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Poincaré’s conjecture proved by G. Perelman

by the isomorphism of Minkowski space and the separable complex Hilbert space

1

The set of all complex numbers, is granted. Then the

corresponding set of all subset of is the separable complex

Hilbert space .

There is one common and often met identification of with

the set of all ordinals of , which rests on the identification

of any set with its ordinal. However, if any ordinal is identified

as a certain natural number, and all natural numbers in Peano

arithmetic are finite1, and should not be equated, for

includes actually infinite subsets2 of 2. Here “actually infinite

subset” means ‘set infinite in the sense of set theory”.

Furthermore, is identified as the set Η of all well-ordered

sets which elements are elements of some set of 2, i.e. in other

words, the elements of 2 considered as classes of equivalency

in ordering are differed in ordering within any class of that

ordering.

Those distinctions can be illustrated by the two basic

interpretations of :(1)as the vectors of n-dimensional

complex generalization of the usual 3D real Euclidean space,

isomorphic to Η, and (2) as the squarely integrable functions,

isomorphic to . The latter adds to the former unitarity (unitary

invariance), which is usually interpreted as energy conservation

in their application in quantum mechanics. Back seen, energy

conservation is a physical equivalent of both (3) equivalence

after ordering and (4) actual infinity, i.e. to (5) the concept of

ordinal number in set theory.

On the contrary, once one does not involves energy

conservation, e.g. generalizing it to energy-momentum

conservation as in the theory of general relativity or that of

entanglement, Η rather than is what should be used unlike

quantum mechanics based on , and actual infinity avoided or

at least precisely thought before utilizing.

Furthermore, (6) the relation between and can be

interpreted as the 3D Euclidean space under (7) the additional

condition of cyclicality (reversibility) of conventionally

identifying the first “infinite” element with the “first” element

of any (trans)finite well-ordering. Indeed, the axiom of

induction in Peano arithmetic does not admit infinite natural

numbers3. If one needs to reconcile both finite and transfinite

induction to each other, the above condition is sufficient.

It should be chosen for Poincaré’s conjecture [34] proved by

G. Perelman [35-37]. If that condition misses, the topological

structure is equivalent to any of both almost disjunctive

domains4 of Minkowski’s space of special relativity5 rather

than to a 4D Euclidean ball. The two domains of Minkowski

space can be interpreted as two opposite, “causal directions”

resulting in both reversibility of the 3D Euclidean space and

topological structure of the above 4D ball.

The relation between and generates any of the two

areas of as follows. Both unitarity of and non-unitarity of

for any ordinal and any well-ordering of length are

isomorphic to a 3D Euclidean sphere6 with the radius (). All

those spheres represent the area at issue.

That construction can be interpreted physically as well.

Energy (E) conservation as unitarity represents the class of

1 This is a property implied by the axiom of induction.

2 Here “actually infinite subset” means ‘set infinite in the sense of set

theory”.

3 1 is finite. The successor of any finite natural number is finite.

Consequently, all natural numbers are finite for the axiom of induction.

equivalence of any ordinal . If the concept of physical force

(F) is introduced as any reordering, i.e. the relation between any

two elements of the above class, it can be reconciled with

energy conservation (unitarity) by the quantity of distance (x)

in units of elementary permutations for the reordering so

that. =.

Back seen, both (6) and (7) implies Poincaré’s conjecture

and thus offer another way of its proof.

One can discuss the case where is identified with and

what it implies. Then (8) the axiom of induction in Peano

arithmetic should be replaced by transfinite induction

correspondingly to (4) above, and (9) the statistical ensemble

of well-orderings (as after measurement in quantum

mechanics) should be equated to the set of the same elements

(as the coherent state before measurement in quantum

mechanics) for (3) above.

In fact, that is the real case in quantum mechanics for

unitarity as energy conservation is presupposed. Then (8)

implies the theorems of absence of hidden variables in quantum

mechanics [1], [2], i.e. a kind of mathematical completeness

interpretable as the completeness of quantum mechanics vs.

Einstein, Podolsky, and Rosen’s hypothesis of the

incompleteness of quantum mechanics [3]:

The (8) and (9) together imply the axiom of choice. Indeed,

the coherent state (the unordered set of elements) excludes any

well-ordering for the impossibility of hidden variables implied

by (8). However, it can be anyway well-ordered for (9). This

forces the well-ordering principle (“theorem”) to be involved,

which in turn to the axiom of choice.

Furthermore, can be represented as all sets of qubits.

A qubit is defined in quantum mechanics and information as

the (10) normed superposition of two orthogonal7 subspaces of

: 0+1

0,1 are the two orthogonal subspaces of .

, :||+||= 1.

Then, (11) Q is isomorphic to a unit 3D Euclidean ball, in

which two points in two orthogonal great circles ate chosen so

that the one of them (the corresponding to the coefficient ) is

on the surface of the ball.

That interpretation is obvious mathematically. It makes

sense physically and philosophically for the above

consideration of space as the relation of and .

Now, it can be slightly reformulated and reinterpreted as the

joint representability of and , and thus their unifiablity in

terms of quantum information.

Particularly, any theory of quantum information, including

quantum mechanics as far as it is so representable, admits the

coincidence of model and reality: right a fact implied by the

impossibility of hidden variables in quantum mechanics for any

hidden variable would mean a mismatch of model and reality.

can be interpreted as an equivalent series of qubits for any

two successive axes of are two orthogonal subspaces of :

4 They are almost disjunctive as share the light cone.

5 Indeed, special relativity is a causal theory, which excludes the

reverse causality implied by cyclicality.

6 This means the surface of a 3D Euclidean ball.

7 Any two disjunctive subspaces of are orthogonal to each other.

; then (12) any successive pair ,=;

under the following conditions:

(13) =

()(); =

()();

(14) = 0;=

||;

(15) If both , = 0, = 0 , = 1.

(14) and (15) are conventional, chosen rather arbitrarily only

to be conserved a one-to-one mapping between and .

is intendedly constructed to be ambivalent to unitarity for

any qubit is internally unitary, but the series of those is not.

Furthermore, one can define n-bit where a qubit is 2-bit

therefore transforming unitarily any non-unitary n-series of

complex numbers. The essence of that construction is the

double conservation between the two pairs: “within – out of”

and “unitarity – nonunitarity”.

That conservation is physical and informational, in fact. The

simultaneous choice between many alternatives being unitary

and thus physically interpretable is equated to a series of

elementary or at least more elementary choices. Then, the

visible as physical inside will look like the chemical outside

and vice versa. If a wholeness such as the universe is defined

to contain internally its externality, this can be modeled anyway

consistently equating the non-unitary “chemical” and unitary

“physical” representations in the framework of a relevant

physical and informational conservation.

can be furthermore interpreted as all possible pairs of

characteristic functions of independent probability

distributions and thus, of all changes of probability

distributions of the state of a system, e.g. a quantum system.

Practically all probability distributions and their

characteristic functions of the states of real systems are

continuous and even smooth as usual. The neighboring values

of probability implies the neighborhood of the states. Thus the

smoothness of probability distribution implies a well-ordering

and by the meditation of it, a kind of causality: the probability

of the current state cannot be changed jump-like.

This is an expression of a deep mathematical dependence (or

invariance) of the continuous (smooth) and discrete. The

probability distribution can mediate between them as follows:

can be defined as the sets of the ordinals of where a

representative among any subset of the permutations (well-

orderings) of elements is chosen according a certain and thus

constructive rule. That rule in the case in question is to be

chosen that permutation (well-ordering), the probability

distribution of which is smooth. Particularly, the homotopy of

can identified with, and thus defined as that mapping of

into conserving the number of elements, i.e. the

dimensionality of the vector between and . If is

interpreted as the set of types on , this implies both “axiom of

univalence” [4] and an (iso)morphism between the category of

all categories and the pair of and .

That consideration makes obvious the equivalence of the

continuous (smooth) and discrete as one and the same well-

ordering chosen as an ordinal among all well-orderings

(permutations) of the same elements and it by itself

accordingly. In other words, the continuous (smooth) seems to

be class of equivalence of the elements of a set (including finite

as a generalization of continuity as to finite sets).

Furthermore, the same consideration can ground (3) and (9)

above, i.e. the way, in which a coherent state before

measurement is equivalent of the statistical ensemble of

measured states in quantum mechanics. The same property can

be called “invariance to choice” including the invariance to the

axiom of choice particularly.

This means that the pure possibility, e.g. that of pure

existence in mathematics, also interpretable as subjective

probability should be equated to the objective probability of the

corresponding statistical ensemble once unitarity (energy

conservation) has already equated and .

Indeed, the set or its ordinal can be attributed to the elements

of and the statistical mix of all elements of corresponding

to a given element of . Any measurement ascribes randomly

a certain element of the corresponding subset of to any given

element of . Thus measurement is not unitary, e.g. a collapse

of wave function.

Then, and can be interpreted as two identical but

complementary dual spaces of the separable complex Hilbert

space. Initarity means right their identity, and the non-unitarity

of measurement representing a random choice means their

complementarity.

That “invariance to choice” can ground both so-called Born

probabilistic [5] and Everett (& Wheeler) “many-worlds”

interpretations of quantum mechanics [6], [7], [8]. The former

means the probability for a state to be measured or a “world” to

take place, and the former complement that consideration by

the fact that all elements constituting the statistical ensemble

can be consistently accepted as actually existing.

One can emphasize that the Born interpretation ascribes a

physical meaning of the one component (namely the square of

the module as probability) of any element of the field of

complex numbers underlying both and . After that, the

physical meaning of the other component, the phase is even

much more interesting. It should correspond to initarity, and

then, it seems to be redundant, i.e. the field of real numbers

would be sufficient, on the one hand, but furthermore, to time,

well-ordering, and choice implied by it. In other words, just the

phase is what is both physical and mathematical “carrier” and

“atom” of the invariance of choice featuring the separable

complex Hilbert space.

REFERENCES

[1] Neuman, J. von. Mathematische Grundlagen der Quantenmechanik.

Berlin: Springer, pp. 167-173 (1932).

[2] Kochen, S., Specker, E. The Problem of Hidden Variables in

Quantum Mechanics. Journal of Mathematics and Mechanics

17(1): 59‐87 (1968).

[3] Einstein, A., Podolsky, B., Rosen, N. Can Quantum‐Mechanical

Description of Physical Reality Be Considered Complete? Physical

Review 47(10): 777‐780.

[4] (Institute for Advanced Study, Princeton, NJ, Univalent

Foundations Program) Homotopy type theory univalent

foundations of mathematics. Princeton, NJ: Lulu Press, Univalent

Foundations Program (2013).

[5] Max Born - Nobel Lecture: The Statistical Interpretations of

Quantum Mechanics". Nobelprize.org. Nobel Media AB 2014.

Web. 13 Jul 2016.

<http://www.nobelprize.org/nobel_prizes/physics/laureates/1954/born

-lecture html>

[6] Everett III, H. "Relative State" Formulation of Quantum Mechanics.

Reviews of Modern Physics 29(3): 454-462 (1957).

[7] DeWitt, B. S., Graham, N. (eds.) The many-worlds interpretation of

quantum mechanics a fundamental exposition. Princeton, NJ:

University Press (1973)

[8] Wheeler, J. A., Zurek, W. H (eds.). Quantum theory and

measurement. Princeton, N.J.: Princeton University Press, 1983.