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Vasil Penchev. Gravity as entanglement, and entanglement as gravity

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  • 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
  • Transcript

    • 1. Gravity as Entanglement Entanglement as gravity 1
    • 2. Vasil Penchev, DSc, Assoc. Prof, 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 2
    • 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 3
    • 4. 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
    • 5. 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 of matter and vacuum in the universe tend to increase with time, like entropy …
    • 6. Background Jae-Weon Lee, Hyeong-Chan Kim, Jungjai Lee’s “Gravity as Quantum Entanglement Force” : …, and there is an effective force called quantum entanglement force associated with this tendency. It is also suggested that gravity and dark energy are types of the quantum entanglement force …” Or: Mark Van Raamsdonk’s “Comments on quantum gravity and entanglement”
    • 7. Background: For the gauge/gravity duality “The gauge/gravity duality is an equality between two theories: On one side we have a quantum field theory in d spacetime dimensions. On the other side we have a gravity theory on a d+1 dimensional spacetime that has an asymptotic boundary which is d dimensional” • Dr. Juan Maldacena is the recipient of the prestigious Fundamental Physics Prize ($3M) 4
    • 8. Background: Poincaré conjecture The third (of 7 and only solved) Millennium Prize Problem proved by Gregory Perelman ($1M refused): Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere The corollary important for us is: 3D space is homeomorhic to a cyclic 3+1 topological structure like the 3-sphere: e.g. the cyclically connected Minkowski space 5
    • 9. The gauge/gravity duality & Poincaré conjecture 3D (gauge) /3D+1 (gravity) are dual in a sense 3D & a 3D+1 cyclic structure are homeomorphic “What about that duality if 3D+1 (gravity) is cyclic in a sense?” – will be one of our questions
    • 10. Background: The Higgs boson  It completes the standard model 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: 6
    • 11. 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:
    • 12. Background: The Higgs boson 1. The theory of gravity which is sure is general relativity, and it is not quantum: This is not a random fact 2. If the standard model is completed by the Higgs boson but without gravity, then the cause for that is: The standard model is quantum. It cannot include gravity in principle just being a quantum theory 3. Of course, a non-universality of quantum theory is a big surprise and quite incomprehensible at present, but all scientific experience of mankind is full of surprises
    • 13. General relativity vs. the standard model Interaction, Force, Energy (mass) Their mechanical action Gravitational mass Gravitational ones Inertial mass The weak, electromagnetic, strong ones The standard model, which is quantum General relativity, which is smooth Inertial mass is the measure of resistance vs. the action of any force field. Gravitational mass is the measure of gravity action And what about entanglement and inertial mass? 7
    • 14. 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
    • 15. If we sense gravity as another and equivalent interpretation of any movement, then ... The standard model repre- sents any quantum force field: strong, electromag- netic, or weak field It does not and cannot re- present gravity because it is not a quantum field at all: It is the smooth image of all quantum fields Space-timeEnergy-momentum Complex Banach (Hilbert) Space Complex probability distribution = Two probability distributions trajectoryquantum force field entangle- ment Pseudo- Riemanian basis 8
    • 16. 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 in the language of the standard model as a theory of unified quantum field
    • 17. 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 or in other words, they are not entangled
    • 18. Quantum field is the only possible field 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”
    • 19. Then we can outline the path to gravity from 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 its physical interpretation
    • 20. 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
    • 21. More about the “cure” The quantity of subadditivity (which can be zero, too) is the degree (or quantity) of entanglement 𝑒: 𝑒 = 𝐴1 + 𝐴2 𝑖𝑛: 𝐴1 + 𝐴2 − 𝐴1 + 𝐴2 , where 𝐴1, 𝐴2 are as quantities in the two entangled quantum systems 1 and 2. To recall that any quantity 𝐴 in quantum mechanics is defined as mathematical expectation, i.e. as a sum or integral of the product of any possible value and its probability, or as a functional: 𝐴 = −∞ ∞ 𝑎𝑝 𝑎 𝑑𝑎 = −∞ ∞ 𝚿 𝐴 (𝚿 ∗) 9
    • 22. More and more about the “cure” (!!!) 𝑒 cannot be quantized in principle even if 𝐴1, 𝐴2 are quantum or quantized, because as expectation as probability are neither quantum, nor quantizable since wave function is smooth (a “leap” in probability would mean infinite energy) (!!!) Granted entanglement and gravity are the same or closely connected, this explains:  (1) why gravity cannot be quantized;  (2) why gravity is always nonnegative (there is no antigravity)
    • 23. 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 Its representation as a whole (or from the “viewpoint of eternity”) is entanglement
    • 24. Then, in a few words, what would gravity be in terms of generalized quantum field? ... a smooth space-time DoF constraint imposed on 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 spheres in all moments constitutes the pseudo-Riemannian space of general relativity
    • 25. The language of quantum field theory: the conception of “second quantization” What does the “second quantization” mean in terms of the “first quantization”? If the “first quantization” gives us the wave function of all the quantum system as a whole, then the “second quantization” divides it into the quantum subsystems of “particles” with wave functions orthogonal between each other; or in other words, these wave functions are not entangled. Consequently, the “second quantization” excludes as entanglement as gravity in principle
    • 26. 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 point to another in different ways, all subspaces share it
    • 27. 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” symmetries
    • 28. 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 relation between two or more space-time points but outside and outside of them as wholenesses
    • 29. Need to add an interpretation of quantum 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 Cusa's 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
    • 30. And now, from the philosophical to the mathematical and physical ...: Minkowski space A space-time trajectory Hilbert space A wave function A Yin-Yang mathematical structure 1 0
    • 31. However ...: Have already added à la Nicolas of Cusa’s interpretation to that Yin-Yang structure, so that ... The “biggest” of the space-time whole is inserted within the “smallest” of any space-time point The “biggest” of the Hilbert-space whole is inserted within the “smallest” of any Hilbert-space point 1 1
    • 32. In last analysis we got a cyclic and frac-tal Yin-Yang mathematical structure ... Will check whether it satisfies our requirements: 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 being cyclic need not be infinite! Need only two entities, “Yin and Yang”, and a special structure tried to be described above 1 2
    • 33. Will interpret that Yin-Yang structure in terms of the standard model & gravity Our question is how the gravity being “outside” space-time points as a curving of a smooth trajectory, to which they belong, will express itself inside, i.e. within space-time points representing Hilbert space divided into subspaces in different 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- time and vice versa
    • 34. 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
    • 35. 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
    • 36. ... where “&” means ... ... equivalence ... relativity ... invariance ... conservation
    • 37. 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)
    • 38. 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 ...
    • 39. 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”
    • 40. The problem of Lorentz invariance Try to unite the following facts: Relativity Quantum theory The Lorentz noninvariant are: Newton’s mechanics Schrödinger’s quantum mechanics The Lorentz invariant are: Maxwell’s theory of electromagnetic field Einstein’s special relativity Dirac’s quantum mechanics of electromagnetic field The locally Lorentz invariant (but noninvariant globally) are: Einstein’s general relativity Our hypothesis of entanglement & gravity 1 3
    • 41. ... whether gravity is not a “defect” of electromagnetic field... However mass unlike electric (or Dirac’s magnetic) charge is a universal physical quantity which characterizes anything existing A perfect, “Yin-Yang” symmetry would require as the locally “flat” to become globally “curved” as the locally “curved” to become globally “flat” as the “biggest” to return back as the smallest and locally “flat” For example this might mean that the universe would have a charge (perhaps Dirac’s “monopole” of magnetic charge), but not any mass: the curved Banach space can be seen as a space of entangled spinors
    • 42. Electromagnetic field as a “Janus” with a global and a local “face” Such a kind of consideration like that in the previous slide cannot be generalized to the “weak” and “strong” field: They are always local since their quanta have a nonzero mass at rest unlike the quantum of electromagnetic field: the photon As to the electromagnetic field, both global and local (the latter is within the standard model) consideration is possible
    • 43. Electromagnetic field as a “Janus” with a global and a local “face” Conclusion: gravity (& entanglement) is only global (external), weak & strong interaction is only local (internal), and electromagnetic field is both local and global: It serves to mediate both between the global and the local and between the external and the internal Consequently, it conserves the unity of the universe
    • 44. More about the photon two faces: • It being global has no mass at rest • It being local has a finite speed in spacetime In comparison with it: o Entanglement & gravity being only global has no quantum, thus neither mass at rest nor a finite speed in spacetime o Weak & strong interaction being only local has quanta both with a nonzero mass at rest and with a finite speed in spacetime
    • 45. Lorentz invariance has a local and a global face, too: In turn, this generates the two faces of photon The local “face” of Lorentz invariance is both within and at any spacetime point. It “within” such a point is as the “flat” Hilbert space, and “at” it is as the tangential, also “flat” Minkowski space Its global “face” is both “within” and “at” the totality of the universe. It is “within” the totality flattening Banach space by the axiom of choice. It is “at” the totality transforming it into a spacetime point
    • 46. It is about time to gaze that Janus in details in Dirac’s brilliant solving by spinors In terms of philosophy, “spinor” is the total half (or “squire root”) of the totality. In terms of physics, it generalizes the decomposition of electromagnetic field into its electric and magnetic component. The electromagnetic wave looks like the following: 1 4
    • 47. That is a quantum kind of generalization. Why on Earth? First, the decomposition into a magnetic and an electric component is not a decomposition of two spinors because the electromagnetic field is the vector rather than tensor product of them Both components are exactly defined in any point time just as position and momentum as to a classical mechanical movement. The quantity of action is just the same way the vector than tensor product of them Consequently, there is another way (the Dirac one) quantization to be described: as a transition or generalization from vector to tensor product
    • 48. Well, what about such a way gravity to be quantized? The answer is really quite too surprising: General relativity has already quantized gravity this way! That is general relativity has already been a quantum theory and that is the reason not to be able to be quantized once again just as the quant itself cannot be quantized once again! What only need is to gaze at it and contemplate it to see how it has already sneaked to become a quantum theory unwittingly
    • 49. Cannot be, or general relativity as a quantum theory Of course the Dirac way of keeping Lorentz invariance onto the quantum theory is the most obvious for general relativity: It arises to keep and generalize just the Lorentz invariance for any reference frame However the notion of reference frame conserves the smoothness of any admissable movement requiring a definite speed toward any other reference frame or movement Should see how the Dirac approach generalizes implicitly and unwittingly “reference frame” for discrete (quantum) movements. How?
    • 50. “Reference frame” after the Dirac approach • “Reference frame” is usually understood as two coordinate frames moving to each other with a relative speed 𝑣(𝑡) • However we should already think of it after Dirac as the tensor product of the given coordinate frames. This means to replace 𝑣(𝑡) with 𝛿(𝑡) (Dirac delta function) in any 𝑡 = 𝑡0 . • Given a sphere 𝑺 with radius 𝒙 𝟐 + 𝒚 𝟐 + 𝒛 𝟐 + 𝒗 𝟐 𝒕 𝟐, it can represent any corresponding reference frame in Minkowski space. 𝑺 can be decomposed into any two great circles 𝑺 𝟏⨂𝑺 𝟐 of its, perpendicular to each other, as the tensor product ⨂ of them 1 5
    • 51. “Reference frame” after the Dirac approach Given a sphere 𝑺 with radius 𝒙 𝟐 + 𝒚 𝟐 + 𝒛 𝟐 + 𝒗 𝟐 𝒕 𝟐 decomposed into any two great circles 𝑺 𝟏⨂𝑺 𝟐 of its, 𝑺, 𝑺 𝟏, 𝑺 𝟐 are with the same radius. We can think of 𝑺 𝟏, 𝑺 𝟐 as the two spinors of a reference frame after Dirac If we are thinking of Minkowski space as an expanding sphere, then its spinor decomposition would represent two planar, expanding circles perpendicular to each other, e.g. the magnetic and electric component of electromagnetic wave as if being quantumly independent of each other 1 6
    • 52. The praising and celebration of sphere The well-known and most ordinary sphere is the crosspoint 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!
    • 53. More about the virtues of the sphere It is the “atom” of Fourier transform: The essence of Fourier transform is the (mutual) replacement between the argument of a function and its reciprocal: 𝑓 𝑡 ↔ 𝑓 1 𝑡 = 𝑓(𝜔), or quantumly: 𝑓(𝑡) ↔ 𝑓(𝐸), As such an atom, it is both: - as any harmonic in Hilbert space: 𝑓𝑛 𝜔 = 𝑒 𝑖𝑛𝜔 - as any inertial reference frame in Minkowski space: 𝑓 𝑡 = 𝑟(𝑡) = 𝑐2 𝑡2 − 𝑥2 − 𝑦2 − 𝑧2 1 7
    • 54. Again about the spinor decomposition Since the sphere is what is “spinorly” decomposed into two orthogonal great circles, the spinor decomposition is invariant to Fourier transform or to the mutual transition of Hilbert and Minkowski space In particular this implies the spinor decompsition of wave function and even of its “probabilistic interpretation”: Each of its two real “spinor” components can be interpreted as the probability both of a discrete quantum leap to, and of a smooth reaching the corresponding value
    • 55. 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 need choice (axiom) Probabilistic (mathematical) approach Mechanical (physical) approach Minkowski space: from the “Earth” to the “Paradise“ by the “stairs” of time Hilbert space: from the “Paradise” to the “Earth “ by the “stairs” of energy 1 8
    • 56. 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
    • 57. 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
    • 58. The sense of quantum movement represented in Hilbert space From classical to quantum movement: the way of 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
    • 59. 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
    • 60. The spin statistics theorem about fermions If one swaps the places of any two quantum particles, this means to swap the places between “particle” and “field”, or in other words to reverse the direction “from time to energy” into “from energy to time”, or to reverse the sign of wave function The following set-theory explanation may be useful: If there are many things, which are the same or “quantumly indistinguishable”, 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 of a set (bosons). Though indistinguishable, the swap of their corresponding ordinal (serial) number is distinguishable in the former case unlike the latter one
    • 61. However that “positive-integers analogy” is limited The well-ordering of positive integers has “memory” in a sense: One can distinguish two swaps, too, rather than only being one or more swaps available (as the fermions swap) The well-ordering of fermions has no such memory. The axiom of choice and well-ordering theorem do not require such a memory However if all the choices (or the choices after the well-ordering of a given set) constitute a set, then such a memory is posited just by the axiom of choice
    • 62. Positive integers vs. fermions vs. bosons illustrated Fermions Positive integers Bosons Initial state Swap After a time Naming True indistinguish- ability Quantum indistinguishability “Weak” (in)distinguish- ability ... ........ ... ........ ... ........ 1, 2, 3, 4, 5, ... 1, 5, 3, 4, 2, ... 1, 2, 3, 4, 5, ... 1, 2, 3, 4, 5, ... 1, 5, 3, 4, 2, ... 1, 5, 3, 4, 2, ... True distinguish- ability 1 9
    • 63. Quantum vs. classical movement in terms of (quantum in)distinguishability Quantum movement Classical movement Dynamic to static aspect: one to one Dynamic to static aspect: much to many Dynamic (boson) aspect: true in- distinguishability Static (fermion) aspect: weak in- distinguishability Quantum indistinguishability 𝑴𝒖𝒄𝒉 ⟺ 𝑴𝒂𝒏𝒚 Distinguishability 𝒕 Dynamic (momentum) aspect Static (position) aspect Wave function as the characteristic function of a random complex quantity Wave function as a (well- ordered) vector Hilbert space Pseudo- Riemannian space 2 0
    • 64. 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 “many”, which is correct only as to the latter
    • 65. 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”
    • 66. What will “spin” be in our interpretation? In particular, a new, specifically quantum quantity, namely “spin”, is added to distinguish between the well-ordered (fermion) and the unorderable (boson) state in a well-ordered way However this makes any quantum understanding of gravity (or so-called “quantum gravity”) impossible, because “quantum” gravity requires the spin to be an arbitrary real number In other words, gravity is the process in time (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” (entanglement out of time, or “boson”) interpretation
    • 67. Our interpretation of fermion vs. boson wave function In 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 quantity The former represents the static aspect of quantum movement, the latter the dynamic one. The static aspect of quantum movement comprises both the static (position) and dynamic (momentum) aspect of classical movement, because both are well-ordered, and they constitute a common well-ordering
    • 68. 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
    • 69. The two parallel phases of quantum movement Quantum field (the bosons) can be thought of as the one phase of quantum movement parallel 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 merely as many quanta
    • 70. Why be “quantum gravity” a problem of philosophy rather than of physics? 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 2 1
    • 71. 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
    • 72. Hilbert vs. pseudo-Riemannian space as actual vs. potential infinity Two oppositions are enough to represent that comparison from the viewpoint of philosophy: 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 potential infinity
    • 73. Hilbert vs. pseudo-Riemannian space: completing the puzzle Oppo- sition Process in time Actual infinity Curve Pseudo- Riemannian space Gravity, General relativity Banach space Entanglement Quantum information Flat Minkowski space Electromag- netism Special relativity Hilbert space Electromagnetic, weak and strong interaction Quantum mechanics The standard model 2 2
    • 74. Our thesis in terms of that table Curve Pseudo- Riemannian space Gravity, General relativity Banach space Entanglement Quantum information Entanglement is gravity as a complete process Gravity is entanglement as a process in time 2 3
    • 75. 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
    • 76. 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 entanglement Consequently a contradiction arises according to our intuition: Gravity should be as “less” in the first relation as much “bigger” in the second relation An obvious, but inappropriate way out of it is to emphasis the difference between the relations
    • 77. Why is such a way out inappropriate? The first relation links the mathematical models of entanglement and gravity, and the second one does 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 that For that our way out of the contradiction must not be such a one
    • 78. Cycling is about to be our way out of the contradiction Should 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 the two sides conformably of the “big or small” as well as of the “more or less” but only a single one like this: 2 4
    • 79. Once again the pathway is ...: 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 strip to an inseparable whole of a merely “much” to the last one as the “second”side of the Möbius band cyclically passing into the other 2 5
    • 80. Holism of the East vs. linear time of the West The edge of gluing the Möbius strip is a very special kind: It is everywhere and nowhere. We can think of it in terms of the 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 and bars quantum mechanics: It feels good in the Chinese Yin-Yang holism. (In the West, to be everywhere and nowhere is God's property) 2 6
    • 81. 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, or in other words whole as time
    • 82. Being people of the West, we should realize 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
    • 83. The axiom of choice self-referentially The choice of all the choices is to choose the choice itself, i.e. the axiom of choice itself , or in philosophi- cal terms to choose between the West and the East However it is a choice already made for all of us and instead of all of us, we being here (in the West) and now (in the age of the West). Consequently we doom to think whole as movement and time, i.e. linearly The mathematical notions and conceptions can aid us in uniting whole and linearity (interpreted in physics and philosophy as movement and time), though In particular, just this feature of mathematics determines its leading role in contemporary physics, especially quantum mechanics
    • 84. Boson – fermion distinction in terms both of whole and movement The two version of any fermion with different spin can be explain in terms of the whole as the same being correspondingly insides and outsides the whole since the outsides of the whole has to be inside it in a sense As an illustration, a fermion rotated through a full 360° turns out to be its twin with reversed spin: In other words, it turns “outsides” after a 2𝜋 rotation in a smooth trajectory passing along the half of the universe. Look at it on a Möbuis strip:
    • 85. A “Möbius” illustration of how a smooth trajectory can reverse the spin 𝟏) 𝟎; 𝟎° 𝟐) 𝝅; 𝟏𝟖𝟎° 𝟑) 𝟐𝝅; 𝟑𝟔𝟎° 𝟒) 𝟑𝝅; 𝟓𝟒𝟎° 𝟓) 𝟒𝝅; 𝟕𝟐𝟎° + 𝟏 𝟐 fermion − 𝟏 𝟐 fermion + 𝟏 𝟐 fermion a the same fermion “outside”“inside” the universe 2 7
    • 86. Exactly the half of the universe between two electrons of a helium atom Here is a helium atom. Exactly the half + 𝟏 𝟐 fermion − 𝟏 𝟐 fermion The universe of the universe is inserted between its two electrons which differ from each other only with reversed spin: The West thinks of the universe as the extremely immense, and of the electrons and atoms as the extre- mely tiny. However as quantum mechanics as Chinese thought shows that they pass into each other everywhere and always 2 8
    • 87. + 𝟏 𝟐 fermion − 𝟏 𝟐 fermion The West's single pathway along or through Taiji is mathematics, though Taiji 太極 is the Chinese transition between the tiniest and the most immense A fortunate exception is Nicolas of Kues 2 9
    • 88. 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 function)
    • 89. + 𝟏 𝟐 fermion − 𝟏 𝟐 fermion Taiji 太極 in the language of mathematics He The common and universal Hilbert (Banach) space Wave function interpreted as a characteristic function Wave function as a vector One single boson!!!! 3 0
    • 90. + 𝟏 𝟐 fermion − 𝟏 𝟐 fermion Taiji 太極 in the language of mathematics He The common and universal Hilbert (Banach) space Wave function interpreted as a characteristic function Wave function as a vector One single boson!!!! The axiom of choice Scolem’s “paradox” 3 1
    • 91. 3 2
    • 92. 𝟎 𝟏 Wuji 無極 as the Kochen-Specker theorem one single bit The common and universal Hilbert (Banach) space Its point interpreted as a characteristic function Its point as a vector The axiom of choice Scolem’s “paradox” One single qubit!!!! The universe of (or as) sundry Turing algorithms Quantum computer A most and most ordinary bit Taiji 太極
    • 93. The mapping between numbers and a sundry A few simplifying assumptions: 1. The sundry constitutes a set, 𝑆1 as well as the numbers, 𝑆2 2. Two smooth functions can substitute for the state of that mapping in any moment 3. Those two functions 𝑓1, 𝑓2 are correspondingly:  a probability distribution: 𝑆1 𝑓1 𝑆2  a “field”: 𝑆2 𝑓2 𝑆1 3 3
    • 94. Quantum mechanics solves the general problem under those assumptions The general problem is the quantitative description of the universe: too complicated! All the universe as a sundry Well-orderable numbers The general problem in terms of Taiji and Wuji The simplifying solving of quantum mechanics Wave function as a fieldWave function as a probability distribution 3 4
    • 95. The solving of quantum mechanics in terms of gauge theories  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 3 5
    • 96. Möbius strip as a fiber bundle “radius” for fiber, F “circle” for bundle the same “radius” from the “other side” for base 3 6
    • 97. The definition of “fiber bundle” by the example of a Möbuis strip The fiber bundle is determined and defined precisely by the topological transform from it to base space or vice versa: i.e. correspondingly as unfolding from a flat sheet (base space) to the Möbius strip (fiber strip), or folding vice versa, in our example: By its unfolding Or By its folding 3 7
    • 98. More precise definition of fiber bundle yet using the "Möbius" illustration Let us 𝑨 and 𝑩 are two “radiuses” of the two sides of a Möbius strip, and 𝑨 𝒔, 𝑩 𝒔 are the same “radiuses” on the sheet. Then the fiber bundle is described as the triangle of mappings for any 𝑨, 𝑩, 𝑨 𝒔, 𝑩 𝒔 as follows: 𝑨 𝑩 𝑨⨂𝑩 „𝑨⨂𝑩“ means Cartesian product 3 8
    • 99. The definition without any illustration Arbitrary neighborhoods of arbitrary topological spaces for the “radiuses” of the illustration However the topological spaces are usual Hilbert spaces or subspaces in the physical interpretation of fiber bundle in the gauge theories In other words, Hilbert spaces substitute for the “radiuses” of Möbius strip, in gauge theories 3 9
    • 100. The leading idea of gauge theory Let us fancy the two “radiuses” or Hilbert spaces 𝐴 and 𝐵 correspondingly as the reference and gauge mark of an uncalibrated indicator, and 𝐴 𝑠 and 𝐵𝑠 are the same after the precise calibrating: 𝑨 𝒔 ≡ 𝑩 𝒔 𝑨 𝑩 0 0 an uncalibrated indicator the indicator calibrated Fiber bundle Cartesian product The Standard Model 4 0
    • 101. The universality of calibration The calibration should be identical for any indication, and this is true as to weak, electromagnetic, and strong interaction, but not as to gravity For that the Standard Model comprises the former three but not the latter A necessary condition is quantization, which guarantees the two vectors A and B to exist Our conjecture will be: It is quantization that gravity cannot satisfy and in principle, there can be no gauge theory of gravity, as a corollary
    • 102. More about Dirac’s spinors Can think of them both ways: - As two electromagnetic waves - As the complex (=quantum) generalization of electromagnetic wave The latter is going to show us the original Dirac theory However the former is much more instructive and useful for our objectives: It is going to show us the connection and unity of gravity and electromagnetism, and hence then the links of gravity and quantum theory by the mediation of electromagnetism
    • 103. 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 scientific problem becomes a philosophical one is: When it cannot be solved in the gestalt of the dominating at present picture of the world despite all outrageous efforts 
    • 104. 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.“
    • 105. An equivalent reformulation of Einstein’s principle of relativity: All physical laws must be invariant to any smooth movement (space-time transformation) Comment: However all quantum movements are not smooth in space-time at all: Even they are not continuous in it Besides: the relativity movements are not “flat” in space-time in general while all quantum movements are “flat” in Hilbert space Definition: A movement is flat when it is represented by a linear operator in the space of movement
    • 106. 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 and our generalization is that the word “smooth” is excluded so the movement can already be quantum However such a kind of invariance (in fact, an invariance as with the discrete as with the continuous) meets a huge obstacle in set theory: consequently, in the true fundament of mathematics requiring to change gestalt
    • 107. The huge obstacle in set theory: The invariance of the discrete and continuous cannot be any isometry in principle since the standard measure of any discrete set is zero (while the measure of a continuum can be as zero as nonzero) Moreover, the obstacle is deeper situated in set theory since the power of any discrete set is less than that of any continuum even if its measure is zero Fortunately Skolem’s paradox offer’s a solution, however, “transgressing boundaries” of the “gestalt”: Unfortunately Skolem’s paradox is based on, and necessarily requires the axiom of choice alleged sometimes as “unacceptable”
    • 108. The inevitability of the axiom of choice in quantum mechanics The 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 quantum mechanics (based on Hilbert space), quantum randomness is not equivalent to any statistical ensemble: Its members or their quantities would be the alleged “hidden variables”
    • 109. 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 it The 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
    • 110. The “quantum wholeness” of the axiom of choice and the “no hiddenness” theorems Preliminary notes: If there is an algorithm, which leads to the choice, the axiom needn’t: Consequently, the axiom core is the opportunity of choice without any algorithm − be guaranteed Given the choice without any algorithm is a random choice in definition, the axiom of choice postulates that a random choice can always be made even if a rational choice by means of any algorithm cannot
    • 111. The “quantum wholeness” of the axiom of choice and the “no hiddenness” theorems: The “no hidden variables” theorems state that any choice of a definite value in measuring is random: Thus, they postulate the axiom of choice in quantum mechanics How, however, can we explain intuitively the randomness of choice in quantum mechanics? The apparatus “chooses” randomly a value among all probable values by the mechanism of decoherence, e.g. a “time” interpretation of coherent state and decoherence is possible:
    • 112. The “time” interpretation of coherent state and decoherence: The de Broglie wave periods of the measuring apparatus 𝑻 𝒂 and of the measured quantum entity (𝑻 𝒆) correspondingly: 𝑻 𝒂 = ħ 𝒄 𝟐 𝟏 𝒎 𝒂 ; 𝑻 𝒆 = ħ 𝒄 𝟐 𝟏 𝒎 𝒆 ; ∴ 𝑻 𝒂 𝑻 𝒆 = 𝒎 𝒆 𝒎 𝒂 ≈ 𝟎 Consequently, coherent state corresponds to 𝑻 𝒆, and decoherence to 𝑻 𝒆 𝑻 𝒂, i.e. − to a random choice of a (≈) point among the continual interval of 𝑻 𝒆 Now, we can explain the difference between a coherent state and a statistical ensemble so: 4 1
    • 113. The “time” interpretation of the difference between a coherent state and a statistical ensemble A discrete (quantum) leap of any function in a point (an argument value) generates a coherent state. For the so-called time interpretation we may accept the argument be time A continuous function (e.g. of time) generates a statistical ensemble (e.g. of the measured values in different time points) The transformation between a discrete leap and a continuous function implies the corresponding transformation between a coherent state and a statistical ensemble
    • 114. The chain of sequences from Skolem’s paradox to our generalization of Einstein’s relativity principle : Scolem’s paradox The axiom of choice “No hidden variables” 𝑇ℎ𝑒 𝐾𝑜𝑐ℎ𝑒𝑛−𝑆𝑝𝑒𝑐𝑘𝑒𝑟 𝑡ℎ𝑒𝑜𝑟𝑒𝑚 Wave- particle duality 𝑇ℎ𝑒 "𝑡𝑖𝑚𝑒" 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛 The invariance of discrete and continuous morphisms (functions) The invariance of discrete and smooth space-time movements Our generalization of Einstein’s relativity principle (GRP) ∴ Skolem’s paradox is a weaker formulation of GRP 4 2
    • 115. A few comments: the first one: wave-particle duality as invariance After Niels Bohr we are keen to understand duality as complementarity: The two dual aspects or quantities cannot be together (e.g. measured simultaneously) However according to the true formalism of quantum mechanics − based on complex Hilbert space, they should be equal: Hence, the dual aspect of quantity is merely redundant In fact, the “no hidden variables” theorems imply the same: So we should speak of wave-particle invariance. In particular, our intuition distinctly separating waves from particles misleads us: They are the same in principle
    • 116. A second comment: wave-particle inva- riance embedded in complex Hilbert space Two important features of complex Hilbert space allow of such embedding in it: (1) It and its dual space are anti-isomorphic (Riesz representation theorem); So (1) allows the following: The four pairs can be identified: (1.1) the two corresponding points of the two dual space; (1.2-3) the Fourier transformation and its reverse one of the probability distribution of a random quantum quantity and its reciprocal one (these are two pairs); (1.4) any quantum quantity and its conjugate one. Besides, (1.5) any point in Hilbert space can be interpreted as a function as a vector
    • 117. A necessary gloss about the probability distribution of a random quantum quantity: The probability distribution of a “classical” random quantity is a real function of a real argument. If however any point in Hilbert space is interpreted as a probability distribution of a random quantum quantity, we need a complement gloss about the meaning both of a complex probability and of a complex value as to a physical quantity. Our postulate: any quantum quantity and its probability distribution is composed by two “classical” ones and their probability distributions sharing a common physical dimension: one for the discrete and another for continuous aspect
    • 118. 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
    • 119. More about the embedding of wave- particle invariance in complex Hilbert space That multiple identification can be complemented more: It identifies a generalized (e.g. ∆-function) and “ungenerelized” function 𝑓 (e.g. a constant). We can interpret it as 𝑓−1 ↔ 𝑓, or as the interchange between the set of arguments and that of values, or as the interchange of the “axes” of Cartesian product. Note that is an anti-isometric 𝜋 2 rotation. The same physically interpreted is the wave-particle invariance in question. The really necessary condition of it is only Skolem’s “paradox”. However whether is not the last also a sufficient condition for it?
    • 120. A set-theory generalization of wave- particle invariance Let us introduce the set of qubit integers ℚ: Any integer is generalized as a numbered qubit: The set of qubit integers ℚ is isomorphic to complex Hilbert space ℍ. According to the well-ordering theorem (an equivalent of the axiom of choice) Hilbert space ℍ is isomorphic to the set of integers 𝕀 by means of the set of qubit integers ℚ: Now already, the equivalence of Skolem’s paradox and wave-particle invariance can be considered as that isomorphism: ℍ ℚ 𝕀 4 3
    • 121. Another useful, now physical interpretation 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:  as a (“covariant”) set of harmonics as a (“contravariant”) set of points, the two sets being anti-isomorphic (anti- isometric measurable) 4 4
    • 122. Another useful, now physical interpretation of the invariance (duality) Formally, we can yield that interpretation by another physical interpretation of a function and its Fourier transformation: 𝑓 𝑥 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 F 1 𝑥 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛 𝑓(𝑡) 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝐹 1 𝑡 4 4
    • 123. Is there any mathematical model, which can coincide with the modeled reality? 𝑓 𝑥 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 F 1 𝑥 𝑎 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛 𝑓(𝑡) 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝐹 1 𝑡 𝑓 𝑥 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 F 1 𝑥 𝑎𝑛𝑜𝑡ℎ𝑒𝑟 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛 𝑝(𝐴) 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝐹 1 𝐴 4 5
    • 124. A philosophical interlude about the logical equivalence of two physical interpretations 𝑓 𝑥 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 F 1 𝑥 𝑎 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛 𝑓(𝑡) 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝐹 1 𝑡 − 𝒕𝒊𝒎𝒆 𝒊𝒏𝒕𝒆𝒓𝒑𝒓𝒆𝒕𝒂𝒕𝒊𝒐𝒏 𝑓 𝑥 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 F 1 𝑥 𝑎𝑛𝑜𝑡ℎ𝑒𝑟 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛 𝑝(𝐴) 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝐹 1 𝐴 − 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛 4 6
    • 125. Let the former (any quantity) be physically interpreted as the argument in the latter: 𝑓 𝑥 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 F 1 𝑥 𝑎 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛 𝐴(𝑡) 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝐹 1 𝑡 𝑓 𝑥 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 F 1 𝑥 𝑎𝑛𝑜𝑡ℎ𝑒𝑟 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛 𝑝(𝐴) 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝐹 1 𝐴 4 7
    • 126. Besides, let the same argument be physically interpreted as time: 𝑓 𝑥 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 F 1 𝑥 𝑎 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛 𝐴(𝑡) 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝐹 1 𝑡 𝑓 𝑥 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 F 1 𝑥 𝑎𝑛𝑜𝑡ℎ𝑒𝑟 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛 𝑝[𝐴 𝑡 ] 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝐹 1 𝐴(𝑡) 4 8
    • 127. A gloss on physical dimensions: First of all, what is the physical dimension of the products, 𝒑 𝑨 . 𝑨 𝒕 and 𝒑[𝑨 𝒕 ]. 𝑨 𝒕 ? Since 𝑨 whatever is is reduced, 𝒑 𝑨 . 𝑨 𝒕 = 𝑯𝒛 ~𝑬. And about 𝒑 𝑨 𝒕 . 𝑨 𝒕 ? 𝑯𝒛 . 𝑨(𝒕) : 4 9
    • 128. A gloss on physical dimensions: For example, if A is distance − 𝑯𝒛. 𝒎 𝒔 = 𝒎 𝒔 𝟐 ~𝒂 ∴ 𝑮~ 𝒂 𝑨 ~ 𝒑 𝑨 𝒕 .𝑨(𝒕) 𝒑 𝑨 .𝑨(𝒕) = 𝒑[𝑨 𝒕 ] 𝒑(𝑨) = = 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑎𝑙 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑞𝑢𝑎𝑛𝑡𝑢𝑚 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 ~ ~ 𝑎𝑠 𝑎 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑎𝑙 𝑒𝑛𝑠𝑒𝑚𝑏𝑙𝑒 𝑎𝑠 𝑎 𝑞𝑢𝑛𝑡𝑢𝑚 𝑤ℎ𝑜𝑙𝑒𝑛𝑒𝑠𝑠 ~ ~ 𝑎𝑠 𝑎 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑎𝑠 𝑖𝑡𝑠 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 = = 𝒇𝒐𝒓 𝑯𝒊𝒍𝒃𝒆𝒓𝒕 𝒔𝒑𝒂𝒄𝒆 = 0 4 9
    • 129. Parseval’s theorem 𝑓 𝑥 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚 𝐹 𝑦 𝑔 𝑥 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚 𝐺 𝑦 𝑢𝑛𝑑𝑒𝑟 𝑎 𝑓𝑒𝑤 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠 ⟺ −∞ ∞ 𝑓(𝑥) . 𝑔 𝑥 . 𝑑𝑥 = −∞ ∞ 𝐹 𝑦 . 𝐺(𝑦). 𝑑𝑦 5 0
    • 130. Parseval’s theorem about the generalization of a quantum quantity 𝒀 and of its conjugate quantity 𝑿 𝐹 𝑦 𝑎 𝑠𝑒𝑙𝑓𝑎𝑑𝑗𝑜𝑖𝑛𝑡 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟 𝐺 𝑦 𝐹(𝑦) 𝑎𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟 𝐺 𝑦 𝑢𝑛𝑑𝑒𝑟 𝑎 𝑓𝑒𝑤 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠 ⟺ −∞ ∞ 𝑓 𝑥 . 𝑔 𝑥 . 𝑑𝑥 = −∞ ∞ 𝐹 𝑦 . 𝐺 𝑦 . 𝑑𝑦 ⟺ ⟺ 𝑋 = 𝑌 (the so-called wave-particle invariance) 5 1
    • 131. Parseval’s theorem simply illustrated as a “cross rule” 𝒇 𝒙 = 𝑭(𝒚) 𝒇 𝒙 = 𝐅(𝐲) 𝒈 𝒙 = 𝑮(𝒚) 𝒈 𝒙 = 𝑮(𝒚) 5 2
    • 132. Obviously Parseval’s theorem is due to the “flatness” of Hilbert space. To get it “curved” into Banach one? 𝒇 𝒙 𝒇 𝒙 𝒈 𝒙 𝒈 𝒙 𝑭(𝒚) 𝐅(𝐲) 𝑮(𝒚) 𝑮(𝒚) 5 3
    • 133. Fourier transform by 3D Cartesian product 𝒙 𝑭 𝑭𝒐𝒖𝒓𝒊𝒆𝒓 𝒇(𝒙) 𝟑𝑫 𝑪𝒂𝒓𝒕𝒆𝒔𝒊𝒂𝒏 𝒑𝒓𝒐𝒅𝒖𝒄𝒕 𝑭 𝒇 𝒇(𝒙) 𝒙(𝑭) 𝑫𝒖𝒂𝒍 𝒔𝒑𝒂𝒄𝒆 𝑭𝒐𝒖𝒓𝒊𝒆𝒓 𝑺𝒑𝒂𝒄𝒆 5 4
    • 134. Riesz representation theorem by 3D Cartesian product 𝟑𝑫 𝑪𝒂𝒓𝒕𝒆𝒔𝒊𝒂𝒏 𝒑𝒓𝒐𝒅𝒖𝒄𝒕 𝑭 𝒇 𝒇(𝒙) 𝒙(𝑭)𝑭 𝒇 𝑭, 𝒙, 𝒇 𝒂𝒓𝒆 𝒕𝒉𝒆 𝒄𝒐𝒏𝒋𝒖𝒈𝒂𝒕𝒆 𝒄𝒐𝒎𝒍𝒆𝒙 𝒓𝒂𝒕𝒉𝒆𝒓 𝒕𝒉𝒂𝒏 𝒏𝒆𝒈𝒂𝒕𝒊𝒗𝒆 𝒂𝒙𝒆𝒔 5 5
    • 135. About 𝒇(𝑭) and the coincidence of 𝒇(𝒙), 𝒙(𝑭), and 𝒇(𝑭) in form 𝒇 𝒙𝑭 𝒙(𝑭) A functional ∀ 𝒙 𝑭 , 𝒇 𝒙 , 𝒇 𝑭 : 𝒇(𝑭) ≡ 𝒇[𝒙 𝑭 ] The zest is what about Banach space! The plane determined by the three “points” 𝑭, 𝒙, 𝒇, is getting curved into … (please imagine it ) 5 6
    • 136. That is: 𝑭 𝒇 𝒙 𝒙 𝒇The “surface” of Banach space The planes (𝑭, 𝒙, 𝒇) (𝒇, 𝒙, 𝒇, 𝒙) (𝑭 ≡ 𝑭, 𝒇 𝒙) represent three Hilbert spaces ℍ1 ℍ2 ℍ3 tensor product such as: ℍ1⨂ℍ2 = ℍ3 Now the case is: No entanglement ⟺ ℍ 𝟏⨂ℍ 𝟐 = ℍ 𝟑 ⟺ No gravity (ℍ 𝟑 is the Hilbert space of the compound system ℍ 𝟏&ℍ 𝟐) 5 7
    • 137. However the case in general is: 𝑭 𝒇 𝒙 𝒙 𝒇 The “surface” of Banach space The “planes” ℍ 𝟏 = (𝑭, 𝒙, 𝒇) ℍ 𝟐 = (𝒇, 𝒙, 𝒇, 𝒙) ℍ3 = (𝑭 ≡ 𝑭, 𝒇 𝒙) form an arbitrary triangle: Such that ℍ 𝟏,ℍ 𝟐 are not orthogonal to each other in general (i.e. they may be in particular) Entanglement ⟺ ℍ 𝟏⨂ℍ 𝟐 ≠ ℍ 𝟑 ⟺ No gravity (ℍ 𝟑 is the Hilbert space of the compound system ℍ 𝟏&ℍ 𝟐) 5 8
    • 138. The different perspectives on Hilbert and Minkowski space In fact the two spaces are the same space seen in different perspectives:  As Hilbert space by frequency, 𝝎 = 𝟐𝝅 𝒕 ,  As Minkowski space by time, 𝒕 Indeed, we can compare the “atoms” of their bases: (countable) expanding in time Minkowski space Continuous perspective: ??? Discrete perspective: Hilbert space (countable) expanding in frequency 5 9
    • 139. The different perspectives on an impulse − a trajectory: Hilbert − Minkowski space (countable) expanding in time Minkowski space Continuous perspective: 𝑭𝒐𝒖𝒓𝒊𝒆𝒓 𝒕𝒓𝒂𝒏𝒔𝒇𝒐𝒓𝒎 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝒅𝒖𝒂𝒍 𝒔𝒑𝒂𝒄𝒆𝒔 Discrete perspective: Hilbert space (countable) expanding in frequency a trajectory an impulse 𝑡𝑡 a world line a quantum leap 6 0
    • 140. Hilbert − Minkowski space: wave-particle duality Minkowski space 𝑭𝒐𝒖𝒓𝒊𝒆𝒓 𝒕𝒓𝒂𝒏𝒔𝒇𝒐𝒓𝒎 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝒅𝒖𝒂𝒍 𝒔𝒑𝒂𝒄𝒆𝒔 Hilbert space a trajectory an impulse 𝑡𝑡 a world line a quantum leap a particle moving continuously in that trajectory well-ordered by time a wave function simultaneous in all the space 6 1
    • 141. Hilbert − Minkowski space: a perfect symmetry of positions and probabilities a trajectory an impulse 𝑡𝑡 a particle moving continuously in that trajectory well-ordered by time a wave function simultaneous in all the space but well- ordered in frequency However the particle trajectory is a singular mix of frequencies However the wave function is a singular mix of positions 6 2
    • 142. The quadrilateral: Hilbert – Banach – Minkowski – pseudo-Riemannian space Hilbert space Banach space Minkowski space Pseudo-Riemannian space Fourier transform 𝑐𝑢𝑟𝑣𝑖𝑛𝑔 𝑓𝑙𝑎𝑡𝑡𝑒𝑛𝑖𝑛𝑔 𝑐𝑢𝑟𝑣𝑖𝑛𝑔 𝑓𝑙𝑎𝑡𝑡𝑒𝑛𝑖𝑛𝑔 6 3
    • 143. The known sides of the quadrilateral: as Hilbert – Banach space as Minkowski – pseudo-Riemannian space Banach space 𝒂𝒅𝒅𝒆𝒅 𝒔𝒄𝒂𝒍𝒂𝒓 𝒑𝒓𝒐𝒅𝒖𝒄𝒕 𝒊𝒏𝒗𝒂𝒓𝒊𝒂𝒏𝒕 𝒕𝒐 𝒕𝒉𝒆 𝒔𝒑𝒂𝒄𝒆 𝒑𝒐𝒊𝒏𝒕𝒔 Hilbert 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 6 4
    • 144. The known sides of the quadrilateral: as Minkowski – pseudo-Riemannian space as Hilbert – Banach space Minkowski space 𝒂𝒅𝒅𝒆𝒅 𝒔𝒄𝒂𝒍𝒂𝒓 𝒑𝒓𝒐𝒅𝒖𝒄𝒕 𝒊𝒏𝒗𝒂𝒓𝒊𝒂𝒏𝒕 𝒕𝒐 𝒕𝒉𝒆 𝒔𝒑𝒂𝒄𝒆 𝒑𝒐𝒊𝒏𝒕𝒔 pseudo− 𝐑𝐢𝐞𝐦𝐚𝐧𝐢𝐚𝐧 space varying scalar product depending on the space points Pseudo-Riemannian space as curved Minkowski space: The change of the scalar product in each point can be interpreted as a function of the curvature in that point 6 5
    • 145. The close analogy of the two transforms as different views on the same transform: Hilbert space 𝒔𝒄𝒂𝒍𝒂𝒓 𝒄𝒖𝒓𝒗𝒊𝒏𝒈 𝒔𝒄𝒂𝒍𝒂𝒓 𝒇𝒍𝒂𝒕𝒕𝒆𝒏𝒊𝒏𝒈 Banach space Minkowski space 𝒔𝒄𝒂𝒍𝒂𝒓 𝒄𝒖𝒓𝒗𝒊𝒏𝒈 𝒔𝒄𝒂𝒍𝒂𝒓 𝒇𝒍𝒂𝒕𝒕𝒆𝒏𝒊𝒏𝒈 pseudo− 𝐑𝐢𝐞𝐦𝐚𝐧𝐧𝐢𝐚𝐧 space We can use the two perspectives mentioned above, on Hilbert − Minkowski space: frequency − time: 6 6
    • 146. The two transforms as the same transform Hilbert space 𝒔𝒄𝒂𝒍𝒂𝒓 𝒄𝒖𝒓𝒗𝒊𝒏𝒈 𝒔𝒄𝒂𝒍𝒂𝒓 𝒇𝒍𝒂𝒕𝒕𝒆𝒏𝒊𝒏𝒈 Banach space Minkowski space 𝒔𝒄𝒂𝒍𝒂𝒓 𝒄𝒖𝒓𝒗𝒊𝒏𝒈 𝒔𝒄𝒂𝒍𝒂𝒓 𝒇𝒍𝒂𝒕𝒕𝒆𝒏𝒊𝒏𝒈 pseudo− 𝐑𝐢𝐞𝐦𝐚𝐧𝐧𝐢𝐚𝐧 space 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 𝑭𝒐𝒖𝒓𝒊𝒆𝒓−𝒍𝒊𝒌𝒆 𝒕𝒓𝒂𝒏𝒔𝒇𝒐𝒓𝒎 𝒕𝒊𝒎𝒆 6 7
    • 147. The curving or flattening in both cases: one dual space space time frequency … …… … …… 1 2 n 1’ 2’ n’ Shifting& rotating of each corresponding sphere in the dual space 6 8
    • 148. The curving or flattening in the first case: two comments 1) It is the first case what one knows till now: time frequency The “flat” Hilbert space of quantum mechanics The “curved” pseudo- Riemannian space of general relativity 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 frequency series and by means of the latter as an impulse, i.e. as a quantum leap: Consequently, quantum mechanics is an empirical knowledge of actual infinity ! 6 9
    • 149. The curving or flattening in both cases: two dual space space time … …… … …… 1 2 n 1’ 2’ n’ Shifting& rotating of each corresponding sphere in the dual space frequency 7 0
    • 150. The curving or flattening in the second case: two comments 1) It is the second case what one would emphases: time frequency The “flat” Minkowski space of special relativity The “curved” Banach space of entanglement 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 be curved, and a partial degree of entanglement might be combi- ned with a corresponding partial degree of gravity 7 1
    • 151. The unknown sides of the quadrilateral: as Hilbert – Minkowski space as Banach – pseudo-Riemannian space 𝐇𝐢𝐥𝐛𝐞𝐫𝐭 𝐨𝐫 𝐁𝐚𝐧𝐚𝐜𝐡 𝐬𝐩𝐚𝐜𝐞 𝒕𝒊𝒎𝒆 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 𝐌𝐢𝐧𝐤𝐨𝐰𝐬𝐤𝐢 𝐨𝐫 𝐩𝐬𝐞𝐮𝐝𝐨 − 𝐑𝐢𝐞𝐦𝐚𝐧𝐧𝐢𝐚𝐧 𝐬𝐩𝐚𝐜𝐞 Discreteness 𝒂𝒔 𝒂 𝒕𝒓𝒂𝒋𝒆𝒄𝒕𝒐𝒓𝒚 𝑨𝒙𝒊𝒐𝒎 𝒐𝒇 𝒄𝒉𝒐𝒊𝒄𝒆 𝒂𝒔 𝒂𝒏 𝒊𝒎𝒑𝒖𝒍𝒔𝒆 Continuity 𝒂𝒔 𝒅𝒊𝒔𝒄𝒓𝒆𝒕𝒆 − 𝒄𝒐𝒏𝒕𝒊𝒏𝒖𝒊𝒕𝒚 𝒂𝒔 𝒘𝒂𝒗𝒆 − 𝒑𝒂𝒓𝒕𝒊𝒄𝒍𝒆 ⊠ 𝒂𝒔 𝒅𝒖𝒂𝒍𝒊𝒕𝒚 𝒂𝒔 𝒊𝒏𝒗𝒂𝒓𝒊𝒂𝒏𝒄𝒆 7 2
    • 152. The sides of the quadrilateral one by one: 1| Minkowski – pseudo-Riemannian space dual space space time … …… … …… 1 2 n 1’ 2’ n’ Shifting& rotating of each corresponding sphere in the dual space frequency(energy) momentum position A body …⍟⍟… in the gravitational field 7 3
    • 153. The quadrilateral one by one: Minkowski – pseudo-Riemannian space: conclusion dual space space The “flat” Minkowski space includes the space-time trajectory of the body The “curved” Minkowski space as pseudo-Riemannian one represents all the universe as a gravitational field of the whole, or of all the rest to the body 7 4
    • 154. The sides of the quadrilateral one by one: 2| Hilbert – Banach space dual space space position … …… … …… 1 2 n 1’ 2’ n’ Shifting& rotating of each corresponding sphere in the dual space position probability probability The wave function of anything…⍟ ⍟… in entanglement 7 5
    • 155. The quadrilateral one by one: Hilbert – Banach space: conclusion dual space space The “flat” Hilbert space includes the wave function of the quantum anything The “curved” Hilbert space as Banach one represents all the universe as an entanglement of the quantum anything with all the rest 7 6
    • 156. The quadrilateral “two by two”: Hilbert – Banach, and Minkowski – pseudo- Riemannian space: conclusion The close analogy between those two sides of the “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
    • 157. A few important notes: on the conclusion Banach (Hilbert) space represents the same as functions globally, and pseudo-Riemannian (Minkowski) space as point trajectories locally The 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
    • 158. A few important notes: on the conclusion Banach (Hilbert) space represents the same as functions globally, and pseudo-Riemannian (Minkowski) space as point trajectories locally 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?
    • 159. A few important notes: on the conclusion Banach (Hilbert) space represents the same as functions globally, and pseudo-Riemannian (Minkowski) space as point trajectories locally The answer is the third earnest note: Abandoning the axiom of choice in all the cases eo 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
    • 160. The fourth earnest note on the conclusion Banach (Hilbert) space represents the same as functions globally, and pseudo-Riemannian (Minkowski) space as point trajectories locally Function space The “curved” case The “flat” case “Line” space “Projection in probabilities” space A point in Banach space A point in Hilbert space A line in pseudo- Riemann. space t f A trajec- tory in a force field A tra- jec- tory A line in Minkow- ski space −∞ ∞ 𝒑𝒅𝒙 < 𝟏 p x A normed probability distribution −∞ ∞ 𝒑𝒅𝒙 = 𝟏 p x A defected probability distribution being due to entanglement (the force field) 7 7
    • 161. A “homily” about negative probability The defected probability distribution being due to entanglement (i.e. to an interrelation) can be also interpreted as an 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 entanglement as a new kind of substance: the substance of quantum information
    • 162. The sides of the quadrilateral one by one: 3) Hilbert – Minkowski space Both spaces are “flat”, well-ordered, expressing the same, but: A real difference: Hilbert space is a function space, while Minkowski space is an ordinary, “point” space An alleged difference: Besides, Hilbert space is interpreted (but incorrectly) only as a “frequent” space representing discrete impulses, while Minkowski space (but also incorrectly) only as a “time” space representing smooth trajectories. In fact, both spaces are equally interpretable as a “time”, as a “frequent” space connected by a Fourier or Fourier-like transform
    • 163. The sides of the quadrilateral one by one: 3| Hilbert – Minkowski space videlicet: Hilbert space is a function space, while Minkowski space is an ordinary, “point” space: A very important corollary from the real difference, 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 infinity The two views mentioned before on a single “Hilbert- Minkowski” space represent it correspondingly as a potential infinity and as an actual infinity
    • 164. 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
    • 165. The quadrilateral, one by one: 4|Banach – pseudo-Riemannian space: the curvature represented in each case by the two dual spaces BA NAC H PRSI EEUMDA ON. N PR OBA BIL ITY SPACEDUAL SPACE orthogonality “A”varying“angle&distnance” 𝐅(𝐲) 𝒇 𝒙 𝒇 𝒙 𝑭(𝒚) …… …… …… n' n p x p x p p x 7 8
    • 166. The dual-spaces representation of mechanical movement in a force field The juxtaposition of Lagrange and Hamilton approach to mechanical movement Hamilton (dual spaces) approach Lagrange (derivatives) approach 𝒑 𝑭(𝒕),𝑬 𝑭(𝒕) forcefield In both cases, three 4-vectors 𝒙, 𝒕; 𝒑, 𝑬; 𝒑 𝑭, 𝑬 𝑭 determines the movement in any point, but … and here as a smooth trajectory … here as three discrete corresponding 4-points …… …… …… n n' x,t spacedual p,E space 7 9
    • 167. The juxtaposition of Lagrange and Hamilton approach to mechanical movement: conclusion A. Both approaches are equivalent in classical mechanics – a well-known fact B. If we accept the equivalence of gravity (Lagrange) & entanglement (Hamilton), both approaches will be immediately equivalent in quantum mechanics, too C. The universal equivalence of both approaches origins from discrete-continuous invariance, or from wave-particle dualism, or from Skolem’s “paradox”, or in last analysis – from the axiom of choice
    • 168. A little philosophical digression about gravitational field and force field
    • 169. A new conjecture: entanglement field If any ordinary field acts to the values of certain physical quantities, the entanglement field acts to the probabilities of those values: So it can be called 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?
    • 170. How can entanglement field act? Its origin is rather mathematical and universal for that: Any discrete, or jump-like change is equivalent to a probability field in a sense: Since a definitive speed of change is impossible to determine, it is substituted by all the values with certain probabilities or in other words, by the probability field of all the values. If there are two or more discrete changes, they can share some values with different probabilities in each probability field generated by a quantum leap. In the last case, a common and equal probability calculable appears instead of the two or more different ones
    • 171. How can entanglement field act? Next: If and only if the probability is zero for each other field where the probability of one of them is nonzero, then the probability fields do not interact, they are “orthogonal” and no entanglement 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
    • 172. The probability field of all the momenta The probability f of all the positio How can entanglement field act? Interpreting the connection between the two dual spaces … tf(E) SpaceDual space A quantum leap inAnother (or the same??) quantum leap in energy (frequency) Heisenberg’s uncertainty Fourier transforms Any momentum Any positio 8 0
    • 173. The complex probabi-lity field of all as posi-tions as momenta The complex probability field of all as momenta as positionsThe probability field of all the momenta The probability f of all the positio How can entanglement field act? Interpreting the connection between the two dual spaces … P(x) P(x,p) P(p) P(p,x) SpaceDual space Heisenberg’s uncertainty Fourier transforms Anymomentum (& position) Anyposition (& mom Complex Hilbert space 8 1
    • 174. The same complex probability field of all as positions as momenta The same complex probability field of all as momenta as positions How can entanglement field act? Interpreting the connection between the two dual spaces … P(x,p)P(p,x) SpaceDual space Complex Hilbert space View from 𝒆𝒕𝒆𝒓𝒏𝒊𝒕𝒚 𝒕𝒊𝒎𝒆 𝑵𝒐 𝒄𝒉𝒐𝒊𝒄𝒆 𝑨𝒙𝒊𝒐𝒎 𝒐𝒇 𝒄𝒉𝒐𝒊𝒄𝒆 view 𝑬~𝒇 = 𝟏/𝒕 𝒕 𝒙𝒑 8 2
    • 175. A digression about the “arrow of time” The “arrow of time” is a fundamental, known to eve- ryone, but partly explainable fact about time unlike all other physical quantities, which are isotropic Our 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” are merely full synonyms expressing the same Consequently, the axiom of choice, which is equivalent with well-ordering, means that any set can be represented as a physical quantity in time or as a trajectory in a special space corresponding to that set: Or in other words, the set can always be transformed into another set. The theory of categories states generalizing that even if the “set” is not a set, but a “category”, it can be transformed
    • 176. 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
    • 177. The same restrictions of choice for the same trajectory point as a field point Any trajectory point undergoes a force being due to the field in the same space-time point That force represents merely a second and different trajectory but only in the dual space of energy and momentum. Such a second energy-momentum trajectory is determined to any possible space-time trajectory There is a single difference: The first restriction is absent: Position and momentum are independent of each other for the second trajectory: However the other two restrictions are valid!
    • 178. 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
    • 179. 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
    • 180. 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
    • 181. Einstein's general relativity revolution 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
    • 182. 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)
    • 183. Following Einstein’s lesson beyond him: … we introduce a second representation, namely that “from eternity” rather for a new equivalence (or “relativity”) than only for it 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 mechanics
    • 184. A view on a system in a force field in terms of eternity instead of time … OK, but we have already introduc ed it a little above 8 3
    • 185. 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 complementarity is a good example for that kind of self-referentality
    • 186. 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, mathematics and philosophy: The ensemble of 'things' and their 'movements' is dually complete in the sense above
    • 187. 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 movement
    • 188. The most essential remark on the dual self-referentiality of eternity and time Our problem is the dual self-referentiality of: View from 𝒆𝒕𝒆𝒓𝒏𝒊𝒕𝒚 𝒕𝒊𝒎𝒆 𝑵𝒐 𝒄𝒉𝒐𝒊𝒄𝒆 𝑨𝒙𝒊𝒐𝒎 𝒐𝒇 𝒄𝒉𝒐𝒊𝒄𝒆 view Our solving is going to be: Eternity and time are merely two different interpretations of the same mathematical structure: namely, Hilbert (Banach) space
    • 189. 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; ... frequency (energy) interprets them as representations of a discrete impulse; …
    • 190. 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
    • 191. 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 entanglement is a quite open question
    • 192. Be eternity and time two different interpretations, then … Well-ordering in time and frequency (energy) by the axiom of choice Complex Hilbert (Banach) Space Wave function as a Fourier transform of two (conjugate) probability distributions eternity time& frequency for entanglement for gravita- tional field The Same!!! 8 4
    • 193. 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)
    • 194. A set-theory interpretation of the links between functional and physical space 𝒙 𝑭 𝑭𝒐𝒖𝒓𝒊𝒆𝒓 𝒇(𝒙) 𝟑𝑫 𝑪𝒂𝒓𝒕𝒆𝒔𝒊𝒂𝒏 𝒑𝒓𝒐𝒅𝒖𝒄𝒕 𝑭 𝒇 𝒇(𝒙) 𝒙(𝑭) 𝑫𝒖𝒂𝒍 𝒔𝒑𝒂𝒄𝒆 𝑭𝒐𝒖𝒓𝒊𝒆𝒓 𝑺𝒑𝒂𝒄𝒆 𝒙, 𝒇, 𝑭 𝒂𝒓𝒆 𝒔𝒆𝒕𝒔 𝒔𝒖𝒄𝒉 𝒂𝒔: 𝒇 = 𝟐 𝒙 , 𝒙 = 𝟐 𝑭 ; 𝒇 𝒙 ⊂ 𝒇, 𝒙 𝑭 ⊂ 𝒙 8 5
    • 195. The set-theory interpretation being continued 𝒙, 𝒇, 𝑭 𝒂𝒓𝒆 𝒔𝒆𝒕𝒔 𝒔𝒖𝒄𝒉 𝒂𝒔: 𝒇 = 𝟐 𝒙 , 𝒇 𝒙 ⊂ 𝒇 𝒙 = 𝟐 𝑭 , 𝒙 𝑭 ⊂ 𝒙 𝑰𝒇 𝒇(𝒙) 𝒄𝒂𝒓𝒅𝒊𝒏𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓 𝒙 𝒇 𝑨𝒏𝒅 𝒙(𝑭) 𝒄𝒂𝒓𝒅𝒊𝒏𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓 𝑭 𝒙 𝒕𝒉𝒆𝒏: 𝒇 𝒙 , 𝒙 𝑭 𝒂𝒓𝒆 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔 8 5
    • 196. Links between function space and physical space 𝒙 𝑭 𝑭𝒐𝒖𝒓𝒊𝒆𝒓 𝒇(𝒙) 𝟑𝑫 𝑪𝒂𝒓𝒕𝒆𝒔𝒊𝒂𝒏 𝒑𝒓𝒐𝒅𝒖𝒄𝒕 𝑭 𝒇 𝒇(𝒙) 𝒙(𝑭) 𝑫𝒖𝒂𝒍 𝒔𝒑𝒂𝒄𝒆 𝑭𝒐𝒖𝒓𝒊𝒆𝒓 𝑺𝒑𝒂𝒄𝒆 𝟑𝑫 𝑬𝒖𝒄𝒍𝒊𝒅𝒆𝒂𝒏 𝒔𝒑𝒂𝒄𝒆 𝒚 ≡ 𝒇 𝒛 ≡ 𝑭 𝝋 = 𝝋 𝒙, 𝒚, 𝒛 = 𝟎 𝒙, 𝒛 𝒑𝒓𝒐𝒋𝒆𝒄𝒕𝒊𝒐𝒏 𝒐𝒇𝝋𝒙, 𝒚 𝒑𝒓𝒐𝒋𝒆𝒄𝒕𝒊𝒐𝒏 𝒐𝒇𝝋 𝑻𝒊𝒎𝒆 𝒑𝒂𝒓𝒂𝒎𝒆𝒕𝒓𝒊𝒛𝒂- 𝒕𝒊𝒐𝒏 𝒎𝒂𝒌𝒆𝒔 𝝋 = 𝜱 𝒕 𝒊𝒏𝒕𝒐 𝒂 𝒔𝒑𝒂𝒄𝒆𝒕𝒊𝒎𝒆 𝒕𝒓𝒂𝒋𝒆𝒄𝒕𝒐𝒓𝒚 8 7
    • 197. Two very intriguing philosophical conclusions from that ℍ ℚ 𝕀 : (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