Transcript Gravity as Entanglement Entanglement as gravity Vasil Penchev, DSc, Assoc. Prof, Bulgarian Academy of Science • • • • [email protected] http://vasil7penchev.wordpress.com http://www.scribd.com/vasil7penchev CV: http://old-philosophy.issk-bas.org/CV/cvpdf/V.Penchev-CV-eng.pdf.

Gravity as Entanglement
Entanglement as gravity
Vasil Penchev, DSc, Assoc. Prof,
Bulgarian Academy of Science
•
•
•
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[email protected]
http://vasil7penchev.wordpress.com
http://www.scribd.com/vasil7penchev
CV: http://old-philosophy.issk-bas.org/CV/cvpdf/V.Penchev-CV-eng.pdf
Our objectives are:
• To investigate the conditions under which
the mathematical formalisms of general
relativity and of quantum mechanics go
over each other
• To interpret those conditions meaningfully
and physically
• To comment that interpretation
mathematically and philosophically
Scientific prudence, or what are not
our objectives:
• To say whether entanglement and gravity are
the same or they are not: For example, our
argument may be glossed as a proof that any
of the two mathematical formalisms needs
perfection because gravity and entanglement
really are not the same
• To investigate whether other approaches for
quantum gravity are consistent with that if any
at all
Background
• Eric Verlinde’s entropic theory of gravity (2009):
“Gravity is explained as an entropic force caused
by changes in the information associated with
the positions of material bodies”
• The accelerating number of publications on the
links between gravity and entanglement, e.g.
Jae-Weon Lee, Hyeong-Chan Kim, Jungjai Lee’s
“Gravity as Quantum Entanglement Force” :
“We conjecture that quantum entanglement of
matter and vacuum in the universe tend to increase
with time, like entropy …
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”
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)
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:
3D space is homeomorhic to a
cyclic 3+1 topological structure
like the 3-sphere: e.g. the
cyclically connected Minkowski
space
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
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:
Background: The Higgs boson
"No quantum gravity!" and it is a new very
strange and amazing principle of nature
If the best minds tried a century to invent
quantum gravity and they did not manage to
do it, then it merely means that quantum
gravity does not exist in principle
So that no sense in persisting to invent the
"perpetuum mobile" of quantum gravity,
however there is a great sense to build a new
theory on that new principle:
Background: The Higgs 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
General relativity vs. the standard model
Interaction, Force,
Energy (mass)
Inertial mass
Gravitational mass
Gravitational ones
Inertial mass is the measure General relativity,
of resistance vs. the action
which is smooth
Their mechanical
action
of any force field.
Gravitational mass is
The weak,
the measure of gravity action electromagnetic,
And what about
entanglement and
inertial mass?
strong
ones
The standard model,
which is quantum
Our strategy on that background is...
1. ... to show that entanglement is another and
equivalent interpretation of the mathematical
formalism of any force field (the right side of the
previous slide)
2. ... to identify entanglement as inertial mass (the
left side)
3. ... to identify entanglement just as gravitational
mass by the equality of gravitational and inertial
mass
4. ... to sense gravity as another and equivalent
interpretation of any quantum-mechanical
movement and in last analysis, of any mechanical
(i.e. space-time) movement at all
If we sense gravity as another and equivalent
interpretation of any movement, then ...
Complex probability distribution = Two probability
entangledistributions
ment
Complex
Energy-momentum
Banach
(Hilbert)
Space
Space-time
trajectory
quantum PseudoRiemanian
force
basis
field
It does not and cannot reThe standard model repre- present gravity because it
sents any quantum force is not a quantum field at
field: strong, electromag- all: It is the smooth image
of all quantum fields
netic, or weak field
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 standard model
language as a theory of unified quantum field
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
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”
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
Which are the troubles?
• The “cure” for them is to be generalized
correspondingly the notion of quantity in
quantum mechanics. If the operator is in
Banach space (correspondingly, yet no
selfadjoint operator), then its functional is a
complex number in general. Its modulus is the
value of the physical quantity. The expectation
of two quantities is nonadditive in general
(more exactly, it is not superadditive).
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 like quantities in the two entangled
quantum systems 1 and 2. It will be recalled 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=
−∞
∞
𝑎𝑝𝑎 𝑑𝑎 =
𝚿𝐴 (𝚿 ∗)
−∞
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)
More and more about the “cure”
Then what is gravity (for God’s sake! )?
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
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
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 and in our terms, the
“second quantization” excludes as entanglement as
gravity in principle
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: It 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
A philosophical interpretation both of
quantum (I) and of quantized (II) field
Quantum vs. quantized field means for any spacetime point to juxtapose the Hilbert space vs. its
division into subspaces
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
group of the subspaces mappings of Hilbert space
The standard model describes the general and
complete group including all the “strong”,
“electromagnetic” and “weak” symmetries
A philosophical interpretation of the
standard model closedness
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
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
And now, from the philosophical
to the mathematical and physical ...:
A wave
function
Hilbert space
A space-time
trajectory
Minkowski space
A Yin-Yang mathematical structure
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
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
be described above
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 spacetime and vice versa
Back to the philosophical interpretation of
quantum (I) or quantized (II) field
The principle is: The global change of a spacetime trajectory (or an operator in pseudoRiemannian space) is equivalent to, or merely
another representation of a mapping between two
local Hilbert spaces of Banach space
(entanglement)
The same principle from the viewpoint of quantum
mechanics and information looks like as follows:
Entanglement in the “smallest” returns and comes
from the “outsides” of the universe, i.e. from the
“biggest”, as gravity
Back to the philosophical interpretation,
or more and more miracles
Turns out the yet “innocent” quantum duality
generates more and more already “vicious”
dualities more and more extraordinary from each
to other, namely:
... of the continuous (smooth) & discrete
... of whole & part
 ... of the single one & many
... of eternity & time
... of the biggest & smallest
... of the external & internal
... and even ... of “&” and duality
... where “&”
means ...
...
equivalence
... relativity
... invariance
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
Hilbert space with its points, which is the
definition of entanglement
However the next “however” is that we can always
determine a self-adjoint operator (i.e. a physical
quantity) between any two points in Banach space
(i.e. between the two corresponding tangential
Hilbert spaces mapping by the operator)
The second quantization in
terms of Banach space
If we can always determine a self-adjoint operator
(i.e. a physical quantity) between any two points in
Banach space, then follows the second
quantization is invariant (or the same) from
Hilbert to any smooth Banach space, and vice
versa, consequently between any two smooth
Banach spaces
As entanglement as gravity is only external, or
both are “orthogonal” to the second quantization:
It means that no any interaction or unity between
both gravity and entanglement, on the one hand ...
The second quantization in
terms of Banach space
As entanglement as gravity is only external, or
both are “orthogonal” to the second quantization:
It means that no any interaction or unity between
both gravity and entanglement, on the one hand,
and the three rest, on the other, since the latters
are within Hilbert space while the formers are
between two (tangential) Hilbert spaces
However as entanglement as gravity can be
divided into the second-quantized parts
(subspaces) of the Hilbert space, which
“internally” is granted for the same though they
are at some generalized “angle” “externally”
The problem of Lorentz invariance
Try to unite the following facts:
The Lorentz noninvariant are:
Schrödinger’s
Newton’s mechanics
quantum mechanics
The Lorentz invariant are:
Maxwell’s theory of
Dirac’s
electromagnetic field
quantum mechanics
Einstein’s special relativity of electromagnetic field
The locally Lorentz invariant
(but noninvariant globally) are:
Our hypothesis
Einstein’s general relativity
of entanglement &
gravity
Relativity
Quantum theory
... whether gravity is not a “defect” of
electromagnetic field...
However mass unlike electric (or Dirac’s magnetic)
charge is a universal physical quantity which
characterizes anything existing
However the next however is that a perfect, “YinYang” symmetry would require as the locally “flat” to
become globally “curved” as the locally “curved” to
become globally “flat” before as the “biggest” to
return back as the smallest and locally “flat”
For example this might mean 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
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:
photon.
As to the electromagnetic field, both global and
local (in the standard model) consideration is
possible
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
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:
Entanglement & gravity being only global has no
quantum, thus neither mass at rest nor a finite
speed in spacetime
Weak & strong interaction being only local has
quanta both with a nonzero mass at rest and with a
finite speed in spacetime
Lorentz invariance has
a local and a global face, too:
In turn, this generates the two faces of photon
The local “face” of Lorentz invariance is as within as
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
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:
That is a quantum kind of
generalization. Why on Earth?
First, the decomposition into a magnetic and a
electric component is not a decomposition of two
spinors because the electromagnetic field is rather
the vector 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
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
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 as it arose keeping and
generalizing just 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 on the next slide:
“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 𝒙𝟐 + 𝒚𝟐 + 𝒛𝟐 + 𝒗 𝟐 𝒕𝟐 ,
it can represent any corresponding reference frame
in Minkowski space. 𝑺 can be decomposed into any
two great circles 𝑺𝟏 ⨂𝑺𝟐 of its, perpendicular to
each other, as the tensor product ⨂ of them
“Reference frame” after the Dirac
approach
Given a sphere 𝑺 with radius 𝒙𝟐 + 𝒚𝟐 + 𝒛𝟐 + 𝒗 𝟐 𝒕𝟐
decomposed into any two great circles 𝑺𝟏 ⨂𝑺𝟐 of
its, 𝑺, 𝑺𝟏 , 𝑺𝟐 are with the same radius. We can think
of 𝑺𝟏 , 𝑺𝟐 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
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!
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
1
𝑡
and its reciprocal: 𝑓 𝑡 ↔ 𝑓
= 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) = 𝑐 2 𝑡 2 − 𝑥 2 − 𝑦 2 − 𝑧 2
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 smooth
entering the corresponding value
A necessary elucidation of the connection
between probabilistic (mathematical) and
mechanical (physical) approach
No axiom of choice (the Paradise)
Probabilistic (mathematical) approach
Totality aka eternity aka infinity
Mechanical (physical) approach
Hilbert space:
from the “Paradise”
to the “Earth “
by the “stairs”
of energy
Both
need
choice
(axiom)
Minkowski space:
from the “Earth”
to the “Paradise“
by the “stairs”
of time
Coherent state, statistical ensemble, and
two kinds of quantum statistics
• The process of measuring transforms the
coherent state into a classical statistical
ensemble. Consequently, it requires the axiom of
choice
• However yet the mathematical formalism of
Hilbert space allows two materially different
interpretations corresponding to the two basic
kinds of quantum statistics, of quantum
indistinguishability, and of quantum particles:
bosons and fermions
The axiom of choice as the boundary
between bosons and fermions
The two interpretations of a coherent state
mentioned above are:
As a nonordered ensemble of complex (= two
real ones) probability distribution after
missing the axiom of choice – aka boson
As a well-ordered series either in time or in
frequency (energy) equivalent to the axiom of
choice – aka fermion
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 quantuties
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 unlike the case of boson
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
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 wellordering 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
Positive integers vs. fermions vs.
bosons illustrated
Initial
state
...
Swap
...
After
a time
...
Naming
.....
.....
.....
...
1, 2, 3, 4, 5, ... 1, 2, 3, 4, 5, ...
...
1, 5, 3, 4, 2, ... 1, 5, 3, 4, 2, ...
...
1, 2, 3, 4, 5, ... 1, 5, 3, 4, 2, ...
True
“Weak”
indistinguish- (in)distinguishability
ability
Quantum indistinguishability
Bosons
Fermions
True
distinguishability
Positive
integers
Quantum vs. classical movement in terms
of (quantum in)distinguishability
𝒕
Dynamic (boson) Static (fermion)
aspect: true in- aspect: weak inDynamic
distinguishability distinguishability (momentum)
aspect
Wave function as Wave function
the characteristic as a (wellStatic
function of
ordered)
(position)
a random
aspect
complex quantity vector
Pseudo𝑴𝒖𝒄𝒉 ⟺ 𝑴𝒂𝒏𝒚
Riemannian
space
Hilbert space
Quantum indistinguishability Distinguishability
Dynamic to static
Dynamic to static
aspect: one to one
aspect: much to many
Quantum movement
Classical movement
Our interpretation of fermion
antisymmetry vs. boson symmetry
The usual interpretation suggests that both
the fermion and boson ensembles are wellordered: 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
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
“fermion” image of it
In turn this hides the essence of quantum
movement, which is “much – many”,
substituting it with a semi-classical “many
– many”
What will “spin” be in our interpretation?
In particular, a new, specifically quantum quantity,
namely “spin”, is added to distinguish between 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 wellorders 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
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
Entangled observable in terms of
“spin”distinction
The standard definition of quantum quantity as
“observable” allows its understanding:
 as a “fermion – fermion” transform,
as a “boson – boson” one
as well as “fermion – boson” and
“boson – fermion” one
Only entanglement and gravity can create
distinctions between the former two and the latter
two cases. Those distinctions are recognizable only
in Banach space, but vanishing in Hilbert space
The two parallel phases of quantum
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
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
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 pseudoRiemannian space as well as one, already started,
of quantum and classical movement
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
Hilbert vs. pseudo-Riemannian space:
completing the puzzle
OppoProcess
in
time
sition
PseudoRiemannian
Curve space
Gravity, General
relativity
Minkowski
space
Flat Electromagnetism
Special relativity
Actual infinity
Banach space
Entanglement
Quantum information
Hilbert space
Electromagnetic, weak
and strong interaction
Quantum mechanics
The standard model
Our thesis in terms of that table
PseudoRiemannian
space
Curve
,
General relativity
Gravity
Banach space
Entanglement
Quantum information
Entanglement is gravity as a complete process
Gravity is entanglement as a process in time
A fundamental prejudice needs
elucidation not to bar:
The complete wholeness of any process is „more“
than the same process in time, in development
Actual infinity is “more” than potential infinity
The power of continuum is “more” than the power
of integers
The objects of gravity are bigger than the objects
of quantum mechanics
The bodies of our everyday world are much
“bigger” than the “particles” of the quantum
world, and much smaller than the universe
Why is that prejudice an obstacle?
According to the first three statements
entanglement should be “more” than gravity
intuitively
However according to the second two statements
gravity should be much “smaller” than
entanglement intuitively
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
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 an one
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:
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
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)
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
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 though, it would sound as Wiji in
“terms” of Taiji, or Yin in “terms” of Yang
Fortunately, the very well developed mathematics
of the West includes enough bridges to think of
whole linearly: The most essential and important
link among them is the axiom of choice
The axiom of choice self-referentially
The choice of all the choices is to choose the choice
itself, i.e. the axiom of choice itself , or in philosophical 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
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 of reversed spin: In
other words, it does “outsides” after a 2𝜋 rotation
in a smooth trajectory passing along the half of the
universe. Look at it on a Möbuis strip
A “Möbius” illustration of how a smooth
trajectory can reverse the spin
𝟏
+
𝟐
fermion
𝟏) 𝟎; 𝟎°
𝟐) 𝝅; 𝟏𝟖𝟎°
𝟏
−
𝟐
fermion
𝟑) 𝟐𝝅; 𝟑𝟔𝟎°
𝟒) 𝟑𝝅; 𝟓𝟒𝟎°
𝟏
+
𝟐
fermion
𝟓) 𝟒𝝅; 𝟕𝟐𝟎°
a the same
fermion
“inside” “outside”
the universe
Exactly the half of the universe between
two electrons of a helium atom
Here is a helium atom. Exactly the half
of the universe is inserted between its
two electrons differ from each other
only with reversed spin:
The West thinks of the universe as
the extremely immense, and of the+ 𝟏 fermion − 𝟏 fermion
𝟐
𝟐
electrons and atoms as the extremely tiny. However as quantum
mechanics as Chinese thought
shows that they pass into each other
everywhere and always
The universe
Taiji 太極 is the Chinese transition between the
tiniest and the most immense
The West's single pathway
along or through Taiji
is mathematics, though
𝟏
+
𝟐
fermion
𝟏
−
𝟐
fermion
A fortunate
exception is
Nicolas of Kues
How on Earth is it possible?
Mathematics offers the universe to be considerer 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(s)
(quantum field, bosons), and (2) as a vector (or a
square integrable function)
Taiji 太極 in the language of mathematics
𝟏
+
𝟐
fermion
He
One single
boson!!!!
𝟏
−
𝟐
fermion
Wave function interpreted
Wave function
as a characteristic function
as a vector
The common and universal Hilbert (Banach) space
Taiji 太極 in the language of mathematics
The axiom of choice
𝟏
Scolem’s
+ fermion
𝟐
“paradox”
He
One single
boson!!!!
𝟏
−
𝟐
fermion
Wave function interpreted
Wave function
as a characteristic function
as a vector
The common and universal Hilbert (Banach) space
Wuji 無極 as the Kochen-Specker theorem
Taiji 太極
The axiom of choice
Scolem’s
Quantum
computer
“paradox”
𝟎
One single
qubit!!!!
The universe of
(or as) sundry
Tu r i n g a l g o r i t h m s
𝟏
A most and most
ordinary bit
one single bit
Its point interpreted
Its point
as a characteristic function
as a vector
The common and universal Hilbert (Banach) space
The mapping between numbers and a
sundry
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
 a “field”: 𝑆2
𝑓2
𝑆1
𝑓1
𝑆2
Quantum mechanics solves the general
problem under those assumptions
The general problem is the quantitative description
of the universe: too complicated!
Well-orderable
Wave
numbers
function
as a field
Wave
function
as a
probability
distribution
All the universe
as a sundry
The general problem
in terms of Taiji and Wuji
The simplifying solving
of quantum mechanics
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
Locally
Metrically
flat
Topologically
two-side
Globally
curved
one-side
Möbius strip as a fiber bundle
“circle” for bundle
“radius” for fiber, F
the same “radius”
from the “other
side” for base
The definition of “fiber bundle” by the
example of a Möbuis 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
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 𝑨𝒔 , 𝑩𝒔 are the same “radiuses” on
the sheet. Then the fiber bundle is described as the
triangle of mappings for any 𝑨, 𝑩, 𝑨𝒔 , 𝑩𝒔 as follows:
𝑨
𝑨⨂𝑩
„𝑨⨂𝑩“ means
Cartesian product
𝑩
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
The leading idea of gauge theories
Let us fancy the two “radiuses” or Hilbert spaces 𝐴
and 𝐵 correspondingly as the reference and gauge
mark of an uncalibrated indicator, and 𝐴𝑠 and 𝐵𝑠 are
the same after the precise calibrating:
Fiber bundle
Cartesian product
The Standard
0
0
Model
𝑨
𝑨𝒔 ≡ 𝑩𝒔
𝑩
an uncalibrated indicator
the indicator calibrated
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
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
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 
Our suggestion to change the gestalt:
the physical picture of the world
• Its essence is: a new invariance of discrete and
continual (smooth) mechanical movements and
their corresponding morphisms in mathematics
• This means a generalization of Einstein’s (general)
principle of relativity (1918): “Relativitätsprinzip:
Die Naturgesetze sind nur Aussagen über
zeiträumliche Koinzidenzen; sie finden deshalb
ihren einzig natürlichen Ausdruck in allgemein
kovarianten Gleichungen.“
An equivalent reformulation of
Einstein’s principle of relativity:
All physical laws must be invariant to any 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
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 “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 true
fundament of mathematics requiring the need of
changing gestalt mentioned above
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 more 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”
The inevitability of the axiom of choice
in quantum mechanics
The axiom of choice in quantum mechanics is wellknown 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”
The Kochen − Specker theorem is the most
general “no hidden variables” theorem:
Its essence: wave-particle duality in quantum mechanics 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
A quantum wholeness is not equivalent to the set or
sum of its parts; quantum logic is not a classical one
The “quantum wholeness” of the axiom 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
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:
The “time” interpretation of coherent
state and decoherence:
The de Broglie wave periods of the measuring
apparatus 𝑇𝑎 and of the measured quantum entity
(𝑇𝑒 ) correspondingly:
𝑇𝑎 =
ħ 1
; 𝑇𝑒
2
𝑐 𝑚𝑎
=
ħ 1
;∴
2
𝑐 𝑚𝑒
𝑇𝑎
𝑇𝑒
=
𝑚𝑒
𝑚𝑎
≈0
Consequently, coherent state corresponds to 𝑇𝑒 , and
decoherence to 𝑇𝑒 𝑇𝑎 , i.e. − to a random choice of a
(≈) point among the continual interval of 𝑇𝑒
Now, we can explain the difference between a
coherent state and a statistical ensemble so:
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
The chain of sequences from Skolem’s
paradox to our generalization of Einstein’s
relativity principle :
Scolem’s paradox
hidden variables”
The axiom of choice
“No
𝑇ℎ𝑒 𝐾𝑜𝑐ℎ𝑒𝑛−𝑆𝑝𝑒𝑐𝑘𝑒𝑟 𝑡ℎ𝑒𝑜𝑟𝑒𝑚
𝑇ℎ𝑒 "𝑡𝑖𝑚𝑒" 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛
Wave-
particle duality
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
A few comments: the first one: waveparticle 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
A second comment: wave-particle invariance 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
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 of both a complex probability and 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
A short comment on the postulate:
Consequently when we measure a quantum
quantity, we lose information. Any quantum
probability distribution is reduced to a statistical
ensemble. The principle of complementary
forbids the question about the lost information.
The most natural hypothesis is that as the two
components as their corresponding probability
distributions coincide. This conjecture founded by
the axiom of choice in quantum mechanics adds
wave-particle invariance to wave-particle duality.
More about the embedding of waveparticle 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.
𝜋
Note that is an anti-isometric rotation. The same
2
physically interpreted is the wave-particle invariance
in question. Its really necessary condition is only
Skolem’s “paradox”. However whether is not the last
also a sufficient condition for it?
A set-theory generalization of waveparticle 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:
ℚ
ℍ
𝕀
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 interpretation: as a
(“covariant”) set of harmonics as a (“contravariant”)
set of points, the two sets being anti-isomorphic (antiisometric measurable). Formally, we can yield that
interpretation by another physical interpretation of a
function and its Fourier transformation:
1
𝑓 𝑥
F
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
𝑥 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛
1
𝑓(𝑡)
𝐹
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
𝑡
Is any mathematical model there, which
can coincide with the modeled reality?
𝑓 𝑥
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
1
F
𝑥
𝑎 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛
1
𝑓(𝑡)
𝐹
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
𝑡
𝑓 𝑥
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
1
F
𝑥
𝑎𝑛𝑜𝑡ℎ𝑒𝑟 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛
1
𝑝(𝐴)
𝐹
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
𝐴
A philosophical interlude about the logical
equivalence of two physical interpretations
1
𝑓 𝑥
F
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
𝑥 𝑎 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛
1
𝑓(𝑡)
𝐹
− 𝑡𝑖𝑚𝑒 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
𝑡
1
𝑓 𝑥
F
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
𝑥 𝑎𝑛𝑜𝑡ℎ𝑒𝑟 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛
1
𝑝(𝐴)
𝐹
− 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
𝐴
Let be the former physically interpreted as
any quantity − the argument in the latter:
𝑓 𝑥
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
1
F
𝑥
𝑎 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛
1
𝐴(𝑡)
𝐹
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
𝑡
𝑓 𝑥
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
1
F
𝑥
𝑎𝑛𝑜𝑡ℎ𝑒𝑟 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛
1
𝑝(𝐴)
𝐹
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
𝐴
Besides, let be functions of the same
argument physically interpreted as time:
𝑓 𝑥
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
1
F
𝑥
𝑎 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛
1
𝐴(𝑡)
𝐹
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
𝑡
𝑓 𝑥
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
1
F
𝑥
𝑎𝑛𝑜𝑡ℎ𝑒𝑟 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑝𝑟𝑒𝑡𝑎𝑡𝑖𝑜𝑛
1
𝑝[𝐴 𝑡 ]
𝐹
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
𝐴(𝑡)
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) , e.g. if A is distance −
𝑚
𝐻𝑧.
𝑠
=
𝑚
𝑠2
~𝑎 ∴
𝑎 𝑝 𝐴 𝑡 .𝐴(𝑡)
𝐺~ ~
𝐴
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
Parseval’s theorem
𝑓 𝑥
𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚
𝑔 𝑥
𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚
𝐹 𝑦
𝐺 𝑦
∞
⟺
𝑢𝑛𝑑𝑒𝑟 𝑎 𝑓𝑒𝑤 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠
∞
𝑓(𝑥) . 𝑔 𝑥 . 𝑑𝑥 =
−∞
𝐹 𝑦 . 𝐺(𝑦). 𝑑𝑦
−∞
Parseval’s theorem about the
generalization of a quantum quantity
𝑌 and of its conjugate quantity 𝑋
𝐹 𝑦
𝑎 𝑠𝑒𝑙𝑓𝑎𝑑𝑗𝑜𝑖𝑛𝑡 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟
𝐹(𝑦)
𝑎𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟
𝐺 𝑦
𝐺 𝑦
∞
⟺
𝑢𝑛𝑑𝑒𝑟 𝑎 𝑓𝑒𝑤 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠
∞
𝑓 𝑥 . 𝑔 𝑥 . 𝑑𝑥 =
−∞
𝐹 𝑦 . 𝐺 𝑦 . 𝑑𝑦 ⟺
−∞
⟺ 𝑋 = 𝑌 (the so-called wave-particle invariance)
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)
Obviously Parseval’s theorem is due to the
“flatness” of Hilbert space. To get it
“curved” into Banach one?
𝑭(𝒚)
𝐅(𝐲)
𝒇 𝒙
𝒇 𝒙
𝒈 𝒙
𝑮(𝒚)
𝑮(𝒚)
𝒈 𝒙
Fourier transform by 3D Cartesian product
𝒙 𝑭
𝑭𝒐𝒖𝒓𝒊𝒆𝒓
𝒇(𝒙)
𝒇
𝒇(𝒙)
𝑫𝒖𝒂𝒍
𝒔𝒑𝒂𝒄𝒆
𝑭𝒐𝒖𝒓𝒊𝒆𝒓
𝑺𝒑𝒂𝒄𝒆
𝒙(𝑭)
𝑭
𝟑𝑫 𝑪𝒂𝒓𝒕𝒆𝒔𝒊𝒂𝒏 𝒑𝒓𝒐𝒅𝒖𝒄𝒕
Riesz representation theorem by 3D
Cartesian product
𝒇
𝒇(𝒙)
𝒙(𝑭)
𝑭
𝑭
𝑭, 𝒙, 𝒇 𝒂𝒓𝒆 𝒕𝒉𝒆 𝒄𝒐𝒏𝒋𝒖𝒈𝒂𝒕𝒆
𝒄𝒐𝒎𝒍𝒆𝒙 𝒓𝒂𝒕𝒉𝒆𝒓 𝒕𝒉𝒂𝒏
𝟑𝑫 𝑪𝒂𝒓𝒕𝒆𝒔𝒊𝒂𝒏 𝒑𝒓𝒐𝒅𝒖𝒄𝒕
𝒏𝒆𝒈𝒂𝒕𝒊𝒗𝒆 𝒂𝒙𝒆𝒔
𝒇
About 𝒇(𝑭) and the coincidence of 𝒇(𝒙),
𝒙(𝑭), and 𝒇(𝑭) in form
𝒇
The zest is
what about
Banach space!
𝑭
𝒙(𝑭)
A functional
∀ 𝒙 𝑭 ,𝒇 𝒙 ,𝒇 𝑭 :
𝒙 𝒇(𝑭) ≡ 𝒇[𝒙 𝑭 ]
The plane determined by the three “points”
𝑭, 𝒙, 𝒇, is getting curved into …
(please imagine it )
For those not imaging it:
Now the case is:
No entanglement ⟺ ℍ𝟏 ⨂ℍ𝟐 = ℍ𝟑 ⟺ No gravity
(ℍ𝟑 is the Hilbert space of the compound system ℍ𝟏 &ℍ𝟐 )
The “surface” of Banach space
𝒇
𝒇
𝒙
𝒙
𝑭
(𝑭, 𝒙, 𝒇)
ℍ1
The planes (𝒇, 𝒙, 𝒇, 𝒙) represent three Hilbert spaces ℍ2
ℍ3
(𝑭 ≡ 𝑭, 𝒇𝒙)
tensor product
such as: ℍ1 ⨂ℍ2 = ℍ3
However the case in general is:
Entanglement ⟺ ℍ𝟏 ⨂ℍ𝟐 ≠ ℍ𝟑 ⟺ No gravity
(ℍ𝟑 is the Hilbert space of the compound system ℍ𝟏 &ℍ𝟐 )
The “surface” of Banach space
𝒇
𝑭
𝒙
𝒇
𝒙
ℍ𝟏 = (𝑭, 𝒙, 𝒇)
The “planes” ℍ𝟐 = (𝒇, 𝒙, 𝒇, 𝒙) form an arbitrary triangle:
ℍ3 = (𝑭 ≡ 𝑭, 𝒇𝒙)
Such that ℍ𝟏 ,ℍ𝟐 are not orthogonal to each other in general
(i.e. they may be in particular)
The different perspectives on Hilbert
and Minkowski space
In fact the two spaces are the same space seen in
different perspectives:
𝟐𝝅
 As Hilbert space by frequency, 𝝎 = ,
𝒕
 As and Minkowski space by time, 𝒕
Indeed, we can compare the “atoms” of their bases:
Continuous perspective:
Discrete perspective:
(countable)
expanding
in time
Minkowski space
???
(countable)
expanding
in frequency
Hilbert space
The different perspectives on an impulse −
a trajectory: Hilbert − Minkowski space
a trajectory
an impulse
𝑡
𝑡
a world
line
𝑭𝒐𝒖𝒓𝒊𝒆𝒓 𝒕𝒓𝒂𝒏𝒔𝒇𝒐𝒓𝒎
𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝒅𝒖𝒂𝒍 𝒔𝒑𝒂𝒄𝒆𝒔
Continuous perspective:
(countable)
expanding
in time
Minkowski space
a quantum
leap
Discrete perspective:
(countable)
expanding
in frequency
Hilbert space
Hilbert − Minkowski space:
wave particle duality
a trajectory
an impulse
𝑡
𝑡
a world
line
𝑭𝒐𝒖𝒓𝒊𝒆𝒓 𝒕𝒓𝒂𝒏𝒔𝒇𝒐𝒓𝒎
𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝒅𝒖𝒂𝒍 𝒔𝒑𝒂𝒄𝒆𝒔
a particle moving
continuously
in that trajectory
well-ordered by time
Minkowski space
a quantum
leap
a wave function
simultaneous in all
the space
Hilbert space
Hilbert − Minkowski space: a perfect
symmetry of positions and probabilities
a trajectory
an impulse
𝑡
𝑡
a particle moving
continuously
in that trajectory
well-ordered by time
a wave function
simultaneous in all
the space but wellordered in frequency
However the particle
trajectory is a singular
mix of frequencies
However the wave
function is a singular
mix of positions
The quadrilateral: Hilbert – Banach –
Minkowski – pseudo-Riemannian space
Banach space
Hilbert space
𝑐𝑢𝑟𝑣𝑖𝑛𝑔
Fourier transform
𝑓𝑙𝑎𝑡𝑡𝑒𝑛𝑖𝑛𝑔
𝑐𝑢𝑟𝑣𝑖𝑛𝑔
𝑓𝑙𝑎𝑡𝑡𝑒𝑛𝑖𝑛𝑔
Pseudo-Riemannian
space
Minkowski space
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
varing scalar product
depending on the space points
Banach space as curved Hilbert space:
The change of the scalar product in each point
can be interpreted as a function of
the curvature in that point
The known sides of the quadrilateral:
as Minkowski – pseudo-Riemannian space
as Hilbert – Banach space
Minkowski
space
𝒂𝒅𝒅𝒆𝒅 𝒔𝒄𝒂𝒍𝒂𝒓 𝒑𝒓𝒐𝒅𝒖𝒄𝒕
𝒊𝒏𝒗𝒂𝒓𝒊𝒂𝒏𝒕 𝒕𝒐 𝒕𝒉𝒆 𝒔𝒑𝒂𝒄𝒆 𝒑𝒐𝒊𝒏𝒕𝒔
pseudo−
𝐑𝐢𝐞𝐦𝐚𝐧𝐢𝐚𝐧
space
varing 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
The close analogy of the two transforms
as different views on the same transform:
Hilbert
space
Minkowski
space
𝒔𝒄𝒂𝒍𝒂𝒓 𝒄𝒖𝒓𝒗𝒊𝒏𝒈
𝒔𝒄𝒂𝒍𝒂𝒓 𝒇𝒍𝒂𝒕𝒕𝒆𝒏𝒊𝒏𝒈
𝒔𝒄𝒂𝒍𝒂𝒓 𝒄𝒖𝒓𝒗𝒊𝒏𝒈
𝒔𝒄𝒂𝒍𝒂𝒓 𝒇𝒍𝒂𝒕𝒕𝒆𝒏𝒊𝒏𝒈
Banach
space
pseudo−
𝐑𝐢𝐞𝐦𝐚𝐧𝐧𝐢𝐚𝐧
space
We can use the two perspectives mentioned
above, on Hilbert − Minkowski space:
frequency − time:
The two transforms as the same transform
Hilbert
space
𝒔𝒄𝒂𝒍𝒂𝒓 𝒄𝒖𝒓𝒗𝒊𝒏𝒈
𝒔𝒄𝒂𝒍𝒂𝒓 𝒇𝒍𝒂𝒕𝒕𝒆𝒏𝒊𝒏𝒈
𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚
𝑭𝒐𝒖𝒓𝒊𝒆𝒓 − 𝒍𝒊𝒌𝒆
𝒕𝒓𝒂𝒏𝒔𝒇𝒐𝒓𝒎
𝒕𝒊𝒎𝒆
Minkowski
space
Banach
space
𝒔𝒄𝒂𝒍𝒂𝒓 𝒄𝒖𝒓𝒗𝒊𝒏𝒈
𝒔𝒄𝒂𝒍𝒂𝒓 𝒇𝒍𝒂𝒕𝒕𝒆𝒏𝒊𝒏𝒈
pseudo−
𝐑𝐢𝐞𝐦𝐚𝐧𝐧𝐢𝐚𝐧
space
2’
1’
dual space
…
Shifting&
rotating
of each
corresponding
sphere
in the
dual
space
n
…
n’
time
frequency
… …
… …
The curving or flattening in both cases: one
space
2
1
The curving or flattening in the first case:
two comments
time
frequency
1) It is the first case what one knows till now:
The “curved” pseudoRiemannian space
of general relativity
The “flat” Hilbert space
of quantum mechanics
2) A philosophical reflection on the quantum
mapping of infinity: The actual infinity of a time
series is mapped as the actual infinity of a 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 !
2’
1’
dual space
…
Shifting&
rotating
of each
corresponding
sphere
in the
dual
space
n
…
n’
frequency
time
… …
… …
The curving or flattening in both cases: two
space
2
1
The curving or flattening in the second
case: two comments
The “curved”
Banach space
of entanglement
frequency
time
1) It is the second case what one would emphases:
The “flat” Minkowski
space of special relativity
2) A methodological reflection on the equavalence of
both cases: “No need of quantum gravity!”, or: Entanglement represents quantum gravity fully. Of course,
does one wish, both spaces could be curved, and a
partial degree of entanglement might be combined
with a corresponding partial degree of gravity
The unknown sides of the quadrilateral:
as Hilbert – Minkowski space
as Banach – pseudo-Riemannian space
𝐇𝐢𝐥𝐛𝐞𝐫𝐭
𝐨𝐫 𝐁𝐚𝐧𝐚𝐜𝐡
𝐬𝐩𝐚𝐜𝐞
𝒕𝒊𝒎𝒆
𝐌𝐢𝐧𝐤𝐨𝐰𝐬𝐤𝐢 𝐨𝐫
𝐩𝐬𝐞𝐮𝐝𝐨 − 𝐑𝐢𝐞𝐦𝐚𝐧𝐧𝐢𝐚𝐧
𝐬𝐩𝐚𝐜𝐞
𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚
𝒂𝒔 𝒂 𝒕𝒓𝒂𝒋𝒆𝒄𝒕𝒐𝒓𝒚
Discreteness 𝑨𝒙𝒊𝒐𝒎 𝒐𝒇 𝒄𝒉𝒐𝒊𝒄𝒆 Continuity
𝒂𝒔 𝒂𝒏 𝒊𝒎𝒑𝒖𝒍𝒔𝒆
𝒂𝒔 𝒅𝒊𝒔𝒄𝒓𝒆𝒕𝒆 − 𝒄𝒐𝒏𝒕𝒊𝒏𝒖𝒊𝒕𝒚
𝒂𝒔 𝒅𝒖𝒂𝒍𝒊𝒕𝒚
⊠
𝒂𝒔 𝒘𝒂𝒗𝒆 − 𝒑𝒂𝒓𝒕𝒊𝒄𝒍𝒆
𝒂𝒔 𝒊𝒏𝒗𝒂𝒓𝒊𝒂𝒏𝒄𝒆
n’
2’
1’
dual space
…
position
A body …⍟
Shifting&
rotating
n
of each
corresponding
sphere
in the
dual
2
space
…
… …
… …
momentum
⍟… in the gravitational
field
frequency (energy)
time
The sides of the quadrilateral one by one:
1| Minkowski – pseudo-Riemannian space
space
1
The quadrilateral one by one: Minkowski –
pseudo-Riemannian space: conclusion
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
dual space
The “flat”
Minkowski space
includes
the space-time
trajectory
of the body
space
n’
2’
1’
dual space
…
probability
The wave function of
Shifting& anything…⍟
rotating
of each
corresponding
sphere
in the
dual
space
n
…
… …
… …
probability
⍟… in entanglement
position
position
The sides of the quadrilateral one by one:
2| Hilbert – Banach space
space
2
1
The quadrilateral one by one: Hilbert –
Banach space: conclusion
The “curved”
Hilbert space
as Banach one
represents
all the universe
as an entanglement
of the quantum
anything
with all the rest
dual space
The “flat”
Hilbert space
includes
the wave function
of the quantum
anything
space
The quadrilateral “two by two”: Hilbert –
Banach, and Minkowski – pseudoRiemannian 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
A few important notes: on the conclusion
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, as Banach (Hilbert)
as pseudo-Riemannian (Minkowski) space
are invariant to time – frequency, or continuous –
discrete interpretation, or wave – particle duality
as mere mathematical formalisms
Banach (Hilbert) space represents the same as
functions globally, and pseudo-Riemannian
(Minkowski) space as point trajectories locally
A few important notes: on the conclusion
The second earnest note:
As pseudo-Riemanian (Minkowski) as Banach (Hilbert)
space both pairs either time or frequency interpreted
are well-ordered in the parameter of either time or
frequency in (geodesic) line. However what is up
if the well-ordering is abandoned in all cases eo ipso
abandoning the axiom of choice?
Banach (Hilbert) space represents the same as
functions globally, and pseudo-Riemannian
(Minkowski) space as point trajectories locally
A few important notes: on the conclusion
The answer is the third earnest note:
Abandoning the axiom of choice in all the 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
Banach (Hilbert) space represents the same as
functions globally, and pseudo-Riemannian
(Minkowski) space as point trajectories locally
The fourth earnest note on the conclusion
A point in
A point in
Function space
Banach space Hilbert space
t f A trajec- A line in
A tra-A line in
tory in pseudojec- Minkow“Line” space
a force Riemann. tory ski space
field
space
∞
p
p
𝒑𝒅𝒙 < 𝟏
∞
−∞
x
𝒑𝒅𝒙 = 𝟏
A defected
−∞
probability
A normed
distribution
probability
being due to
distribution
entanglement
(the force field)
The “curved” case The “flat” case
Banach (Hilbert) space represents the same as
functions globally, and pseudo-Riemannian
(Minkowski) space as point trajectories locally
“Projection in
probabilities”
space
x
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
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
The sides of the quadrilateral one by one:
3| Hilbert – Minkowski space
A very important corollary from the real difference,
videlicet: Hilbert space is a function space, while
Minkowski space is an ordinary, “point” space:
So that a trajectory in Minkowski space represents
a potentially infinite, current process in time or
“in frequency”, while a point in Hilbert space
represents the same process as complete or
as an actual infinity
The two views mentioned before on a single “HilbertMinkowski” space represent it correspondingly as a
potential infinity and as an actual infinity
The sides of the quadrilateral one by one:
4|Banach – pseudo-Riemannian space
Both spaces are “curved”, and all the rest
said about Hilbert – Minkowski space is valid to
their pair, too:
Both spaces express the same in different
perspectives:
Both spaces can be interpreted as a time
as a frequency space, but the Minkowski space
represents a process in potential infinity as a
world line in an ordinary, “point” space, while
Hilbert space an actual infinity as a complete
result, namely as a point in a function space
The quadrilateral, one by one: 4|Banach –
pseudo-Riemannian space: the curvature
𝐅(𝐲)
𝒇 𝒙
…
n
…
… …
… …
B
A
N
𝒇 𝒙
A
C
𝑭(𝒚)
H
PR
SI
n'
EE
UM
DA
ON
. N
p
P
R
O
p
p
B
A
B
IL
x
x
IT
DUAL SPACE
Y
“A” varying “angle&distnance ”
orthogonality
represented in each case by the two dual spaces
p
x
SPACE
The dual-spaces representation of
mechanical movement in a force field
…
n
𝒑𝑭 (𝒕),𝑬𝑭 (𝒕)
…
… …
… …
n'
force field
The juxtaposition of Lagrange and Hamilton
approach to mechanical movement
In both cases, three 4-vectors 𝒙, 𝒕; 𝒑, 𝑬; 𝒑𝑭 , 𝑬𝑭
determines the movement in any point, but
… here as three discrete
corresponding 4-points
… and here
as
a
smooth
dual p,E space x,t space
trajectory
Hamilton (dual spaces) approach
Lagrange (derivatives) approach
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
A little philosophical digression about
gravitational field and force field
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: probability field
The source of probability or entanglement field
can be any discrete, jumplike change
of the same quantity in any point of space-time.
It can act upon any other discrete change of that
quantity anywhere:
However how?
How can entanglement field act?
Its origin is rather mathematical and universal for
that: Any discrete, or jumplike 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
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
How can entanglement field act?
Interpreting the
connection between
the two dual
spaces
…
t
f(E)
A quantum
Another (or the
same??) leap
quantum leap in energy
(frequency)
The probability
Thefield
probabil
of all the momenta
of all the p
Any Heisenberg’s uncertainty Any
momentum Fourier transforms posit
Dual space
Space
How can entanglement field act?
Interpreting the connection
Complex
Hilbert
between the
two dual
spacesspace
…
P(p)P(x)
P(x,p)
P(p,x)
The complex
The probabil
probability
of all the p
field of all
as momenta
The
as
Thepositions
probability
field
complex
of all the momenta
probabi-lity
field
of
all
Any
Any
Heisenberg’s
uncertainty
as
posi-tions
momentum
positi
Fourier
transforms
(& position) as momenta
(& mom
Dual space
Space
How can entanglement field act?
Interpreting the connection
Complex
Hilbert
between the
two dual
spacesspace
…
P(p,x)
P(x,p)
The same complex
probability field of all as
momenta as positions
𝒆𝒕𝒆𝒓𝒏𝒊𝒕𝒚
View from
𝒕𝒊𝒎𝒆
𝑬~𝒇 = 𝟏/𝒕
The same complex
probability field of all as
positions as momenta
𝑵𝒐 𝒄𝒉𝒐𝒊𝒄𝒆
view
𝑨𝒙𝒊𝒐𝒎 𝒐𝒇 𝒄𝒉𝒐𝒊𝒄𝒆
𝒕
𝒙
𝒑
Dual space
Space
A digression about the “arrow of time”
The “arrow of time” is a fundamental, known to everyone, 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 quantity. 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 the “set” is
not a set, but a “category”, it can be transformed
Three restrictions of choice for a trajectory point
The dependence of momentum on position: The
value of momentum in a moment is proportional
the position derivative in the same moment, i.e. to
the value of speed
The “smooth choice” of both momentum and
position: The choice of the trajectory following
point is restricted to an infinitely small
neighborhood of the point, so that the trajectory
and its derivative are smooth in any point
The exact correspondence of the measure of the
same value set with the value probability
The same restrictions of choice for the
same trajectory point as a field 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
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 the
source of the force field and of potential
mechanical energy
The interaction of a system with a
force field in terms of whole and part
The energy-momentum of the system interacts with
the energy-momentum of the field in the same
space-time point as adding 4-vectors in Minkowski
space
We can interpret that as the forming of 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 “settheory curving” as inverse to a “flattening” choice
according to the axiom of choice
A view on a system in a force field in terms
of frequency (energy) instead of time
The energy-momentum representation is that
viewpoint. Any force field, which comprises a
system, represents a mismatch of the discrete and
continuous aspect of the system
By tradition that mismatch is embedded in energymomentum 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
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 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 spacetime expression of the way, in which any whole
comprises any part of its.
The deep meaning is not in the geometrization of
physics, i.e. not in the representation of a force
field as a “curved” space-time, namely pseudoRiemannian 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, for
tradition)
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 wellordering in space-time (energy-momentum) eo ipso
removing the axiom of choice, and eo ipso the choice
itself
That second representation is … quantum
mechanics
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
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
A few remarks on that amazing kind
of self-referentiality
Totality, infinity, and wholeness should posses
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
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
The most essential remark on the dual
self-referentiality of eternity and time
Our problem is the dual self-referentiality of:
𝒆𝒕𝒆𝒓𝒏𝒊𝒕𝒚
View from
𝒕𝒊𝒎𝒆
𝑵𝒐 𝒄𝒉𝒐𝒊𝒄𝒆
view
𝑨𝒙𝒊𝒐𝒎 𝒐𝒇 𝒄𝒉𝒐𝒊𝒄𝒆
Our solving is going to be:
Eternity and time are merely
two different interpretations
of the same mathematical structure:
namely, Hilbert (Banach) space
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 discrete impulse;
…
Be eternity and time two different
interpretations, then …
 … we should admit the equivalent curvature (i.e.
the nonorthogonality) as between eternity and
time as between time and frequency (energy) as
between frequency (energy) and eternity, and as
between all of them;
 … as entanglement (from the particular view of
eternity) as gravity (from the particular view of
time and energy) as any equivalent combination
of them expresses the same;
 … we should admit even an interaction between
entanglement and gravity
Be eternity and time two different
interpretations, then …
… that which is the same but expressed
differently by gravity (in terms of time and
energy) and entanglement (in terms of two
probability distributions) represents the same
interaction between a system and the
universe (environment), in which it is
included, from the two viewpoints of time
(and energy) and eternity
… whatever about the eventual interaction of
gravity and entanglement is a quite open
question
Be eternity and time two different
interpretations, then …
Wave function as
a Fourier transform
of two (conjugate)
probability distributions
Complex
Hilbert
(Banach)
Space
eternity
for entanglement
The Same!!!
for gravitational field
Well-ordering in time
and frequency (energy)
by the axiom of choice
time&
frequency
Consequently, our conclusion is ... !!!
Entanglement is a
view on a system in a
force field in terms of
eternity instead of
time (or frequency,
energy
A set-theory interpretation of the links
between functional and physical space
𝒙 𝑭
𝑫𝒖𝒂𝒍
𝒔𝒑𝒂𝒄𝒆
𝑭𝒐𝒖𝒓𝒊𝒆𝒓
𝑭𝒐𝒖𝒓𝒊𝒆𝒓
𝒇(𝒙)
𝑺𝒑𝒂𝒄𝒆
𝒇
𝒇(𝒙)
𝒙(𝑭)
𝑭
𝟑𝑫 𝑪𝒂𝒓𝒕𝒆𝒔𝒊𝒂𝒏 𝒑𝒓𝒐𝒅𝒖𝒄𝒕
𝒙, 𝒇, 𝑭 𝒂𝒓𝒆 𝒔𝒆𝒕𝒔 𝒔𝒖𝒄𝒉 𝒂𝒔:
𝒙
𝑭
𝒇 = 𝟐 ,𝒙 = 𝟐 ;
𝒇 𝒙 ⊂ 𝒇, 𝒙 𝑭 ⊂ 𝒙
The set-theory interpretation
being continued
𝒙
𝒇 = 𝟐 ,𝒇 𝒙 ⊂ 𝒇
𝒙, 𝒇, 𝑭 𝒂𝒓𝒆 𝒔𝒆𝒕𝒔 𝒔𝒖𝒄𝒉 𝒂𝒔:
𝒙 = 𝟐𝑭 , 𝒙 𝑭 ⊂ 𝒙
𝑰𝒇 𝒇(𝒙)
𝒄𝒂𝒓𝒅𝒊𝒏𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓
𝒙
𝒇
𝒕𝒉𝒆𝒏:
𝑭
𝑨𝒏𝒅 𝒙(𝑭)
𝒙
𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔
𝒇 𝒙 , 𝒙 𝑭 𝒂𝒓𝒆
𝒈𝒆𝒏𝒆𝒓𝒂𝒍𝒊𝒛𝒆𝒅 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔
𝒄𝒂𝒓𝒅𝒊𝒏𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓
Links between function space
and physical space
𝒙 𝑭
𝑫𝒖𝒂𝒍
𝒔𝒑𝒂𝒄𝒆
𝑭𝒐𝒖𝒓𝒊𝒆𝒓
𝑭𝒐𝒖𝒓𝒊𝒆𝒓
𝒇(𝒙)
𝒇
𝒇(𝒙)
𝑺𝒑𝒂𝒄𝒆
𝒙(𝑭)
𝑭
𝟑𝑫 𝑬𝒖𝒄𝒍𝒊𝒅𝒆𝒂𝒏
𝒚 ≡ 𝒇 𝟑𝑫 𝑪𝒂𝒓𝒕𝒆𝒔𝒊𝒂𝒏 𝒑𝒓𝒐𝒅𝒖𝒄𝒕
𝒔𝒑𝒂𝒄𝒆
𝒙, 𝒚 𝒑𝒓𝒐𝒋𝒆𝒄𝒕𝒊𝒐𝒏 𝒐𝒇𝝋
𝒙, 𝒛 𝒑𝒓𝒐𝒋𝒆𝒄𝒕𝒊𝒐𝒏 𝒐𝒇𝝋
𝑻𝒊𝒎𝒆 𝒑𝒂𝒓𝒂𝒎𝒆𝒕𝒓𝒊𝒛𝒂𝒕𝒊𝒐𝒏 𝒎𝒂𝒌𝒆𝒔 𝝋 = 𝜱 𝒕
𝒊𝒏𝒕𝒐 𝒂 𝒔𝒑𝒂𝒄𝒆𝒕𝒊𝒎𝒆
𝝋 = 𝝋 𝒙, 𝒚, 𝒛 = 𝟎
𝒕𝒓𝒂𝒋𝒆𝒄𝒕𝒐𝒓𝒚
𝒛≡𝑭
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