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Vasil Penchev. Princeton ideas renewed to interpret wave function in the terms of quantum information. Curved Hilbert spaces

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Vasil Penchev. Princeton ideas renewed to interpret wave function in the terms of quantum information. Curved Hilbert spaces Powered By Docstoc
					            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).
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        Conclusions:
        1. The law of conserving energy distribution integral is a universal law, but not
that of the distribution itself
        2. 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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