Orevkov’s 1972 Results Can This Result Help . . . Problem Revisited Orevkov, Khalfin, and Explanation Possible Applications Quantum Field Theory: How Bibliography Constructive Mathematics Acknowledgments Home Page Can Help Physics Title Page Olga Kosheleva and Vladik Kreinovich ◭◭ ◮◮ ◭ ◮ University of Texas at El Paso El Paso, Texas 79968, USA Page 1 of 19 olgak@utep.edu, vladik@utep.edu http://www.cs.utep.edu/vladik/olgavita.html Go Back http://www.cs.utep.edu/vladik Full Screen Close Quit
Orevkov’s 1972 Results 1. Orevkov’s 1972 Results Can This Result Help . . . • In 1972, Vladimir Orevkov presented a talk on con- Problem Revisited structive complex analysis at LOMI. Explanation Possible Applications • The main results from this talk were published in 1974. Bibliography • In that paper, he provided new more explicit construc- Acknowledgments tive proofs of basic results of complex analysis: Home Page – that a function is differentiable iff it can be ex- Title Page panded in Taylor series at each point, ◭◭ ◮◮ – that two such (analytical) functions are equal if ◭ ◮ they coincide on a non-finite compact set, and Page 2 of 19 – that it is possible to constrictively find all the roots of such function on each bounded domain. Go Back • These results were previously proved by Vladimir Lif- Full Screen schitz in a more implicit way. Close Quit
Orevkov’s 1972 Results 2. Orevkov’s 1972 Results Can This Result Help . . . • As usual, many results from classical (non-constructive) Problem Revisited mathematics turned out to be constructively true. Explanation Possible Applications • Some results from classical mathematics turned out to Bibliography be constructively false, in the sense that: Acknowledgments – while there is a classical existence theorem, Home Page – no general algorithm for constructing the corre- Title Page sponding object is possible. ◭◭ ◮◮ ◭ ◮ Page 3 of 19 Go Back Full Screen Close Quit
Orevkov’s 1972 Results 3. Can This Result Help Physics? Can This Result Help . . . • This talk attracted attention of Leonid Khalfin, Orevkov’s Problem Revisited LOMI colleague interested in physics applications. Explanation Possible Applications • Khalfin asked whether constructive mathematics can Bibliography solve a problem related to physics use of complex #s. Acknowledgments • On macro-level, we observe many non-smooth and even Home Page discontinuous phenomena: Title Page – earthquakes, ◭◭ ◮◮ – phase transitions, etc. ◭ ◮ • However, on the micro-level, all equations and all phe- Page 4 of 19 nomena are smooth – and even analytical. Go Back • Some of these phenomena are very fast – so we perceive Full Screen them as discontinuous. Close Quit
Orevkov’s 1972 Results 4. Can This Result Help Physics (cont-d) Can This Result Help . . . Problem Revisited • For complex numbers, smoothness means analyticity. Explanation • Analyticity has been successfully used in quantum field Possible Applications theory. Bibliography • For example, to compute the values of some integral Acknowledgments expressions, it is convenient to use the fact that: Home Page – for an analytical function, Title Page – a contour integral over a closed loop is 0: ◭◭ ◮◮ � ◭ ◮ f ( z ) dz = 0 γ Page 5 of 19 – or it is equal to an explicit expression in terms of Go Back the poles. Full Screen Close Quit
Orevkov’s 1972 Results 5. Can This Result Help Physics (cont-d) Can This Result Help . . . • Thus, by using a loop [ − N, N ] ∪ γ ′ , we can: Problem Revisited Explanation – replace a difficult-to-compute integral over real num- � N Possible Applications bers − N f ( x ) dx Bibliography – with an easier-to-compute integral over the com- Acknowledgments � plex values γ ′ f ( z ) dz . Home Page • This idea – mostly pioneered by Nikolai Bogolyubov – Title Page led to many successful applications. ◭◭ ◮◮ • This “macro” analyticity has been confirmed by many ◭ ◮ experiments and makes perfect physical sense. Page 6 of 19 Go Back Full Screen Close Quit
Orevkov’s 1972 Results 6. Can This Result Help Physics (cont-d) Can This Result Help . . . Problem Revisited • The problem is that in traditional mathematics: Explanation – such “macro” analyticity is equivalent to “micro” Possible Applications one, Bibliography – that the corresponding dependencies can be ex- Acknowledgments panded in Taylor series: Home Page f ( z ) = a 0 + a 1 · ( z − z 0 )+ a 2 · ( z − z 0 ) 2 + . . . + a n · ( z − z 0 ) n + . . . Title Page • In the opinion of physicists, however: ◭◭ ◮◮ ◭ ◮ – this “micro” analyticity does not make direct phys- ical sense, Page 7 of 19 – since on the micro level, quantum uncertainty makes Go Back exact measurements impossible. Full Screen Close Quit
Orevkov’s 1972 Results 7. Can This Result Help Physics (cont-d) Can This Result Help . . . Problem Revisited • From this viewpoint, it is desirable to come up with a model in which: Explanation Possible Applications – physically meaningful macro analyticity is present, Bibliography but Acknowledgments – physically meaningless micro analyticity is not. Home Page • Khalfin hoped that: Title Page – this “thornless rose” effect can be achieved ◭◭ ◮◮ – if we consider constructive mathematics instead of ◭ ◮ the traditional one. Page 8 of 19 Go Back Full Screen Close Quit
Orevkov’s 1972 Results 8. Can This Result Help Physics (cont-d) Can This Result Help . . . Problem Revisited • In the early 1970s, this hope did not materialize, since: Explanation – as Errett Bishop has shown in his 1967 book (and Possible Applications as Vladimir Lifschitz pointed to Khalfin), Bibliography – the fact that macro analyticity implies micro one Acknowledgments can be proven in constructive mathematics as well. Home Page • Indeed, once we know f ( z ), we can determine all the Title Page coefficients a n as ◭◭ ◮◮ 1 f ( z ) � a n = 2 π · i · ( z − z 0 ) n +1 dz. ◭ ◮ γ Page 9 of 19 • And there are known algorithms for computing an in- Go Back tegral of a computable function. Full Screen Close Quit
Orevkov’s 1972 Results 9. Problem Revisited Can This Result Help . . . • Bishop’s derivation is based on the usual constructive Problem Revisited mathematics. Explanation Possible Applications • In this approach, existence of an object means, in ef- Bibliography fect: Acknowledgments – the existence of an algorithm producing more and Home Page more accurate approximations to this object, Title Page – irrespective to how long this algorithm may take. ◭◭ ◮◮ • A more realistic idea is to only allow feasible (= polynomial- ◭ ◮ time) algorithms are allowed. Page 10 of 19 • It turns out that in this case, Khalfin’s dream can be materialized. Go Back Full Screen Close Quit
Orevkov’s 1972 Results 10. Problem Revisited (cont-d) Can This Result Help . . . Problem Revisited • Indeed: while there exists an algorithm computing: Explanation – for each computable macro analytical function, Possible Applications – all the terms in its Taylor series expansion. Bibliography • However, the computation time of this algorithm seems Acknowledgments to grow exponentially with the number n of the term. Home Page • Let us provide arguments for this conclusion. Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 19 Go Back Full Screen Close Quit
Orevkov’s 1972 Results 11. Explanation Can This Result Help . . . • We have a computable function f ( z ). Problem Revisited Explanation • This means that we can, given z , compute f ( z ). Possible Applications • For simplicity, we can also assume that we know the Bibliography upper bound D on | f ′ ( z ) | ≤ D . Acknowledgments • Computation of the n -th Taylor coefficient a n is based Home Page on the formula Title Page 1 f ( z ) � a n = 2 π · i · ( z − z 0 ) n +1 dz. ◭◭ ◮◮ γ ◭ ◮ • Here, the simplest possible loop γ around the point z 0 is a circle of some small radius r < 1. Page 12 of 19 • For this loop, | z − z 0 | = r . Go Back • We want to compute a n with a given accuracy ε > 0. Full Screen • This means that we need to compute the corresponding Close integral with accuracy ε ′ = 2 π · ε . Quit
Orevkov’s 1972 Results 12. Explanation (cont-d) Can This Result Help . . . � Problem Revisited • A natural way to compute an integral g ( z ) dz is to consider the corresponding integral sum Explanation Possible Applications � g ( z i ) · ∆ z, with | z i +1 − z i | = h for some small h. Bibliography Acknowledgments • In this approximation, we approximate g ( z ) with g ( z i ) Home Page on each arc of length h for which | z − z i | ≤ h/ 2. Title Page • The inaccuracy of this approximation is ◭◭ ◮◮ � � | g ′ ( z ) | · | z − z i | | g ( z ) − g ( z i ) | ≤ max ≤ z ◭ ◮ | g ′ ( z ) | · ( h/ 2) . max Page 13 of 19 z Go Back ( z − z 0 ) n +1 ≈ f ( z ) f ( z ) • Here, g ( z ) = r n +1 . Full Screen | g ′ ( z ) | ≤ max | f ′ ( z ) | D Close • Thus, max = r n +1 . r n +1 z Quit
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