Lorenz, Gödel and Penrose

NINTH DENNIS SCIAMA MEMORIAL LECTURE

Lorenz, Gödel and Penrose: New perspecves on determinism and unpredictability: from climate predicon 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

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/arcles/ams_arcle/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. “Halng 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 compung abilies, 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 drascally diﬀerent theory from the present quantum mechanical theory and for which there may be a paral 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 predicons 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 acon 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 inequalies are violated. This is almost universally considered implausible. Bell inequalies violated

Big Bang Big Bang′ Bell inequalies 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 Foundaons 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 me singularity is … destroyed. …A beer 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 stascs emerge naturally? Such a construcon would have to explain the appearance of … a)  b) i

c) 0 + 1 or

…in the Schrödinger equaon? A: Yes, using symbolic labelling techniques: htp://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

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 me – quasi-local in space me. 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 hp://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 Unbesmmtheitsprinzip?

π π Complex Hilbert cosθ ' = cos q (=cos ) Space as a 2 8 “compleon” of the ∉ for 0

Big Bang ∈I Bell inequalies U violated.

Big Bang Big Bang′ Bell inequalies 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 drascally compung abilies, 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.” paral 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

Predicon: 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 primive expression in terms of state-space geometry IU

Quantum General Mechanics Relavity