A possible interpretation of the equivalence of the Schrödinger equation and the Einstein field equation

1. The universe in a quantum
A possible interpretation of the equivalence of
the Schrödinger equation and the Einstein field equation

2. Prehistory and background

3. Vasil Penchev
• Bulgraian Academy of Sciences:
Institute for the Study of Societies and Knowledge:
Department of Logical Systems and Models
vasildinev@gmail.com
• The conference “Modes of Thinking, Ways of Speaking
School of Philosophy, National Research University Higher
School of Economics: Miasnitskaya, 20, Moscow, Russia
26 – 29 April 2017

4. The Schrödinger equation (SE) and the Einstein
field equation (EFE)
• The essence of two fundamental physical theories, quantum
mechanics and general relativity is concentrated in their
basic equations:
The Schrödinger equation (SE)
• And the Einstein field equation (EFE)

5. The problem of consistency
• The mutual consistency of both theories is one of the biggest
open problems in physics and its philosophy
The philosophical essence of that problem might be
represented as follows:
• General relativity refers to smooth motion in a curved space,
which can be thought as constituted by the propagation of a
deformed spherical wave in the usual Euclidean 3D space
Quantum mechanics needs to represent any quantum leap
as a discrete motion and uses the separable complex Hilbert
space, which is never curved in principle

6. The axiom of choice
• The mathematical equivalence of SE and EFE is provable under the
axiom of choice
The philosophical sense of the axiom of choice in the present
context is that:
• It can transform any smooth trajectory in the curved space-time of
general relativity in a discrete trajectory (wave function) in the
fundamentally uncurved Hilbert space of quantum mechanics
For example, the curved trajectory represents a smooth motion in
Euclidean space, and the wave function a “straight” leap in it, too.
The axiom of choice can equate both according to a certain rule

7. The underlying mathematical structure
• SE and EFE can be considered correspondingly as the
nonstandard and standard interpretations (in the sense of
Robinson's nonstandard analysis) of one and the same
mathematical structure
The philosophical sense of that structure is the unification of
smooth, but curved motion and discrete, but straight leap:
• Both can be consider as two equally possible, alternative or
“complementary” interpretations of a common abstract
motion in that underlying structure
That underlying motion should be considered as processing
quantum information rather than mechanical motion whether
smooth or discrete, though

8. The physical sense
• The physical sense of their equivalence consists in the
equivalent transformation between:
A smooth continuum of inertial reference frames (as what
space-time is considered in general relativity by pseudo-
Riemannian space), and:
• Complementary pairs of qubits, the one member of which
corresponds to the velocity (or dynamically, to the
momentum)of a single reference frame, and the other to its
position
The latter can be thought as a physical motion, the former
as the information about it, and both turn out to be
equivalent to each other

9. The mathematical sense of time
• The quantity of time is reduced to the well-ordering of those
pairs for the axiom of choice
Two inconsistent kinds of time distinguishes further general
relativity and quantum mechanics:
• Reversible time in general relativity as in classical mechanics
Irreversible time in quantum mechanics as in
thermodynamics
• They can be compatible to each other only under the same
condition to be considered as the standard and non-standard
interpretation of unifying underlying time

10. A “cinematographic” metaphor borrowed
from Bergson
• Any present moment of time is a frame in the "movie" of the
universe
What conserves between two frames is (quantum) information,
i.e. what is mapped on any given frame, rather than the "matter"
of frame, which is different segments of the film of about one inch
length
• However, unlike the movie, the smooth motion is not an illusion in
our brains, but really equivalent to the images frame by frame
Thus the reversible time of general relativity would be that of the
film tape, and the irreversible time of quantum mechanics would
be that within the movie

11. The reconciling axiom of choice
• Thus, the axiom of choice is what reconciles the space-Iike
time in general relativity to the arrow-like time in quantum
mechanics and allows of their equating
Using again the cinematographic metaphor, the axiom of
choice can be likened to the camera making a series of shots
at regular intervals, e.g. 24 FPS
• The shots, to which the smoot motion is decomposed, can be
always enumerated by a series of natural numbers starting
from 1 to an arbitrary, but corresponding natural number

12. The relativeness of the small and the big
• However, the “camera” of the axiom of choice can be
directed to an infinitely small interval of smooth motion, and
it will decompose it into a series of frames not less
successfully
Thus, a series of frames can represent as an infinitely small
segment as a finite segment, thus equating them to each
other
• Thus the axiom of choice opens the road to a cyclic ontology
of both small and big in the manner of Nicolas of Cusa

13. The physical sense of the axiom of choice
• The axiom of choice, itself is interpreted physically as the choice
of a single value after measurement by the measurement itself
from the coherent whole of values before measurement
Consequently, measurement is what realizes the axiom of
choice physically in quantum mechanics, and any single
measurement is a random choice
• However, the collection of all possible measurements of the
same quantum state is absolutely determined rather than
random
It represents a statistical ensemble, the probability distribution
of which coincides of the probability distribution of the
measured coherent state

14. The problem of “hidden variables” in
quantum mechanics
• Indeed, the theorems of the absence of hidden variables in
quantum mechanics (Neumann 1932; Kochen, Specker 1968)
exclude any well-ordering before measurement
On the contrary, the results after measurement are always
well-ordered according to the moments of registration
• Using again the “cinematographic” metaphor, measurement
corresponding to the axiom of choice is the “camera”, which
extends into a series of frames, each of which “shoots” a
random small part of the entire picture unlike a real camera

15. The “shot” of a quantum leap by
the cinematographic metaphor
• At last, all frames are collected and all frames are superposed to
represent the coherent superposition measured
The more shots of the same part, the brighter will be on the
joint picture of all parts
• Thus, if one “shoots” a quantum leaps by a camera of that kind,
the picture will represent a blur having vague and fuzzy outlines
reaching gradually infinity … such as a wave function
Then, the sense of that underlying structure would be to see
both leap and blur … as the same unlike common sense

16. The quantum state before and after
measurement
• The coherent whole of values before measurement can be
mapped into the results after measurement unambiguously
only under the well-ordering principle equivalent to the axiom
of choice
By means of the cinematographic metaphor, this means: the
quantum leap can be transformed into, and thus “seen” as a
blur only by the camera described above:
• Any human being cannot see the leap itself, but only its blur
• Unlike the metaphor, quantum mechanics states that both
leap and blur are the same for the mathematical formalism
states it , unlike our senses and common sense

17. From the mathematical to physical
equivalence
• The purely mathematical equivalence of SE and EFE might be
as an occasional, eccentric, and curious, but meaningless
coincidence (as the "channels on Mars") as an expression of
a certain deep ontological essence underlying both theories
So, we have a mathematical structure as an answer as “42”
in Duglas Adams’s “The hitchhiker’s guide in galaxy”, but we
know neither which is the ontology, to which it can be
referred, nor even that ontology exists at all
• So, the condition of the presentation is the conjecture of
existing that ontology

18. The address of the talk
• The talk is intended to the research of the latter alternative,
namely a certain deep ontological essence underlying both
theories: general relativity and quantum mechanics, as well
as and first of all, ontological conclusions implied by it
The structure, for which a relevant ontology will be sought,
needs the axiom of choice inherently
• Thus ‘choice’ can be a track to that ontology
The second track can be information as the quantity of
choices for information seems to be a relevant link between
a mathematical formalism, for which an ontology of choice is
searched

19. Thesis

20. The thesis: The whole of both universe and
quantum
• The standard (EFE) and nonstandard (SE) interpretations can be
embodied in an ontology equating the whole of the universe to
the whole of a quantum
Indeed, both universe and quantum are a whole
• This implies the conjecture to be unified for wholeness
interpreted as totality should be a single one
Furthermore, both universe and quantum exclude to be
transcended in principle but in “opposite directions” figuratively
said:
• One cannot “leave” the universe as one cannot “enter” a
quantum

21. The thesis: The whole and its … externality
• Then, EFE represents reality inside that whole, and SE
outside it
In other words, the observer is suggested to be inside the
universe as to general relativity, but outside the quantum as
to quantum mechanics
• Then the unification of EFE and SE would correspond to that
transition “inside – outside” and vice versa
However, the identification of both universe and quantum
implies that we observe outside observing the smallest (the
quantum), and inside, observing the biggest (the universe)

22. The thesis: The universe within a single
quantum
• The universe is situated with in a quantum
That statement seems to be shocking at first glance because those
are the two ends of the physical world, between which all existing
physically is situated
• However, one can suggest a cycle, in which the one end is glued to
the other as the idea of wholeness or totality hints
Then our step outside the biggest return us magically to the other
end, outside the smallest, and vice versa: inside the smallest, into
inside the biggest
• The shared underlying mathematical structure describes physically
that ontology and that transition

23. Possible objections

24. A possible objection: too different in scale
• (i) A quantum is the smallest, and the universe the biggest in our
physics and its metaphysics
Indeed, that suggestion seems to be ridiculous or just kidding!
• Only the visible universe consists of a huge number of galaxies or
similar in size astronomical mega-objects, each of which consists
of a huge number stars or similar objects, each of which consist of
huge, huge number elementary particles, only to each the
concept of quantum is meaningful and thus applicable
The idea is obviously too crazy, but is it too “crazy enough to be
true”?

25. A possible objection: too different in mass
and energy
• (ii) Both mass and energy of what is "alleged" to be inside
the universe, are about hundreds of exponents bigger than
what "as if" is outside a quantum
The law of energy conservation is fundamental for all
branches of physics
• So, the quantity of energy or mass is one of the main
indicators for any physical system to be the same
Thus the universe and any quantum entity are obviously
absolutely different after the difference in their masses and
energies is so impressing

26. A possible objection: What is “outside” is
“inside”, too
• (iii) What is outside, a quantum, is inside, too, even as many,
many, many quanta within the universe
The universe is a single one (excluding the conjecture of
multiverses as irrelevant in the present context)
• The quanta are many, many, many
Even if one accepts that the universe can be considered
somehow as a single quant, the transformation one-to-many
should be explained anyway:
• How can the single entity transform into many, many entities
after the biggest has “glued” to the smallest?

27. A possible objection: Then, what about the
“Big Bang”?
• (iv) The experimentally very well confirmed conceptions of the "Big
Bang" and the expanding universe seem to be nonsense in the
internality of a quantum
A quantum is absolutely unchangeable, but the universe is
changing permanently
• Even the concept of smooth change in time is inapplicable to it:
Indeed its quantity of action is constant in definition; then its
energy as the derivative to time is zero: either it is not a physical
object (with zero energy) or the concept of smooth change
(derivative) is inapplicable to it

28. A few arguments in favor of
the thesis

29. The ontology of the universe in a single
quantum
• (1) That ontology is consistent to the mathematical
equivalence of SE and EFE though it is not implied by their
equivalence
Indeed, both ontology and SE – EFE equivalency share the
same structure of a standard – nonstandard interpretation
• This was already demonstrated for the pair of SE – EFE
All quanta “outside” the universe represent “non-standardly”
any single whole within it or even it itself only under the
condition of the axiom of choice, accepted for the SE – EFE
equivalency already

30. Both quanta and continua within the universe
• Thus, a one-to-one mapping of any continuum in a subset of all
quanta in the universe is implied: the main problem of quantum
mechanics, how to describe uniformly smooth motion and
quantum leaps, is resolved just as by SE
That ontology of the universe within a quantum adds cyclicity,
whose consistency to general relativity and EFE will be
demonstrated a little further by Poincaré's conjecture
• However, the conception of time is different in both theories:
reversible and thus space-like in general relativity as in classical
mechanics, but irreversible in quantum mechanics as in
thermodynamics

31. Time in quantum mechanics and SE
• However, time in SE seems to be reversible after its derivative is
involved
The time of measurement in quantum mechanics is irreversible for
measurement is a random choice in definition, though, generating
an element of a statistical ensemble
• In fact, SE (obviously after Born’s interpretation) equates the
coherent state before measurement, whose time is reversible, to the
characteristic function of the probability distribution of the
statistical ensemble after measurement, whose time is irreversible
Thus, SE equates and even identifies reversible and irreversible time
to each other

32. Time in general relativity and EFE
• Time in both special and general relativity is space-like and thus
reversible. It is able to be both extended and shortened just as
space distance is
Independently of this, both relativities do not violate causality
and thus, the twin, space traveler is just so old as the twin,
homebody after returning to earth
• Consequently, time in general relativity is reversible as a
“differential” (e.g. in relative velocity), but irreversible as any
finite period of time therefore not violating causality
Both quantum mechanics and general relativity identify
reversible and irreversible time but in different ways

33. A comparison between the two methods for
identifying both times
• The method of quantum mechanics: by the series of choices
representing the continuum of coherent state as a well-ordering of
measurements
The method of general relativity: by situating the infinitely small
and reversible differential as a special kind of part within any
irreversible finite interval
• Robinson’s non-standard analysis equating the potentially infinitely
small differential in a Cauchy sense to the actually infinitely small
differential in a Leibniz sense implies the identification of both
methods to each other and to the ontology at issue

34. Information as the concept
unifying reversible and irreversible time
• If the unification of both standard and non-standard interpretation is
well-exemplified physically by the pair of reversible and irreversible
time, an “atom” of that unification can be considered as the natural
unit representing both above unities
That “atom” is a single choice, and that unit is an elementary choice,
i.e. a bit of information, the choice between two equally probable
alternatives
• Thus the concept and quantity of information unifies not only the
mathematical pair “standard – nonstandard interpretation” or the
physical pair “reversible – irreversible time”, but both pairs to each
other: information is the “bridge” between physics and mathematics

35. A conclusion from Poincaré's conjecture
• (2) Poincaré's conjecture proved by Grigori Perelman implies
the necessity of cyclic closure of space-time of general
relativity.
This due to the topological equivalence (physically
interpretable as equivalence in causality) of:
(A) The usual three-dimensional Euclidean space, and
(B) Pseudo-Riemannian space of general relativity, only under
(C) The condition of Poincaré's conjecture

36. The sense of Poincaré's conjecture
as to pseudo-Riemannian space and Euclidean space
• To be space-time of general relativity isomorphic to Euclidean
space topologically, one needs a link between its “last” and its
“first” element in time, figuratively said, in order to be
topologically equivalent to a 3-ball
That link transforms the irreversible time of any finite time
into reversible “through infinity” “where” the link is situated
• Thus, the totality of space-time turns out to be as reversible as
any differential in it opening the road for the isomorphism of
the totality and any differential corresponding to isomorphism
of the isomorphism of the universe and quantum

37. About the “past”
in the space-time of general relativity
• After having been closed, the (curved) cone of the past turns out
to be the opposite direction of reversible time, “trough the
negative infinity”
However, the pair of both standard and nonstandard (equivalent
to each other) interpretations of the transition trough infinity as if
splits time into two branches: reversible (for the standard
interpretation) and irreversible (for the nonstandard
interpretation)
• The latter branch corresponds right to the past as irreversible
time: general relativity means just it as the time of any finite
interval

38. The past in space-time of quantum mechanics
• The term “space-time” is relevant to quantum mechanics as far
as it refers to general relativity
Its formalism by the separable complex Hilbert space manages
to join the additive (discrete) and multiplicative (continuous)
group in two complementary levels
• A field (such as that of the complex numbers) joins them in the
same level after which the former is a subgroup of (as if
dissolves in) the latter.
Thus, that Hilbert space manage to conserve a one-to-one
mapping between the two groups just as by means of the
axiom of choice

39. The past in space-time of quantum mechanics (II)
• That one-to-one mapping of both discrete and continuous
groups is able to represent the link between correspondingly
irreversible and reversible time consistently
The past turns out to be just and only information
• It transforms into physical action in quanta by the Planck
constant from the past into the present
Energy conservation means the flux of quanta (thus
information), constant in time
• Quantum correlations are the physical way for the
transforming information into action

40. The fundamental meditation of quantum
correlations
• Unlike a single quantum of action, whose energy is zero, quantum
correlations possess energy due to the interaction of fluxes of
quanta
That interaction means non-orthogonality of the corresponding
wave functions or thus an overlap of probability distributions
• It is equivalently represented as curving the space-time of general
relativity and thus as gravity
The closure of time bifurcates trough infinity into the irreversible
discrete time of the past and the reversible continuous time of
the present and future
• Quantum correlations are what unifies both branches of time and
thus the course itself of time

41. References to the philosophical tradition
• (3) Many examples of philosophical doctrines equating the
biggest and the smallest (as Nicolas of Cusa) or introducing
cyclicality (as Nietzsche) can be referred
Time in the philosophical tradition of Christianity is granted
as an “arrow” and thus irreversible, non-cyclic
• However philosophy of science and the physical theories are
divided to time into the group of irreversible time such as
thermodynamics and that of reversible time such as
mechanics where irreversibility is added secondarily to the
mathematical formalism by the principle of causality

42. Poincaré's conjecture and the thesis of
Nicolas of Cusa
• Poincaré's conjecture means right the equivalence of both biggest and
smallest
The already proved conjecture can be considered as a physical and
mathematical equivalent of the philosophical thesis of Nicolas of Cusa
• It has furthermore the advantage to link physical causality to Nicolas’s
idea
The meant topological equivalency implies reversible time however
“trough infinity”
• The transition “trough infinity” is able to bifurcate time into two
disjunctive but equivalent (complementary) branches: either reversible
or reversible

43. The action of both universe and quantum
• (4) Both mass and energy of a quantum cannot be directly compared
with those of the universe because the physical dimension of a
quantum (the Planck constant) is action, which depends of both energy
(mass) and time
What is transferred from the past to the present and future is just and
only information equivalently transformed into action into the past and
future
• That transformation is inherently irreversible right for the transition
from irreversible to reversible time
So, the quantum and universe can be equated to each other from the
information viewpoint of the past
• They might be distinguished from each other only from the viewpoint
of the present by the “differential” of time

44. Time: discrete or continuous?
• Time is right that physical quantity, which should not be
compared directly between the non-standard interpretation
outside of the whole and the standard interpretation inside of it
for the time is continuous (smooth) inside and discrete outside
in definition
If one needs to discuss both aspects of time simultaneously,
the concept of information is what is necessary (but not
sufficient)
• It represents the transition from continuous to discrete time as
a choice
The Hegelian triad in philosophy might represent the opposite
transition: from both disjunctive alternatives of “thesis” and
“antithesis” to a single “synthesis”

45. The ratio of discrete and continuous time
• (5) An option is a certain and unknown dimensionless fundamental
constant to determine the smooth length of a unit of time “inside”
per a discrete unite of time “outside”
Another option is the reciprocal ratio to be interpreted as
frequency and further, as the frequency of De Broglie’s wave,
different for each physical entity and equivalent to its energy or
better, to its action
• Then, both Landauer limit (L = kTln2, k – the Boltzmann constant,
T – absolute temperature) and Planck constant can be involved to
determine a single equivalent temperature in thermodynamics for
De Broglie’s frequency
One can coin a dimensionless kind of temperature, which per unit
of time to be equivalent to the temperature defined as usual

46. About the “absolute zero” of temperature
• The “absolute zero” of temperature would correspond to the zero
ratio of discrete and continuous time
Thus, that “absolute zero” would mean right the factual non-
existence of discrete time as commensurable with the continuous
one
• Also vice versa: the prohibition of the “absolute zero” state (as e.g.
in the so-called third law of thermodynamics) means the existence
of discrete time as a nonzero interval commensurable with that of
continuous time: thus, the finite ratio of them
This mean both kinds of time: discrete and continuous or
irreversible and reversible are inseparable from each other
• The equivalency of both standard and nonstandard interpretation
turns out to imply … the third law of thermodynamics

47. Again about that unknown constant
• More conjectures: e.g. that constant can be defined implicitly just
postulating the equality of the action (information) of the universe and
that of a single quantum (right the Planck constant), or:
It can be recognized as a certain one among the known fundamental
physical constants (first of all, among the thermodynamic ones, such as
Boltzmann's)
• The Boltzmann constant can be interpreted as dimensionless as the
natural unit of entropy
Indeed, entropy is an indirect measure for the relation or ratio of a
whole and its equal or uniform discrete parts as what the time
intervals might be considered: the maximal entropy means just the
case where the parts are well-ordered, and the whole is their sum

48. What “the totality is total” implies
• (6) The totality such as the universe right being just 'totality'
should contain itself in the final analysis for any externality of
it contradicts to its definition
In fact, the universe should be thought as the totality
therefore containing its externality in itself
• Indeed, there exist physical hypotheses about “many
universes or “multiverse”
However, they only rerefer to the totality named otherwise
than the universe
• So, that argument addresses the totality as what we mean
the universe according to the contemporary knowledge

49. Transcending the “boundary of the universe”
• The transition from the externality to the internality of the
universe implies the parallel one-to-many transformation:
one single quantum "outside" into many, many, many quanta
"inside“
Indeed that transition is inherent for the transition from
standard to non-standard transformation at all
• The axiom of choice implies that for it is able to enumerate
the whole of any continuum
So, the quantum seen “outside” can be interpreted as a
continuum, but it “inside” as its countable equivalent

50. About those many quanta inside
• The set of all quanta inside can be interpreted as "many worlds"
or "many universes" as the different states in the whole of the
universe (also in the thermodynamic sense of Gibbs)
So, the totality as what we have accepted the universe forces:
• The many worlds in the interpretation of the same name in
quantum mechanics to be inside the universe itself
The many universes of the multiverse to be inside the universe
itself
• The many states of the system in the Gibbs thermodynamics to
be identified as the parts of the same system in the Boltzmann
thermodynamics

51. Many Hilbert spaces in a single one
• Any quantum "inside" generates a separable complex Hilbert
space, which can be non-collinear to those of the others and
thus, entangled to them
One can assign a certain value of mass to any entangled
entity, even to a single “axis” (entangled to another) of the
separable complex Hilbert space
• That assign mass would depend only on the velocity in time
for the change of entanglement reciprocally
One can assume furthermore that the origin of any mass
might be explained so or at least equated to an equivalent
mass of entanglement

52. Gravity by entanglement
• That general entanglement transferred from the non-
standard to standard interpretation right represent gravity
just as it is described by general relativity
If any mass can be described by entanglement, it will be able
to underlie gravity
• The transition from the viewpoint of quantum mechanics to
that of general relativity substitutes the description in terms
of Hilbert space by those of pseudo-Riemannian space
This means the discrete description to be replaced with the
smooth one

53. The boundary of the universe and infinity
• (7) The ontological borders of the universe as a quantum can
be thought mathematically as those between finiteness and
infinity
Indeed, the boundary between any standard interpretation
and its non-standard counterpart is right the boundary
between infinity and finiteness
• Thus the boundary of the universe or that of a quantum
would exemplify the same boundary between infinity and
finiteness for that boundary is only a physical example of the
relation between standard and nonstandard interpretation

54. The relativity of both standard and
nonstandard interpretations
• Indeed, the visible universe is necessary finite though
exceptionally immense, while the nonstandard interpretation
physically interpreted as quantum reality needs necessarily
infinity
Thus, the universe being finite can be placed into a single
quantum being infinite even according to common sense for any
finiteness seems to be “inside” infinity again according to it
• However, the concept of both standard and nonstandard
interpretations involves rather the relativity of the finite and
infinite just as the cyclic philosophical conceptions address the
relativity of the big and small or the future and past

55. The consistency with the “Big Bang”
• The conception of both beginning (the "Big Bang") and
expansion (the "expanding universe") of a finiteness such as the
universe within infinity such as a quantum is not nonsense
We may think of the Bing Bang as a singular (“starting”) point of
time, in which is concentrated its irreversibility
• Thus, the time after it can be granted as reversible as it is
available in the mathematical formalism of the pfundamental
theories
However, we may deconctrate its irreversibility from that point
and clutter uniformly over all moments so that time to botjh
reversible and irreversible in each moment
• Then, the metaphor of the “big Bang” would not necessary

56. The universe in terms of Skolem’s “paradox”
• Skolem's conception about the "relativity of 'set’” implies
particularly one-to-one mappings of infinite sets into finite
ones
It can explain the transformation of the quantum "outside"
into quanta "inside" as existing only "purely mathematically“
• Indeed, the axiom of choice can represent the same
transition representable constructively and thus physically
only as a probability distribution just as quantum mechanics
does

57. Conclusions

58. 1. EFE and SE are isomorphic to each other as the standard
and nonstandard interpretations of the same mathematical
structure
2. One can offer a coorsponding “ontology of the universe in
a single quantum” consistent to that mathematical
equivalency
3. Though counterintuitive and challenging commons sense,
that onology is not contradictory
4. It implies cyclicity as well as the relativeness of the small
and big as in the philosophical doctrines of Nicolas of Cusa,
Nietzsche, etc.
5. It allows of interesting hypotheses as corollaries from it

59. Спасибо за внимание!