11

A Numerical Philosophy of the A Numerical Philosophy of the Universe Universe and the Fundamental Nature of and the Fundamental Nature of

Computer ScienceComputer Science

Michael Nicolaidis Michael Nicolaidis TIMA LaboratoryTIMA Laboratory

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OutlineOutline

-- Motivation Motivation

-- Vision Vision

-- Emergence of relativistic space-time Emergence of relativistic space-time

-- Representation of quantum mechanics by Representation of quantum mechanics by stochastic computationsstochastic computations

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OutlineOutline

-- MotivationMotivation

-- Vision Vision

-- Emergence of relativistic space-time Emergence of relativistic space-time

-- Representation of quantum mechanics by Representation of quantum mechanics by stochastic computations stochastic computations

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Space-TimeSpace-TimeDuring millenniums, we considered that the world was made of During millenniums, we considered that the world was made of objects immerged in a space and evolving with the progress of objects immerged in a space and evolving with the progress of a time a time

Objects, space and time would be the primary ingredients Objects, space and time would be the primary ingredients that engender our world. that engender our world.

This representation reflects a perception of the world that we This representation reflects a perception of the world that we realize through our sensorial and mental structures.realize through our sensorial and mental structures.

In this harmonious perception:In this harmonious perception:-- space would be immutable and identical for all observers, space would be immutable and identical for all observers, only objects evolve with time.only objects evolve with time.- - each object occupies a precise position in space at any given each object occupies a precise position in space at any given time.time.

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Metaphysics, Paradoxes, Logic Metaphysics, Paradoxes, Logic Antinomies, Fracture of PhysicsAntinomies, Fracture of Physics

Relativity and quantum mechanics conserved this vision but Relativity and quantum mechanics conserved this vision but disrupted its harmonious perception by disrupted its harmonious perception by introducing introducing variousvarious paradoxes:paradoxes:

A dilemma: A dilemma: - - In a In a 3D world3D world it it existsexists a division of events into a division of events into pastpast,, present present, , and and futurefuture, but it is not consistent with relativity. , but it is not consistent with relativity. - In the alternative view – - In the alternative view – timetime is a is a 4th dimension4th dimension – there is (i) – there is (i) nono objective time flowobjective time flow (all events of space-time are equally (all events of space-time are equally existent), (ii) absolute determinism (at the macro scale), and existent), (ii) absolute determinism (at the macro scale), and (iii) no free will. These consequences make most physicists (iii) no free will. These consequences make most physicists and philosophers agree that this world view is undoubtedly and philosophers agree that this world view is undoubtedly wrong. wrong. But so far, no one has succeeded in formulating a view that But so far, no one has succeeded in formulating a view that avoids the above dilemma and is compatible with relativity.avoids the above dilemma and is compatible with relativity.

From theFrom the foundation text foundation text of International Conference on the of International Conference on the Nature and Ontology of Space-timeNature and Ontology of Space-time

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Metaphysics, Paradoxes, Logic Metaphysics, Paradoxes, Logic Antinomies, Fracture of PhysicsAntinomies, Fracture of PhysicsMetaphysics:Metaphysics: structure of space-time structure of space-time length contraction of length contraction of

objects (by which objects (by which metaphysicalmetaphysical mean space constraints mean space constraints objects to contract). objects to contract). Idem for time dilatation effects on the pace of processes.Idem for time dilatation effects on the pace of processes.

Logic antinomy:Logic antinomy: in the state of quantum superposition, an in the state of quantum superposition, an object at a given instant can be at several positions (following object at a given instant can be at several positions (following certain probabilities)certain probabilities)

Paradox:Paradox: an action on a particle can influence instantaneously an action on a particle can influence instantaneously the state of another particle, and at any distance (entangled the state of another particle, and at any distance (entangled particles). Space looses its essence: separate objects.particles). Space looses its essence: separate objects.

Fracture of physics:Fracture of physics: since space, time and elementary objects since space, time and elementary objects are primary ingredients of the universe, two theories are are primary ingredients of the universe, two theories are necessary to describe the world (structure of space-time, necessary to describe the world (structure of space-time, behavior of particles).behavior of particles).

Is there a vision which eliminates these paradoxes and Is there a vision which eliminates these paradoxes and unifies the physics ?unifies the physics ?

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OutlineOutline

-- Motivation Motivation

-- VisionVision

-- Emergence of relativistic space-time Emergence of relativistic space-time

-- Representation of quantum mechanics by Representation of quantum mechanics by stochastic computations stochastic computations

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Insert your pin code

Objects and BehaviorsObjects and Behaviors

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Objects and ObserversObjects and Observers(external to the observable)(external to the observable)

Observer external to Observer external to the observed object: the observed object: he can have access he can have access on information on information concerning the concerning the nature of the object nature of the object producing the producing the observed behavior.observed behavior.

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Objects and InteractionsObjects and InteractionsThe The nextnext state state of an of an object is determined by object is determined by its its present statepresent state and by and by informations coming informations coming from the objects with from the objects with which it interacts.which it interacts. Since Since interactions interactions may may only only reflect the behaviorreflect the behavior of the objects of the objects the the statestate of an object can of an object can only reflect the behavior only reflect the behavior of other objects: canof other objects: can not not contain information contain information concerning their concerning their profound natureprofound nature

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Observers being part of the Observers being part of the observableobservable (the Universe and its (the Universe and its

structures)structures)

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Observers being part of the Observers being part of the UniverseUniverse

Ultimate Ultimate limitlimit ofof our our knowledgeknowledge: as we are observers : as we are observers being part of the Universe, we can have being part of the Universe, we can have access onaccess on information concerning the information concerning the behavior of elementarybehavior of elementary particles, but particles, but notnot concerning concerning the profound naturethe profound nature of of the objects producing this behavior.the objects producing this behavior.

Particles: meta-objects producing a behavior Particles: meta-objects producing a behavior through the process of evolution of their state (like a through the process of evolution of their state (like a computation)computation)

Universe:Universe: engendered by engendered by a meta-system a meta-system composed of these meta-objectscomposed of these meta-objects

We can, at our convenience, associate to this We can, at our convenience, associate to this meta-system any structure (or architecture), as far as meta-system any structure (or architecture), as far as it produces the behavior of the Universe that we it produces the behavior of the Universe that we observe.observe.

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A Universe engendered by a processA Universe engendered by a process

Illustration:Illustration: (meta)cellular network(meta)cellular network

Each cell is a meta-object that engenders the behavior of Each cell is a meta-object that engenders the behavior of an elementary particle. an elementary particle.

Comports a set of state variables which determine: particle Comports a set of state variables which determine: particle kind, its position, its speed, etc. kind, its position, its speed, etc.

Changes its state at the pace of a meta-time: Changes its state at the pace of a meta-time:

- - present state of the cell + present state of the cells in present state of the cell + present state of the cells in interaction + computation rules = next state interaction + computation rules = next state

- - computation rules = laws of interactionscomputation rules = laws of interactions

Local interactions : only between particles whose position Local interactions : only between particles whose position variables have close values.variables have close values.

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A Universe engendered by a processA Universe engendered by a processMeta-Cellular-Network:Meta-Cellular-Network: The position variable of a The position variable of a cell determines the position cell determines the position of the particle. of the particle. The set of all position The set of all position variables determines the variables determines the form of space. form of space. Communication Communication (interaction) between (interaction) between particles having close particles having close positions positions Value of Position Value of Position Variable:Variable:

-- frequency of modulation to frequency of modulation to emit its stateemit its state

-- frequency of frequency of demodulation to receive demodulation to receive states of close particlesstates of close particles

c’

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Universe

b

a

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b’

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A Universe engendered by a processA Universe engendered by a process

If the rules computing the If the rules computing the cell states are identical to cell states are identical to the laws which govern the the laws which govern the evolution of the state of the evolution of the state of the particles in the universe, the particles in the universe, the state of these cells will state of these cells will evolve/move "identically" evolve/move "identically" with the particles of the with the particles of the universe and will create the universe and will create the similar structures.similar structures. If some observers emerge If some observers emerge in this universe, an image of in this universe, an image of a world similar to our will be a world similar to our will be formed in their mental formed in their mental structures.structures.

p

p’

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Emergence of timeEmergence of timeLet us imagine a world in which: Let us imagine a world in which: -- certain times the zebras are incomparably faster than certain times the zebras are incomparably faster than the lions and certain times it is the opposite,the lions and certain times it is the opposite, -- a car being at several km from a person suddenly a car being at several km from a person suddenly covers this distance in a fraction of a second and covers this distance in a fraction of a second and crushes him, crushes him, -- the earth carries out hundreds of revolutions around the earth carries out hundreds of revolutions around the sun without your biological age being increased, the sun without your biological age being increased, while several generations of other people already while several generations of other people already passed, and suddenly you age of a hundred years in a passed, and suddenly you age of a hundred years in a fraction of a secondfraction of a second......,......,

-- a world in which there is no stable correlation a world in which there is no stable correlation between the paces of evolution of the various between the paces of evolution of the various processes. processes. In such a world the notion of time has no In such a world the notion of time has no

meaning.meaning.

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Emergence of timeEmergence of timeOn the other hand:On the other hand:

In a system where the relationship between the paces of In a system where the relationship between the paces of evolution of any two processes is the same each time evolution of any two processes is the same each time they take place, we can speak about time, since:they take place, we can speak about time, since:

-- we can choose a process as reference for measuring we can choose a process as reference for measuring time, time, and and -- after having observed once the correspondence after having observed once the correspondence between the events of the reference process and the between the events of the reference process and the events of another process, we can:events of another process, we can:

• use the reference process to predict the instant (event of use the reference process to predict the instant (event of the reference process) in which each event of the second the reference process) in which each event of the second process occurs.process occurs.

• measure the duration of a process, by observing the measure the duration of a process, by observing the events of the reference process in which starts and finishes events of the reference process in which starts and finishes the process under measurement. the process under measurement.

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Emergence of timeEmergence of timeIn a universe engendered by the evolution of the states of a set In a universe engendered by the evolution of the states of a set of meta-objects composing a meta-system, there is emergence of meta-objects composing a meta-system, there is emergence of time of time ifif:: -- the laws governing the evolution of the states of the meta-the laws governing the evolution of the states of the meta-objects are invariant (independent of the values of the position objects are invariant (independent of the values of the position variables, and constant throughout the evolution of the meta-variables, and constant throughout the evolution of the meta-system)system),, In the engendered universe, the correspondence between the In the engendered universe, the correspondence between the events of two processes will be always the same. events of two processes will be always the same.

A

B

c’

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Universe

b

a

a’

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Destruction of time in case of lost of Destruction of time in case of lost of laws’ invariabilitylaws’ invariability

In a universe engendered by the evolution of the states of a In a universe engendered by the evolution of the states of a set of meta-objects composing a meta-system, there is set of meta-objects composing a meta-system, there is emergence of time emergence of time only ifonly if:: -- the laws governing the evolution of the states of the meta- the laws governing the evolution of the states of the meta-objects are invariable objects are invariable otherwise the correspondence between the events of two otherwise the correspondence between the events of two processes will vary arbitrarily.processes will vary arbitrarily.

c’

c

Universe

b

a

a’

b’A

B

2020

Time and Meta-timeTime and Meta-timeMeta-time:Meta-time: the emergence of time presupposes that the the emergence of time presupposes that the state of the meta-objects (cells of the meta-network) state of the meta-objects (cells of the meta-network) evolves. For example, new values of state variables are evolves. For example, new values of state variables are computed at each step of a meta-clock. computed at each step of a meta-clock.

Does it means that time is a simple translation of this Does it means that time is a simple translation of this meta-time?meta-time?

c’

c

Universe

b

a

a’

b’

c’

c

Universe

b

a

a’

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T

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Independence between time and meta-Independence between time and meta-timetimeThe period The period TT of the meta-clock takes two different values of the meta-clock takes two different values T1T1

and and T2T2 in two different cycles of computation: in two different cycles of computation: -- the same interval the same interval tthh of time corresponds to two different intervals of time corresponds to two different intervals

T1T1 and and T2T2 of meta-time. of meta-time.

-- freezing, decelerating, or accelerating the meta-clock does not freezing, decelerating, or accelerating the meta-clock does not have any effect on the time of the universe.have any effect on the time of the universe. Time is not a category having an autonomous existence, Time is not a category having an autonomous existence, independent of the laws of the universe, because it is not a independent of the laws of the universe, because it is not a translation of meta-time.translation of meta-time. TimeTime is is determineddetermined by the relations between the paces of by the relations between the paces of processes, thus processes, thus by the lawsby the laws which govern the evolution of the which govern the evolution of the universe. It is a universe. It is a byproductbyproduct ofof the the evolutionevolution ofof the the statestate ofof the the particlesparticles governed by these laws. governed by these laws.

th

T1 T2

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Nature and Emergence of TimeNature and Emergence of Time timetime, like everything else in the universe, cannot , like everything else in the universe, cannot not exist not exist without changewithout change: an : an engine of changeengine of change (meta-clock or meta-time) (meta-clock or meta-time) is is mandatorymandatory

timetime does does not havenot have a a per se existenceper se existence (it is not a translation (it is not a translation of a preexisting meta-time). of a preexisting meta-time).

timetime is is determineddetermined byby the the lawslaws which which governgovern the evolution of the evolution of the the state of the particlesstate of the particles (or of the universe): (or of the universe): -- qualitativelyqualitatively: : the the invarianceinvariance of the of the lawslaws is the is the necessary and necessary and sufficient conditionsufficient condition for its existence for its existence -- quantitativelyquantitatively: it is determined by the : it is determined by the ratios of ratios of thethe paces paces ofof the the evolution ofevolution of the the processesprocesses. These ratios are determined in . These ratios are determined in their turn by the their turn by the laws governinglaws governing the evolution of the the evolution of the state of state of thethe particlesparticles (or of the universe) (or of the universe)

2323

Emergence of Relativistic Space-Emergence of Relativistic Space-Time in a Computational Universe Time in a Computational Universe

Next talkNext talk

2424

OutlineOutline

-- MotivationMotivation

-- VisionVision -- Emergence of relativistic space-time Emergence of relativistic space-time

-- Representation of quantum mechanics by Representation of quantum mechanics by stochastic computations stochastic computations

2525

Quantum Mecanics Quantum Mecanics (state superposition)(state superposition)

Observable A

State superposition

,...,...,, 21 n,...,...,, 21 nppp

particle

A: operator of a physical observable

1, 2, n eigen-vectors of A

1, 2, …, n: eigen-values of A

p1= |c1|2, p2 = |c2|2, …, pn = |cn|2

ci = ‹i|› (inner product of i and ).

= solution of Schrödinger’s equation

Measurement of physical observable

i {1, 2, …, n}

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Computations over stochastic Computations over stochastic signals: signals: theoremtheorem of of existenceexistence

f(_)g=f(w)w

Probability density function

s(w)

Probability density function

p(g)

s(w) p(g)

Theorem: p(g) f(_): p(f(w)) = p(g), discreet and continue spectrum

Input signal Output signal

w g

ervalsintby:Fp(a)da)g(F],s(b)db[)w(fg 1gw

gw ii

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State superposition State superposition deterministic deterministic transformation of a stochastic variablestransformation of a stochastic variables

• particle = meta-object performing two computations:- compute deterministic functions f(_) according to previous slide: to give the statistical distributions of observables defined by QM. - transform stochastic states w into stochastic states g, by means of functions f(_).

• A measurement of a physical observable: value g = f(w) corresponding to the current value of w.

• The particle realizes a state superposition determined by S.E.

• Measurement of physical observable

i {1, 2, …, n}

Wave function + Rules Algebra

Operators => statistical distributions :

P( r )P( p )

E1, E2, …, En

pe1, pe2, …, pen

s1, s2

ps1, ps2

Interaction with environnement

wr r

wp p

we E

ws s

fr(wr)

fp(wp)

fe(we)

fs(ws)

Compute functions

fr, fp, fe, fs

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State superposition State superposition deterministic deterministic transformation of a stochastic variabletransformation of a stochastic variable

Observable A

State superposition

,...,...,, 21 n,...,...,, 21 nppp

• Elimination of the logical antinomy: in state of superposition an object can be at the same time in several positions following certain probabilities.

• But this interpretation is incompatible with the vision of the particles immerged in a true space (veritable position not computable).

The position must be a state variable.

w f(_)g=f(w)

s(w) p(g)w

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Entangled ParticlesEntangled ParticlesParticle 1

,...,...,, 21 n,...,...,, 21 nppp

Particle 2

Measure = i Measure = f(i)

spin = sx spin = - sx

Creation of entangled states

,...,...,, 21 n,...,...,, 21 nppp

Particle 1 Particle 2

How the result of a measurement on particle 1 can determine How the result of a measurement on particle 1 can determine instantaneously the result of measurement on the very distant instantaneously the result of measurement on the very distant particle 2?particle 2? => Paradox in a veritable space=> Paradox in a veritable space

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Entangled Particles versus Entangled Particles versus Computing Meta-objects: an Computing Meta-objects: an

exampleexampleCompute functions

fa, fp, fe, fs

Particle j:

IDj

signalswa

signals

EVj=IDi

fa(wa) signalswa

signals

EVi=IDJ

fa(wa

)

Particle i:

IDiCompute functions

fa, fp, fe, fs

• Each computing meta-object: ID number and entanglement variable Each computing meta-object: ID number and entanglement variable EVEV

• During entanglement between particles i and j meta-object i sets EVi During entanglement between particles i and j meta-object i sets EVi = IDj. Idem for particle j.= IDj. Idem for particle j.

• At each computation step:At each computation step:-- meta-object meta-object j emits its state at frequency = IDjj emits its state at frequency = IDj-- meta-object meta-object i uses demodulation frequency = EVi (= IDj). It receives i uses demodulation frequency = EVi (= IDj). It receives state of particle j. state of particle j. -- meta-object meta-object i receives at each computation step the state of particle j i receives at each computation step the state of particle j and can adapt its state immediately. and can adapt its state immediately. - - idem for particle jidem for particle j