quantization causes waves smooth finitely computable functions are affine

Quantization causes wavesSmooth finitely computable functions are affine

Vladimir Anashin

Faculty of Computational Mathematics and CyberneticsFaculty of Physics

Lomonosov Moscow State University*****************************

Institute of Informatics ProblemsRussian Academy of Sciences

Vladimir Anashin (MSU-RAS) Quantization causes waves September 8, 2015, Belgrade 1 / 27

Outline

1 Experiments and problems

2 Representations of automata maps in real and complex spacesGeneral automataFinite automata

3 Relations to quantum theoryPhysical measurements, information, and quantum theoryTorus windings and wave functions


Experiments and problems

The first theme






Systems, automata and word maps

Notion: system (R. E. Kalman, 1969)

A (discrete) system (or, a system with a discrete timet ∈ N0 = {0, 1, 2, . . .}) is a 5-tuple A = 〈I, S,O,S,O〉 where






I is a non-empty finite set, the input alphabet;







O is a non-empty finite set, the output alphabet;








S is a non-empty (possibly, infinite) set of states;









S: I× S→ S is a state transition function;










O: I× S→ O is an output function.










O: I× S→ O is an output function.

During the talk, an automaton A(s0) (or, a transducer) is a systemwhere one of the states, s0 ∈ S, is fixed; it is called the initial state.




si∙ ∙ ∙χi+1χi

S

O

si+1 = S(χi , si)

state transition

input

ξi = O(χi , si) ξiξi−1 ∙ ∙ ∙ ξ0

output

Figure: Schematics of an automaton




Figure: Example state diagram of automaton (I = O = {0, 1};#S = 5, s0 = 1)




Any automaton transforms (left-) infinite words to (left-) infinite words;corresponding mapping of the set of all such words into itself is calledan automaton function.





Therefore given an automaton A with p-letter input/output alphabets I =O = Fp = {0, 1, . . . , p − 1} (p a prime), under a natural one-to-onecorrespondence between the set of all left-infinite words and the spaceof p-adic integers Zp, automaton function fA : Zp → Zp is a 1-Lipschitzmap w.r.t. p-adic metric on Zp. Vice versa, given a 1-Lipschitz mapf : Zp→ Zp, the map f is an automaton function for a suitable automatonA.





Automaton function fA : Zp→ Zp is a 1-Lipschitz map w.r.t. p-adic metricon Zp. Vice versa, given a 1-Lipschitz map f : Zp→ Zp, the map f is anautomaton function for a suitable automaton A.

Each letter of output word depends only on those letters of input wordwhich have already been fed to the automaton. Letters of the inputword can naturally be ascribed to ‘causes’ while letters ofcorresponding output word can be regarded as ‘effects’. ‘Causality’just means that effects depend only on causes which ‘already havehappened’. Therefore automata serve as adequate mathematicalformalism for causality principle .



Experimenting with automataGiven an automaton A with a p-letter input/output alphabets Fp ={0, 1, . . . , p− 1}, let f = fA be the automaton function. For k = 1, 2, . . .let Ek(fA) be the set of all points of the Euclidean unit square I2 =[0, 1]× [0, 1] ⊂ R2 s.t.:

Ek(fA) =

{(x mod pk

pk ;f (x)mod pk

pk

)

: x ∈ Zp

}

where given z =∑∞

j=0χjpj ∈ Zp, we denote z mod pk a non-negativeinteger whose base-p expansion is χk−1pk−1+ ∙ ∙ ∙+ χip+ χ0.

This way we define the automaton function on numbers n/pk ∈ [0, 1]

Low order digits are feeded to/outputted from the automaton prior tohigher order digits!



Experimenting with automataGiven an automaton A with a p-letter input/output alphabets Fp ={0, 1, . . . , p− 1}, let f = fA be the automaton function. For k = 1, 2, . . .let Ek(fA) be the set of all points of the Euclidean unit square I2 =[0, 1]× [0, 1] ⊂ R2 s.t.:

Ek(fA) =

{(x mod pk

pk ;f (x)mod pk

pk

)

: x ∈ Zp

}

where given z =∑∞

j=0χjpj ∈ Zp, we denote z mod pk a non-negativeinteger whose base-p expansion is χk−1pk−1+ ∙ ∙ ∙+ χip+ χ0.

A

w= = fA(w)χk−1 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙χ1χ0 ξk−1 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ξ1ξ0︸︷︷︸

(0.χk−1 . . . χ1χ0; 0.ξk−1 . . . ξ1ξ0)



Experimenting with automata

A

w= = fA(w)χk−1 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙χ1χ0 ξk−1 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ξ1ξ0︸︷︷︸

(0.χk−1 . . . χ1χ0; 0.ξk−1 . . . ξ1ξ0)

Ek(fA) = {(0.w; 0.f (w)) : w runs over words of length k}

Experimentally it can be observed that Ek(fA) when k → ∞ basicallyexhibits behaviour of two kinds only:




A

w= = fA(w)χk−1 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙χ1χ0 ξk−1 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ξ1ξ0︸︷︷︸

(0.χk−1 . . . χ1χ0; 0.ξk−1 . . . ξ1ξ0)



1 Ek(f ) is getting more and more dense so that at k→∞ they fill theunit square completely




A

w= = fA(w)χk−1 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙χ1χ0 ξk−1 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ξ1ξ0︸︷︷︸

(0.χk−1 . . . χ1χ0; 0.ξk−1 . . . ξ1ξ0)



1 Ek(f ) is getting more and more dense2 Ek(f ) is getting less and less dense and with pronounced straight

lines that constitute windings of a torus




A

w= = fA(w)χk−1 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙χ1χ0 ξk−1 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ξ1ξ0︸︷︷︸

(0.χk−1 . . . χ1χ0; 0.ξk−1 . . . ξ1ξ0)





The goal of the talk is to explain the following:

what really happens (=mathematical results), and




A

w= = fA(w)χk−1 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙χ1χ0 ξk−1 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ξ1ξ0︸︷︷︸

(0.χk−1 . . . χ1χ0; 0.ξk−1 . . . ξ1ξ0)





The goal of the talk is to explain the following:

what really happens (=mathematical results), and

how the results could be related to quantum theory interpretation.


Representations of automata maps in real and complex spaces

The main mathematical part





Representations of automata maps in real and complex spaces General automata

The next topic:






The automata 0-1 lawDenote via α(fA) Lebesgue measure of the plot of A, i.e., of the closureP(fA) = P(A) of the union of sets Ek(fA), k = 1, 2, 3, . . ..

Theorem (The automata 0-1 law; V. A., 2009)

Given an automaton function f = fA, either α(f ) = 0, or α(f ) = 1.






These alternatives correspond to the cases P(f ) is nowhere dense in I2

and P(f ) = I2, respectively.






These alternatives correspond to the cases P(f ) is nowhere dense in I2

and P(f ) = I2, respectively.

We will say for short that an automaton A is of measure 1 if α(fA) = 1,and of measure 0 if otherwise.

Finite automata (=automata with a finite number of states) are all ofmeasure 0

Therefore if A is a finite automaton then P(fA) is nowhere dense in I2

and thus P(fA) cannot contain ‘figures’, but it may contain ‘lines’.


Representations of automata maps in real and complex spaces Finite automata

Next topic:






Plots of linear finite automataGiven an automaton function is fA, denote via AP(A) = AP(fA) the setof all accumulation points of the plot P(A) ⊂ T2 on the torus T2.



Plots of linear finite automataGiven an automaton function is fA, denote via AP(A) = AP(fA) the setof all accumulation points of the plot P(A) ⊂ T2 on the torus T2.

Important example: linear automata

Let A be a finite automaton, and let fA(z) = f (z) = az+ b (a, b ∈ Zp ∩Qthen). Considering I2 as a surface of the torus T2, we have that

AP(f ) ={(x mod 1; (ax+ b)mod 1) ∈ T2 : x ∈ R

}

is a link of Nf torus knots either of which is a cable(=winding) withslope a of the unit torus T2: If a= q/k,b= r/s are irreducible fractions,d = gcd(k, s) then Nf is multiplicative order of p modulo s/d. Each cablewinds q times around the interior of T2 and k times around Z-axis.

Given a, b ∈ Zp, the mapping z 7→ az+ b is an automaton function of asuitable finite automaton if and only if a, b ∈ Zp ∩Q.

set of all rational p-adic integers is denoted via Zp ∩Q.Vladimir Anashin (MSU-RAS) Quantization causes waves September 8, 2015, Belgrade 10 / 27


Plots of linear finite automata




}

is a link of Nf torus knots either of which is a cable(=winding) withslope a of the unit torus T2: If a= q/k,b= r/s are irreducible fractions,d = gcd(k, s) then Nf is multiplicative order of p modulo s/d. By usingcylindrical coordinates (T2 : (r0− R)2+ z2 = A2; R= A= 1 for unittorus) we get:

r0

θ

z

=

R+ A ∙ cos

(ax− 2πb ∙ p`

)

xA ∙ sin

(ax− 2πb ∙ p`

)

, x ∈ R, ` = 0, 1, 2, . . .



Plots of linear finite automata




}

is a link of Nf torus knots either of which is a cable(=winding) withslope a of the unit torus T2: If a= q/k,b= r/s are irreducible fractions,d = gcd(k, s) then Nf is multiplicative order of p modulo s/d. Thereforethe plot of f (=of the automaton A) can be described by Nf

complex-valued functions:

AP(A) = AP(az+ b)←→ ei(ax−2πb∙p`); (x ∈ R, ` ∈ N0)



The affinity of smooth finitely computable functions

Q: What smooth curves are finitely computable; i.e. what smoothcurves may lie in the plot P(A) of a finite automaton A?





A: Only straight lines (=cables of torus with rational p-adic slopesand rational p-adic constant terms).





A: Only straight lines (=cables of torus with rational p-adic slopesand rational p-adic constant terms).

Theorem (V.A., in pNUAA, 2015, vol. 7, No 3, pp. 169–227)

Let g be a two times differentiable function (w.r.t. the metric in R)defined on [α, β] ⊂ [0, 1) and valuated in [0, 1); let g′′ be continuous on[α, β]. If (x; g(x)) ∈ P(A) for all x ∈ [α, β] then there exist a, b ∈ Zp ∩Qsuch that g(x) = ax+ b for all x ∈ [α, β] and AP(az+ b) ⊂ P(A). Given afinite automaton A, there are no more than a finite number ofa, b ∈ Zp ∩Q such that AP(az+ b) ⊂ P(A).






Actually this means that smooth curves in the plot of a finite automatonconstitute a finite union of torus links, and every link consists of a finitenumber of torus knots with the same slope.






The theorem can be considered as a contribution to the theory of com-putable functions since the main result means that a finite automatoncan compute only a very restricted class of smooth functions; namely,only affine ones.






The theorem holds for automata with m inputs and n outputs: Smoothsurfaces in the plot (in multidimensional torus) constitute a finitenumber of families of multidimensional torus windings, and each familyis a finite collection of windings with the same matrix A.

AP(zA+ b)←→ ei(xA−2πb∙p`); (x ∈ Rm; b ∈ Rn; ` ∈ N0)






The theorem holds for automata with m inputs and n outputs: Smoothsurfaces in the plot (in multidimensional torus) constitute a finitenumber of families of multidimensional torus windings, and each familyis a finite collection of windings with the same matrix A.

Thus the theorem can be applied to study hash functions since they allare automata functions of finite automata with multiple inputs/outputs.


Relations to quantum theory

Final part





Relations to quantum theory Physical measurements, information, and quantum theory

The next topic:






On physical measurements

A physical law is a mathematical correspondence betweenquantities of impacts a physical system is exposed to andquantities of responses the system exhibits.





The measured experimental values of physical quantities lie in Q.






People usually are trying to find a physical law as acorrespondence between accumulation points (w.r.t. the metrics inR) of experimental values.







An experimental curve is a smooth curve (the C2-smoothness iscommon) which is the best approximation of the set of theexperimental points.







An experimental curve is a smooth curve (the C2-smoothness iscommon) which is the best approximation of the set of theexperimental points.

A physical law is a curve which can be approximated by theexperimental curves with the highest achievable accuracy.



On physical measurementsLet physical quantities which correspond to impacts and reactions arequantized; i.e, take only values, say, 0, 1, . . . , p− 1. Then, once the sys-tem is exposed to a sequence of k of impacts, it outputs correspondingsequence of k reactions. Considering these sequences as base-p ex-pansions of natural numbers z= αk−1pr−1 + ∙ ∙ ∙ + α0, after the normal-ization z

pk we obtain experimental points in a unit square of R2.





Our main theorem shows that if the number of states of automatonwhich corresponds to a physical system is much less than the length ofinput sequence of impacts then experimental curves necessarily tendto straight lines (or torus windings, under a natural map of the unitsquare onto a torus). This implies the linearity of a physical law.






Q: Once Prof. A. Yu. Khrennikov asked a question: “Why mathemati-cal formalism of quantum theory is linear?”






Q: Once Prof. A. Yu. Khrennikov asked a question: “Why mathemati-cal formalism of quantum theory is linear?”A: Quantization + Small set of states⇒ Linearity



Information in physics: Wheeler’s ‘it from bit’ doctrineJohn Archibald Wheeler (July 9, 1911 – April 13, 2008) was a promi-nent American physicist known for his contribution to general relativityand quantum theory. In 1990, Wheeler suggested that information isfundamental to the physics of the universe and started developing theinformational interpretation of physics. He wrote:




‘It from bit’ symbolizes the idea that every item of the physicalworld has at bottom — a very deep bottom, in most instances— an immaterial source and explanation; that which we call re-ality arises in the last analysis from the posing of yes-no ques-tions and the registering of equipment-evoked responses; inshort, that all things physical are information-theoretic in originand that this is a participatory universe.





...it is not unreasonable to imagine that information sits at thecore of physics, just as it sits at the core of a computer.





...it is not unreasonable to imagine that information sits at thecore of physics, just as it sits at the core of a computer.

Let’s give some mathematical reasoning why ‘it’ is indeed ‘from bit’.Vladimir Anashin (MSU-RAS) Quantization causes waves September 8, 2015, Belgrade 15 / 27

Relations to quantum theory Torus windings and wave functions

The next topic:






Speculation: Automata as models of quantum systemsAccumulation points of a plot of a finite linear automaton (whose au-tomaton function is then f : z 7→ az+ b for suitable a, b ∈ Zp ∩ Q) looklike a finite collection of waves with the same wavenumber a (up to anormalization s.t. ~ = 1), where x stands for position, 2πb for angularfrequency ω and p` for time t.




Speculation: Automata as models of quantum systemsAccumulation points of a plot of a finite linear automaton (whose au-tomaton function is then f : z 7→ az+ b for suitable a, b ∈ Zp ∩ Q) looklike a finite collection of waves with the same wavenumber a (up to anormalization s.t. ~ = 1), where x stands for position, 2πb for angularfrequency ω and p` for time t. Consider a special case when a ∈ Z andb= 0:

AP(A) = AP(az)←→ ei(ax); (x ∈ R, ` ∈ N0)



Speculation: Automata as models of quantum systems

AP(A) = AP(az)←→ ei(ax); (x ∈ R, ` ∈ N0)

It is reasonable to suggest that indeed a is the wavenumber (cf.left and right pictures); thus wavelength λ = 1

a. Moreover, thenω = λ−1 = a; and note that AP(az+ at) = AP(az) for every t ∈ Z.





The “time-looking” multiplier p` is a proper time of the automaton.





The “time-looking” multiplier p` is a proper time of the automaton.Namely, multiplying by p corresponds to on step of the automaton:(p`x)mod 1 is an `-step shift of base-p expansion of x ∈ R.





The “time-looking” multiplier p` is a proper time of the automaton. Canp` be treated a physical time?





The “time-looking” multiplier p` is a proper time of the automaton. Canp` be treated a physical time?Yes!





The “time-looking” multiplier p` is a proper time of the automaton. Canp` be treated a physical time?Yes!Just take p close to 1 (forgetting for a moment that p is a base of a radixsystem); i.e., p= 1+ τ where τ is small.For instance, assume that τ is a Planck time (=a quant of time) or othertime interval which is less then the accuracy of measurements and thuscan not be measured.





The “time-looking” multiplier p` is a proper time of the automaton. Canp` be treated a physical time?Yes!Just take p close to 1 (forgetting for a moment that p is a base of a radixsystem); i.e., p= 1+ τ where τ is small.

Then p` ≈ 1+ `τ and therefore for large ` we see that `τ = t is just a

time. And here we are:

ei(ax−2πb∙p`) ≈ ei(ax−2πb∙(1+t)) = c ∙ ei(ax−2πb∙t) ← the wave!!!









Is it mathematically correct to put p= 1+ τ where 0< τ � 1?









Is it mathematically correct to put p= 1+ τ where 0< τ � 1?Yes, if one uses β-expansions (introduced by Renyi—Parry in 1957–1960) rather than base p-expansions.



Using β-expansions of numbersTo construct a plot of an automaton over a p-letter alphabet we usebase-p expansions of numbers: To every pair of input/output words

input word χk−1 . . . χ1χ0 −→ output word ξk−1 . . . ξ1ξ0

we put into the correspondence the point on the torus

(χk−1p−1+ ∙ ∙ ∙χ1p−k+1+ χ0p−k; ξk−1p−1 ∙ ∙ ∙+ ξ1p−k+1+ ξ0p−k)







We may take β ∈ R+ such that bβc = p− 1 and use β-expansionsrather than base-p expansions; that is, we put into the correspondenceto the pair of input/output words the following point on the torus:

(χk−1β−1+ ∙ ∙ ∙χ1β

−k+1+ χ0β−k; ξk−1β

−1 ∙ ∙ ∙+ ξ1β−k+1+ ξ0β

−k)








(χk−1β−1+ ∙ ∙ ∙χ1β

−k+1+ χ0β−k; ξk−1β

−1 ∙ ∙ ∙+ ξ1β−k+1+ ξ0β

−k)

We must take β s.t. arithmetics of numbers represented by β-expansions can be performed by a finite automaton.








(χk−1β−1+ ∙ ∙ ∙χ1β

−k+1+ χ0β−k; ξk−1β

−1 ∙ ∙ ∙+ ξ1β−k+1+ ξ0β

−k)

We must take β s.t. arithmetics of numbers represented by β-expansions can be performed by a finite automaton.Such β do exist; e.g., we may for instance take β = N

√2.



Using β-expansions of numbersWe may take β ∈ R+ such that bβc = p−1 and use β-expansions ratherthan base-p expansions; that is, we put into the correspondence to thepair of input/output words the following point on the torus:

(χk−1β−1+ ∙ ∙ ∙χ1β

−k+1+ χ0β−k; ξk−1β

−1 ∙ ∙ ∙+ ξ1β−k+1+ ξ0β

−k)


√2. In general

to ensure the finiteness, if β = 1 + τ with 0 < τ � 1, then β mustsatisfy equation 2 = u(β) where u is a polynomial with coefficients 0, 1(for instance, we may take β = N

√2 with N large enough).




(χk−1β−1+ ∙ ∙ ∙χ1β

−k+1+ χ0β−k; ξk−1β

−1 ∙ ∙ ∙+ ξ1β−k+1+ ξ0β

−k)


√2. In general



Then

necessarily the input/output alphabets are binary, and




(χk−1β−1+ ∙ ∙ ∙χ1β

−k+1+ χ0β−k; ξk−1β

−1 ∙ ∙ ∙+ ξ1β−k+1+ ξ0β

−k)


√2. In general



Then


any torus link will be dense and can be ascribed to a matter wavecei(ax−2πbt) where x, t are space and time coordinates accordingly.




(χk−1β−1+ ∙ ∙ ∙χ1β

−k+1+ χ0β−k; ξk−1β

−1 ∙ ∙ ∙+ ξ1β−k+1+ ξ0β

−k)

Then (as β = 1+ τ with τ small)


any torus link can be ascribed to a matter wave cei(ax−2πbt) wherex, t are space and time coordinates accordingly.

Therefore our main theorem can serve a mathematical reasoning whya specific ‘it’ — the matter wave, which is a core of quantum theory —is indeed ‘from bit’; that is, from sufficiently long binary inputs of anautomaton with a relatively small number of states.



Using β-expansions of numbersWe may take β ∈ R+ such that bβc = p−1 and use β-expansions ratherthan base-p expansions;Then (as β = 1+ τ with τ small)



The approach seemingly reveals more correspondences between phys-ical entities and mathematical properties of automata, for instance:

helicity corresponds to the sign of a;








probability of finding a particle at a certain point of spacecorresponds to the average amplitude when t ∈ [0,∞);









automata with multiple inputs/outputs correspond tofinite-dimensional Hilbert spaces (to include into the considerationinfinite-dimensional Hilbert spaces one needs to considerautomata of measure 0 rather then just finite ones);








automata with multiple inputs/outputs correspond to Hilbertspaces ;

pure states correspond to ergodic linear subautomata, mixedstates correspond to automata states leading to more than 1ergodic subautomata; entagled states correspond to states fromergodic subautomata of automata with multiple inputs/outputs.



Using β-expansions of numbers

pure states correspond to ergodic linear subautomata, mixedstates correspond to automata states leading to more than 1ergodic subautomata; entagled states correspond to states fromergodic subautomata of automata with multiple inputs/outputs.




Interpretation of the case β = 1+ τ where 0< τ � 1

In our model, β j may be interpreted as a time which is needed toacquire the next j-th bit; so the time Tk needed to acquire a k-bitinformation is exponential in k, namely Tk =

1τ (β

k − 1).






1τ (β

k − 1).

Note that then Tk ≈ k for large k.






1τ (β

k − 1).


The case Tk ∼ k implies that the time needed to acquire the nextj-th bit does not depend on j.






1τ (β

k − 1).



But E. Lerner recently has shown that in the latter case the plotwill be a polygon (=”body”) rather than a torus winding (=”wave”).






1τ (β

k − 1).



But E. Lerner recently has shown that in the latter case the plotwill be a polygon (=”body”) rather than a torus winding (=”wave”).

So in QT acquiring of information actually is exponential in time, in acontrast to classical case when this is linear. In our model, classicalcase appears as a limit case at τ → 0; i.e., as it should be: In a contrastto QT, there are arbitrarily small time intervals in classical case.




In our model, classical case appears as a limit case at τ → 0; i.e., as itshould be: In a contrast to QT, there are arbitrarily small time intervalsin classical case.


Messages of the talk

Discreteness+Causality+Finiteness⇒Waves




Waves, the its, are indeed from bits




Waves, the its, are indeed from bits

Acquisition of information in QT during ameasurement is exponential in time (though baseis close to 1)


Thank you!


p = 2: f (x) = 1+ x+ 4x2;α(f ) = 1.


f (z) = 1115z+ 1

21, p= 2. (Therefore Nf = mult7 2= 3)


f1(z) = −2z+ 13; f2(z) = 3

5z+ 27, (p= 2).


quantization causes waves smooth finitely computable functions are affine

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