The universe in a quantum: A possible interpretation of the equivalence of the Schrödinger equation

The universe in a quantum A possible interpretation of the equivalence of the Schrรถdinger equation and the Einstein field equation

Prehistory and background

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

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)

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

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

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

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 pseudoRiemannian 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

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

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

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 ďƒ˜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

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

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

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

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

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

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

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

Thesis

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

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)

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

Possible objections

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”?

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

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?

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

A few arguments in favor of the thesis

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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”

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

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 nonexistence 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

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

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

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

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

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

Gravity by entanglement • That general entanglement transferred from the nonstandard 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

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

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

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

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

Conclusions

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

Спасибо за внимание!