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Lorenz, Gödel and Penrose NINTH DENNIS SCIAMA MEMORIAL LECTURE Lorenz, Gödel and Penrose: New perspec5ves on determinism and unpredictability: from climate predic5on to fundamental physics Tim Palmer University of Oxford X = −σ X +σY Y = − XZ + rX − Y Z = XY − bZ But what about… ? X = −σ X +σY IL Y = −XZ + rX −Y Z = XY − bZ “We conclude that there is an infinite complex of surfaces each extremely close to one or the other of two merging surfaces….” Lorenz 63 Differenal Equaons IL Number Logic Theory R L LRLRL a trefoil knot LRLRLRL a type (3,5) torus knot LRLRRRLRRR a type (-3,-7, 2) pretzel knot Lorenz knots = Modular knots (Ghys 2000) A ∈PSL(2,) =SL(2,) /{I,−I} The Modular Group ⎛ 164 133 ⎞ eg A= = LRRRLLRRRLLLLR ⎝⎜ 127 103 ⎠⎟ where ⎛ 1 1 ⎞ ⎛ 1 0 ⎞ L= ; R= ⎝⎜ 0 1 ⎠⎟ ⎝⎜ 1 1 ⎠⎟ hp://www.josleys.com/ar5cles/ams_ar5cle/Lorenz3.htm A Lace Λ = ω = nω + mω :n,m ∈ { 1 2 } Modular Forms Ellipc Curves G L = ω −2k 2 3 2k ( ) ∑ y = 4x − 60G4 x −140G6 ω∈L\{0} ⋅ p ∈I L ? Is there a (finite) No. “Hal5ng Sets algorithm for must have integral deciding whether p Hausdorff belongs to IL? dimension” Post Correspondence Problem Given a collection of dominos, eg ⎧⎫bacaabc ⎡⎤⎡⎤⎡⎤⎡⎤,,, Emil ⎨⎬⎢⎥⎢⎥⎢⎥⎢⎥ Post ⎩⎭⎣⎦⎣⎦⎣⎦⎣⎦ca ab a c can we make a list of dominos (repetitions allowed) so that the string on the top matches the string on the bottom? In this case yes, ie ⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤abcaaabc ⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥ab ca a ab c The PCP is to determine whether any given collection of dominos has a match. Not only is PCP unsolvable by algorithms, it is (Dube 1992) equivalent to asking whether a given line intersects the fractal attractor of an iterated function system. X = −σ X +σY IL Y = −XZ + rX −Y Z = XY − bZ X = −σ X +σY IL Y = −XZ + rX −Y Z = XY − bZ “According to quantum physics, no maer how much informaon we obtain or how powerful our compu5ng abili5es, the outcomes of physical processes cannot be predicted with certainty because they are not determined with certainty.” “One can always hope that there will be future developments which will lead to a dras5cally different theory from the present quantum mechanical theory and for which there may be a par5al return of determinism.” “It seems to me to be quite plausible that the correct theory of quantum gravity might be a deterministic but non- computable theory.” … X = F[X] dF δ X = δ X dX ∂ρ + ∇.(ρv) = 0 ∂t Liouville Equaon for a Hamiltonian System ∂ρ ∂ρ + ∇.(ρv) = 0 ⇒ = − ⎡ρ, H ⎤ ⎣ ⎦PoissonBracket ∂t ∂t Schrödinger Equaon ∂ρ i = − ⎡ρ, H ⎤ ⎣ ⎦OperatorCommutator ∂t Bell’s Theorem No physical theory based on locally-causal “hidden variables” can ever reproduce all of the predic5ons of quantum mechanics. 1+ C(b,c) ≥ C(a,b) − C(a,c) Superdeterminism “There is a way to escape the inference of superluminal speeds and spooky ac5on at a distance. But it involves absolute determinism in the universe…” John Bell Determinism and Superdeterminism X = F(X) X(0)• Determinism Initial State + Dynamics ⇒ Future State free choice fixed Superdeterminism Initial State + Dynamics ⇒ Future State fixed fixed fixed The problem: X(0) appears to have to conspire (with dX/dt=F[X]) to produce an incredibly special “fine-tuned” universe in which the Bell inequali5es are violated. This is almost universally considered implausible. Bell inequali5es violated Big Bang Big Bang′ Bell inequali5es sasfied Why “Big Bang” and not “Big Bang Prime”? Fractal Determinism (i) C = Ck k∈ ∈IC C ⇒ IFS{f , f } 1 2 X(0) f1(X) = X / 3 X′(0) f2 (X) = (X + 2) / 3 2 C f C = i ( ) ∉IC i=1 eg X(0) = .02202002.. ∈IC X′(0) = X(0) +δ X = .02202002 + .00001201.. ∉IC "geometrically unconstrained" perturbation X(0) looks incredibly fine-tuned when viewed from the outside Fractal Determinism (ii) C = Ck k∈ ∈IC C ⇒ IFS{f , f } 1 2 X(0) f1(X) = X / 3 X′(0) f2 (X) = (X + 2) / 3 2 C f C = i ( ) ∈IC i=1 X(0) = .02202002.. ∈IC X′(0) = X(0) +δ X = .02202002 + .00000220.. ∈IC "geometrically constrained" perturbation Nothing fine tuned about X(0) when viewed from the inside! Fractal Determinism in Lorenz ‘63 ⎛ Evolution ⎞ ⎛ X (0) ⎞ ⎫ ⎛ X (t) ⎞ ⎫ ⎪ ⎜ along ⎟ ⎪ ⎜ ⎟ ⎪ ⎜ ⎟ ⎜ ⎟ ⎪ Y(0) ⎬ ∈I L + ⇒ Y(t) ⎬ ∈I L ⎜ ⎟ ⎜ trajectories⎟ ⎜ ⎟ ⎜ Z(0) ⎟ ⎪ ⎜ Z(t) ⎟ ⎪ ⎝ ⎠ ⎭⎪ ⎜ of I ⎟ ⎝ ⎠ ⎭⎪ ⎝ L ⎠ ⎛ X (0) ⎞ ⎫ A geometrically ⎜ ⎟ ⎪ unconstrained I ⎜ Y +δY(0) ⎟ ⎬ ∉ L (“physically ⎜ ⎟ ⎪ unreasonable”) ⎝ Z +δ Z(0) ⎠ ⎭⎪ perturbaon Determinism on Fractals: A Summary • X(0) and dX/dt=F[X] are not independent, but are both dependent on the underlying invariant set geometry . • X(0) appears incredibly fine tuned, but only with respect to geometrically unconstrained (“physically unreasonable”) perturbaons. • X(0) is not at all fine tuned with respect to (the equally numerous) geometrically constrained (“physically reasonable”) perturbaons. Fractal determinism may provide a conspiracy- free loophole for the Bell Theorem. THE INVARIANT SET POSTULATE A New Geometric Framework for the Founda3ons of Quantum Theory and the Role Played by Gravity Proc.Roy.Soc. A 465:3165-3185 (2009) ISP: ∈IU For Lorenz ‘63 X = −σ X +σY Y = − XZ + rX − Y Z = XY − bZ ∂X ∂Y ∂Z ∇.v = ∂X + ∂Y + ∂Z = −σ − 1 − b < 0 Hawking Radiaon Black Body Radiaon “..informaon…directed into such a space 5me singularity is … destroyed. …A be]er way of describing this is as a loss of degrees of freedom, so that … the phase space [of the universe] has actually become smaller than it was before.” (Penrose 2010: Black Hole Cycles of Time) Phase-space convergence at the Planck scale Penrose, 2010 From a Postulate to a Theory. Can we construct a fractal invariant set in state space, from which quantum stas5cs emerge naturally? Such a construc5on would have to explain the appearance of … a) b) i c) 0 + 1 or …in the Schrödinger equaon? A: Yes, using symbolic labelling techniques: h]tp://arxiv.org/abs/1210.3940 t1 t2 t0 Use to determine whether neighbouring trajectories (each a space-time) are gravitationally distinct E G = gravitational interaction energy Cf Diósi, Penrose, Kibble, Percival…. t1 Trajectories on IU Is E dt > ? ∫ G NO t0 t0 t1 Use to determine whether neighbouring trajectories (each a space-time) are gravitationally distinct E = gravitational interaction energy G 0 t2 Trajectories on IU Is E dt > ? ∫ G t0 t0 YES Cf Diósi, Penrose, Kibble, Percival…. t2 1 Palmer, T.N., 1978: Phys.Rev., D18, 4399‑4407. Gravitaonal energy-momentum as a tensor field on the tangent bundle to space 5me – quasi-local in space 5me. i ct′ = γ (ct − βx) x′ = γ (x − βct) Hilbert Space – Unitary Transformaons 0 . 0′ ψ ψ 1 θ′ θ′ ψ = 0 + eiφψ1 = cos 0′ + sin eiφ′ 1′ 2 ( ) 2 2 The Symbolic Skeleton hp://arxiv.org/abs/1210.3940 π φ = q q ∈ cosθ ∈2 {0,0,0,0.....} 2 2 {0,0,0,1,.....} 2 θ 1,0P,0(,01),..... = cos {0,1,1,0,.....} { } 2 Symbolic strings defined by families of self- {1,1,1,1...} similar quaternionic operators (isomorphic to Dirac matrices). See h]p://arxiv.org/abs/1210.3940 for details S = {UDUUDUDDUDDD....} S = {(UD)(UU)(DU)(DD)(UD)(DD)....} iS = {(UU)(DU)(DD)(UD)(UU)(UD)....} i2S ={(DU)(DD)(UD)(UU)(DU)(UU)....} = {DUDDUDUUDUUU....} = −S Das Unbes5mmtheitsprinzip? . π π Complex Hilbert cosθ ' = cos q (=cos ) Space as a 2 8 “comple5on” of the ∉ for 0<q < 1∈2 Symbolic Skeleton on the sphere. • = indeterminate wrt rotated pole! • = not one of the allowed states wrt rotated pole! So What? Using Fractal Determinism, and looking from the inside…there is no fine tuning Big Bang ∈I Bell inequali5es U violated. Big Bang Big Bang′ Bell inequali5es violated. Big Bang′ ∈IU Bell’s “implausible conspiracy” is all in the mind!! “According to quantum “One can always hope that physics, no maer how “It seems to me to there will be future much informaon we obtain be quite plausible developments which will or how powerful our that the correct lead to a dras5cally compu5ng abili5es, the theory of quantum different theory from the gravity might be a outcomes of physical present quantum processes cannot be deterministic but mechanical theory and for predicted with certainty non-computable which there may be a because they are not theory.” par5al return of determined with certainty.” determinism.” The Hopf Fibraons S1 3 2 S → S In quantum theory, the 3 S Hopf fibraons allow 1-, 2-, S 7 S4 3- qubit states to be → classified as separable or entangled. What about 4- S7 qubits and more? S15 → S8 An experimental test? • Can one prepare an n≥4 qubit system in a state which carries informaon about the topology of the system’s state space: can the state be measured in such a way that the measurement outcome is YES if the system’s state space is a fibre bundle, and NO if it is not? • If such a state can be prepared, quantum theory and Invariant Set theory would predict different measurement outcomes (NO/YES) respecvely). The Graviton Predic5on: No such thing as a graviton! The very expression “quantum gravity” may be a misguided one – pung the quantum “cart” before the gravitaonal “horse” No such thing as a graviton! IL Chaos Theory The laws of physics will have their most primi5ve expression in terms of state-space geometry IU Quantum General Mechanics Rela3vity .
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