Beyond and across space:

From space to entanglement

and back

Vasil Penchev Bulgarian Academy of Science: Institute for the Study of Societies and Knowledge: Dept. of Logical Systems and Models; vasildinev@gmail.com

July 7, 2015, 13:00, Lecture hall: 1/2 floor, “Velika dvorana” (Split, Teslina 12)

The Fourth Physics & Philosophy Conference: "Time, Space and Space-Time", University of Split, Croatia, July 6-7, 2015

I History

I History

Einstein, Podolsky, and Rosen

(1935)

hey suggested a thought experiment in order to demonstrate that quantum mechanics was ostensibly incomplete

ne can try to complement and elucidate its sense newly by Einstein’s criticism (1909, 1910) commented by Haubold, Mathai, and Saxena (2004) about the quantity of thermodynamic probability (W) and “Boltzmann’s principle”, i.e. the proportionality of entropy (S) and log W:

𝑆 = 𝑘. 𝑙𝑜𝑔𝑊 (k – the Boltzmann constant)

Furthermore, they showed:

f the mathematical formalism of quantum mechanics had been granted as complete, it would imply

instein called it “spooky” robability “W” implies some uncertainty, lack of

knowledge about the macrostate (the one “ocular”) in terms of the microstates (the other “ocular”)

hus it reproduces “binocularly” the cognitive space of possible solutions, after which that space can be merely observed, and the “events” in it described in “Gedankenexperiments”.

The “spooky” acTion: ince that kind of action contradicted the principle of

physics ostensibly, quantum mechanics should be incomplete in their opinion

ne can say that quantum mechanics turns out to be a thermodynamic theory seen “binocularly” in that space

his originates from its fundamental principles formulated yet by Bohr:

nlike classical mechanics, it is a “binocular” or “dualistic” theory about both quantum entities and “apparatus” and thus about both microstate and macrostate implying a fundamental counterpart of W

Edwin Schrödinger (1935)

e also pointed out that quantum mechanics implies some special kind of interactions between quantum systems, « » called by him

ollowing Einstein’s tradition of “Gedankenexperiments”, let us begin shrink the “apparatus” more and more

he shrink of the apparatus causes some diminution of all microstates, and the microstates remain constant

his results into increasing W, thermodynamic probability and decreasing S, entropy

John Bell (1964)

e suggested a real experiment t was apt to distinguish quantitatively and observably

between: he classical case without that “spooky action at a

distance”, and he quantum one involving a special kind of

correlation between physical systems hen the size of the macrostates becomes

commeasurable with that of the microstates, W begins to converge to 1, and S to 0

his happens when the size of the apparatus has become commeasurable with that of the measured quantum entities

Bell’s inequalities (1964)

his kind of correlation, quantum correlations can exceed the maximally possible limit of correlation in classical physics

hat exceeding, the so-called violation of Bell’s inequalities can be measured experimentally

hus one can test the incompleteness of quantum mechanics according to the literal EPR argument

hen their sizes become equal to each other, W is

just 1, and S is 0

“ ne microstate, one macrostate, one theory,

probability one, but zero freedom (entropy)”, rather

“totalitarian”: Classical mechanics is deterministic

Aspect, Grangier, and Roger

(1981, 1982)

heir experiments as well as all later ones showed unambiguously that the forecast quantum correlations are observable phenomena

ome would stop the “thought experiment” here. Not we!

he apparatus continues to shrink and its size is already less than that of the measured entities

he microstate is correspondingly bigger than that of the macrostate, and W > 1: an extraordinary kind of probability, and S changes sign from plus to minus transforming itself into negative

II Concepts

II Concepts

«Spooky» quantum mechanics

hus, that “spooky action at a distance” exists and thus quantum mechanics should be complete

he case of probability bigger than 1 can be equivalently represented as that of negative probability if one considers the system of two independent events, the probability of the one of which is negative (Penchev 2012)

he negative probability implies the complex values of entropy:

he room of the macrostate is already so tiny that a part of the microstate is already forced to go out in the space of the microstate

Entanglement

he new phenomenon was called “entanglement” and a separate branch of quantum mechanics

he theory of quantum information, studying that kind of phenomena, has appeared and blossomed out since the 90th of the past century

ts probability is negative and its entropy is complex adding some purely imaginary entropy for the parts of the microstate remained outside of the macrostate

his is the world of quantum information and entanglement

Entanglement and space

he concept of entanglement restricts that of space hat restriction refers to the coherent states in

quantum mechanics et us exchange the inscriptions “MACROSTATE”

and “MICROSTATE” to each other: uddenly , we turn out to be in the starting point of the

“Gedankenexpereiment”, i.e. in our world his is the quantum world if one exchanges the

inscriptions “MACROSTATE” and “MICROSTATE” owever, one cannot even exchange them, but may

look to the sky at night and to see the “microstates” as big as stars and nebulas …

Space versus coherent state

pace is a well-ordered set of points in relation to any observer or reference frame in it

oherent state in quantum mechanics is the whole of those points:

t is inseparable and thus unorderable in principle

oth concepts of space and coherent state are initial elements of cognition mutually restricting their applicability

n the ground of that “Gedankenexperiment”, one can reflect Einstein’s criticism to both “Boltzmann’s principle” and quantum mechanics newly

Experience and science

« pace» refers to our everyday experience, and he concepts of coherent state and entanglement, to

scientific cognition in an area inaccessible to our senses he quantity of our “ignorance”, W*= 1 − 𝑊, about

any physical quantity of any microstate makes physical sense in quantum mechanics as the thermodynamic probability W* of the conjugate of the physical quantity at issue

he necessary condition is: 𝑙𝑜𝑔 1 − 𝑊 ≅ 𝑙𝑜𝑔 1 − 𝑙𝑜𝑔𝑊 = −𝑙𝑜𝑔𝑊 , which is true only if 𝑊 ≅ 0, i.e. the “size” of the microstate is much, much less than that of the microstate:

ight the case in quantum mechanics!

The limits of «space»

he concept of space should be limited to the relations between physical bodies of commeasurable mass

owever, the above thought experiment demonstrates that quantum mechanics should be approximately valid if Boltzmann principle holds and the Boltzmann – Gibbs – Shannon definition of entropy is relevant

n fact, the theorem about the absence of hidden variables (Neumann 1932, Cochen and Specker 1968) demonstrate that quantum mechanics is complete:

hus Boltzmann’s principle and entropy should be only approximately valid right just to that limit of macrostates much, much bigger than microstates

De Broglie wave (1925)

he concept of space is being diluted gradually to and the beyond the limits

de Broglie wave can be attached to any physical entity according to quantum mechanics

he theorems about the absence of hidden variables in quantum mechanics (Neumann 1932; Kochen and Specker 1968) can be interpreted as both:

bsolute exact coincidence of model and reality, and

nversing the relation between the model and reality in comparison to classical physics

ere is how:

The period of de Broglie wave

ts period is reciprocal to its mass (or energy)

ne can interpret this period as the length of the present moment specific to the corresponding physical entity of this mass (energy)

he model in quantum mechanics equates the degree of our ignorance about any physical quantity (i.e. the mismatch of the model to reality) to its conjugate

owever the conjugate is merely another physical quantity and therefore EPR’s “element of reality”

uantum mechanics transforms our ignorance in a exactly measurable quantity though in another experiment

III Interpretations

III Interprettaions

For the mass of an observer

uman beings are granted as observers in space

he range of masses comparable with their mass (or energy) determines fussily and roughly a domain

ithin its scale, the concept of space is just applicable

fter the difference between the model and reality is included in both model and reality, this implies formally their necessary coincidence

his corresponds rather directly to the axiom of choice in mathematics

The measure of

an oBserver’s mass

f the masses (energies) of the interacting physical entities are commeasurable, they can share approximately a common enough present

hen one can postulate that “ridiculous principle”:

here is a special theory, right quantum mechanics, which is always and forever true, i.e. in any reality

instein’s general relativity seems to be an apparent exclusion of the “ridiculous principle”, though

The masses of the apparatus

and quantum entity

f their masses (energies) are incommensurable, the lengths of their present moments (the corresponding periods of de Broglie waves) are also incommensurable

ust that is the case in quantum mechanics

ndeed it studies the system of a macroscopic device, which measures one or more microscopic quantum systems

nd vice versa: if the “ridiculous principle” holds even to it, entanglement and gravitation should be linked to each other

A point on a segment

he present of the entity of much bigger mass (energy) can be idealized as a point

t should be somewhere on the segment representing the length of the present of the entity with much less mass (energy)

he relation (or even ratio) of the macro- and microstate as is variable in our “Gedanken-experiment”

hen energy conservation should be generalized to action conservation for the essentially different “lengths of now”

Future and the past

within the present

he present of the measured quantum systems is an approximately common segment

t will include also as the past as future rather than only the present of the device

uantum mechanics is forced to invent the relevant way to describe both quantitatively and uniformly future and the past along with the present

lassical mechanics is restricted only to the present

The past and future of the device

he past of the device is all points of the segment, which are before the point of the present

ts future will be those after this point

owever the way of being for both sub-segments above is radically different, even opposed to each other

he points of the past are always a well-ordered series

n the contrary, the points of future constitute an inseparable, coherent whole

Time, space and coherent state

he concept of coherent state in quantum mechanics refers to both future and past as well as to the present of the investigated system, though

owever the interpretation of «coherent state» is absolutely different: It is:

nseparable in future

well-ordered series in the past

statistical ensemble of states and the choice of a trajectory in the present

« pace» will refer only to its present shared by both apparatus and quantum entity

Unforecastable future

ndeed the future of any entity is unorderable in principle

ust this property is rigorously and thus quantitatively represented by the concept of coherent state

ny wave function is some state of some quantum system

t means the so-called superposition of all possible states of the system at issue as to future

the always well-ordered past

owever the past of any entity is always the well-ordered series of all past moments in time

he concept of wave function needs a not less relevant interpretation as to the past

hen it is equivalent to a transfinite series of bits, i.e. to an “infinitely long” tape of a Turing machine

he concept of Hilbert space is just that relevant mathematical structure, which is able to describe uniformly both unorderable (future) and well-ordered (the past)

The two elements:

future and the past

herefor the description in quantum mechanics has to provide the invariance to both unorderable future and well-ordered past

ne can thought of them as two opposite media being reconciled by the “phase transition” of the present

ilbert space is:

hat manages to provide the relevant tool for a general theory of phase transition

nother viewpoint to phase space

mathematics enters ...

econciling both “elements” means:

he so-called well-ordering theorem equivalent to the axiom of choice is necessarily involved

owever we have already demonstrated:

ilbert space is able to represent both “elements” and thus even the phase transition of the present between them uniformly

ilbert space is invariant to the axiom of choice in the sense above

The present between

the inconsistent two elements

he present always is situated and intermediates between the past and future

hoice in the present is just what transforms future into the past

hen:

ilbert space consists of choices in final analysis

nd the phases of choice are:

uture, before choice

he past, after choice

he present, the choice properly

Hilbert space

as the space of information

o, Hilbert space consists of choices in final analysis

nformation is the quantity of choices

hat implies is:

ilbert space is the space of information

he units of information are:

it : the choice between two equally probable alternatives (classical information about finite entities)

ubit : the choice between infinite alternatives (quantum information about infinite entities)

The service of space

pace in turn is what makes possible choice and thus the transformation of future into the past

pace unlike Hilbert space refers only to the present

pace unlike Hilbert space represents any motion only continuously

evertheless space and Hilbert space are topologically equivalent by virtue of the Poincaré conjecture proved by Grigori Perelman (2002;2003)

n fact, this is implied by that Hilbert space is the mathematical structure unifying the description of both continuous (even smooth) and discrete motion

The unity of time, and

what entanglement serves

ntanglement transcending space should be defined as temporal interaction

t involves future and the past of the macroscopic device

hus it demonstrates quantum correlation

lassical correlation is only within space and thus the present

uantum correlation is in Hilbert space adding correlation due to future

he past being already well-ordered seems not to allow of any correlation in principle

The temporal secret of

entanglement

ny classical correlation should refer only to the present of the correlating entities

hus refers only to the space, in which they are and which they share

ny quantum correlation transcends the present and space

It involves future and the past

nly so, it can exceed the maximal possible bound of any classical correlations

Quantum information

ntanglement involves the concept of quantum information

uantum information as well as its unit of qubit is shared by both single Hilbert space and two or more entangled Hilbert spaces

he former is the case where the system is considered as a whole

he latter is the case where the system is considered as composed by subsystems

he two cases are equivalent to each other

Quantum information

as a generalization

t is a generalization of the classical concept of information

n it, the units of elementary finite choice are merely substituted by ones among an infinite set of alternatives

n fact, the fundamental equation of quantum mechanics, the Schrödinger equation means:

nergy of quantum information is conserved in the course of time: from future via the present to the past

Conclusions:

Conclusions:

The reflection on

philosophy of space and time

ll those studies in quantum mechanics and the theory of quantum information reflect on the philosophy of space and its cognition

pace is the space of realizing choice

pace unlike Hilbert space is not able to represent the states before and after choice or their unification in information

owever space unlike Hilbert space is:

he space of all our experience, and thus

he space of any possible empirical knowledge

What should space be

philosophically?

pace should be discussed as:

“transcendental screen”

necessary condition of visualization or objectification

n it, all phenomena are represented by masses comparable with those of observers granted as human beings

ilbert space relevant to physical reality anyway can be exhaustedly projected on the screen of space as well-ordered series of “frames” (Bergson 1908)

ntanglement seems to be gravity after that projection

The limits

of our sensual experience

ur sensual experience as well as classical physics observes and studies only phenomena within the framework of space

herefore it cannot transcend its limits

owever our knowledge is able to transcend them by means of:

oing consistent any series of “frames” in space

dding those elements hidden for sensual experience but necessary for the series of frames to “make sense”, to be consistent

The breakthrough of

quantum theory

uantum theories can also transcend those limits

he general quantum principle of knowledge is:

ilbert space to be restored by any empirical or experimental series of frames in space

hus quantum mechanics allows of interpreting space newly:

t is the domain of interaction of bodies of both commeasurable mass and energy

hus it is the area of choice transforming future into the past

References (quantum mechanics):

spect, A, Grangier, P., Roger, G. (1981) Experimental tests of realistic local theories via Bell’s theorem. Physical Review Letters, 47(7), 460-463.

spect, A, Grangier, P., Roger, G. (1982) Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedanken Experiment: A New Violation of Bell’s Inequalities. Physical Review Letters, 49(2), 91-94.

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

roglie, L. de (1925) Recherches sur la théorie des quanta (Researches on the quantum theory), Thesis (Paris), 1924. Annales de Physique (Paris, 10-ème série) 3, 22-128.

References (quantum mechanics):

instein, A., Podolsky, B., Rosen, N. (1935) Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review, 47 (10), 777-780.

ochen, S. and E. Specker (1968) The problem of hidden variables in quantum mechanics. Journal of Mathematics and Mechanics, 17 (1): 59-87.

eumann, J. von (1932) Mathematische Grundlagen der Quantenmechanik, Berlin: Verlag von Julius Springer.

chrödinger, E. (1935) Die gegenwärtige situation in der Quantenmechanik. Die Naturwissenschaften, 23(48), 807-812; 23(49), 823-828; 23(50), 844-849.

instein, A. Theorie der Opaleszenz von homogenen Flüssigkeiten und Flüssigkeitsgemischen in der Nähe des kritischen Zustandes. Annalen der Physik (Leipzig) 33: 1275–1298 (1910).

instein, A. Zum gegenwärtigen Stand des Strahlungsproblems. Physikalische Zeitschrift 10: 185–193 (1909).

aubold, H. J. A. M. Mathai, R. K. Saxena. Boltzmann-Gibbs Entropy Versus Tsallis Entropy: Recent Contributions to Resolving the Argument of Einstein Concerning “Neither Herr Boltzmann nor Herr Planck has Given a Definition of W”? Astrophysics and Space Science, 290(3-4): 241-245 (2004).

enchev, V. A Philosophical View on the Introduction of Negative and Complex Probability in Quantum Information. Philosophical Alternatives, 2012(1): 63-78.

References (einsTein’s Thermodynamics):

Other References:

ergson, H. (1908) L'évolution créatrice. Paris: Félix Alcan

erelman, G. (2002) The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159 .

erelman, G. (2003) Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109 .

erelman, G. (2003) Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math.DG/0307245 .

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