sto=expect 0 We note that there isand no physical aAcoming our construction is that a connection o Monday, March 7, 2011

Three aspects of geometry, which can be measured: pa ⊕ q a = pa + q a +

•Torsion:

bc Γa pb qc

+ ...

measures non-commutativity of interactions. bc Ta

=

bc Γa

−

cb Γa

•Curvature: measures non-associativity of interactions. abc R d

=

a bc ∂ Γd

−

b ac ∂ Γd

+ ΓΓ

•Non-metricity: if the connection defined by interactions is not the metric connection defined from propagation. N Monday, March 7, 2011

abc

=∇ g

a bc

Dynamics •Spacetime emerges from the dynamics on momentum space. •In our limit, we study first classical particle dynamics •Each process has an action principle S

process

=

!

trajectories,I

Monday, March 7, 2011

f ree SI

+

!

interactions,α

Sαint

Emergence of space-time J 2Given 2 we can The geometry of momentum spa these define the free partic C (k) ≡ D (k) − mJ . (18)

onnections and non linear composition ! !a∗ H

J Laurent I a ˙J a J Freidel Spacetime isthat an auxiliary concept that from the dynamics emphasize the contraction xJa k˙emerges does not involve ds xJ ka + NJ Car( a S f ree = ! 0 a of particles Perimeter Institute for Theoretical Physics, Caroline St. N, ON N2 ric, and the dynamics is otherwise given 31 by constraints a˙ pe S = (x k − N C(k)) a dynamics chFree are particle functions only of coordinates P and depend only where s is anon arbitrary time parameter and −∞ February 28, 2011 ex geometry of P . This leadsmultiplier to the Poisson brackets, imposing the mass shell conditio fr

M ×P

mass shell b b conjugate partUcoordinates (p)a ≡ ∂q (p ⊕ q)a |q=0 a J a J J 2 2 C (k) ≡ D (k) − m . {x , k } = δ δ (19) J I I b b ∗ T0 P Abstract ak J ˙ We emphasize that the contraction x (p ⊕Γ(p ⊕ ofq)particle, = p ⊕ which ((p ⊕inp)more note Ispace, describe the momentum space haveInathis phase of⊕geometry ap)) single a J⊕

We then The free particle action makes no reference to dynamics a metric forisspacetime. metric, and the otherwise gi heSpacetime cotangent bundle of P . We note that there is neither geometry is inferred from the geometry of momentum space. (p ⊕ p) ⊕ q = p ⊕ (p ⊕ q) which are functions only of coordinates on nvariant projection from Γand to anon spacetime, nor is there 1 Connections linearMcomposition the geometry of P . This leads to the Poisso ned any invariant spacetime metric. YetΛ(q)) this structure is D(Λ(p), = D(p, q) 2 2 2 ! in the usual case the metric is flat and D(k) = k − k 0 ficient to describe the dynamics of free !particles. a The J facta J W {xI , kb } = δb δI 0 (p), Tr (q)) = D(p,toq) va there is no invariant projectionD(T to aSrspacetime a ˙is related = (x ka − N C(k)) non linearity of momentum We space. non linthenIndeed have−∞ a under phase aspace, Γ of a sin

r⊕p

Monday, March 7, 2011

from the integration by parts. aretranslated 0, ∞, −∞combination de1,2the where ⊕k sis rule torsion and ne whether thenonterm is incoming outgoing. Worldline action Here weorexpand moreBefore on how Lorentz i while curvag/m thep ,boundary terms we confirm we havewith the curved desiredmomentum made compatible of the worldline action gives ofThe thevariation equations of motion well known that around the identity 0 we ca

that the combinaJ ˙ olors charges kof a = bulkoreom n for identical para operational way x˙to J = e have Bose statisJ C any (k) = o not allow for this princiJrticles, = 0 at s = ±∞ and a etric and hence the

set of coordinates: The Riemannian norma 0 coordinates the distance from 0 is given this J space formula hence the mass shell conditi δC simplified by Riemannian NJ isJ simply7 normal coordinates nates δk a

0

2 2 2 C (k) = k0(26) − ki − m

= 0.

The Lorentz that preserves t examine thetransformations remainder of the

and the metric therefore acts in the usual ma # dinate systems.& the metric and the a a Moreover b δKbunder Jthe hypothesis of homog R = K (k) δz − xJ (0) − z δk (27) a nnection is ametric J extended to cover almost al dinates can δkbe a ance between two The Lorentz generators therefore satisfy the erns the mass shell we assume in turn that the Lorentz transform nnection determine ical transformations preserving the Poisson Monday, March 7, 2011

Emergence of space-time

ical Physics, 31 Caroline St. N, ON N2L 2Y5, Waterloo,Canada interaction contribution to the action is simply a lagrange

tiplier times the conservation law (7). The interaction imposes March 2, 2011the conservation law S

int

= K (k(o))a z

a

(23) Lagrange multiplier

We have set the interaction to take place at affine parameter z becomesAbstract the location of the interaction: a 0the forinteraction each of the particles. At this point z can be just concoordinate q geometry of momentum space in more detail red to be a lagrange multiplier to enforce the conservationk momentum (7) at the interaction where for each particle 0. on linear composition We vary the total action. After an integration by parts in p h e.g. of the K a free = (pactions ⊕ (q ⊕ we k))ahave =

a bc ∂ Γd

b [(pMarch ⊕7, q) c qMonday, 2011

−

b ac ∂ Γd

+

ai bc ai bc Γd Γi total − Γd Γi δS =

− (q ⊕ p)c ]|p,q=0 =

ab Tc (0)

et the four conservation laws of the interaction (7). From Worldline action J a are takenrelations for each particle at the parameter Here xJ and ka These ∂K /∂k evalua (29), (30) illustrate concis J anishingtimeofs =the coefficient δk we find 4n 0. This has to vanishof if the variational principle isconditions tor, more prec a of locality. For some fortunate observers the a The variation of the worldline action of gives to have solutions. From the vanishing the coefficient of δz h hold atwethe interaction get the four conservation of the interaction (7). From place at thelaws origin of their coordinates, so th boundary eom of the coefficient of δkaJ we find 4n conditions the vanishing where τ(k) is in which case the interaction is observed to be which hold at the interaction to k introduce a b δKb xJ (0) = z δKb J with respect to these, (28) observer, translated has operator of co a b δk xJ (0) = z (28) a a J z and hence sees δka the interaction to take plac a of events. These are centered particle coordinates interaction coordinates around z but B. Using (17) this gives conditions g (17) this gives conditions atxa (0) the= zsame a b valuesacof L the coordinates. That i − z ∑ CJ,L Γb kc + ... (29) Is this a rea J a aparticles b L∈involved ac L interaction of in an remove J (J) dinate artifact xJ (0) = z − z ∑ CJ,L Γb kc + ... (29) a becauseover the ∆ of the observer by a vector z are spread This x=0 tells us that to leading we ignore at thethe origin L∈isJ order, (J) in if z=0 then : interaction local forwhich an observer

lation to the c curvature of momentum space, all of the worldlines involved formulation, t a in the interaction meet at a single spacetime event, z . The vation of ene For a distant observer there is a dispersion ∆x ≈ |z||Γ|k a z is leading not constrained and cannot for its variation tells uschoice thatof to order, in be,which we ignore servablethe in ph gives the conservation laws (7). Thus, we have recovered the we can constr ature of usual momentum space, all of the worldlines involved Locality is relative notionThe that interactions of particles takepossess place at single relationship (28) a very nice ma structed by an a events in meet spacetimeatfrom the conservation of energy and mo- z . The e interaction a single spacetime event, translation lab ingis good too.because Sincein quantum the momentum space is in ge mentum. This field theory cona ceMonday,ofMarchz7, 2011servation is not implies constrained cannot for its variation locality, andand it is good to have be, a formulation

d non linear composition

x = z U (k) Two kinds of spacetime coordinates n linear composition tions and non linear composition

x =a z a

a b x = z U (k)b particle coordinates 2 D(k) ∗ ∗ Tk P T0 P

!

0

b

a U (k)b

a

x

a = z U (k) b interaction coordinates b

2 2 ! =Tk∗0P − Tk∗ P 0 k

a

b

U (k)ba

a b

δKa = δkb

2 The geometry D(k) = k02 − "k 2of momen

Paralell transport operator ! 0 a ˙ Laurent Freidela∗ S= (x ka − N C(k))

a Perimeter Institute for Theoretical Physics, 31 Caroline S −∞ δK a b δK a aU˙(k) = b a N C(k)) b b U (k) = S= (x k − a a U (p) ≡ ∂ (p ⊕ q)a |q=028, 2011 δk February b T* a q k a δk

P

b k −∞ a ∗

x 2 (0) ∗ 2

2 T P=M × P T= 0 Pk − " k D(k) 2 2 2 0 " (p ⊕ (p ⊕ p)) ⊕ q) = p ⊕ ((p ⊕ p) ⊕ q) − k∂!b (p ⊕ q) | D(k)U (p) = kb0 ≡ a q=0 a q 0 Abstract U ! 0 (p ⊕ p) ⊕ q = p ⊕ (p ⊕ q) S= (xa k˙ a − N In C(k)) this note I describe the geometry of momentum s a˙ a −∞ S = ka=0 (x ka − NzC(k)) D(Λ(p), Λ(q)) = D(p, q) Phase space = T*P −∞ 0 b b 1 T* non linear compo (p)a=≡p∂q⊕ (p((p ⊕ Connections q)⊕ (p ⊕ (p ⊕ p)) ⊕Uq) p) ⊕D(T q)and a |q=0 r (p), Tr (q)) = D(p, q) b b projection from phase space to space time. T ∗ P != M × P No canonical U (p)a ≡ ∂q (pT⊕ q)=ar|q=0 ∗P (p) ⊕ p r 0 (p ⊕ p) ⊕ q = p ⊕ (p ⊕ q) 1 1 1 xa =1z b U (k)a K = %p ⊕ (q %k ) b (p ⊕ (p ⊕ p)) ⊕ q) = p ⊕ ((p ⊕ p) ⊕ q)

each particle carries its own momentum dependent spacetime ∗ ∗ µ α µ 1 µ T P T x1 = z1 ∂p1 Kα = (z1kUx1 ) 0 P by to interaction spacetime D(Λ(p), Λ(q)) = D(p, q)⊕ q) ⊕ p)⊕ ⊕ p) q= pthe ⊕ (p p related ⊕ p)) ⊕ q)parallel = p (p ⊕transport ((p ⊕ q) δKa Monday, March 7, 2011

Ux1 :

T0∗ P

→

∗(k)b = U T Pa p1

δkb

d non linear composition

x = z U (k) Two kinds of spacetime coordinates n linear composition tions and non linear composition

x =a z a

a b x = z U (k)b particle coordinates 2 D(k) ∗ ∗ Tk P T0 P

!

0

b

a U (k)b

a

x

a

U (k)ba

a = z U (k) b interaction coordinates b

2 2 ! =Tk∗0P − Tk∗ P 0 k

b

a b

δKa = δkb

D(k)2 = k02 − "k 2 Paralell transport operator ! 0 S= (xa k˙ a − N C(k))

−∞ δK a b δK a aU˙(k) = b a N C(k)) b b If theUconservation law is linear then U =1 and x=z (k) = S= (x k − a a U (p) ≡ ∂ (p ⊕ q)a |q=0 δk b a q δk b −∞ the interaction is local

2 T0∗ P 2 T ∗P = M × P "k 2 − D(k) = k 2 2 2 0 If the conservation law is non linear interaction relatively " (p ⊕ (p ⊕ p)) ⊕ q)is=only p ⊕ ((p ⊕ p) ⊕ q) − k∂!b (p ⊕ q) | the D(k)U (p) = kb0 ≡ local! i-e x=0 ifa z=0 q 0 a a q=0 0 S= (x k˙ a − N C(k)) (p ⊕ p) ⊕ q = p ⊕ (p ⊕ q) a˙ −∞ S= (x ka − N C(k)) D(Λ(p), Λ(q)) = D(p, q) −∞ x is a commutative coordinate b (p)ba=≡p∂q⊕ (p((p ⊕ q)⊕ (p ⊕ (p ⊕ p)) ⊕Uq) p) ⊕D(T q)r (p), Tr (q)) = D(p, q) a |q=0 bnon commutative b z is a coordinate U (p)a ≡ ∂q (pT⊕ q) | ∗P a q=0 (p) = r ⊕ p r 0 (p ⊕ p) ⊕aq = p ⊕ (p ⊕ q) 1 1 1 1 K = %p ⊕ (q % k ) d ((p ⊕ abc (p ⊕ (p ⊕ p))b ⊕ q) =abp ⊕ p) ⊕ q)d

{z , z } = Td z + R

= D(p, ⊕ p)⊕ ⊕ p) q =⊕ pq) ⊕ (pq) ⊕ q) p ⊕ p)) ⊕D(Λ(p), q) = p (p ⊕Λ(q)) ((p Monday, March 7, 2011

d pµc z

+ · · · α µ 1

x1 = z1 ∂p1 Kα = (z1 Ux1 )µ Ux1 : T0∗ P → Tp∗1 P

Vertex looks local to local observers for which z=0 Vertex looks non-local to distant observers 4 3 3 4 p p 3 4 p3 x4 p4 x3 z

1

z=0

x2

2 p

p2

2

2

local observer a δxI Monday, March 7, 2011

= ±{b

distant observer c

a Kc , xI }

=b + a

ac a I Γb b pc

+ ...

Specialising the geometry The correspondence principle Special relativity describe accurately all phenomena involving momenta smaller than a mass scale m to be determined exp. Torsion is of order 1/m curvature of order 1/m^2 The dual equivalence principle The algebra of interactions is independent on the nature ( color, charge) of the particles Torsionless if no modification of statistics Maximal symmetry Momentum space has as many symmetries as flat space time its an AdS or dS spacetime with radius of curvature Mp connexion is metric Monday, March 7, 2011

Experimental test Theorists propose but experiments decide. The geometry of momentum space should be measured rather than assumed A new phenomenological sets of questions opens up Two types of search: theoretical or purely phenomelogical Given the maximally symmetric model find a clean measure of the dual cosmological constant Test the 4 principles: Torsion, non metricity, Lorentz invariance, homogeneity

Monday, March 7, 2011

momen of co-linear velocities but is measured in Thomas procession princip [6]. This suggest an experiment inspired by the Thomas pretask fo cession experiment [7]. The idea is to follow a system in orbit on the (an electron in an or a particle circling in thespace LHC). Such Measure of atom the curvature of momentum a system is enclosing a loop in momentummotivated space atbyeach peA thomas precession analogy Girelli, Livine metries riod of revolution, which enclose the curvature in momentum Thes A system in orbit ( electron ,part at the LHC) space. At each period the localization of the orbiting partiopen a encloses a loop in momentum space at each cle will be shifted compared to the localization of aperiod particleofatrevolution retical, localisation of the orbiting particle will be shifted with rest. The Effectively, the particle will experience an infinitesimal to the particle boostrespect Ni at each period givenatbyrest, it experiences a boost

Experimental test

∆Acd ∆Acd cda Ni = 2 R i pa ≈ 2 mRcd0 i mP mP

(47)

time, cumulative. wheresmall ∆Acd displacement is the area of in thespace loop and in momentum space and m the mass of the particle. One should be able to observe then a spacetime due to relative locality effect. Even It pullsdisplacement itself by the bootstraps! if this effect is tiny this type of observation possess a huge potential since it is a cumulative effect and we can use the large number of orbits that develop over time. Monday, March 7, 2011

The cality r from a implies sured. might b the rela

Gamma ray exp The process of localising a distant object is momentum dependent The experiment: a distant star emits two photons One of low energy and one of high energy If the photons are emitted at the same time for an observer local to the star are they observed arriving at the same time by us ? T S

photon 2, S

Note that in Riemannian normal coordinates the speed of light is constant D(p) = η pa pb ab

T photon 1,

atom in Monday, March 7, 2011

atom in

∂ g |p=0 = 0 → Γ = T + N b bc

Gamma ray exp The process of localising a distant object is momentum dependent The experiment a distant star emits two photons One of low energy and one of high energy If the photons are emitted at the same time for an observer local to the star are they observed arriving at the same time by us ? T S

photon 2, S

T photon 1,

atom in Monday, March 7, 2011

atom in

NO ! even if the photons have the same speed

b Ua∗(p)a

b ≡ ∂q (p ⊕ q)a |q=0 4 Laurent Freidel Abstract Idea ofN, the derivation: from The first photon travels for a time Tcompute of the 1 in the rest frame r Theoretical Physics, 31 Caroline St. ON N2L 2Y5, Waterloo,Canada Gtime → 0 byspace In this note I describe the geometry of momentum more emitter, at !, which it is absorbed a detector,in which is at detail

Gamma ray exp

The setting

r2

tworestdifferent the momentumtransport with respect to perspective the emitter and has initially energy, q2a and, after the detection, momentum ka2 . This abfrom z1kofto 4≡ using parallel February 28, 2011 L (p) k⊕ p detector sorption thezfirst photon bythe the is eventtransport E3 . The u4

z4

y4

x4

k2

r1

u2

Connections and non linear composition p2,

T2

interaction coordinate of this third emission is za3 , the photon z is absorbed at event ya3 , the detector jumps from position x3a to y ua3 when it absorbs the photon. 01 x q Then, after a proper time s2 in− the state ka2 , the detector aba E4 . This leaves sorbs the asecond photon at a fourth event, the detector with momentum ra2 . The fourth interaction coorAbstract abc bc b the acdetectionaiof the bc second ai bcis dinatea associated with photon d za , the photon i i d dissapearsdat event yad, the detector d from jumps 4 4 of momentum detail coordinatein x4a more to ua4 when its momentum changes from ka2 to a b space ab +++ 2 c p,q=0 p q2 ra . c 1 c The time the second photon traveled was T2 . u3

3

z2

y2

3

2

x2

2

k1 u1

p1,

T1

y1

z1 x1

cribe the geometry q1

Results: ∆θ = T ∆ET K order = (p ⊕ (q ⊕ k)) The leading effect is due to non-metricity. R = ∂ Γ ∆S − ∂= Γ 0+ Γ Γ − Γ Γ

−∂ S ∂ [(p (q ⊕ p) ]| − ⊕S q)=−−T ∆EN

= T (0)

b b ++ U (p) ≡ ∂ ⊕ q)a |q=0 composition =aN q (p ∆E

and linear DYNAMICS For non a metric connection the next effect due to the torsion −IV.is THE

!,experiment G → 0in terms of solutions to the vari∗ the We describe but no time delay ∆S = 0 T principle, P %=which Mare× Pform ational of the ++ ++ f ree p ≡ total photons of different energies appear different locations k (p) Sto =come S⊕ (25) S2 − S1 = Γ ∆E = N ∆EL α + ∑ k from ∑ SIint −

−

++ N− ∆E

The free part of the action is given by +a

x =z a

b

a U (k) !b

worldlines,α

interaction,I

1 = ∆θ = (E1 + EZ 2 ) T− ηab T−+b " 2T ∗ PSIf ree = Tds∗!P a ˙J J xJ ka + NJ C (k) dual gravitational∗ lensing 0 k T P "= M × P ∆S = 0

Monday, March 7, 2011

(26)

(S2 − Sball · pˆ 1 ) = ∆z Soccer issue

(S − S ) = z (U − U )k − z (U − U )k 2 1 em x u rec x u 2 1 4 3 If one modifies the law of addition of momenta with a scale mp µ αβ for soccer balls? how come we do∼not see strange effects (z − z ) T p k rec

em

µ

α β

1 T (p1 , p2 ) + · · · p1 ⊕ p2 = p 1 + p 2 + mP

The main point is that the effective mass scale for the interaction institute.ca of two soccer balls of size N is N mp

1

Monday, March 7, 2011

pp1 + − q #ppΓ(p Γ(q , q ) # pq 2 = #m q,Pp= )q =#qp + q − + mP mP

mP

Soccer ball issue 1 1 1 1 2 2 1 2 2 2 1 2 1 1 p + pn − Γ(p , p ) = q + q − + n+1

11 1 1 1 2 2 1 2 1 Γ(q , q ) 1 ) =21 p + p − Γ(p , p ) = q + q − + Γ(q , q ) 1 2 − Γ(p , p q + q − + Γ(q , q ) (1 1 1 1 or order p 2 The 2 2 /m 1 2 1 1 m m 2 2 1 2 1 1 argument is as follows: P ,− P ,q ) p m m P P − Γ(p , p ) = q + q + Γ(q , q ) (17) p + − Γ(p p ) = q + q − + Γ(q m m 2 2 P 1 P m m m m n+1 n Supose that two rigid bodies 1, 2 both composed P P P P of N particles n+1 n tands for order p /m 1 /m 1 p order p n p ⊕ k = p p /m n+1 n 1 1 1 exchanges out in interact such that each atom of a photon with an n p for order p /m ⊕ k = p p /mp p out in 1 1 atom of 2 2 2 p 1 1in = pout ⊕ k k ⊕ pin = p out pin 1= pout1 ⊕2k 2 1 1 1 in = 1pout ⊕ k p k⊕ p2in pout ⊕ k= 2pout pin = pout ⊕ k pin 2= in = pout Pin = N p2 in k2 ⊕ pink2=⊕ pp2out 2 k ⊕ pin = p2out 2pthat 2p atoms k ⊕ = ∞ Lets also assume all of 1, have the same momenta in out k ⊕ p = p P = N p A left invertible law on in composition outP = Nin in a manifold P is a C pin in Pout = N pout Pin = N pin ∞ map p ⊕ p = 2p P = N p P = N p ble composition law on a manifold P is a C out out in in 1 2 1 out = 2N pout P = N (p ⊕ p ) = N (p⊕ :⊕Pp× P ) → P

outPout = outN pout P = N (p ⊕ p ) ∞ 1 2 1 2 tot in pout le composition law on a manifold P is a C map Ptot = N (p ⊕ p ) = N (p ⊕ p ) ⊕:P × P → P in pin out out Ptot = N(p, (pin q)⊕ p→ 1 pout ) (p ⊕ q) Pon Γ(P1composition , P2 ) +1· · · law on a manifold 1.PA left invertible 1+ 2− (p, q) → (p ⊕ q) 1 Γ(P , P ) + · · · m P ⊕ : PN× P → P =P + P − 1 1 , 2P2 ) + · · · =1P1 +2P2 − Γ(P m the following three properties: P×PP → P ⊕N: N Pm in

pin

normal coord. Definition

P is a C ∞ map

(1

(p, q) → (p ⊕ q) (18) ∞ hree properties: ∞map left law on a manifold P is a C ses ainvertible unit 0 such that (0 ⊕ p) = p (p ⊕ which satisfi A left invertible law on a manifold P is a C map (p, q) → (p ⊕ q) mPelement →composition Ncomposition mp denoted There is no soccer ball prob ! = ment denoted 0 such that (0 ⊕ p) = p = (p ⊕ 0) 1- It pos hree properties: s a left inversion⊕#: P: P× → P such that (#p ⊕ p) = 0 P → P ∞ ⊕ : P × P → P 2- It pos tisfies the following three properties: on # : P → P such that (#p ⊕ p) = 0 omposition law on a manifold P is a C map ment denoted 0 such thatinverse (0 ⊕ p) = p= (p composition ⊕ 0) rsion provides a left for the i-e possesses a unit for element denoted 0→ such that 0) in (p, q)q) → (pi-e ⊕ q)q)(0 ⊕ p) = p = (p ⊕3-This (p, (p ⊕ a left inverse the composition Monday, March 7, 2011

Conclusion New framework in which we can relax the notion of absolute locality in a controlled manner Momentum space possess a non trivial geometry (metric, connection) that can and should be probed experimentally Under general principles a preferred class of momentum space geometries can be proposed Many interesting and extremely surprising experimental consequences Interesting new math: connection between algebra and geometry No soccer ball problem Monday, March 7, 2011