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I. Princeton ideas renewed to interpret \If -function in terms of quantum information Vasil Penchev, Institute for philosophical re·s earch- Sofia, vaspench@abv.bg Abstract: Princeton ideas (A. Einstein, J. Wheeler, H. Everett III, B . De Witt, Yang Ch., R. Mills) are generalized for interpreting \If-function. Dynamics is geometry and topology . liJ-function and operators in Hilbert space are kinds of mapping ofMinkowski space (MS) into itself. The parameter ·natural (proper, historical) time' T is introduced (Vl. Fok, J. Schwjn~). It is interpreted as the parameter of homotopy by m~pping MS into itself. The main conclusions are: (1) the law of energy distribution integral con servation is universal law but not that of itself distribution. (2) The conditions of (1) are : (A) to consider operators, which conserve whatever measure but not the measure itself; (B)" to consider operators which conserve the energy distribution integrability. The only requirement is that of existing of the integral. It is not mandatory to observe the conditions ofvon Neumann'· ergodic theorem (and even more those of the Birkhofi's) but its reduced variant of integral converging to a limit (the integral of conserving measure operators of the function to the integral of function projective operator). The phenomena of (A) quantum gravity are identical with those of (B) quantum information but they are represented from clifferent points of view : (A) Minkowski space (separable topology, non-isotropic objectc;); (B) Hilbert space (inseparable topology, isotropic objects). ,- c ,·,?.51 t 2 . I t7 ~ S t. cl e a ( 1' 7 ; s) ' ,, . •' ·. I I .. /1 ;>, 11 c... :· c of t , /?'7 c- - ..sj-' a c c I J I Wv ~ hl? r:. i .5 /) .I J_ , ,h t:> c.J t ? (~ 1 {/ t1 l"' 2 l!v /::JCJ. ;?' t.. I. ' J :: p . 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The conditions for (1) are: (A) to consider only operators, which conserve the presence of measure ~ (B) to consider only operators, which conserve the integrability of energy distribution. 3.We call intrinsic gauging the transformation of energy distribution within (1), because of(2A). 4. We call base space gauging the transformation of energy distribution within (1), because of (2B). 5. We call external gauging the final gauging, which includes intrinsic gauging and base space gauging. 6. It is not mandatory to observe the conditions of von Neumann's ergodic theorem (and even more those of the Birkhoff's), but its reduced variant of integral converging to a limit (the integral of conserving measure operators of the function to the integral of projector of the function). 7. The strong, weak, and electromagnetic coupling only approximates the intrinsic gauging, i.e. the measure gauging. 8. «The grand gauging unification" is in principle impossible, since: (A) gravity gauges the base space, but not the fiber of fibered space (fig. 31, fjg. 32)~ (B) the energy distribution is not conserve. The presence of specific gravity objects - like gravitational waves and gravitons - is ·ruled out in principle. It means that base space do~ains with nonzero measure and with comprising no fiber points do not exist. I 9. The phenomena of(A) quantum gravity are identical with those of (B) quantum information but they are represented from different points ofview: (A) Minkowski space (separable topology, non-isotropic objects); (B) Hilbert space (inseparable topology, isotropic objects). We continue further the next lecture "Curved Hilbert spaces. Quantum information & quantum gravity" (fig. 29, fig. 30). ------ I. Princeton ideas renewed to interpret w-function in terms of quantum infonnatiou Some Princeton ideas: In 1935, it is the year when Einstein published his famous work together with Podolski and Rosen, he published together only with Rosen an article devoted to geometric interpretation of the particles in general relativity as "bridges" between two sheets of space (fig. 1). John Wheeler, as well from Princeton, developing further Einstein' s ideas of physics as geometry, he supposed. "As another possibility, we can represent two different parts of one and the same Euclidean space in fact so remote one from another that any mutual influence never manifests itself throughout "an outside space". In that case the joint can be called 'wormhole' for demonstrativeness or by using the usual topological term 'handle' (fig. 2). His are also the ideas of ' 3-sphere' with n- handles as an acceptable topology (ftg. 3); to the effect that 'every particle represents a . \ space resonant between two various topologies (fig. 4), to the effect that spin represents relative orientation of two spaces (fig. 5). However we have to reject his assertion that the super-space, which generalizes and subsumes Minkowski space, bas Hausdorff topology. Following Wheeler beyond him, '¥-function is a metric model of the topology of two points with infmity of handles between them (fig. 6). Hugh Everett ill defended his PhD thesis also in Princeton developed a new interpretation of quantum mechanics, which is later called 'many worlds' interpretation. According it, apparatus and world are represented by two 'W-functions, respectively by two Hilbert spaces. Hilbert space of the jointed system is Cartesian product of them. According Wheeler and De Witt who developed further that idea, the theory of quantum mechanics includes its own interpretation in it itself. Instead of 'reduction of wave packet by measuring' , it is supposed that the universe is split up really into alternatives in superposition~ then every measurable value is realized in one of the parallel universes (fig. 7, flg. 13, fig. 14). In addition, John von Neumann had already proven his 'quantum-statistic ergodic theorem' generalized by Birkhoff, according which the arithmetical mean of sequence of isometric operators in complex Hilbert space converges to the projector on the subspace, which is invariant space in relation to the sequence of isometric operators. Everett's interpretation demands isometric operators to be reversal, that's why they have to be 1 unitary operators. The energy conservation law is valid 'nearly everywhere' among the universes according to B irkhoffs generalization (f'.g. 8, fig. 12). In 1954 Yang Ch. and R Mills supposed that all internal symmetries connected with charges and quantum numbers could consider as gauge symmetries and by means of gauge transformations connected With them. The mathematical theory of fiber space (fig. 9, fig. 10, fig. 11) considers every point of base space as a space, transformation of which from pomt to point represents just gauge transformation. In our context, we consider them as transformations of measure from point to point, i.e. as gauging measure or gauging of measure (fig. 15). The point of base is gauged in relation to measure by a gauge operator to be conserved the isometric operator guaranteeing the conservation laws in base space (fig. 16). In my work I am going to generalize all those Princeton ideas. Its essence is that sequential operation of a gauge and an isometric (possibly unitary) operator is replaced by an united operator, which is neither Hermitian nor unitary operator but conserving only measure one (It means: it is conserved only existing of measure, but not a concrete measure.) Respectively, it is conserved not an energy distribution, but only its distribution integral (fig. 17, fig 18). The only requirement is that of existing the integral, i.e. energy distribution integrability. The nondimensional correspondence of that integral (i.e. as though in units proportional of the fundamental constants) we call information. That approach is more general since (1) it is not restricted to well-known tree basic gauge symmetries and couplings: week, strong and electromagnetic coupling but it permits a continuum of equivalent gauge transformations; (2) it considers gravity not as a gauge field but as base space deformation (consequently, the strategy of the Grand unification is in principle a misconception); (3) on that base, it is deduced the homogeneity of quantum gravity and quantum information. Some preliminary correspondences necessary to interpreting llf -function Figure 19: 1.1 Complex number (a, b) 1.2 Double number (alternion) (a, ib) 2 2.1. IP' -fi.mction 2.2. Self-mapping of Minkowski space (self-mapping of path (or world lines) group) 3 .1. Operators in Hilbert space 3.2. Operators in the space ofMinkowski space self-mapping (2.2) The correspondences above permit '¥-function to be interpreted as a transformation of the future and the past of a micro-object in relation to apparatus reference (fig. 20). The advantages of such an interpretation are as follows: 1. Since path group generalizes groups of translations and rotations towards the axes ofMinkowski space then accordingly conservation laws (or integrals of motion) are generalized. What is conserved in that case? 2. The local and global properties of a micro-object are divided and united: the local properties are inside of Minkowski space; the global properties are within <I.e- function ( = Minkowski space transformation) and consequently within Hilbert space. The local and global properties of a micro-object are united by means of the parameter 'natural time' (also 'proper or historical time') T. It can be interpreted as well as the parameter of homotopy, by which a lf'-function passes continually to another. Global properties and big mass correspond to small values ofT and great values ofT correspond to local properties and small mass. 3. The local and global properties have a natural interpretation accordingly in separable (metric) and inseparable properties of a topology. So, John Wheeler's idea of 'geometrodynamics' is generalized to such one of 'topologodynamics'. 4. It makes clear that entanglement and quantum gravity are manifested for the sake of global properties of a particle. By means of such an interpretation, the connection between quantum information and quantum gravity clear up. Generalization of conservation laws We have to elucidate: (1) what is conserved? (2) From which group (or, in relation to which symmetry) does that conservation come? The conservation laws come from the translation and rotation groups within Minkowski space. Path group united and generalized those (and maybe also CPT) symmetries. 3 The new parameter is that of transition from local to global properties, i.e. proper timeT. The physical meaning of proper time can be illustrated by means of uncertainty relation (fig 21). If the last is represented as a constant square of a forbidden rectangle, which sides vary, then proper time is the relation of those sides. T characterizes the measuring but not the measured micro-object and it exhibits the proportion between the local and the global properties by that measuring. It makes clear that the searched symmetry is such one in relation to measurement group. Consequently, measurement has to be accepted as a physical interaction. It is a question if path group and measurement group are one and the same. It means that an isomorphism between measurement group and path group exists. It also follows by the adduced visualizing that the proper time 1s a nondimensional quantity. Energy conservation law is valid when the proper time is a constant. It is realized if the quotient of the apparatus mass and the micro-object mass converges to infm.ity. Then the proper time is the constant of infinity (and according the «fourth" uncertainty relation, the energy is defined). "Proper time is the quotient of the micro-object and apparatus de Broglie's waves periods. Consequently, it should be possible to observe the violation of energy conservation law with de-coherence processes. A very interesting possible interpretation of proper time is that of probability at that of the probability to measure the corresponding value of energy, i.e. E. l"ll! j2. The energy conservation law in quantum mechanics means conservation of energy distribution. If we generalized that what is conserved, it is not energy but the product of energy and probability, it means that the conservation of energy distribution integral is ( conserved (i.e. the distribution itself can be deformed arbitrarily). At that, isometric operators are generalized as those, which conserve only the existing of whatever measure. Which are they? These are operators, which can be represented as a product of a Hermitian and an isometric operator. Accordingly, it is to generalize also the ergodic theorem as converging of integrals. If it is accepted that the one micro-object enters as though as an apparatus in relation to another or to others with the phenomena of entanglement, then the law of energy conservation is violated also with them. 4 However what is conserved? According the adduced visualizing such a universally conserving quantity has to have dimension of energy. The physical meaning represented by uncertainty relation is that of a square projected on a flat rectangle with a fixed square. From this it follows that a non-dimensional quantity 'physically existing information' or simply 'information', I, can be ascribed to any micro-obJect. Consequently, universally conservmg quantity bas meaning of energy multiplied by information and information itself imports a quantity of elementary choices, alternatives, bits, or a quantity of de Broglie wave oscillations, or a quantity of alternatives 'local- global' . Then the concrete value of proper time gets meaning of a de Broglie wave phase. Proper time unites local and global propet1:ies of a micro-object Local properties of a micro-object are related to a world line within Minkowski space but global properties concerns self-transformation of Minkowski space (i.e. '¥- function) and after that transformation of '¥-function, to which transformation corresponds a point ofMinkowski space (observable) as a value of functional. This is a circle: Figure22: (1) A point of(or within) Minkows.ki space- (2) A trajectory (world line, path) within Minkowski space - (3) A transformation ofMinkows.ki space ('¥-function) - (4) Hilbert space of'¥-functions- (5) A linear transformation of Hilbert space (a ~inear operator)- ( (6) A functional of operator, which values are points ofMinkowski space - (7) A point ofMinkowski space - (1) = (7) Proper time, t , is the parameter of that circle. It displays where is the current point on it. In fact the circle is not closed (1) =f (1) and consequently this implies (3). But (3) =f (9) - that's why (5) is implied. This lead to (7) =f (13) and the cycle (but not the circle) is closed. 5 The process can be represented as a continuous wave with two components. The one component of wave propagates within the Minkowski space, and the other advances within the Hilbert space. Using analogy with the two components of electromagnetic wave, we can unite the Minkowski space and the Hilbert space as perpendicular components of a topological wave. At first glance, it seems to be an artificial unification. The unified space has a transfinite dimension ( oo + 4) and it has two kinds, heterogeneous dimensions. But it is only at :first glance! Properly, any dimension of Hilbert space is a two-dimensional, 'flat' oscillator m a four-dimensional complex Euclidean space (the last is not Minkowsk.t space). So, the unified space can be thought as a four-dimensional Euclidean space and, at the same time, as a four-dimensional complex Minkowski space. At that, the two are isomorphic spaces in relation to Lorentz invariance. The four-dimens10nal complex Minkowskl space can be represented as two real four-dimensional Minkowski spaces connected by the formula: A-iB. A can be interpreted as time-space. and B as energy-momentum. Uncertainty relation connects the two types dimension. There is another possibility, at that simple possibility, of visualizing of a so sketched ''wave'' process, namely as a generalized rotation m Minkowski space. Properly, although it concerns two Minkowski spaces and two Hilbert spaces, the transfiguration method as well as using only of the imaginary domain of Minkowski space suggest.Sthat the two Minkowski space can be combined as the real and the imaginary domain of one and the same Minkowski space (fig. 21) and the two Hilbert space to be represented as ( the "two" side of the isotropic surface of light cone (fig 23). Then the wave process is a rotation in the whole Minkowski space, whereas the Lorentz rotation is only in its imaginary domain without the light cone. From this it follows that invariance is in relation to the full rotation, i.e. in relation to the Poincare group. Also general invariance in relation to operators in Hilbert space, which conserve whatever measure, corresponds to the last invariance, i.e. conserving energy distribution integral but not the distribution itself (fig. 24). Properly, difference between Hilbert space and Mink.owski space is not absolute but relative difference. From a reference frame connected with light, our world represents 6 an isotropic hyper-surface and inertial reference frames with different "velocity" exist internally within the light. These different velocities from our point of view are different possibilities. The electromagnetic constant is interpreted as a coefficient of transformation between velocity and possibility (fig. 25), v = c.( l-p 2Y, 12 or between 112 velocity and proper time, v = c.{ 1-[ft:'t)j2} . Hence, invariance in relation to generalized Poincare group is not more than invariance in relation to operators in Hilbert space, which conserve whatever measure. Two types curves exist in the unified space: open and closed curves, resp. world lines (paths) and loops. The bases of loops (oscillators) are bases of Hilbert space. The two kinds of objects are homologies and co-homologies in topological relation. Quantum mechanics as topology Wheeler's idea of "geometrodynamics" is generalized as "topologodynamics" to a certain extent. However it is impossible to advance by the way of separable topologies. '¥-functionhas to be considered as a metric model of inseparable topology. Spin is essentially connected with the type of separability. If the very weak axiom of separability- To (the Kolmogorov' s)- is held, then the one point is separable, whereas the other is not. There are two cases: the left is separable, whereas the right is not and vice versa. They correspond to the two values of half integer spin. When there are simultaneously left and right separability, then this corresponds to integral spin. In metric model of'¥-function (fig. 26): (1) Case of half integer spin is then, when there are two oriented surfaces. (2) Case of integral spin is then, when there is only one, at that nonoriented surface. On the other hand, spin is mutual orientation of the two tandem Minkowski spaces : being with opposite orientation is half integer spin~ being without orientation is integral spin (i .e. the two opposite orientation are present). The same orientation of the two tandem spaces is impossible, since correspondence between complex and double number is an unrealizable construction in that case. Two pairs, two separable and two inseparable topologies correspond to the two aspects, local and global, represented respectively by Minkowski space and its linear transformation (which is equivalent to a pair ofMinkowski spaces) and by Hilbert space and its linear transformation (which is equivalent to a pair of Hilbert spaces). The two 7 separable topologies are represented as two co-homologies and the two inseparable topologies are represented as two homologies. The universe as though pulses between its two aspects of global inseparability (holism) and oflocal separability (time-space). The universe is one great pulsing heart. Then the transformation of '¥-function (i.e. the operator in Hilbert space) is such one of (1) a micro-object towards an apparatus and (2) of the apparatus towards the universe. At that the micro-object and the universe have to be accepted as identical just as the apparatus in (1) and (2) is accepted as one and the same (fig. 27). Minkowski space in (1) is time-space (i.e. the apparatus towards the micro-object, the apparatus as a 'particle') and in (2) it is energy-momentum (i.e. the apparatus towards the universe, the apparatus as a 'wave'). Their mutual orientation or non-orientation defines the spin of a micro-object. It is already outlined the perspective to be seen phenomena of quantum information and of quantum gravity as identical. The connection between quantum gravity and quantum information Let us interpret EPR by Everett' s consideration. There are an apparatus and two universes, which are dividing (i.e. they are continuously divided). According the standard interpretation we can acknowledge that there are two objects and a process of dividing. According the sketched above interpretation the universe and the micro-object are identical (this may be called generalized principle of relativity, since the universe and micro-object are indistinguishable) and it is the dividing process that is decisive. From Everett's point of view, forces of gravity appear between the dividing universes, from the standard point of view, quantum correlations arise between the ( dividing micro-objects. From the sketched above generalized interpretation, forces of gravity and quantum correlations are one and the same and their common essence is the dividing process (fig. 28). Dividing as micro-object has neither integral spm (absolute topological inseparability) nor half integer spin (absolute topological divisibility, within T 0, only the one point is absolutely divided each from other). If both of the points are absolutely divided each from other, this is Hausdorff topology. If both of the points are inseparable each from other, this is absolutely inseparability. Consequently, the two tandem Minkowski spaces do not exist together. They are tandem also in relation to existing. It 8 means that the uruverse and the micro-object also are tandem in relation to eXIsting. This is the proper meaning of uncertainty relation. The proper time cycle, which can be called 'timing cycle', of topological wave includes: . . . left separability - absolute inseparability - right separability - absolute inseparability - left separability - ... (tlg. 26). All the phenomena of entanglement are intermediate states in that topological wave. The topological wave, wlnch is defined by uncertainty relation, has parameter ''proper time". The usual Newton trme is a subset, namely : ±mTI2, and in its frame, the energy conservation law is valid. Consequently, spin means which moment towards the eternal now (i.e. spin 0, the photons) is chosen particularly also from the set of Newton time. The energy conservation law is not valid in relation to those intermediate states. They are not isometric in terms of von Neumann's ergodic theorem. There are two causes to violate isometry of operators. The first is the presence of gauge field. The corresponding gauge transformation changes ·the measure previously at a given point to be made equal the measure at two different points of base space. That calls internal gauging. By analogy with intrinsic and external curvature of a curved hyper-surface in a curved space, we introduce intrinsic and external gauging. Intrinsic gauging is fiber gauging, whereas external gauging is base space curvature. The last has gravitional and informational character. Conclusions: 1. The law of conserving energy distribution integral is a universal law, but not that of the distribution itself 2. The conditions for (I) are: (A) to consider only operators, which conserve the presence of measure; (B) to consider only operators, which conserve the integrability of energy distribution. 3.We call intrinsic gauging the transformation of energy distribution within (1), because of(2A). 4. We call base space gauging the transformation of energy distribution within (1 ), because of (2B). 5. We call external gauging the fmal gauging, which includes mtrinsic gauging and base space gauging. 9 6. It is not mandatory to observe the conditions of von Neumann's ergodic theorem (and even more those of the Birkhoff's), but its reduced variant of integral converging to a limit (the integral of conserving measure operators of the function to the integral of projector of the function). 7. The strong, weak, and electromagnetic coupling only approximates the intrinsic gauging, i.e. the measure gauging. 8. "The grand gauging unification" is in principle impossible, since : (A) gravity gauges the base space, but not the fiber of fibered space (fig. 31, fig. 32); (B) the energy ..---- ..;-- distribution is not conserve. The presence of specific gravity objects -like gravitational is ct;/ c!"e~ fL' waves and gravitons - 1B8p8 out in principle. It means that base space domains with nonzero measure and with comprising no fiber points do not exist. 9. The phenomena of (A) quantum gravity are identical with those of (B) quantum information but they are represented from different points of view : (A) Mink.owski space (separable topology, non-isotropic objects); (B) Hilbert space (inseparable topology, isotropic objects). We continue further the next lecture "Curved Hilbert spaces. Quantum information & quantum gravity" (fig. 29, fig. 30). 10 II. Curved Hilbert spaces. Quantum information & quantum gravity Abstract : The general idea is: The curved Hilbert spaces are considered as spaces on curved surfaces. This is a generalization of the approach, which is followed by building ofRiemann geometries. The surface itself of a Descartes product is curved in the case of cutved Hilbert spaces. A curve on the Descartes product sutface corresponds to any function of Hilbert space. The orthogonal basis of Hilbert space is a set of synchronized oscillators. Hilbert space curving means de-synchronization of the oscillators. The de-synchronization coefficient An defmes the deformation of the n-th oscillator. The de-synchronization coefficients set {An} defines a basis of almost periodic functions. The main result is that the n-th oscillator deformation ts expressed just by the de- synchronization coefficient An and the corresponding almost periodic function. The question if a basis of ~]most periodical function is completed, i.e.· the functional space on that basis is a Banach space, was investigated by Norbert Wiener. ln that case exists a scalar, but non-invariant from point to point product. The scalar product change is defined by the Hilbert space curving. The physical meaning consists in constructing a bridge between quantum information and quantum gravity. A curved Hilbert space corresponds to an entangled Hilbert space. A Riemann space corresponds to a curved Hilbert space. Just one Riemann space corresponds to any entangled Hilbert space. Quantum information and quan~um gravity are nothing more than different points of view to one and the same phenomenon. ----. ~ ---- - - ---- ----~ ~ ------ - ' Slide 1 Curved Hilbert spaces Quantum information & quantwn gravity (2+1. DSR) A 2+1 DSR quantum gravity interpretation of EPR Towards an entanglement interpretation of gravity Vasil Penchev, Institute for philosophical research- Sofia, vaspench@abv.bg Slide 2 Literature: 1. Freidel, L., J. Kowalski-Glikman, L. Smolin. 2-r I gravity and doubly special relativity. - Physical Review D 69, 044001 (2004). MaguelJO, J., L. Smolin. String theones with deformed energy-momentum relations. and a possible nontachyonic bosonic strmg. - Physical Review D 7L ~ 026010 (2005). - -:3-. Bohr. LT Can Quantum-mechamcal Description of Physical Realit:y be cons1dered complete?- Physical Review, 1935, 48, 696-702. -1-. Bui!ep, H. P. IT:m:n. I1peo6pa3oBa:HHe <l>ypbe B I<OMrmeKCHOH o6nacm. MocKBa: "Hayxa", 1964. Slide 3 . Vvny 2+ 1 DSR quantum gravity is appropriate to be interpreted as entanglement? 1. The bmding energy between pairs of particles does not depend on the distance between them, but depends only on the md!v1dual energie~ and momenta. 2. Nonlinear addition of energy and momentum. It may be supposed that the entanglement itself has its own energy and momentum. 3. The upper mass limit in DSR models is ex-plained as the maximal degree of entanglement. -t The DSR gravitational energy is independent of the distance between the particles because it is a constant angle ("deficit angle") between the two spaces ofDSR. Slide 4 What means DSR quantum gravity? <\ TwoMmkowsk1 spaces (MS) rotated with an angle (deficit angle) a.s 27t. The 1magmary domam of both MS is interpreted as space-time, and~l domain as energy-momentum "" / / 'C<>ia ting X W• ~h r:J. ~21/ i.mqJinauJ ' cio~ma.in Slide 5 Every MS may be interpreted as a '¥-function (my previous lecture) and it exist<; a corresponding Hilbert space (HS) with the same basis: exp(int). Then the second MS is interpreted as another '¥a. -function and another corresponding Hl~bert space with the basis: exp(iant) Slide 6 DSR quantum gravity interpretation ofEPR: 1. Bohr's (1935) interpretation ofEPR - - - - - - f q : , pt) "" [ q2, P2J = €) \ [q1, q2] = [P~> P2]= [P~> q2J = [CJ,<, pJ] = 0 Let us pass to new canonically conjugated quantities rotated with an angle a in the planes (q], q;::), (p], P2): q1=Q1cosa- Q2sma, p,=P 1cosa- P§ma qz=Qicosa + Q2cosa, p2 =P-sma+ P 2cosa Since also [Q~. P 1] = ih and [Q1.P2] = 0, and Ql=q;cosa+q2sina, P2 =-p 1sina+P2cosa, then, by measuring of Q·,P1 and a following measurement of q2, it is predicted q~, respectively: ofp"· p 1• Slide 7 2. DSR quantum gravity interpretation ofEPR The description above corresponds \Vith two MS rotated with an angle a, i.e. it represents DSR quantum gravity model of EPR. It is clear that this is -;;;.-universal model of entanglement. xa ro 1-1a Slide 8 Curved Hilbert space: with a basis: exp(iant). This is a complete basi: ace ding to Wiener's theorem [4. V since: ------nnn a.n /n = a.< 2n; when oo ' ---------. 31] • The theorem permits to reduce the con . on of a..., namely not an = a, but only Gn -- a. The physical meaning of an, resP. a. ts that of a coe:ffictent of de-synchronization e -------------------- - with respect to those of the first osclllators ··flat" Hilbert space w1th the ··f1aC bas1s: 0 exp(int). st't'de s- Slide 9 ~'hat represents a f1at oscillator ofHS in 1v1S? A reduced image of the light cone with the coefficient ofreducmg ~=v/c. The flat basis of oscillators is all the attached de Broghe waves to every n~ertial reference frame (w1th -/:he imaqi1'1a'l;fi l?az ·t an unit energy). tJf a. ~~4~ ,c:s~, ~ea to~ _ ..... / - e,qnf ~oV>'<' / / / e +he -c t'Cl ;:oa l I "j ii ~ de a. i 0 .sci ea fo? / / Slide 10 What represents a curved oscillator ofCHS inMS? A reduced and rotated unage of the light cone with the coefficient of reducing ~=v/c oes and the angle of cone axes rotating a(deformed Lorenz group) The curved basis of oscillatcm< (the case accelerated reference frame. '-... all the a a 'hed de Broglie Waves to every constantly t / / K ouro ca c:: i<,/o)~'-'t'e,,o 11 / .:t;o.o .....uv~ da. ca c / ~;:>f!<' ffl:n·.7eHe r;:u.J8o 0o. <::a.. ~<.~.....Ve.~u't'UCI- "/€ ip.JJBa da.. <X1 ~c.u...Ll'-1! 7.7£$: - A'l ( d0 /lj' - lO jU /La /C. TC) Cl 3 . c.~...u :t<J ;:-osnu/lou.u.JA) / Slide 11 DSR ;s a 2+ 1 quantum gra-v1ty model. ([1] displays that 2+ 1 ts a DSR model, consequentl.y, 2+ 1 and DSR .are equivalentJD.Q~ (7'\ Let us conSJder the set 'l'"~{C,. a,}. 'l'~C,en•', t'·hen ~ Cn are contra-van ant components represented-etrlhe two real <L'Ces of 2+ 1 MS. ao are' co-variant components represented on the tnlaginary axis of2+1 MS. ~-;;: f)ossilifc connectwn between 2+ 1 ana 3-+-1 MS 1S dJ.scu:-;sea m [11 Slide 12 <-rJ.. Conclusion: Towards an entanglement inteipretation of gravity A tangentiall\1S can be drawn through any pomt of a pseudo Riemann space (PRS). This represents a generalization of Legendre transformation. An entanglement exists between any two points ofPRS because any two tangent1al ~rresponding HS) are rotated with some angle <l.ij· The transformation 0 =r_~S;\HJ is not linear operator. 0 deforms the distribution of energy-momentum. •

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