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Vasil Penchev. Gravity as entanglement, and entanglement as gravity

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Vasil Penchev. Gravity as entanglement, and entanglement as gravity Vasil Penchev. Gravity as entanglement, and entanglement as gravity Presentation Transcript

  • 1Gravity as Entanglement Entanglement as gravity
  • Vasil Penchev, DSc, Assoc. Prof, 2 The Bulgarian Academy of Sciences• vasildinev@gmail.com• http://vasil7penchev.wordpress.com• http://www.scribd.com/vasil7penchev• CV: http://old-philosophy.issk-bas.org/CV/cv- pdf/V.Penchev-CV-eng.pdf
  • 3 The objectives are:• To investigate the conditions under which the mathematical formalisms of general relativity and of quantum mechanics go over each other• To interpret those conditions meaningfully and physically• To comment that interpretation mathematically and philosophically
  • Scientific prudence,or what are not our objectives:• To say whether entanglement and gravity are the same or they are not: For example, our argument may be glossed as a proof that any of the two mathematical formalisms needs perfection because gravity and entanglement really are not the same• To investigate whether other approaches for quantum gravity are consistent with that if any at all
  • Background• Eric Verlinde’s entropic theory of gravity (2009): “Gravity is explained as an entropic force caused by changes in the information associated with the positions of material bodies”• The accelerating number of publications on the links between gravity and entanglement, e.g.Jae-Weon Lee, Hyeong-Chan Kim, Jungjai Lee’s“Gravity as Quantum Entanglement Force” :“We conjecture that quantum entanglement ofmatter and vacuum in the universe tend to increasewith time, like entropy …
  • BackgroundJae-Weon Lee, Hyeong-Chan Kim, Jungjai Lee’s“Gravity as Quantum Entanglement Force” :…, and there is an effective force called quantumentanglement force associated with thistendency. It is also suggested that gravity anddark energy are types of the quantumentanglement force …”Or: Mark Van Raamsdonk’s “Comments onquantum gravity and entanglement”
  • Background: For the 4 gauge/gravity duality“The gauge/gravity dualityis an equality between twotheories: On one side we havea quantum field theoryin d spacetime dimensions.On the other side we havea gravity theory on a d+1dimensional spacetime thathas an asymptotic boundarywhich is d dimensional”• Dr. Juan Maldacena is the recipient of the prestigious Fundamental Physics Prize ($3M)
  • 5The third (of 7 and only solved)Millennium Prize Problem provedby Gregory Perelman ($1Mrefused): Every simply connected,closed 3-manifold ishomeomorphic to the 3-sphereThe corollary important for us is:3D space is homeomorhic to acyclic 3+1 topological structurelike the 3-sphere: e.g. thecyclically connected Minkowskispace
  • 3D (gauge) /3D+1 (gravity) are dual in a sense 3D & a 3D+1 cyclic structure are homeomorphic “What about that dualityif 3D+1 (gravity) is cyclic in a sense?” – will be one of our questions
  • Background: The Higgs boson It completes the standard model 6 without gravity, even without leaving any room for it: The Higgs boson means: No quantum gravity! As the French academy declared "No perpetuum mobile" and it was a new principle of nature that generated thermodynamics:
  • Background: The Higgs boson"No quantum gravity!" and it is a new very strange and amazing principle of natureIf the best minds tried a century to invent quantum gravity and they did not manage to do it, then it merely means that quantum gravity does not exist in principleSo that no sense in persisting to invent the "perpetuum mobile" of quantum gravity, however there is a great sense to build a new theory on that new principle:
  • Background: The Higgs boson1. The theory of gravity which is sure isgeneral relativity, and it is not quantum:This is not a random fact2. If the standard model is completed bythe Higgs boson but without gravity, thenthe cause for that is: The standard modelis quantum. It cannot include gravity inprinciple just being a quantum theory3. Of course, a non-universality of quantumtheory is a big surprise and quiteincomprehensible at present, but allscientific experience of mankind is full of
  • General relativity vs. the standard model Their mechanical Interaction, Force, action Energy (mass) 7 Inertial mass Gravitational mass Gravitational onesInertial mass is the measure General relativity,of resistance vs. the action which is smoothof any force field.Gravitational mass is The weak,the measure of gravity action electromagnetic, strong And what about ones entanglement and The standard model, inertial mass? which is quantum
  • Our strategy on that background is... 1. ... to show that entanglement is another and equivalent interpretation of the mathematical formalism of any force field (the right side of the previous slide) 2. ... to identify entanglement as inertial mass (the left side) 3. ... to identify entanglement just as gravitational mass by the equality of gravitational and inertial mass 4. ... to sense gravity as another and equivalent interpretation of any quantum-mechanical movement and in last analysis, of any mechanical (i.e. space-time) movement at all
  • If we sense gravity as another and equivalent interpretation of any movement, then ... 8 Complex probability distribution = Two probability entangle- distributions ment ComplexEnergy-momentum Banach Space-time (Hilbert) Space quantum Pseudo- Riemanian trajectory force basis field It does not and cannot re-The standard model repre- present gravity because itsents any quantum force is not a quantum field atfield: strong, electromag- all: It is the smooth imagenetic, or weak field of all quantum fields
  • The Higgs boson is an answer ... and many questions:What about the Higgs field? The standard model unifies electromagnetic, weak and strong field. Is there room for the Higgs field?What about the Higgs field and gravity?What about the Higgs field and entanglement?... and too many others ... We will consider the Higgs field as a“translation” of gravity & entanglement
  • However what does “quantum field” mean? Is not this a very strange and controversial term?Quantum field means that field whose value in any space-time point is a wave function. If the corresponding operator between any two field points is self- adjoint, then:  A quantum physical quantity corresponds to it, and  All wave function and self-adjoint operators share a common Hilbert space
  • Quantum field is the only possiblefield in quantum mechanics, because: • It is the only kind of field which can satisfy Heisenberg’s uncertainty • The gradient between any two field points is the gradient of a certain physical quantity • However the notion of quantum field does not include or even maybe excludes that of entanglement: If our suspicion about the close connection between entanglement and gravity is justified, then this would explain the difficulties about “quantum gravity”
  • Then we can outline the path to gravityfrom the viewpoint of quantum mechanics: ... as an appropriate generalization of ‘quantum field’ so that to include ‘entanglement’:  If all wave functions and operators (which will not already be selfadjoint in general) of the quantum filed share rather a common Banach than Hilbert space, this is enough. That quantum field is a generalized one. However there would be some troubles with
  • Which are the troubles?The “cure” for them is to be generalized correspondingly the notion of quantity in quantum mechanics.If the operator is in Banach space (correspondingly, yet no selfadjoint operator), then its functional is a complex number in generalIts modulus is the value of the physical quantityThe expectation of two quantities is nonadditive in general
  • More about the “cure” 9
  • More and more about the “cure”
  • More and more about the “cure”Then what is gravity?It cannot be define in terms of “classical” quantum field, but only in those of generalized quantum fieldIt is always the smooth curvature or distortion of “classical” quantum fieldIt is an interaction (force, field) of second order: rather the change of quantum field in space-time than a new quantum fieldThat change of quantum field is neither quantum, nor quantizable:It cannot be a new quantum field in principle
  • Then, in a few words, what would gravitybe in terms of generalized quantum field?... a smooth space-time DoF constraint imposedon any quantum entity by any or all othersEntanglement is another (possibly equivalent) mapping of gravity from the probabilistic rather than space-time viewpoint of “eternity”The smooth space-time DoF constraint in each moment represents a deformed “inwards”3D light sphere of the 4- Minkowski-space light cone (“outwards” would mean antigravity)The well-ordered (in time) set of all such
  • The language of quantum field theory:the conception of “second quantization”What does the “second quantization” meanin terms of the “first quantization”?If the “first quantization” gives us thewave function of all the quantum system asa whole, then the “second quantization”divides it into the quantum subsystems of“particles” with wave functionsorthogonal between each other; or in otherwords, these wave functions are notentangled. Consequently, the “secondquantization” excludes as entanglement as
  • The second quantization in terms of Hilbert spaceThe second quantization divides infinitedimensional Hilbert space into also infinitedimensional subspacesA subspace can be created or annihilated: This means that a particle is created or annihilatedThe second quantization juxtaposes a certain set of Hilbert subspaces with any space-time pointOne or more particles can be created or annihilated from any point to any pointHowever though the Hilbert space is divided into subspaces from a space-time
  • A philosophical interpretation both of quantum (I) and of quantized (II) fieldQuantum vs. quantized field means for any space- time point to juxtapose the Hilbert space and a division into subspaces of itsThe gauge theories interpret that as if the Hilbert space with its division into subspaces is inserted within the corresponding space-time pointAny quantum conservation law is a symmetry or a representation into Hilbert space of the corresponding groupThe standard model describes the general and complete group including all the “strong”, “electromagnetic” and “weak”
  • A philosophical interpretation as to the closedness of the standard modelThe standard model describes the general and complete group including all the “strong”, “electromagnetic” and “weak” symmetries within any space-time pointConsequently the standard model is inside of any space-time point, and describes movement as a change of the inside structure between any two or more space- time pointsHowever gravity is outside and remains outside of the standard model: It is a
  • Need to add an interpretation ofquantum duality à la Nicolas of Cusa:After Niels Bohr quantum duality has been illustrated by the Chinese Yin and YangHowever now we need to juxtapose them in scale in Nicolas of Cusas manner:Yin becomes Yang as the smallest becoming the biggest, and vice versa:Yang becomes Yin as the biggest becoming the smallestBesides moreover, Yin and Yang continue to be as parallel as successive in the same scale
  • And now, from the philosophical 1 to the mathematical and physical ...: 0A wave A space-timefunction trajectory Hilbert space Minkowski spaceA Yin-Yang mathematical structure
  • However ...: Have already added à la Nicolas of Cusa’s interpretation to that Yin-Yang structure, so that ... 1 1The “biggest” of the space-time whole is inserted withinthe “smallest” of any space-time point The “biggest” of the Hilbert-space whole is inserted within the “smallest” of any Hilbert-space point
  • In last analysis we got a cyclic and frac-tal Yin-Yang mathematical structure ...Will check whether it satisfies our requirements: 2 1Yin and Yang are parallel to each otherYin and Yang are successive to each otherYin and Yang as the biggest are within themselves as the smallestBesides, please note: it beingcyclic need not be infinite! Needonly two entities, “Yin and Yang”,and a special structure triedto be described above
  • Will interpret that Yin-Yang structure in terms of the standard model & gravityOur question is how the gravity being“outside” space-time points as a curvingof a smooth trajectory, to which theybelong, will express itself inside, i.e.within space-time points representingHilbert space divided into subspaces indifferent waysWill try to show that:The expression of gravity “outside” looks like entanglement “inside” and vice versaBesides, the expression of entanglement “outside” looks like gravity inside of all the space-
  • Back to the philosophical interpretation of quantum (I) or quantized (II) fieldThe principle is: The global change of a space-time trajectory (or an operator in pseudo-Riemannian space) is equivalent to, or merely another representation of a mapping between two local Hilbert spaces of Banach space (entanglement)The same principle from the viewpoint of quantum mechanics and information looks like as follows:Entanglement in the “smallest” returns and comes from the “outsides” of the universe, i.e. from the “biggest”, as gravity
  • Back to the philosophical interpretation, or more and more „miracles“Turns out the yet “innocent” quantum duality generates more and more already “vicious” dualities more and more extraordinary from each to other, namely:... of the continuous (smooth) & discrete... of whole & part ... of the single one & many... of eternity & time... of the biggest & smallest... of the external & internal... and even ... of “&” and duality
  • &
  • The second quantization in terms of Banach spaceIf the Banach space is smooth, it is locally “flat”, which means that any its point separately implies a “flat” and “tangential” Hilbert space at this pointHowever the system of two or more points in Banach space do not share in general a common tangential Hilbert space, which is another formulation of entanglementOne can always determines a self-adjoint operator (i.e. a physical quantity) between any two points in Banach space (i.e. between the two corresponding tangential Hilbert spaces mapping by the operator)
  • The second quantization in terms of Banach spaceIf we can always determine a self-adjoint operator (i.e. a physical quantity) between any two points in Banach space, then follows the second quantization is invariant (or the same) from Hilbert to any smooth Banach space, and vice versa, consequently between any two smooth Banach spacesAs entanglement as gravity is only external, or both are “orthogonal” to the second quantization: It means that no any interaction or unity between both gravity and entanglement, on the one hand ...
  • The second quantization in terms of Banach spaceAs entanglement as gravity is only external, or both are “orthogonal” to the second quantization: It means that no any interaction or unity between both gravity and entanglement, on the one hand, and the three rest, on the other, since the latters are within Hilbert space while the formers are between two (tangential) Hilbert spacesHowever as entanglement as gravity can be divided into the second-quantized parts (subspaces) of the Hilbert space, which “internally” is granted for the same though they are at some generalized “angle” “externally”
  • The problem of Lorentz invariance Try to unite the following facts: 1 3 The Lorentz noninvariant are: Schrödinger’s Newton’s mechanics quantum mechanics The Lorentz invariant are:Maxwell’s theory of Dirac’selectromagnetic field quantum mechanicsEinstein’s special relativity of electromagnetic field The locally Lorentz invariant (but noninvariant globally) are: Our hypothesisEinstein’s general relativity of entanglement & gravity Relativity Quantum theory
  • ... whether gravity is not a “defect” of electromagnetic field...However mass unlike electric (or Dirac’s magnetic)charge is a universal physical quantity whichcharacterizes anything existingA perfect, “Yin-Yang” symmetry would require as thelocally “flat” to become globally “curved” as the locally“curved” to become globally “flat” as the “biggest” toreturn back as the smallest and locally “flat”For example this might mean that the universe wouldhave a charge (perhaps Dirac’s “monopole” ofmagnetic charge), but not any mass: the curvedBanach space can be seen as a space of entangledspinors
  • Electromagnetic field as a “Janus” with a global and a local “face”Such a kind of consideration like that in theprevious slide cannot be generalized to the“weak” and “strong” field:They are always local since their quanta have anonzero mass at rest unlike the quantum ofelectromagnetic field: the photonAs to the electromagnetic field, both global andlocal (the latter is within the standard model)consideration is possible
  • Electromagnetic field as a “Janus” with a global and a local “face”Conclusion: gravity (& entanglement) is onlyglobal (external), weak & strong interaction isonly local (internal), and electromagnetic field isboth local and global:It serves to mediate both between the globaland the local and between the external and theinternalConsequently, it conserves the unity of theuniverse
  • More about the photon two faces:• It being global has no mass at rest• It being local has a finite speed in spacetimeIn comparison with it:o Entanglement & gravity being only global has no quantum, thus neither mass at rest nor a finite speed in spacetimeo Weak & strong interaction being only local has quanta both with a nonzero mass at rest and with a finite speed in spacetime
  • Lorentz invariance has a local and a global face, too:In turn, this generates the two faces of photonThe local “face” of Lorentz invariance is both withinand at any spacetime point. It “within” such apoint is as the “flat” Hilbert space, and “at” it is asthe tangential, also “flat” Minkowski spaceIts global “face” is both “within” and “at” thetotality of the universe. It is “within” the totalityflattening Banach space by the axiom of choice. It is“at” the totality transforming it into a spacetimepoint
  • It is about time to gaze that Janus in details in Dirac’s brilliant solving by spinorsIn terms of philosophy, “spinor” is the total half (or“squire root”) of the totality. In terms of physics, itgeneralizes the decomposition of electromagneticfield into its electric and magnetic component. Theelectromagnetic wave looks like the following: 1 4
  • That is a quantum kind of generalization. Why on Earth?First, the decomposition into a magnetic and anelectric component is not a decomposition of twospinors because the electromagnetic field is thevector rather than tensor product of themBoth components are exactly defined in any pointtime just as position and momentum as to a classicalmechanical movement. The quantity of action is justthe same way the vector than tensor product of themConsequently, there is another way (the Dirac one)quantization to be described: as a transition orgeneralization from vector to tensor product
  • Well, what about such a way gravity to be quantized?The answer is really quite too surprising:General relativity has already quantized gravitythis way! That is general relativity has alreadybeen a quantum theory and that is the reasonnot to be able to be quantized once again just asthe quant itself cannot be quantized once again!What only need is to gaze at it and contemplateit to see how it has already sneaked to become aquantum theory unwittingly
  • Cannot be, or general relativity as a quantum theory Of course the Dirac way of keeping Lorentzinvariance onto the quantum theory is the mostobvious for general relativity: It arises to keep andgeneralize just the Lorentz invariance for any referenceframe However the notion of reference frame conservesthe smoothness of any admissable movementrequiring a definite speed toward any other referenceframe or movement Should see how the Dirac approach generalizesimplicitly and unwittingly “reference frame” fordiscrete (quantum) movements. How?
  • “Reference frame” 1 5after the Dirac approach
  • “Reference frame” after the Dirac 1 6 approach
  • The praising and celebration of sphereThe well-known and most ordinary sphere is thecrosspoint of:... quantization... Lorentz invariance... Minkowski space... Hilbert space... qubit... spinor decomposition... electromagnetic wave... wave function... making their uniting, common consideration,and mutual conceptual translation – possible!
  • More about the virtues of the sphere 1 7
  • Again about the spinor decompositionSince the sphere is what is “spinorly” decomposedinto two orthogonal great circles, the spinordecomposition is invariant to Fourier transform or tothe mutual transition of Hilbert and MinkowskispaceIn particular this implies the spinor decompsition ofwave function and even of its “probabilisticinterpretation”: Each of its two real “spinor”components can be interpreted as the probabilityboth of a discrete quantum leap to, and of a smoothreaching the corresponding value
  • A necessary elucidation of the connection between probabilistic (mathematical) and mechanical (physical) approach 1 8 No axiom of choice (the Paradise) Probabilistic (mathematical) approach Totality aka eternity aka infinity Mechanical (physical) approach Hilbert space: Both Minkowski space:from the “Paradise” from the “Earth” to the “Earth “ need to the “Paradise“ by the “stairs” choice by the “stairs” of energy (axiom) of time
  • Coherent state, statistical ensemble,and two kinds of quantum statistics• The process of measuring transforms the coherent state into a classical statistical ensemble• Consequently, it requires the axiom of choice• However yet the mathematical formalism of Hilbert space allows two materially different interpretations corresponding to the two basic kinds of quantum statistics, of quantum indistinguishability, and of quantum particles: bosons and fermions
  • The axiom of choice as the boundary between bosons and fermions The two interpretations of a coherent state mentioned above are: As a nonordered ensemble of complex (= two real ones) probability distribution after missing the axiom of choice – aka bosons As a well-ordered series either in time or in frequency (energy) equivalent to the axiom of choice – aka fermions
  • The sense of quantum movement represented in Hilbert spaceFrom classical to quantum movement: the wayof generalization:A common (namely Euclidean) space includes the two aspects of any classical movement, which are static and dynamic one and corresponding physical quantities to each of themAnalogically, a common (namely Hilbert) space includes the two aspects of any quantum movement: static (fermion) and dynamic (boson) one, and their physical quantities
  • Quantum vs. classical movementHowever the two (as static as dynamic) aspects of classical movement are included within the just static (fermion) aspect of quantum movement as the two possible “hypostases” of the same quantum stateThe static (fermion) aspect of quantum movement points at a quantum leap (the one fermion of the pair) or at the equivalent smooth trajectory between the same states (the other)These two fermions for the same quantum state can be seen as two spinors keeping Lorentz invariance
  • The spin statistics theorem about fermionsIf one swaps the places of any two quantum particles, thismeans to swap the places between “particle” and “field”,or in other words to reverse the direction “from time toenergy” into “from energy to time”, or to reverse the signof wave functionThe following set-theory explanation may be useful: Ifthere are many things, which are the same or “quantumlyindistinguishable”, there are anyway two opportunities:either to be “well-ordered” as the positive integers are(fermions), or not to be ordered at all as the elements ofa set (bosons). Though indistinguishable, the swap oftheir corresponding ordinal (serial) number isdistinguishable in the former case unlike the latter one
  • However that “positive-integers analogy” is limitedThe well-ordering of positive integers has “memory”in a sense:One can distinguish two swaps, too, rather than onlybeing one or more swaps available (as the fermionsswap)The well-ordering of fermions has no such memory.The axiom of choice and well-ordering theorem donot require such a memoryHowever if all the choices (or the choices after thewell-ordering of a given set) constitute a set, thensuch a memory is posited just by the axiom of choice
  • Positive integers vs. fermions vs. 1 bosons illustrated 9Initialstate ... ..... ... 1, 2, 3, 4, 5, ... 1, 2, 3, 4, 5, ...Swap ... ..... ... 1, 5, 3, 4, 2, ... 1, 5, 3, 4, 2, ...Aftera time ... ..... ... 1, 2, 3, 4, 5, ... 1, 5, 3, 4, 2, ... True “Weak” indistinguish- (in)distinguish- TrueNaming ability ability distinguish- Quantum indistinguishability ability Bosons Fermions Positive integers
  • Quantum vs. classical movement in terms of (quantum in)distinguishability 2Dynamic (boson) Static (fermion) 0aspect: true in- aspect: weak in- Dynamicdistinguishability distinguishability (momentum)Wave function as Wave function aspectthe characteristic as a (well-function of Statica random ordered) (position)complex quantity vector aspect Pseudo- Riemannian Hilbert space space Quantum indistinguishability Distinguishability Dynamic to static Dynamic to static aspect: much to many aspect: one to one Quantum movement Classical movement
  • Our interpretation of fermion antisymmetry vs. boson symmetryThe usual interpretation suggests that both the fermion and boson ensembles are well-ordered: However any fermion swap reverses the sign of their common wave function unlike any boson swapOur interpretation is quite different: Any ensemble of bosons is not and cannot be well-ordered in principle unlike a fermion one: The former is “much” rather than
  • The well-ordering of the unorderable: fermions vs. bosonsThe unorderable boson ensemble represents the real essence of quantum field unlike the “second quantization”. The latter replaces the former almost equivalently with a well-ordered, as if a “fermion” image of itIn turn this hides the essence of quantum movement, which is “much – many”, substituting it with a semi- classical “many – many”
  • What will “spin” be in our interpretation? In particular, a new, specifically quantum quantity,namely “spin”, is added to distinguish between thewell-ordered (fermion) and the unorderable (boson)state in a well-ordered way However this makesany quantum understanding of gravity (or so-called“quantum gravity”) impossible, because “quantum”gravity requires the spin to be an arbitrary realnumber In other words, gravity is the process intime (i.e. the time image of that process), which well-orders the unorderable The true “much – many”transition permits as a “many” (gravity in time, or“fermion”) interpretation as a “much” (entanglementout of time, or “boson”) interpretation
  • Our interpretation of fermion vs. boson wave functionIn turn it requires distinguishing between: the standard, “fermion” interpretation of wave function as a vector in Hilbert space (a square integrable function), and a new,“boson”interpretation of it as the characteristic function of a random complex quantityThe former represents the static aspect of quantummovement, the latter the dynamic one. The staticaspect of quantum movement comprises both thestatic (position) and dynamic (momentum) aspect ofclassical movement, because both are well-ordered,and they constitute a common well-ordering
  • Entangled observables in terms of “spin” distinction The standard definition of quantum quantity as “observable” allows its understanding: as a “fermion – fermion” transform,as a “boson – boson” oneas well as “fermion – boson” and “boson – fermion” one Only entanglement and gravity can create distinctions between the former two and the latter two cases. Those distinctions are recognizable only in Banach space, but vanishing in Hilbert space
  • The two parallel phases of quantum movementQuantum field (the bosons) can be thoughtof as the one phase of quantum movementparallel to the other of fermion well-ordering:The phase of quantum field requires the universe to be consider as a whole or indivisible “much” or even as a single quantThe parallel phase of well-ordering (usually represented as some space, e.g. space-time) requires the universe to yield the well-known appearance of immense and unbounded space, cosmos, i.e. of an indefinitely divisible “many” or
  • Why be “quantum gravity” a problem of philosophy rather than of physics? 2 1The Chinese "Taiji 太極 (literally "great pole"), the "Supreme Ultimate" can comprise both phases of quantum movement. Then entanglement & gravity can be seen as “Wuji 無極 "Without Ultimate"In other words, gravity can be seen as quantum gravity only from the "Great Pole"This shows why "quantum gravity" is rather a problem of philosophy, than and only then of physics
  • Hilbert vs. pseudo-Riemannian space: a preliminary comparisonAs classical as quantum movement need a common space uniting the dynamic and static aspect: Hilbert space does it for quantum movement, and pseudo-Riemannian for classical movementQuantum gravity should describe uniformly as quantum as classical movement. This requires a forthcoming comparison of Hilbert and pseudo-Riemannian space as well as one, already started, of quantum and classical movement
  • Hilbert vs. pseudo-Riemannian space as actual vs. potential infinityTwo oppositions are enough to representthat comparison from the viewpoint ofphilosophy:Hilbert space is ‘flat’, and pseudo- Riemannian space is “curved”Any point in Hilbert space represents a complete process, i.e. an actual infinity, and any trajectory in pseudo- Riemannian space a process in time, i.e. in development, or in other words, a
  • Hilbert vs. pseudo-Riemannian space: 2 completing the puzzle 2Oppo-sition Process in time Actual infinity Pseudo- Riemannian Banach space Curve space Entanglement Gravity, General Quantum information relativity Minkowski Hilbert space space Electromagnetic, weak Flat Electromag- and strong interaction netism Quantum mechanics Special relativity The standard model
  • Our thesis in terms of that table 2 3 Pseudo- Riemannian space Banach spaceCurve Gravity , Entanglement Quantum information General relativityEntanglement is gravity as a complete process Gravity is entanglement as a process in time
  • A fundamental prejudice needs elucidation not to bar:The complete wholeness of any process is „more“ than the same process in time, in developmentActual infinity is “more” than potential infinityThe power of continuum is “more” than the power of integersThe objects of gravity are bigger than the objects of quantum mechanicsThe bodies of our everyday world are much “bigger” than the “particles” of the quantum world, and much smaller than the universe
  • Why is that prejudice an obstacle?According to the first three statements entanglement should be intuitively “more” than gravityHowever according to the second two statements gravity should be intuitively much “smaller” than entanglementConsequently a contradiction arises according to ourintuition: Gravity should be as “less” in the firstrelation as much “bigger” in the second relation An obvious, but inappropriate way out of it is to emphasis the difference between the relations
  • Why is such a way out inappropriate?The first relation links the mathematical models ofentanglement and gravity, and the second onedoes the phenomena of gravity and entanglement To be adequate both relations to each other, one must double both by an image of the other relation into the domain of the first one. However one can show that the “no hidden parameters” theorems forbid thatFor that our way out of the contradiction must notbe such a one
  • Cycling is about to be our 2 way out of the contradiction 4Should merely glue down both ends to each other:the biggest as the most to the least as the smallest.However there is a trick: There not be anymore thetwo sides conformably of the “big or small” as well asof the “more or less” but only a single one like this:
  • Once again the pathway is ...: 2 5from the two sides of a noncyclic stripto the two cyclic sides of a cylinderto a single and cyclic side of a Möbius strto an inseparable whole of a merely “much”to the last one as the “second”side of tMöbius band cyclically passing into the othe
  • Holism of the East vs. linear time of the West 2 6The edge of gluing the Möbius strip is avery special kind: It is everywhere andnowhere. We can think of it in terms ofthe East, together: as Taiji 太極 (literally "great pole"), or "Supreme Ultimate“ as Wuji 無極 (literally "without ridgepole") or "ultimateless; boundless; infinite“As a rule, the West thought torments andbars quantum mechanics: It feels good inthe Chinese Yin-Yang holism. (In the
  • The “Great Pole” of cycling in terms of the axiom of choice or movementThe “Great Pole” as if “simultaneously” both (1) crawls in a roundabout way along the cycle as Taiji, and (2) comprises all the points or possible trajectories in a single and inseparable whole as WujiBy the way, quantum mechanics itself is like a Great Pole between the West and the East: It must describe the holism of the East in the linear terms of the West,
  • Being people of the West, we shouldrealize the linearity of all western science! Physics incl. quantum mechanics is linear as all the science, too For example we think of movement as a universal feature of all, because of which there is need whole to be described as movement or as time. In terms of the Chinese thought, it would sound as Wiji in “terms” of Taiji, or Yin in “terms” of Yang Fortunately, the very well developed mathematics of the West includes enough bridges to think of whole linearly: The most essential and important link among them is the axiom of choice
  • The axiom of choice self-referentiallyThe choice of all the choices is to choose the choiceitself, i.e. the axiom of choice itself , or in philosophi-cal terms to choose between the West and the EastHowever it is a choice already made for all of us andinstead of all of us, we being here (in the West) andnow (in the age of the West). Consequently we doomto think whole as movement and time, i.e. linearlyThe mathematical notions and conceptions can aid usin uniting whole and linearity (interpreted in physicsand philosophy as movement and time), thoughIn particular, just this feature of mathematicsdetermines its leading role in contemporary physics,especially quantum mechanics
  • Boson – fermion distinction in terms both of whole and movement
  • A “Möbius” illustration of how asmooth trajectory can reverse the spin 2 7 a the same fermion “inside” “outside” the universe
  • Exactly the half of the universe between two electrons of a helium 2 atom 8Here is a helium atom. Exactly the halfof the universe is inserted between itstwo electrons which differ from eachother only with reversed spin:The West thinks of the universe asthe extremely immense, and of theelectrons and atoms as the extre-mely tiny. However as quantummechanics as Chinese thoughtshows that they pass into each othereverywhere and always The universe
  • Taiji 太極 is the Chinese transitionbetween the tiniest and the most immense 2 9The Wests single pathwayalong or through Taijiis mathematics, though A fortunate exception is Nicolas of Kues
  • How on Earth is it possible?Mathematics offers the universe to be considered in two equivalent Yin – Yang aspects corresponding relatively to quantum field (bosons) and quantum “things” (fermions): an unorderable at all set for the former, and a well-orderable space for the latterIt is just the axiom of choice (more exactly, Scolem’s “paradox”) that makes them equivalent or relative. Hilbert space can unite both aspects as two different (and of course, equivalent by means of it) interpretations of it: (1) as the characteristic function of a complex (or two real) quantity(es) (quantum field, bosons), and (2) as a vector (or a square integrable
  • Taiji 太極 in the language of mathematics30 He One single boson!!!! Wave function interpreted Wave function as a characteristic function as a vectorThe common and universal Hilbert (Banach) space
  • Taiji 太極 in the language of mathematics3The axiom of choice 1Scolem’s“paradox” He One single boson!!!! Wave function interpreted Wave function as a characteristic function as a vectorThe common and universal Hilbert (Banach) space
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  • Wuji 無極 as the Kochen-Specker theorem Taiji 太極The axiom of choiceScolem’s Quantum computer“paradox” The universe of (or as) sundry Tu r i n g a l g o r i t h m sOne single A most and most qubit!!!! ordinary bit one single bit Its point interpreted Its point as a characteristic function as a vector The common and universal Hilbert (Banach) space
  • The mapping between numbers and a sundry 3 3
  • Quantum mechanics solves the general problem under those assumptionsThe general problem is the quantitative description of the universe: too complicated! Well-orderable Wave numbers function Wave as a field function as a probability distribution 3 All the universe 4 as a sundryThe general problem The simplifying solvingin terms of Taiji and Wuji of quantum mechanics
  • The solving of quantum mechanics in terms of gauge theories 5 3 The leading notion is “fiber bundle”:The Möbius strip is an as good as simple enough example of fiber bundle:Its as topologic as metric properties are quite different locally vs. globally Möbius strip Metrically Topologically Locally flat two-side Globally curved one-side
  • Möbius strip as a fiber 3 6 bundle “circle” for bundle “radius” for fiber, F the same “radius” from the “other side” for base
  • The definition of “fiber bundle” by the example of a Möbuis strip7 3The fiber bundle is determined and defined preciselyby the topological transform from it to base space orvice versa: i.e. correspondingly as unfolding from aflat sheet (base space) to the Möbius strip (fiberstrip), or folding vice versa, in our example: By its unfolding Or By its folding
  • More precise definition of fiber bundleyet using the "Möbius" illustration 3 8
  • The definition without any illustration Arbitrary neighborhoods of arbitrary topological spaces for the “radiuses” of the illustration 3However the topological spaces 9are usual Hilbert spaces or subspacesin the physical interpretation offiber bundle in the gauge theoriesIn other words, Hilbert spaces substitute for the “radiuses” of Möbius strip, in gauge theories
  • The leading idea of gauge theory 4 Fiber bundle Cartesian product 0 0 The Standard 0 Modelan uncalibrated indicator the indicator calibrated
  • The universality of calibrationThe calibration should be identical for any indication,and this is true as to weak, electromagnetic, andstrong interaction, but not as to gravityFor that the Standard Model comprises the formerthree but not the latterA necessary condition is quantization, whichguarantees the two vectors A and B to existOur conjecture will be: It is quantization that gravitycannot satisfy and in principle, there can be no gaugetheory of gravity, as a corollary
  • More about Dirac’s spinorsCan think of them both ways:- As two electromagnetic waves- As the complex (=quantum) generalization of electromagnetic waveThe latter is going to show us the original Dirac theoryHowever the former is much more instructive anduseful for our objectives:It is going to show us the connection and unity ofgravity and electromagnetism, and hence then thelinks of gravity and quantum theory by the mediationof electromagnetism
  • Why is “quantum gravity” a philosophical problem?• Not for Alan Socal’s "Transgressing the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity“   • But for the need of “transgressing the boundaries” of our gestalt: the gestalt of the contemporary physical “picture of the world”!Thus, our answer when an unsolved scientificproblem becomes a philosophical one is: When itcannot be solved in the gestalt of the dominating atpresent picture of the world despite all outrageousefforts 
  • Our suggestion to change the gestalt: the physical picture of the worldIts essence is: a new invariance of discrete and continual (smooth) mechanical movements and their corresponding morphisms in mathematicsThis means a generalization of Einstein’s (general) principle of relativity (1918): “Relativitätsprinzip: Die Naturgesetze sind nur Aussagen über zeiträumliche Koinzidenzen; sie finden deshalb ihren einzig natürlichen Ausdruck in allgemein kovarianten Gleichungen.“
  • An equivalent reformulation of Einstein’s principle of relativity:All physical laws must be invariant to any smoothmovement (space-time transformation)Comment: However all quantum movements arenot smooth in space-time at all: Even they are notcontinuous in itBesides: the relativity movements are not “flat” inspace-time in general while all quantummovements are “flat” in Hilbert spaceDefinition: A movement is flat when it isrepresented by a linear operator in the space ofmovement
  • Our suggestion the general relativity principle to be generalized:All physical laws must be invariant to any movement(space-time transformation)The difference between Einstein’s formulation andour generalization is that the word “smooth” isexcluded so the movement can already be quantumHowever such a kind of invariance (in fact, aninvariance as with the discrete as with thecontinuous) meets a huge obstacle in set theory:consequently, in the true fundament of mathematicsrequiring to change gestalt
  • The huge obstacle in set theory:The invariance of the discrete and continuous cannotbe any isometry in principle since the standardmeasure of any discrete set is zero (while the measureof a continuum can be as zero as nonzero)Moreover, the obstacle is deeper situated in settheory since the power of any discrete set is less thanthat of any continuum even if its measure is zeroFortunately Skolem’s paradox offer’s a solution,however, “transgressing boundaries” of the “gestalt”:Unfortunately Skolem’s paradox is based on, andnecessarily requires the axiom of choice allegedsometimes as “unacceptable”
  • The inevitability of the axiom of choice in quantum mechanicsThe axiom of choice in quantum mechanics is well-known as its “randomness” in principle or as the“no-go” theorems about the “hidden variables”(Neumann 1932; Kochen, Specker 1967):Given the mathematical formalism of quantummechanics (based on Hilbert space), quantumrandomness is not equivalent to any statisticalensemble: Its members or their quantities would bethe alleged “hidden variables”
  • The Kochen − Specker theorem is the most general “no hidden variables” theorem:Its essence: wave-particle duality in quantum me-chanics is equivalent with “no hidden variables” in itThe most important corollary facts of its:A qubit is not equivalent to a bit or to any finite sequence of bitsBell’s inequalitiesThe inseparability of apparatus and quantum entityThe “contextuality” of quantum mechanicsQuantum wholeness is not equivalent to the set or sum of its parts; quantum logic is not a classical one
  • The “quantum wholeness” of the axiom ofchoice and the “no hiddenness” theoremsPreliminary notes: If there is an algorithm,which leads to the choice, the axiom needn’t:Consequently, the axiom core is the opportunityof choice without any algorithm − beguaranteedGiven the choice without any algorithm is arandom choice in definition, the axiom of choicepostulates that a random choice can always bemade even if a rational choice by means of anyalgorithm cannot
  • The “quantum wholeness” of the axiom ofchoice and the “no hiddenness” theorems:The “no hidden variables” theorems state that anychoice of a definite value in measuring is random:Thus, they postulate the axiom of choice in quantummechanicsHow, however, can we explain intuitively therandomness of choice in quantum mechanics?The apparatus “chooses” randomly a value among allprobable values by the mechanism of decoherence,e.g. a “time” interpretation of coherent state anddecoherence is possible:
  • The “time” interpretation ofcoherent state and decoherence: 4 1
  • The “time” interpretation of the difference between a coherent state and a statistical ensemble A discrete (quantum) leap of any function in apoint (an argument value) generates a coherent state.For the so-called time interpretation we may acceptthe argument be time A continuous function (e.g.of time) generates a statistical ensemble (e.g. of themeasured values in different time points) Thetransformation between a discrete leap and acontinuous function implies the correspondingtransformation between a coherent state and astatistical ensemble
  • The chain of sequences from Skolem’s paradox to our generalization of Einstein’s relativity principle : 4 2
  • A few comments: the first one:wave-particle duality as invariance
  • A second comment: wave-particle inva-riance embedded in complex Hilbert spaceTwo important features of complex Hilbert spaceallow of such embedding in it: (1) It and its dual spaceare anti-isomorphic (Riesz representation theorem);So (1) allows the following: The four pairs can beidentified: (1.1) the two corresponding points of thetwo dual space; (1.2-3) the Fourier transformationand its reverse one of the probability distribution of arandom quantum quantity and its reciprocal one(these are two pairs); (1.4) any quantum quantity andits conjugate one. Besides, (1.5) any point in Hilbertspace can be interpreted as a function as a vector
  • A necessary gloss about the probabilitydistribution of a random quantum quantity:The probability distribution of a “classical” randomquantity is a real function of a real argument. Ifhowever any point in Hilbert space is interpreted as aprobability distribution of a random quantumquantity, we need a complement gloss about themeaning both of a complex probability and of acomplex value as to a physical quantity. Ourpostulate: any quantum quantity and its probabilitydistribution is composed by two “classical” ones andtheir probability distributions sharing a commonphysical dimension: one for the discrete and anotherfor continuous aspect
  • A short comment on the postulate:Consequently when we measure a quantum quantity, we lose informationAny quantum probability distribution is reduced to a statistical ensembleThe principle of complementary forbids the question about the lost informationThe most natural hypothesis is that as the two components as their corresponding probability distributions coincideThis conjecture founded by the axiom of choice in quantum mechanics adds wave-particle invariance to wave-particle duality
  • More about the embedding of wave-particle invariance in complex Hilbert space
  • A set-theory generalization of wave- 4 particle invariance 3
  • Another useful, now physicalinterpretation of the invariance (duality) Given the wave-particle invariance (duality) as the two (possibly coinciding) points of the dual anti-isomorphic Hilbert spaces, it admits one more inter- pretation: 4 4  as a (“covariant”) set of harmonics as a (“contravariant”) set of points, the two sets being anti-isomorphic (anti- isometric measurable)
  • Another useful, now physical 4 4interpretation of the invariance (duality)
  • Is there any mathematical model, which can coincide with the modeled reality?4 5
  • A philosophical interlude about the logicalequivalence of two physical interpretations 4 6
  • Let the former (any quantity) be physicallyinterpreted as the argument in the latter: 4 7
  • Besides, let the same argument be physically interpreted as time: 4 8
  • A gloss on physical dimensions: 4 9
  • A gloss on physical dimensions: 4 9
  • 5Parseval’s theorem 0
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  • 5 Parseval’s theorem simply 2illustrated as a “cross rule”
  • Obviously Parseval’s theorem is dueto the “flatness” of Hilbert space. To get it “curved” into Banach one? 5 3
  • Fourier transform by 3D Cartesian product 5 4
  • Riesz representation theorem by 3D 5 Cartesian product 5
  • 5 6The zest iswhat aboutBanach space! A functional
  • That is: 5The “surface” of Banach space 7
  • However the case in general is: 5 8The “surface” of Banach space
  • The different perspectives on Hilbert 5 and Minkowski space 9 Continuous perspective: Discrete perspective: (countable) (countable) expanding expanding in time in frequencyMinkowski space Hilbert space
  • The different perspectives on an impulse − a trajectory: Hilbert − Minkowski space a trajectory an impulse 6 0 a world a quantum line leap Continuous perspective: Discrete perspective: (countable) (countable) expanding expanding in time in frequencyMinkowski space Hilbert space
  • Hilbert − Minkowski space: 6 wave-particle duality 1 a trajectory an impulse a world a quantum line leapa particle moving a wave functioncontinuously simultaneous in allin that trajectory the spacewell-ordered by time Minkowski space Hilbert space
  • Hilbert − Minkowski space: a perfectsymmetry of positions and probabilities a trajectory an impulse 6 2a particle moving a wave functioncontinuously simultaneous in allin that trajectory the space but well-well-ordered by time ordered in frequencyHowever the particle However the wavetrajectory is a singular function is a singularmix of frequencies mix of positions
  • The quadrilateral: Hilbert – Banach –Minkowski – pseudo-Riemannian space 6 3 Banach space Pseudo-Riemannian space Hilbert space Minkowski space
  • The known sides of the quadrilateral: 6 as Hilbert – Banach space 4as Minkowski – pseudo-Riemannian space varying scalar product depending on the space points Banach space as a curved Hilbert space:The change of the scalar product in each point can be interpreted as a function of the curvature in that point
  • The known sides of the quadrilateral:as Minkowski – pseudo-Riemannian space as Hilbert – Banach space 6 5 varying scalar product depending on the space pointsPseudo-Riemannian space as curved Minkowskispace: The change of the scalar product in each point can be interpreted as a function of the curvature in that point
  • The close analogy of the two transformsas different views on the same transform: 6 6We can use the two perspectives mentioned above, on Hilbert − Minkowski space: frequency − time:
  • The two transforms as the same transform 6 7
  • The curving or flattening in both cases: one … … … 6 … … 8 frequency time Shifting& n n’ rotating … of each corresponding sphere 2’ in the 2 dual space 1’ 1 dual space space
  • The curving or flattening in the first case: two comments 6 9 1) It is the first case what one knows till now: time frequencyThe “curved” pseudo- Riemannian space The “flat” Hilbert space of general relativity of quantum mechanics 2) A philosophical reflection on the quantum mapping of infinity: The actual infinity of a time series is mapped as the actual infinity of a frequencyseries and by means of the latter as an impulse, i.e. asa quantum leap: Consequently, quantum mechanics is an empirical knowledge of actual infinity !
  • The curving or flattening in both cases: two … … … … … 7 0 time frequency Shifting& n n’ rotating … of each corresponding sphere 2’ in the 2 dual space 1’ 1 dual space space
  • The curving or flattening in the second case: two comments 7 1 1) It is the second case what one would emphases: frequency time The “curved” Banach space The “flat” Minkowski of entanglement space of special relativity 2) A methodological reflection on the equavalence of both cases: “No need of quantum gravity!”, or: Entan- glement represents quantum gravity integrally. Of course, does one wish, both spaces could becurved, and a partial degree of entanglement might be combi-ned with a corresponding partial degree of
  • The unknown sides of the quadrilateral: as Hilbert – Minkowski space 7 2 as Banach – pseudo-Riemannian space
  • The sides of the quadrilateral one by one:1| Minkowski – pseudo-Riemannian spacemomentum position⍟… in the gravitational time frequency (energy) A body …⍟ … … …field … … 7 Shifting& 3 rotating n’ of each n corresponding … sphere in the 2’ dual 2 space 1’ 1 dual space space
  • The quadrilateral one by one: Minkowski – pseudo-Riemannian space: conclusion The “curved” 7 4 Minkowski space The “flat” as pseudo-Riemannian Minkowski space one represents includes all the universe the space-time as a gravitational field trajectory of the whole, of the body or of all the rest to the body dual space space
  • The sides of the quadrilateral one by one: 2| Hilbert – Banach spaceprobability probability⍟… in entanglement position position The wave function of … … … Shifting& anything…⍟ n’… … rotating of each corresponding n … sphere 7 5 in the 2’ dual 2 space 1’ 1 dual space space
  • The quadrilateral one by one: Hilbert – Banach space: conclusionThe “curved” 7 6Hilbert space The “flat”as Banach one Hilbert spacerepresents includesall the universe the wave functionas an entanglement of the quantumof the quantum anythinganythingwith all the rest dual space space
  • The quadrilateral “two by two”: Hilbert – Banach, and Minkowski – pseudo- Riemannian space: conclusion The close analogy between those two sides ofthe “quadrilateral” hints their common essence as two different ways for expressing the same: Banach (Hilbert) space as functions globally, and pseudo-Riemannian (Minkowski) space as point trajectories locally
  • A few important notes: on the conclusionThe first earnest note: The time (instead of “frequency”) interpretation of pseudo-Riemanian(Minkowski) space is due only to tradition or from force of habit: In fact, both Banach (Hilbert) and pseudo-Riemannian (Minkowski) space are invariant to time – frequency, or continuous – discrete interpretation, or wave – particle duality as mere mathematical formalisms Banach (Hilbert) space represents the same as functions globally, and pseudo-Riemannian (Minkowski) space as point trajectories locally
  • A few important notes: on the conclusion The second earnest note: Both pseudo-Riemanian (Minkowski) and Banach (Hilbert) space are well-ordered in the parameter of either time or frequency in (geodesic) line. However what is up if the well-ordering is abandoned in all cases eo ipso abandoning the axiom of choice? Banach (Hilbert) space represents the same as functions globally, and pseudo-Riemannian (Minkowski) space as point trajectories locally
  • A few important notes: on the conclusion The answer is the third earnest note: Abandoning the axiom of choice in all the caseseo ipso well-ordering, the whole becomes a coherent mix of all its possible states or parts (well-ordered in time or in frequency before that ). Any possible state or part can be featured by its probability to happen. We can illustrate that probability as the obtained by projection number or measure of the corresponding state or part Banach (Hilbert) space represents the same as functions globally, and pseudo-Riemannian (Minkowski) space as point trajectories locally
  • The fourth earnest note on the conclusionFunction space A point in A point in Banach space Hilbert space t f A trajec- A line in A tra-A line in“Line” space tory in pseudo- jec- Minkow- a force Riemann. tory ski space field space p 7 p 7“Projection in A defected x x probabilityprobabilities” distribution A normed being due to probability space distribution entanglement (the force field) The “curved” case The “flat” case Banach (Hilbert) space represents the same as functions globally, and pseudo-Riemannian (Minkowski) space as point trajectories locally
  • A “homily” about negative probability The defected probability distribution being due to entanglement (i.e. to an interrelation) can be also interpreted asan alleged substance featured by negative probability. However that requires for quantum wholeness to be transformed into an “equivalent” statistical ensemble.If doing so, we can consider entanglementas a new kind of substance: the substance of quantum information
  • The sides of the quadrilateral one by one: 3) Hilbert – Minkowski spaceBoth spaces are “flat”, well-ordered, expressing thesame, but:A real difference: Hilbert space is a function space,while Minkowski space is an ordinary, “point” spaceAn alleged difference:Besides, Hilbert space is interpreted (but incorrectly)only as a “frequent” space representing discreteimpulses, while Minkowski space (but alsoincorrectly) only as a “time” space representingsmooth trajectories. In fact, both spaces are equallyinterpretable as a “time”, as a “frequent” spaceconnected by a Fourier or Fourier-like transform
  • The sides of the quadrilateral one by one: 3| Hilbert – Minkowski spaceA very important corollary from the real difference, videlicet: Hilbert space is a function space, while Minkowski space is an ordinary, “point” space: So that a trajectory in Minkowski space represents a potentially infinite, current process in time or “in frequency”, while a point in Hilbert space represents the same process as complete or as an actual infinityThe two views mentioned before on a single “Hilbert- Minkowski” space represent it correspondingly as a potential infinity and as an actual infinity
  • The sides of the quadrilateral one by one: 4|Banach – pseudo-Riemannian space Both spaces are “curved”, and all the rest said about Hilbert – Minkowski space is valid to their pair, too: Both spaces express the same in different perspectives: Both spaces can be interpreted as a time as a frequency space, but the Minkowski space represents a process in potential infinity as a world line in an ordinary, “point” space, while Hilbert space an actual infinity as a complete result, namely as a point in a function space
  • The quadrilateral, one by one: 4|Banach –pseudo-Riemannian space: the curvature represented in each case by the two dual spacesB “A” varying “angle&distnance ”A 7NA 8C orthogonalityH … … …PRSIEE … … nUMDA nON. N … P p R p O B p p A B I L x x x I T Y DUAL SPACE SPACE
  • The dual-spaces representation of 7 mechanical movement in a force field 9 The juxtaposition of Lagrange and Hamilton approach to mechanical movement … here as three discrete corresponding 4-points … and heredual p,E space x,t space as a smooth trajectory … … … force field … … n n …Hamilton (dual spaces) approach Lagrange (derivatives) approach
  • The juxtaposition of Lagrange and Hamilton approach to mechanical movement: conclusionA. Both approaches are equivalent in classicalmechanics – a well-known factB. If we accept the equivalence of gravity (Lagrange)& entanglement (Hamilton), both approaches will beimmediately equivalent in quantum mechanics, tooC. The universal equivalence of both approachesorigins from discrete-continuous invariance, or fromwave-particle dualism, or from Skolem’s “paradox”,or in last analysis – from the axiom of choice
  • A little philosophical digression about gravitational field and force field
  • A new conjecture: entanglement fieldIf any ordinary field acts to the values of certainphysical quantities, the entanglement field acts tothe probabilities of those values: So it can becalled so: probability field The source of probability or entanglement field can be any discrete, jump-like change of the same quantity in any point of space-time. It can act upon any other discrete change of that quantity anywhere: However how?
  • How can entanglement field act?Its origin is rather mathematical and universal forthat: Any discrete, or jump-like change is equivalentto a probability field in a sense: Since a definitivespeed of change is impossible to determine, it issubstituted by all the values with certainprobabilities or in other words, by the probabilityfield of all the values. If there are two or morediscrete changes, they can share some values withdifferent probabilities in each probability fieldgenerated by a quantum leap. In the last case, acommon and equal probability calculable appearsinstead of the two or more different ones
  • How can entanglement field act?Next: If and only if the probability is zero foreach other field where the probability of one ofthem is nonzero, then the probability fields donot interact, they are “orthogonal” and noentanglement If there is entanglement, it “happens” mathematically by means of the pair of dual spaces: How? Firstly, we should interpret the connection between the two dual spaces
  • How can entanglement field act? 8 0 Heisenberg’s uncertainty Fourier transforms
  • How can entanglement field act? 8 1 Heisenberg’s uncertainty Fourier transforms
  • How can entanglement field act? 8 2 The same complex The same complexprobability field of all as probability field of all as momenta as positions positions as momenta
  • A digression about the “arrow of time”The “arrow of time” is a fundamental, known to eve-ryone, but partly explainable fact about time unlikeall other physical quantities, which are isotropicOur simple and obvious explanation is the following:Time is the well-ordering of any other physical quan-tity. The “arrow of time” and the “well-ordering” aremerely full synonyms expressing the same Consequently, the axiom of choice, which isequivalent with well-ordering, means that any setcan be represented as a physical quantity in time oras a trajectory in a special space corresponding tothat set: Or in other words, the set can always betransformed into another set. The theory ofcategories states generalizing that even if the “set” isnot a set, but a “category”, it can be transformed
  • Three restrictions of choice for a trajectory pointThe dependence of momentum on position: The value of momentum in a moment is proportional the position derivative in the same moment, i.e. to the value of speedThe “smooth choice” of both momentum and position: The choice of the trajectory following point is restricted to an infinitely small neighborhood of the point, so that the trajectory and its derivative are smooth in any pointThe exact correspondence of the measure of the same value set with the value probability
  • The same restrictions of choice for the same trajectory point as a field pointAny trajectory point undergoes a force beingdue to the field in the same space-time pointThat force represents merely a second and differenttrajectory but only in the dual space of energy andmomentum. Such a second energy-momentumtrajectory is determined to any possible space-timetrajectoryThere is a single difference: The first restriction isabsent: Position and momentum are independent ofeach other for the second trajectory: However theother two restrictions are valid!
  • An interpretation of both trajectories in terms of whole and partThe first trajectory represents the case without any force field, including gravitational one. The system is closed as if it was alone in the universe and its mechanical energy is only kinetic. That is the case where a part is considered as the whole.The second trajectory represents the universe, or the whole including the first system as a part (subsystem). It is closed, too, really alone, and which is the source both of the force field and of the potential mechanical energy
  • The interaction of a system with a force field in terms of whole and partThe energy-momentum of the system interacts with the energy-momentum of the field in the same space-time point as adding 4-vectors in Minkowski spaceWe can interpret that as forming a new whole of two previous wholes. The whole of the universe includes the whole of the system in consideration. We have also discussed such an operation as “set-theory curving” as inverse to a “flattening” choice according to the axiom of choice
  • A view on a system in a force field in terms of frequency (energy) instead of timeThe energy-momentum representation is that viewpoint. Any force field, which comprises a system, represents a mismatch of the discrete and continuous aspect of the systemBy tradition that mismatch is embedded in energy- momentum or in other words, in terms of frequency and discrete impulseIn fact, it represents the impact of the whole or of the environment onto the system, and it is equivalently representable as in terms of frequency and discrete impulse as in those of time and smooth trajectory
  • Einsteins general relativityrevolution represented in the same terms Since any force field including gravitational one can be equivalently represented as a second but space-time for and instead of energy- momentum trajectory, that second trajectory can be considered as the basis of a “curved”, namely pseudo-Riemannian space, in which the first trajectory of any partial subsystem happens. The space comprises trajectory as a space- time expression of the way, in which any whole comprises any part of its
  • The deep meaning is not in the geometrization of physics, i.e. not in the representation of a force field as a “curved” space-time, namely pseudo- Riemannian spaceThe real meaning is in the equivalence of the two representation of any force field: as a second energy-momentum space (or trajectory) as a second space-time (or trajectory)However, let us emphasis it, both representations are not only continuous but smooth (in fact, in tradition)
  • Following Einstein’s lesson beyond him:… we introduce a second representation, namelythat “from eternity” rather for a newequivalence (or “relativity”) than only forit itselfThat “relativity” or equivalence is between the discrete and the continuous (smooth) And the second representation, which is from the “viewpoint of eternity” merely removes the well-ordering in space-time (energy- momentum) eo ipso removing the axiom of choice, and eo ipso the choice itselfThat second representation is … quantum
  • A view on a system in a force field in terms of eternity instead of time 8 3 … OK, but we have already introduc ed it a little above
  • Note, please, an amazing property of that “relativity” … self-referentialityParticularly, duality offers a new model of double referentiality as self- referentiality: Both the dual (e.g. spaces) can be considered as a generalization of each other if each of the two dual (e.g. spaces) is equivalent to the ensemble of the two ones: Besides that ensemble is as the generalization as the equivalent of both of themThe “flat” Hilbert space of quantum mechanics with its principle of
  • A few remarks on that amazing kind of self-referentialityTotality, infinity, and wholeness should possess the same property: Consequently, the ensemble of two dual (e.g.) spaces would be an appropriate model of any of them, and quantum mechanics using the same model can be considered as an empirical (note!) science of all of them!There are at least a few important interpretations of the same idea in physics,
  • A few remarks on that amazing kind of self-referentiality... besides, the ensemble of functors and categories in category theory is dually complete; the ensemble of proper (without the axiom of choice) and improper (with the axiom of choice) interpretation in set theory, too;Truly said, we refer to that self- referentiality (again) for the pair of the eternity ("no axiom of choice") and time (by the axiom of choice) view to mechanical
  • The most essential remark on the dualself-referentiality of eternity and timeOur problem is the dual self-referentiality of: Our solving is going to be:Eternity and time are merely two different interpretations of the same mathematical structure:
  • Be eternity and time two different interpretations, then …… frequency (energy), time and eternity are three equivalent interpretations;… eternity interprets Hilbert (Banach) space as a dual (double) probability distribution and its Fourier(-like) transform;... time interprets Hilbert (Banach) space as Minkowski (pseudo-Riemannian) space and movement as a smooth trajectory;
  • Be eternity and time two different interpretations, then …… we should admit the equivalent curvature (i.e. the nonorthogonality) as between eternity and time as between time and frequency (energy) as between frequency (energy) and eternity, and as between all of them;… as entanglement (from the particular view of eternity) as gravity (from the particular view of time and energy) as any equivalent combination of them expresses the same;… we should admit even an interaction between entanglement and gravity
  • Be eternity and time two different interpretations, then …… that which is the same but expressed differently by gravity (in terms of time and energy) and entanglement (in terms of two probability distributions) represents the same interaction between a system and the universe (environment), in which it is included, from the two viewpoints of time (and energy) and eternity… whatever about the eventual interaction of gravity and
  • Be eternity and time two different interpretations, then … 8 4Wave function as eternitya Fourier transform for entanglementof two (conjugate)probability distributions Complex Hilbert The Same!!! (Banach) Space for gravita- tional fieldWell-ordering in timeand frequency (energy)by the axiom of choice time& frequency
  • Consequently, our conclusion is ... !!! Entanglement is a view on a system in a force field in terms of eternity instead of time (or frequency, energy)
  • A set-theory interpretation of the linksbetween functional and physical space 8 5
  • The set-theory interpretation 8 5 being continued
  • Links between function space 8 7 and physical space
  • (1) Quantum mechanics as an interpretation of Hilbert space can be considered as a physical theory of mathematical infinity(2) Reality by means of the physical reality based on quantum mechanics can be interpreted purely mathematically as a class of infinities admitting an internal proof of its completeness; in other words, as that model, which can be identified with reality