atoms, quanta, and qubits: atomism in quantum mechanics and information

8/11/2019 Atoms, quanta, and qubits: Atomism in quantum mechanics and information

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Atoms, quanta, and qubits:

Atomism in quantum mechanics and

information


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Vasil Penchev

Bulgarian Academy of Sciences:Institute for the Study of Societies of

Knowledge

[email protected]

Monday, November 3rdFaculty of Sciences of the University of Lisbon

The international colloquium Atomism and

Imagery (from the 20th century onwards)


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Atomism in original

The original conception of atomism suggests thatthe process of dividing the matter has a limit:the atoms, which cannot be divided more intocomposing parts

The matter itself is defined in turnas what consists of atoms

The initial concept originates from Leucippus and

Democritus, and perhaps earlier. The etymologyof atom, or , indivisible,uncutable, non-cutable, from - ("not") and("I cut") is Ancient Greek


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A few main properties of Leucippus

and Democritus atom:

Nothing else real than atoms and voids Atoms are indivisible and unchangeable

The atoms distinguish from each other in form

Anything in the world is composed by atoms

All things differ from each other in the combination

of atoms

In fact, Leucippus and Democritus concept of atom

is closer to the contemporary notion of atom in

chemistry than in physics


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John Daltons atom

John Dalton (1803), a chemist, was who revived theancient philosophical concept of atom foran experimental science such as chemistry

According to his conception:

The atom is the least unit of any chemical elementdefined by its mass

The atoms cannot be decomposed, neither created

They are identical: All chemical reactions are onlysome recombinations of atoms

In other words, the atoms are granted as the units

of chemical matter


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British physicist

and chemist

John Dalton(1766-1844)

by Charles Turner

(1773-1857)after

James Lonsdale

(1777-1839)


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Matter and motion:

atoms and infinitesimals

The differentials of Leibniz and Newton

rigorously reinterpreted in Robinsons

nonstandard analysis (1966) can be thought as

a kind of infinitely tiny atoms of physical

motion

However, the discovery of quanta and the

fundamental Planck constant in the beginningof the XX century needed different

conceptions about the atomism and unity of

matter and motion in physics


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The view of classical physics

Matter Motion

Atoms

Infinitesimals

The bridge of

energy accordingto relativity

The infinitely small

atoms of motion


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Atomism and the concept of atom

in physics

The name atom in physics is reserved for

entities, which can be divided into electrons,

protons, neutrons and into some of the rest

elementary particles, some of which are

in turn compounded by other, still more

elementary ones

Even more, those ostensibly elementaryparticles turn out to be too much to be able

to be really elementary: more than hundreds

or even more than thousands


12/102http://abyss.uoregon

.edu/~js/i

mages/

at

om.jpg


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The atoms of physical matter

As the matter should be single, the atoms ofchemical matter were granted also as those of

physical one

However about a century later, the atoms weredecomposed into electrons, protons, photons, etc.

Even before that, the physical motion unlike any

chemical reaction could not be ever represented as

some recombinations of atoms

All atoms are discrete, but all physical motion

are continuous and even smooth after Newton


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http://www.wikihow.com/images/e/eb/Find-

the-Number-of-Protons%2C-Neutrons%2C-and-Electrons-Ste -1.


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http://pad3.whstatic.com/images/thumb/5/52

/Find-the-Number-of-Protons%2C-

Neutrons%2C-and-Electrons-Step-2.jpg/629px-

Find-the-Number-of-Protons%2C-

Neutrons%2C-and-Electrons-Step-2.jpg


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The Standard model

After the Higgs boson has confirmed

experimentally in the mid of 2012,

the Standard model consisting of 6 quarks,

6 leptons, and 13 bosons was ultimately

corroborated

Just these can form all elementary

particles are the fundamental and

indivisible atoms both of physical matterand physical motion according to

the Standard model


17/102http://www.pha.jhu.edu/~dfehling/particle.gif


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More about the Standard model

The fermions in it (the quarks and leptons) canbe interpreted as the atoms of matter, andthe bosons (the 8 gluons, photon,

2W-bosons, and Z-boson), as the atoms ofmotion

The Higgs boson describes the relationbetween the two groups in a sense andtherefore, how mass at rest can arise(or motion to be transformed in matter)


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http://upload.wikimedia.org/wikipedia/commons/thumb/0/0

0/Standard_Model_of_Elementary_Particles.svg/819px-Standard_Model_of_Elementary_Particles.svg.png


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A few rather big dark clouds on

the horizon of the Standard model

Kelvins (1900) metaphor about a few littleclouds on the horizon of the classical physicscan be reinterpreted about the Standard

model: These clouds are rather dark andentangled, though:

Dark matter

Dark energy Entanglement

Gravity

General relativity


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Dark

matter

Darkenergy

Entanglement

Gravity

General

relativity


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Dark matter

Trimble (1987) demonstrated that the masscalculated by gravitational effects andthe mass calculated by radiation differconsiderably from each other

An immense part of the mass in the universeshould not emit any radiation therefore beingnamely dark matter

Only the neutrinos could be that dark matteraccording to the Standard model Plank 2013 results estimate the dark matter

to be over 5 times more than the visible one


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http://hetde

x.org/ima

ges/

da

rk_

energy

/particle_

zoo.jpg


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The dark matter versus the Standard model

The mass at rest of the neutrinos is almost zero:So the ascription of more than 5/6 of the mass ofthe universe to them is not convincing enough

If the Standard model does not describe all kinds

of matter, the unknown ones would behavior asthat dark matter

Indeed gravity and thus the gravitational mass,

which are described by general relativity, arebeyond the Standard model: Therefore theysupply an independent evidence, which is rathernot in favor of the Standard model, though


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The visible masThe dark mass

The dark energy withzero mass at rest


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Dark energy Two collectives (1998, 1999) demonstrated that

theexpansion of the universe accelerates This can be explained by dark energy, which should

be 68.3% from the energy in the universe accordingto the Plank 2013 results: The rest belongs to

the bright and dark matter Any energy, which is not any matter with nonzero

mass at rest, can be hardly dark according to

the Standard model: That energy should be visible for the gluons and

photon are only stable bosons. Free gluons arealmost impossible because of the phenomena of

confinement, and the photons are visible


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http://scienceblogs.com/startswithabang/files/2013/03/url.gif

h d k


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The dark energy versus

the Standard model

The dark matter is only not convincinglyenough explainable in the framework of

the Standard model

Unlike it, the dark energy is absolutelyunexplainable within it: no any particle with zero

mass at rest, to which the phenomenon of dark

energy might be due

As dark matter as dark energy becomes visible

only by astrophysical observations interpreted as

usual in terms of general relativity


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http://static.ddmcdn.com/gif/blogs/6a00d834

1bf67c53ef0154353aba0f970c-800wi.jpg


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Entanglement The entanglement was theoretically forecast in

thefamous paper of Einstein, Podolsky, and Rosen(1935)

Schrdinger (1935) also forecast the phenomena ofentanglement called by him verschrnkte

Zustnde

Both papers deduced the phenomena ofentanglement from the mathematical base of

quantum mechanics, namely from the properties ofHilbert space

However, the former three demonstrated theforecast phenomenon as the proof of the allegedincompleteness of quantum mechanics


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Anal

ternative

choic

ebetween

them

easurementof:

Th

isleadsto

two

differentstatesof

thesyst

emI

I:


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The experimental confirmation of

entanglement

John Bell (1964) deduced a sufficientcondition as an experimentally verifiablecriterion in order to distinguish classical from

quantum correlation (entanglement)

Aspect, Grangier, and Roger (1981, 1982)confirmed experimentally the existence of

quantum correlations exceeding the upperlimit of any possible classical correlations

All other experiments until now confirmcategorically the phenomena of entanglement


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http://upload.wikimedia.org/wikipedia/commons/1/1e/Bell-test-photon-analyer.png


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The entanglement versus

the Standard model

The Standard model is linear in principle as far asit is based on Hilbert space and its linear

transformations

Moreover, the Standard model is closed for itcannot be the linear part or approximation of

some future non-linear generalization

Particularly, it cannot incorporate any additionalsymmetry as that would violate its linearity

The Standard model is ultimate knowledge

in a sense and even thus necessarily incomplete


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Linear

Symmetries of

a single Hilbert space:

[U(1)]X[SU(2)]X[SU(3)]

Only symmetries equivalentto the wave-particle image

of an inertial reference frame

Description by

two or more

Hilbert spaces:

Nonlinear

Equivalents tothe wave-particle

image of an arbitrary

reference frame

Gravity


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Gravity Gravity is the only physical force (or interaction),

which remains unexplained by any experimentallyconfirmed theory in the framework of the Standardmodel

Even more, any theory of gravity consistent with

quantum mechanics, i.e. any theory of quantumgravity consistent with all relevant experimentsdoes not exist

However, the hypothesis that quantum gravitycannot exist even as a law of nature would meanthat quantum theory and quantum mechanicsparticularly are not universal doctrines about

nature


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Electromagnetic

interaction

Weak interaction

Strong interaction

The Standard model

Does gravity allowany relevantquantum model at all?


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The gravity versus the Standard model

Gravity is a physical interaction, obviously,

but it is not a part of the Standard model

Even more, it cannot be any part of the Standard

model because it is not and cannot be local and

thus linear in the sense of the Standard model

Even still more, the Standard model itself does not

allow of any room in it at where gravity would stay:

The Standard model is already closed: No rooms!

That closedness originates immediately from

its linearity and thus from the fundamental locality

of all gauge theories


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General relativity

General relativity might be called the Standard

model of gravity

Unfortunately, it seems to be not only non-

quantum in principle but even not allowing of any

consistent quantum generalization of it Furthermore, it supplies the interpretational

framework, in which all existing experimental

results against the Standard model such as the darkmatter and energy can survive

General relativity is the only bastion of theoretical

resistance against quantum mechanics


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Dark matter

and energy


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N i i t


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Non!

Linear Nonlinear

Gaugeable Nongaugeable

Local in space-time Nonlocalin space-time

Invariant to choiceNon-invariant

to choice

A f li i l i


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A few preliminary conclusionsabout atomism in physics

Concerning the divisibility of the physicalatom

Concerning the relation of the concepts of

atoms and elementary particles Concerning the indivisibility of the Plank

constant

Concerning the fundamental problem ofquantum mechanics

Concerning the probabilistic interpretation of

quantum mechanics

E d i


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Units of matter

Matter and motion

independent

of each other

Energy and time

independent

of each other

Units of action

Action unifies

matter and motion

The fundamental

Plank constant

Action unifies

energy and time

No any fundamental

constant


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The divisibility of the physical atom

Even more, there is no physical principle, which canunderlie the alleged indivisibility or atomicity of

the atom itself

The atom can be considered as elementary only in

relation to all chemical properties, but not at all as

to the physical ones such as mass, energy, charge,

momentum, trajectory, and other

The ostensibly indivisible atom turns out to be very

complex composite system consisting of other

elementary particles such as protons, neutrons,

electrons, gluons, quarks and still more others


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http://www.ep.ph.bham.ac.uk/general/outr

each/SparkChamber/img/fundamental.gif

Atoms and elementary particles


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Atoms and elementary particlesThe atoms and all elementary particles are pieces ofmass or energy, which has to be conserved in closedsystems according to the law of its conservation,but there is not any limit of its divisibility

The revolution in physics made by quantum mecha-

nics introduced the Plank constant as the fundamentallimit of divisibility of one physical quantity, action

Thus it managed to discover something, which is

indivisible according to the laws of nature and evenguaranteed by a fundamental constant, the Plank one

Even more, physics and science have not managed tofind until now nothing else also as fundamentally

indivisible as the quantum of action


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http://abyss.uoregon.edu/

~js/images/scale.gif


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The indivisibility of the Plank constant

Quantum mechanics is grounded ontheactually indivisible quanta of actionlimited by the fundamental Planck constant

Any other indivisibility in physics originates

from the quantum of action

The meaning of that quant is so grandiose forknowledge of mankind that quantum is

a synonym of modern or contemporaryand antonym of classical: e.g. quantumversus classical physics and even classicalversus quantum science


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What implies the existing of the indivisible

constant of action (but not its value):

Th f d t l bl f t


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The fundamental problem of quantum

mechanics

Quantum mechanics resolves the problem ofhow both discrete and continuous (evensmooth) to be described uniformly and

invariantly The quantum of action is what forced

thephysicists to generalize the description intheabove way particularly to unify the

quantum leaps and the smooth motions ofclassicalphysics including even those studiedby special and general relativity

The temporal interpretation of the definition


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The temporal interpretation of the definition

of physical quantity (A)in quantum mechanics

*(x) (x)

About the probabilistic interpretation


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About the probabilistic interpretationof quantum mechanics

Max Born (1926) demonstrated that the solutionunifying discrete and smooth motion at the sametime unifies the description of probable and actualstates of quantum systems

Just this is the sense of the so-called probabilisticinterpretation of quantum mechanics

The fermions (having semi-integral spin) can be

referred to the actual and therefore well-orderedstate

The bosons (having integer spin) should describe

the possible and unorderable coherent state


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A necessary elucidation of the connection

between probabilistic mathematical) and

mechanical physical) approach

Totality aka eternity aka infinity

No axiom of choice (the Paradise)

Both

needchoice

(axiom)

Probabilistic (mathematical) approach

Mechanical (physical) approach

Minkowski space:from the Earthto the Paradise

by the stairsof time

Hilbert space:from the Paradise

to the Earth by the stairs

of energy


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Hilb T i


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Hilbert space as a quantum Turing tape

Hilbert space can be represented in a few ways,one of which is that of quantum information as

a tape of qubits instead of bits and processed by

a quantum computer (quantum Turing machine)

instead of a usual computer (standard Turing

machine)

Thus Hilbert space furthermore allows of any

physical process to be interpreted as

computational or informational, and the universe

itself, as an omnipresent quantum computer


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h b b


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Both bit and qubit

As a bit is the unit of classical information, asa qubit is the unit of quantum information

Qubit is: |0 |1where , are twocomplex numbers such that 2+ 2 = 1, and

|0, |1are any two orthonormal vectors (e.g. theorthonormal bases of any two subspaces) in anyvector space (e.g. Hilbert space, Euclidean space,etc.)

Both bit and qubit share some choice, which isbetween two alternatives for the former andamong an infinite set for latter

One bit (a finite choice)


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0

1

0

1

One bit(afinitechoice)

One qubit (an infinite choice)

Choice Well-ordering

Q bit th i fi it li ti f bit


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Qubit as the infinite generalization of bit

The qubit can be realized as a kind of

generalization of a bit as to infinite sets ofalternatives in a choice instead of the twoequiprobable alternatives in a choicedetermining a bit of information

Consequently, the quantity of quantuminformation measured in units of qubits canbe interpreted as the generalization of

the quantity of classical information measuredin units of bits

Thus, the quantity of information whetherclassical or quantum is the quantity of choices


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Finite sets Infinite sets

Classical

information

Quantum

information

Th i di i ibilit b th f bit d f bit


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The indivisibility both of a bit and of a qubit

Both a bit and a qubit are indivisible units

Therefor they can be called atoms of informationeither classical or quantum

The indivisibility or atomicity of choiceunderstood philosophically or mathematicallyunderlies both

The choice can be well understood as an atom of

time for the choice is the only which can transformfuture into past by means of the choice ofthe present

So the choice is an atom of time

I f ti A choice


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Much Many

Information A choice

Finite

(binary)

Infinite

A cell Values

0

1

...

... ...

Turing tapes = well orderings:


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Wave function in terms of qubits

Any wave function or any state of a physicalsystem can be represented as a correspondingseries of those indivisible qubits or atoms ofquantum information

Indeed any physical system should bequantum according to quantum mechanics

Hilbert space can be also considered as

thefree variable of quantum information Then any wave function is a value of

that variable as what the Hilbert spacehas been considered


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1 n n

Little transfinite ordinals

Bits Qubits

1 n

The mostentangledqubit

Entanglement


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Entanglement

Even more, wave functions can interact to each

other immediately, by entanglement In fact, the entanglement cannot violate the

indivisibility of the atoms of quantum

information, that is the qubits constitutingany wave functions

Entanglement is defined as a state of a compositequantum system, when its Hilbert space cannot be

factorized as any tensor product of the Hilbertspaces of the subsystems

Nevertheless, the entanglement does not shatter

the qubits into any parts

Classical correlations


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Quantum system 1Quantum system 2

The violation of Bells inequalities:

The imagery of atomism in physics


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The imagery of atomism in physics The comparison of atoms, quanta, and qubits

demonstrates certain imagery of atomism inmodern physics: from atoms of matter (or energy)via atoms (quanta) of action to atoms (qubits)of quantum information

That series of images of atoms can be understoodas the direction of a generalization from the atomsof matter by itself to quanta shared both by theparticles of matter (the fermions in the Standard

model) and the particles of motion andinteraction (i.e. the bosons there)

Consequently, the concept of quanta is more

fundamental than that of atoms


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Atomism and the cognition of reality


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Atomism and the cognition of reality

The transformation from atoms to quanta is

a conceptual shift in the cognition of reality Reality cannot be understood more as some base

of matter, which is in continuous motion

The atoms cannot refer only to that base ofmatter neither continuity only to the motion ofmatter

Time is what unifies matter and motion bythe choice as a special kinds of temporal atom

The cognition of reality is gradually redirected toquanta, information, choice, and time in final

analysis


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The choice as a kind of temporal atom

Present

Past

The Standard model in terms of


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The Standard model in terms of

quantum information

The number of particles (about fifty togetherwith the antiparticles) though exactly limitedseems to be too big to be willingly accepted for

kinds of atoms They call for some underlying and unifying

principle in turn

The so-called traditionally elementary particlesare rather mathematical than physical entities

Accordingly, their nature should be ratherinformational than only physical or even only

mathematical


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http://www.quantumdiaries.org/wp-

content/uploads/2014/03/SUSY-diagram-

Particle-Fever1.png

The Standard model as the quantum


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q

description a privileged reference frame

The Standard model can be associated with justone privileged reference frame

It should be inertial for both Standard model and

quantum mechanics are necessarily linear It is identified as that of the Big Bang

One can use the metaphor of the following

equation: It equates the nonlinearity of generalrelativity to the linearity of Hilbert space

That equation would have the singularity of the

beginning, i.e. the Bing Bang as the only solution


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The equation

The Big Bang as the solving

of that equation

The Big Bang satisfiesthe equating of the linearity of

the Standard model to

the nonlinearity

of general relativity

Relativity and a privileged position


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y p g p Special relativity arose in fighting against

the absolute reference frame of the alleged

ether and thus against any privileged referenceframe

However Einstein himself declared (1920) that

general relativity unlike special one is consistentboth to the availability and lack of some privilegedreference frame(s)

So the dethroned ether can be restored as togeneral relativity according to him

Thus the Standard model can be seen asthe quantum doctrine on the restored ether and

reference frame linked to it


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Special relativity

General relativity

Quantum mechanics

The Standard model

The absolutereference frameof classical mechanics

Ether

Revolution and Restauration

of the privileged position in physics

The privileged reference frame of


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the Big Bang

The Standard model needs only a privilegedreference frame, which can be only inertial becauseof its linearity

However the Bing Bang requires furthermore still

another condition ostensibly self-obvious:

All physical laws and thus including the Standardmodel itself should be invariant to any reference

frame: This implies the beginning (i.e. the Big Bang) as

the only possible general reference frame asthe only one common to all in the universe


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The equations of the Big Bang

The Big Bang

A privileged

referenceframe

Universality of

the physical laws

+

=

The Big Bang =The Standard

model

Universality of

the Standard model

+

Other interpretations of the privileged


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reference frame However if the Standard model is allowed to be not

so general and valid to any reference frame, other

options appear, too:

For example ,one can suggest that inertial reference

frame linkable to the present position of the earth

to the universe as that privileged by the Standard

model

Then, all different enough reference frames inthe universe would generate a proper Standard

model relevant to each corresponding position to

the universe

One and many standard models


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Oneinertialreference

frame Wave-

particleduality

Onestandardmodel

The inertialreference

frame of

the Big Bang

Our Standard

model

Anyinertial

referenceframe Wave-

particle

duality

Some one

standardmodel

The privileged inertial reference frame


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as a single qubit Indeed the information containing in a single qubit

is exactly equal to that featuring an inertialreference frame

Even more, both are mathematically isomorphic to

each other and still evidence about the directtransition from informational to physical concepts

One can see in detail how:

A qubit = an inertial reference frameA point of the ball = The initial position

A point on the surface (sphere) = The velocity

The superpositional norm, 1 = The light speed, c

l b


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|defines a point of the unit ball and thus it can

be interpreted as the position of a reference frame|and |define a point of the unit sphere and

thus |can be interpreted as the velocity of the

same reference frame, which should be inertial

+ =

|

|

, are two complex numbers:

|, |are two orthonormalvectors or a basis such as two

orthogonal great circles of

the unit ball

Speed as

Positionas

The speed of light, c, as 1


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The multiverse in a single universe


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The multiverse in a single universe

The Bing Bang is one solution of the above

equation however only under the additionalcondition of that the Standard model is universal

at least for our universe

However, if one unifies the many-worlds interpretationof quantum mechanics (Everett III 1957) with

the smooth general relativity, the multiverse should be

equivalent to the universe just as quantum mechanics

unifies discrete and smooth motion

Furthermore, any different enough reference frame in

the universe should possesses a proper standard model

different from any each in general


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Many discretely separated parallel worldsfor any measurement

World 1 World 2 World N... ...

Many smoothly flowing inertial reference frames

for any point in space-time

Referenceframe 1

Referenceframe 2

Referenceframe N... ...

One-to-oneOne-to-one One-to-one

R f


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World 1 World 2 World N... ...

Referenceframe 1

Referenceframe 2

Referenceframe N... ...

Standard

model 1

Standard

model 2

Standard

model N


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Th fThe empty sell

h


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TheThe

The

th

(+n)th

Transfinitebits

The sense ofSchrdingerequation

p yof the nthqubit

The empty sells oftwo equivalent, but

complementaryqubits SU(2)SU(3)

The

The special valueof the qubit = theStandard model

SU(2): as

3-sphere

SU(3): as3 numbers

U(1)

The three privileged sub-choicesi t f th St d d d l


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in terms of the Standard model Each of those three choices can be further

interpreted as the source correspondingly ofthe first, second, and third generation of thefermions (i.e. the alleged particles of matter) of

the Standard model The strong interaction can be considered as

the symmetry of the three single choices

The weak one, as the symmetry of the pairs ofthese three choices or as the symmetry ofa 3-sphere

The electromagnetic one, by means of the single

triple of these choice The Big Bang: The record of a value inthe empty sell of a qubit


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SU(2)SU(3)

SU(2): The symmetryof the chosen value

of the qubit asthe rotation of

a 3-sphere

SU(3): The symmetry ofthe same chosen valueof the qubit as 3independent real

numbers

U(1)

Quarks Leptons

Gluons

Photon

Bosons

U(1): The symmetry of

the empty sell ofthe qubit

the empty sell of a qubit

TheStandardmodel


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Conclusions (2):


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Quantum information allows of understanding theStandard model by means of the symmetries ofexactly three underlying choices

These three choices constitute a privileged qubitequivalent to a privileged inertial reference frame

Quantum information decomposes the quanta tochoices

Choice is the atom of the being in final analysis

Choice in turn originates from time and its coursefrom the unorderble or coherent future to the well-ordered or unchangeable past by the choices ofthe present

References (1):


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Planck Collaboration and Tauber, J., 2013, Planck 2013 results.I. Overview of products and scientific results. Astronomy andAstrophysics (in print, accepted 17 May 2014, DOI:10.1051/0004-6361/201321529). Also available at:http://planck.caltech.edu/pub/2013results/Planck_2013_results_01.pdf accessed 27 August 2014.

Aspect, A. et al, 1981, Experimental Tests of Realistic Local

Theories via Bells Theorem. Physical Review Letters, 47 (7),460-463. Aspect, A. et al, 1982, Experimental Realization of Einstein-

Podolsky-Rosen-Bohm Gedanken Experiment: A NewViolation of Bells Inequalities. Physical Review Letters, 49 (2),91-94.

Bell, J., 1964, On the Einstein Podolsky Rosen paradox.Physics(New York), 1 (3), 195-200.

Born, M., 1926, Physical aspects of quantum mechanics.

Nature, 119, 354-357.

References (2):
http://planck.caltech.edu/pub/2013results/Planck_2013_results_01.pdfhttp://planck.caltech.edu/pub/2013results/Planck_2013_results_01.pdfhttp://planck.caltech.edu/pub/2013results/Planck_2013_results_01.pdfhttp://planck.caltech.edu/pub/2013results/Planck_2013_results_01.pdfhttp://planck.caltech.edu/pub/2013results/Planck_2013_results_01.pdf


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( )

Dalton, J., 1805, Experimental Enquiry into the Proportion of

the several Gases or Elastic Fluids, constituting theAtmosphere. Memoirs of the Literary and PhilosophicalSociety of Manchester, Second Series, 1, 244-258.

Dalton, J., 1805, On the Absorption of Gases by Water andother Liquids. Memoirs of the Literary and PhilosophicalSociety of Manchester, Second Series, 1, 271-287.

Dalton, J., 1808,A New System of Chemical Philosophy (Part1), Manchester, printed by S. Russell for R. Bickerstaff, Strand,London.

Einstein, A., 1920,ther und Relativittstheorie, Berlin,Springer. Einstein, A. et al, 1935, Can Quantum-Mechanical

Description of Physical Reality Be Considered Complete?Physical Review, 47 (10), 777-780.

References (3):


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Everett III, H., 1957, Relative state Formulation ofQuantum Mechanics.Reviews of Modern Physics, 29 (3), 454-462.

Higgs, P., 1964, Broken symmetries and the masses of gaugebosons. Physical Review Letters, 13(16): 508-509.

Kelvin, 1901, Nineteenth-Century Clouds over the Dynamical

Theory of Heat and Light. Philosophical Magazine Series 6, 2(7), 1-40. Leucippus and Democritus (ed. Taylor, C.), 1999,

The atomists, Leucippus and Democritus: fragments: a textand translation with a commentary, Toronto, University ofToronto Press.

Peebles, P. and Ratra, B., 2003, The cosmological constantand dark energy. Reviews of Modern Physics,75 (2), 559606.


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M it


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Muito

obrigado

pela sua

ateno!


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atoms, quanta, and qubits: atomism in quantum mechanics and information

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