PLAN OF MY TALK. 2. A story you might have not heard: Inertial Observers and Relationalism

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4 PLAN OF MY TALK 1. Overview of the workshop 2. A story you might have not heard: Inertial Observers and Relationalism 3. My first timid step towards defining Observers in Quantum Spacetimes 2

5 1. Overview of the Workshop OBSERVERS IN QUANTUM GRAVITY 3

6 OBSERVERS IN QUANTUM GRAVITY 3

7 OBSERVERS IN QUANTUM THEORY OBSERVERS IN GENERAL RELATIVITY 4

8 OBSERVERS IN QUANTUM THEORY she who makes the wavefunction collapse she who has access to gauge-invariant observables OBSERVERS IN GENERAL RELATIVITY inertial frame of reference cosmological observers 4

9 OBSERVERS IN QUANTUM THEORY quantum objectivity (Časlav Brukner) many-world interpretation(s) and measurement problem (Jeremy Butterfield, Julian Barbour, Henrique Gomes) quantum theory local Lorentz invariance (Markus Mueller) light cone and quantum observers (Časlav Brukner, Yours Truly) OBSERVERS IN GENERAL RELATIVITY classical cosmological observers (Jeremy Butterfield) observers in geometrodynamics (Henrique Gomes) 5

10 OBSERVERS IN QUANTUM THEORY + gravity quantum version of Einstein s clocks (Maximilian Lock) OBSERVERS IN GENERAL RELATIVITY + quantum quantum inertial systems (Flaminia Giacomini) 6

11 OBSERVERS IN QUANTUM SPACETIME inertial frames of reference in a quantum spacetime (Angel Ballesteros) accelerated observers in quantum spacetimes (Michele Arzano) observer-dependent vs. universal notion of spacetime (Jose Manuel Carmona, Giovanni Amelino-Camelia) quantum light cone (Yours Truly) 7

12 2. A story I want you to hear A LITTLE HISTORY OF INERTIAL OBSERVERS FROM A RELATIONALIST S PERSPECTIVE 8

13 INERTIAL OBSERVERS Newton s Principia Mathematica Philosophiae Naturalis (1726): A body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it. But what did he mean with rest, uniform motion or right (straight) line? 8

14 In his Scholium to the Principia: I. Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year. II. Absolute space, in its own nature, without relation to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies; and which is commonly taken for immovable space; [... ] he made his system logically consistent, at the cost of tying the First Law to unobservable entities like absolute space and time. 9

15 Newton was aware of this difficulty. Later in the Scholium: It is indeed a matter of great difficulty to discover, and effectually to distinguish, the true motions of particular bodies from the apparent; because the parts of that immovable space, in which those motions are performed, do by no means come under the observation of our senses. Yet the thing is not altogether desperate; for we have some arguments to guide us, partly from the apparent motions, which are the differences of the true motions; partly from the forces, which are the causes and effects of the true motions. [...] But how we are to obtain the true motions from their causes, effects, and apparent differences, and the converse, shall be explained more at large in the following treatise. For to this end it was that I composed it. Thus, he considers that deducing the motions in absolute space from the observable relative motions to be the fundamental problem of Dynamics, and claims that he composed the Principia precisely to provide a solution to it. Except he never mentions the Scholium Problem again in the Principia. 10

16 Peter Tait in his Note on Reference Frames (1884) provides a solution in the case of inertial motion of N bodies: a 11

17 Peter Tait in his Note on Reference Frames (1884) provides a solution in the case of inertial motion of N bodies: 1. Fix the origin at the position of particle 1: then r 1 = (0, 0, 0). 2. t = 0 when particle 2 is closest to particle 1. r 12 (t = 0) = a. 3. y axis parallel and x orthogonal to particle 2 worldline: x 2 = a, z 2 = use particle 2 as Karl Neumann s inertial clock (1870): r 2 = (a, t, 0) 5. The motions of the remaining N 2 particles remain unspecified. All one knows is that they will move along straight lines uniformly with respect to the time t read by Neumann s inertial clock. Their trajectories will therefore be: r a = (x a, y a, z a ) + (u a, v a, w a ) t. 11

18 (x a, y a, z a ) and (u a, v a, w a ) together with a will be 6N 11 unspecified variables. One needs 6N 11 observable data to construct an inertial frame. If we are given the observable inter-particle distances r ab at unspecified instants of time, can only get 3N 7 independent data from each snapshot. 2 snapshots aren t enough. We are short of three data. Lange in 1885 coined the expressions inertial system, in which bodies left to themselves move rectilinearly, and inertial time scale, relative to which they also move uniformly. Basis on modern notion of inertial frame of reference. 12

19 What about Newton s Scholium problem? Henri Poincaré in Science and Hypothesis (1902) What precise defect, if any, arises in Newton s mechanics from his use of absolute space? Angular momentum. Cannot deduce the total angular momentum of the system from observable initial data r ab and their first derivatives alone. These are the 3 missing data in Tait s analysis. 13

20 One can obtain the missing data from the second derivatives of r ab, as demonstrated by Lagrange in 1772 for the 3-body problem, but this remedy looks unnatural (N large: need 3N 6 first + 3 second derivatives). Poincaré found this situation, in his words, repugnant, but had to accept the observed presence of a total angular momentum of the Solar System, and renounce to further his critique. Interestingly, it didn t occur to Poincaré that the Solar System is but a rather small part of the Universe, as was already obvious in

21 This book is about the nature of space and time. It is the first textbook on Shape Dynamics, a new theory which describes the dynamics of gravity as the evolution of conformal 3-dimensional geometry. Shape Dynamics is equivalent to Einstein s General Relativity in those situations in which the latter has been tested experimentally, but the theory is based on different first principles. It differs from General Relativity in certain extreme conditions. Shape Dynamics allows us to describe situations in which the spacetime picture is no longer adequate, such as in the presence of singularities, when the idealization of infinitesimal rods measuring scales and infinitesimal clocks measuring proper time fails. This tutorial book contains both an introduction for readers curious about Shape Dynamics, and a detailed walk-through of the historical and conceptual motivations for the theory, its logical development from first principles and a description of its present status. It includes an explanation of the origin of the theory, starting from problems posed first by Newton more than 300 years ago. The book will interest scientists from a large community including all foundational fields of physics, from quantum gravity to cosmology and quantum foundations, as well as researchers interested in foundations. The tutorial is sufficiently self-contained for students with some basic background in Lagrangian/Hamiltonian mechanics and General Relativity. FLAVIO MERCATI is a postdoctoral researcher at the Perimeter Institute for Theoretical Physics. In 2015 he won the Buchalter Prize for Cosmology for his research on the arrow of time. m e r c at i SHAPE DYNAMICS SHAPE DYNAMICS relativity and relationalism f l av i o mercat i RELATIVITY AND RELATIONALISM Cover image: 1 ISBN

22 QUANTUM INERTIAL SYSTEMS/OBSERVERS Flaminia Giacomini s talk of today: Tait with quantum nonrelativistic point particles. Angel Ballesteros talk of today: how to define an inertial frame of reference in a noncommutative spacetime? What about GR? the total angular momentum of the universe still cannot be determined within the system. It is now encoded in boundary degrees of freedom. A spatially closed universe solves the problem. But observers only have access to a part of the spacetime manifold. Henrique Gomes talk of today: how to define a notion of reference frame in GR through boundary DOFs in geometrodynamics (i.e. Hamiltonian GR). Consequences for quantum geometrodynamics? 15

23 3. My first step towards a definition of Observers in Quantum Spacetime QUANTUM LIGHT CONES Minkowski s transparency at his lecture in Cologne in

24 QUANTUM LIGHT CONES In flat-space QFT: [ˆφ (x), ˆφ(y)] = i P J [g µν ; x, y) 10 5 z 0 -y π π z 1 -y 1 P J (z, y) 17

25 But Minkowski space might not be the exact vacuum of Quantum Gravity... Evidence from 2+1D Quantum Gravity [Matschull Welling, CQG 1998] (well-understood topological theory): D[g]D[φ]e is[g,φ] = D[φ]e is eff[φ] S eff [φ] = nonlocal theory (infinite derivatives of φ). Can be reformulated as ordinary QFT on a noncommutative spacetime [Freidel Livine PRL 2006] [x µ, x ν ] L p x ρ symmetric under Hopf algebra/quantum Group deformation of ISO(3, 1): x µ = Λ µ ν x ν +a µ 1, s.t. [x µ, x ν ] L p x ρ [Λ µ ν, Λ ρ σ] [Λ µ ν, a ν ] [a µ, a ν ] 0 18

26 Can t repeat that in 3+1D... but we can study the possible Hopf algebras generalizing the Poincaré group. Most studied example: κ-poincaré, leaving the following algebra A invariant: [x 0, x i ] = i L p x i, [x i, x j ] = 0, x µ A. Leads to rich new physics (curved momentum space, deformed statistics, deformed CPT, deformed second quantization). In with Matteo Sergola we introduced the Pauli-Jordan function for κ-poincaré invariant complex scalar fields: [ˆφ (ẑ), ˆφ(ŷ)] = i P J (ẑ, ŷ). 19

27 Now P J (ˆx, ŷ) is an element of the algebra A A (generalizing the concept of function of two points ): ẑ µ = x µ 1, ŷ µ = 1 x µ. Introducing a representation for A A: ẑ 1 = L p q (q R), ẑ 0 = i L p q q + il p 2, ŷ 1 = L p r (r R), ŷ 0 = i L p r r + il p 2, we have a Hilbert space of geometric states: ψ = ψ(q, r) L 2 (R 2 ), φ ψ = dqdr φ(q, r)ψ(q, r) 20

28 What does ψ mean? Imagining that this model is the effective description of QG+matter when graviton contributions are averaged out, the state of two points ψ should determine the amplitude of finding observables (like P J ) which depend on ˆx and ŷ with a certain value. Then the expectation value ψ P J (ẑ, ŷ) ψ gives expected value of Pauli Jordan function on the geometric state ψ. 21

29 There are many states ψ. Most of them excluded by everyday experience: δz 1 = L p, δz T p = 62 days all states satisfy the uncertainty relation: δz 1 δz 0 L p 2 z1 so we focus on the states that most closely resemble points on a classical Minkowski spacetime: (δz 1 ) 2 + (δz 0 ) 2 L p z 1 and ψ(q) is localized, i.e. its support is concentrated in a neighbourhood of z µ. 22

30 Then we can calculate ψ P J (ẑ, ŷ) ψ = f( z µ, y µ ) as a function of the expectation values z µ and y µ : 10 5 κ( z 0 - y 0 ) κ( z 1 - y 1 ) κ L 1 p 23

31 Zooming out: κ( z 0 - y 0 ) κ( z 1 - y 1 ) κ( z 1 - y 1 ) κ( z 1 - y 1 ) 24

32 Taking a constant-time cross-cut: 1000 Δ PJ (z,y) π 2 κ z 0 = π 8 π 8 0 κ ( z 1 - y 1 ) κ z commutative κ-minkowski 25

33 The profile is the same at all distances from the origin (in units of z 0 ): Δ PJ (z,y) π 2 κ z 0 = π 8 κ ( z 1 - y 1 ) κ z π 8 0 commutative κ-minkowski 26

34 The profile is the same at all distances from the origin (in units of z 0 ): Δ PJ (z,y) π 2 κ z 0 = π π

35 The profile is the same at all distances from the origin (in units of z 0 ): Δ PJ (z,y) κ z 0 = π 2 3 π 8 κ ( z 1 - y 1 ) κ z 0 π

36 The Pauli Jordan functions falls off exponentially away from the classical light-cone, with a falloff radius of order: L p z 0 = geometric mean between L p and distance from the origin. A pointlike source one billion light-years away, z y will be detected with a time uncertainty of L p z 0 = s = 10 femtoseconds. space (T p <z 0 >) 1/2 c<z 0 > detector << (T p <z 0 >) 1/2 source <z 0 > time 27

37 Časlav Brukner s talk of today: light cones can be indefinite in non-classical spacetimes. Angel Ballesteros talk of today: a proposal for definition of inertial observers in noncommutative spacetimes. Noncommutative spacetime noncommutative Poincaré transformations x µ = Λ µ ν x ν + a u 1, [x µ, x ν ] 0 [Λ µ ν, Λ ρ σ] [Λ µ ν, a ν ] [a µ, a ν ] 0. Does that avoid causality violations that might follow from a leaky lightcone? Jose Manuel Carmona s talk tomorrow: is spacetime an objective notion when Poincaré group deformed? 28

38 THANKS, AND WELCOME TO THE WORKSHOP! 29