The Butterfly and the Photon:
New Perspectives on Unpredictability in Classical and Quantum Physics
Tim Palmer, Department of Physics, Oxford University [email protected] THE INVARIANT SET POSTULATE A New Geometric Framework for the Foundations of Quantum Theory and the Role Played by Gravity
Proc.Roy.Soc. A 465:3165-3185 (2009) Three revolutions in 20th century physics Relativity Theory
Quantum Theory Chaos Theory “My own view is that to understand quantum non- locality, we shall require a radical new theory. This new theory will not just be a slight modification of quantum mechanics but something as different from standard quantum mechanics as General Relativity is different from Newtonian Gravity.” (Penrose 1997: The Large, The Small and the Human Roger Mind) Penrose
XXY Y XZ rX Y Z XY bZ XXY Y XZ rX Y Z XY bZ “I think there is some reason to believe…that the true quantum theory of gravity might be non- computable.” (Penrose 1997: The Large, The Small and the Human Mind)
Roger Penrose ? Alice
Bob
Can Alice conceive of an alternative weather state for14 Jan, which improves her weather, but a) leaves Bob’s weather unchanged, and b) is consistent with the laws of weather? pI . .
Alice’s Weather Alice’s pI . pI Bob’s Weather 1) The weather is a causal fluid dynamical system – no fluid dynamical influence travels faster than the speed of sound
2) Observationally, fluctuations in Alice’s weather and Bob’s weather are (more or less) statistically uncorrelated.
3) Nevertheless, Alice’s weather and Bob’s weather are interdependent: constrained to the invariant set, hypothetical (or counterfactual) alterations to Alice’s weather would “instantaneously” have an affect on Bob’s weather
In this context the notion of “time” has no meaning, hence no violation of 1) above XXY Y XZ rX Y Z XY bZ Invariant Set Postulate (ISP)
States of physical reality lie on a non-computable subset I of the euclidean state space of the universe “At the heart of the problem is… the question of realism”
Einstein 1950 Some two dimensional cuts through the (putative) state space of universe
Position of Position of “particle “particle here” there” . .
Momentum of Momentum of “particle here” “particle there”
Position of Position of . “particle “particle here” . here”
Position of Momentum of “particle there” “particle there” Position of “particle here” .
Position of “particle there”
In conventional classical physics, position “here” can be varied independently of position “there”, and vice versa.
What happens if states of physical reality lie on the noncomputable subset I? Elementary Cantor Set
Generated by 2 elements and reduction of 1/3. D=log 2/log 3. Fractal Lacunarity Each row is a self similar 1D Cantor set based on the unit interval, and made up by a generator of 2k intervals and a reduction factor 1/4k with the same D=1/2, but different lacunarity.
k 6 Large lacunarity. No point in the finite sized neighbourhood belongs to the fractal
k 1
k 5
Small lacunarity or “plentiful”. Many points in the same neighbourhood belong to the fractal Suppose in “position here/position there” space, I has the structure
Position of “particle .pI here” I “gappy” .pI “plentiful”
“Position of particle there”
consistent with (cf uncertainty principle)
lacunarity12 lacunarity k Position of pI “particle . . pI here”
.pI
“Position of particle there”
On the other hand, there do exist states p’’ where the position of “particle here” differs (more or less arbitrarily) from what it is at p. However, for these states, the position of the “particle there” is necessarily also different.
Nb This conclusion is independent of whether the dynamics DI on I is causal (ie relativistically invariant).
Assume DI is causal. Mach-Zehnder Interferometer
C
A B
“The only real mystery” - A B i C Feynman
This situation is amenable to a “realistic” interpretation. Can we move the block into the lower beamHow keeping does thisthe particle properties“know” ofthat this the lower particlebeam unchanged isn’t blocked? (cf Alice/Bob Weather)
If we assume local realism, how do we explain this? pIpI ? Position of pI obstruction . . pI
.pI
If I in the “λ - position of block” space is as before, then the perturbation pp’ very likely takes p off I to a state of physical unreality. A state p’’ of reality exists for the new position of the block but with λ→λ’’ Nb Dynamics are assumed causal pIpI
Riddled Basins of Attraction
Chaos +Multi-Potential Well
Attractors with Basins which contain no open sets d2 x dx V( x , y ) p sin t dt2 dt x d2 y dy V(,) x y dt2 dt y 2 V( x , y ) 1 x22 x x y
Palmer, T.N.: Proc Roy Soc, (1995), 451, 585-608 Cantor Set generated by 23 intervals with reduction factor 1/24. D=3/4 Coloured red, blue to represent two basins of attraction with a specific riddled structure ai red , blue 1/4 a,,,,,,, a a a a a a a (red )=blue ( blue )= red i 1 2 3 4 5 6 7 8 (a8 ), a 7 , a 5 , a 6 , a 1 , a 2 , a 3 , a 4 i
2 i 1 (a2 ), a 1 , ( a 4 ), a 3 , ( a 6 ), a 5 , ( a 8 ), a 7
i i4 1
(a1 ), ( a 2 ), ( a 3 ), ( a 4 ), ( a 5 ), ( a 6 ), ( a 7 ), ( a 8 )
Permutation representation of complex multiplication. Works for bit strings of a1, a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 length 2N, for any N i1/4
i2 1
Easily extendible to the quaternions: i Palmer, T.N.: Proc Roy Soc, (2004), 460, 1039-1056
pI Fraction red cos2 /2 . . 2 pI
0 . pI pIpI
1st Objection.
No underlying causal reality because of Bell’s theorem! Bell’s Theorem
"Quantum mechanics cannot be embedded in a locally-causal theory."
Consider a pair of spin-1/2 particles formed in the singlet spin state. Assume a hidden variable model S ( , ) 1 for particle 1 and 2 respectively. Then P ( , ) d ( ) S ( , ) S ( , ) Alice Bob Alice Bob cannot equal the quantum expectation value.
"The vital assumption " is that the result 1 for Bob's particle does not depend on the setting Alice for Alice's particle, and vice versa. This “vital assumption” is violated with ISP. But this implies neither a breakdown of Alice local causality, nor a conspiracy!
pI Fraction red cos2 /2 . . 2 pI
0 . pI Bob
Assume Bob and Alice’s measurements disagree
Bob 0 Bob and Alice’s measurements agree Bell’s theorem does not imply that “quantum mechanics cannot be embedded in a locally- causal theory”.
The Invariant Set postulate provides a major loophole in Bell’s theorem to allow quantum theory to be embedded in a locally-causal deterministic theory”. 2nd Objection.
But the Schrödinger equation is linear! .(v ) 0 t Liouville equation. Linear in ρ even though underlying dynamics are nonlinear Liouville Equation
, H t PoissonBracket
Cf Schrödinger Equation for density operator
iH, t OperatorCommutator Quantum Theory vs ISP
• Eigenstates • Basins of attraction on I • Unpredictability of • Riddled-basin dynamics – measurement basins contain no open sets
• Uncertainty Principle • Lacunarity1 x Lacunarity2 • Complex numbers • Structure of riddled basins • Schrödinger equation • Liouville Equation on I • Nonlocality • Constraint from global state space geometry • Abstract Hilbert Space • Non-computability of I I and the role of gravity
• Existence of noncomputable dynamically invariant sets require the dynamics to be unstable, nonlinear, with some mechanism for collapse onto a zero-volume subset of euclidean space (eg dissipation). • All of these can be provided by gravity: – Instability (Ubiquitous cf 3-body problem) – Nonlinearity (General Relativity) – Irreversibility (Black Hole No Hair Theorem) “What does information loss at the singularity actually mean? A better way of describing this is as a loss of degrees of freedom, so that some of the parameters describing the phase space have disappeared and the phase space has actually become smaller than it was before.” (Penrose 2010: Cycles of Roger Time) Penrose Penrose, 2010 Invariant Set Postulate So What?!
Conjecture:
Quantum Theory (inc. QFT) will be emergent from a causal but non-computational theory of gravity which is an extension of GR in the sense of being geometric in both space time and state space. In particular the Schrödinger (and Dirac) equations will emerge as Liouville equations for this extended theory of gravity
If true:
Standard approaches to “quantum gravity” – string theory etc, which apply standard quantisation rules to GR or some variant thereof – are misguided. Fractals and Symbolic Dynamics as Invariant Descriptors of Chaos in General Relativity
by
N.J.Cornish
The study of dynamics in general relativity has been hampered by a lack of coordinate independent measures of chaos. Here I review a variety of invariant measures for quantifying chaotic dynamics in relativity that exploit the coordinate independence of fractal dimensions and symbolic entropies.
arXiv:gr-qc/9709036 Relativity Theory
States of physical reality lie on a non- computable subset I of euclidean state space
Quantum Theory Chaos Theory
“Despite impressive progress…towards the intended goal of a quantum theory of gravity, there remain fundamental problems…some of the basic principles of quantum theory may need to be called into question.” (Penrose 1976: The Nonlinear Graviton) Roger Penrose Post Correspondence Problem
Given a collection of dominos, eg b a ca abc ,,, 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 a b ca a abc ab ca a ab c
The PCP is to determine whether a collection of dominos has a match. The problem is unsolvable by algorithms. “The state of a system is defined to be that thing represented by any mathematical object which can be used to predict the probability associated with every measurement that may (conceivably) be performed on the system.” Hardy 2004 p. We could define:
State=P(RED) if State = undefined otherwise. But there exists no algorithm for deciding whether or not pI
Hence there exists no algorithm for knowing theoretically whether the measurement with some φ=φ0 lies on the invariant set or not.
If we are to define the notion of state as in Hardy’s definition ie for all φ, State must be mathematically well-defined irrespective of whether or not
Hence define State using the frequentist notion of probability for and by the abstract algebraic structure of probability (without reference to a sample space), for pI