Elements of Santilli s Lie - isotopic time evolution theory

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1 Elements of Santilli s Lie - isotopic time evolution theory Svetlin Georgiev Georgiev Sorbonne University, Paris, France Department of Differential Equations, Faculty of Mathematics and Informatics, University of Sofia, Bulgaria sgg2000bg@yahoo.com, svetlingeorgiev@gmail.com October, 205 Abstract In this article we define Santilli s Lie isotopic powe series and investigate it for absolutely and uniformly convergence and differentiability. Index Terms: evolution, isotopic, Lie. Introduction Numerous aspects help the bradening of the scattering theory to incorporate non - Hamiltonian effects, effects which can not be represented using the conventional Hamiltonian. Following decades of research this incorporated required the construction by various authors of a new mathematics, well known as isomathematics and proposed by Santilli [2] in 978, subsequently studied by the same author and numerious pure and applied mathematicians as: S. Okubo, H. Myung, M. Tomber, Gr. Tsagas, D. Sourlas, J. Kadeisvili, A. Aringazin, A. Kirhukin, R. Ohemke, G. Wene, G. M. Benkart, J. Osborn, D. Britten, J. Lohmus, E. Paal, L. Sorgsepp, D. Lin, J. Voujouklis, P. Broadbridge, P. Chernooff, S. Guiasu, E. Prugovecki, A. Sagle, C. Jiang, R. Falcon Ganfornina, J. Nunez Valdes, A. Davvaz, and others. As a result of these efforts, the new mathematics can be constructed via the systematic application of axiom - preservibg liftings, called isotopies, of the totality of the mathematics of quantum mechanics, inluding all its operators and all its operations, icluding the isotopic lifting of numbers, functinal analysis, differential calculas, geometries, topologies, Lie theory and others [3], [4], [5]. In [6] is shown that the isotopies can be very easly constructed using the application of nonunitary transforms to the totality of the formalism of the conventional scattering theory. The physical needs for isomathematics have been indicated in [6], and consists in the necessity for a representation of non-hamiltonian scattering effects in a form that is invariant over time so as to admit the same numerical predictions under the same conditions at different times. Following the study of all possible alternatives, the latter condition required the representation of non- Hamiltonian scattering effects with an axiom preserving generalization of the trivial positivede nite unit of quantum mechanics ĥ = into the most general possible positive-deffinite as a condition to characterize an isotopy, integrodifferential operator Î which is as positive - definite as +, functional depending of local variables, that is assumed to be the inverse of the isotopic element ˆT + ˆ>0 Ît, r, p, a, E,... = ˆT ˆ>0 and it is called Santilli isounit. Santilli introduced a generalization called lifting of the conventional associative product ab into the form ab a ˆ b = a ˆT b

2 called isoproduct for which: Î ˆ a = ˆT ˆT a = a ˆ Î = a ˆT ˆT = a for every element a of the field of real numbers, complex numbers and quaternions. The Santilli isonumbers are defined as follows: for given real number or complex number or quaternion a, with isoproduct â ˆ ˆb = â ˆTˆb = a ˆT â = aî, ˆT b ˆT = ab ˆT = âb. If a 0 the corresponding isoelement of a will be denoted with â or Î â. On more technical grounds, the nonconservative character of the events implies the inapplicability of Lie s theory with the familiar time evolution of a Hermitean operator A i da = [A, H] = AH HA in favor of Santilli s Lie-isotopic time evolution, which is presented firstly in [7] of 978 i da = [A, H] = AT H HT A, where T is a second operatorgenerally independent from, and non - commutating with H, characterizing the nonunitarity of the theory. Let X and Y be complex Banach spaces. With LX, y we will denote the space of linear bounded operators C : X Y. Let A, T and H LX, y and da = iat H HT A. Our aim here is to be investigated the serties A0 + da 0w + d 2 A 2! 2 0w2 +.. The series. will be called Santilli s Lie isotopic power series. 2 Convergence of Santilli s Lie isotopic power series Firstly we will deduct the general term of 99.. We have d 2 A 2 = d = i da da da T H HT = i iat H HT AT H HT iat H HT A = i 2 AT H HT AT H HT AT H HT A, d 3 A 3 = d d 2 A 2 d = i 2 A T H HT d2 A 2 2 = i i 2 AT H HT AT H HT AT H HT AT H HT i 2 AT H HT AT H HT AT H HT A = i 3 AT H HT AT H 2 HT AT H HT AT H HT AT H HT AT H +HT 2 AT H HT A = i 3 AT H HT AT H 2 2HT AT H HT AT H +HT 2 AT H HT A, 2

3 d 4 A = d d 3 A 4 3 d = i 3 A T H HT d3 A 3 3 = i i 3 AT H HT AT H 2 2HT AT H HT AT H +HT 2 AT H HT AT H i 3 HT AT H HT AT H 2 2HT AT H HT AT H +HT 2 AT H HT A = i 4 AT H HT AT H 3 2HT AT H HT AT H 2 +HT 2 AT H HT AT H HT AT H HT AT H 2 +2HT 2 AT H HT AT H HT 3 AT H HT A = i 4 AT H HT AT H 3 3HT AT H HT AT H 2 +3HT 2 AT H HT AT H HT 3 AT H HT A = i k HT AT H 3 k. HT k AT H We suppose that for some natural number n we have d n A n = i n n HT AT H n k. We will prove that d n+ A n+ = i n+ n HT AT H n k. Really, d n+ A n+ = d n d n A n n n k d = i n A T H HT dn A n n k HT k AT H k HT k AT H = i i n n n k HT AT H n k i n HT n k n HT AT H n k = i n+ AT H HT AT H n HT AT H HT AT H n n HT n AT H HT AT H HT AT H HT AT H n HT k AT H HT k AT H HT AT H HT AT H n n HT n AT H HT A 3

4 = i n+ AT H HT AT H n n n + HT AT H HT AT H n + n HT n AT H HT A = i n n k n n k HT AT H n k. HT k AT H From here and induction principle follows that for every natural n we have For r > 0 and x 0 Ω we will denote with S r x 0 the ball S r x 0 = {x C : x x 0 < r}. Theorem 2.. Let w 0 0 and w 0 Ω. Then S w0 0 Ω and in every ball S r 0, 0 < r < w 0, the series 2. is absolutely and uniformly convergent. Proof. Since w 0 Ω then the series An w0 n is convergent. From the properties of convergent series we have that lim n A n w0 n = 0. From here we conclude that the sequence {A n w0 n } n= is bounded. Therefore there exists a constant M > 0 such that A n w n 0 M for n N. d n A n = i n n k n HT AT H n k. Let A n = in n! n k n HT AT H n k 0, A 0 = A0.. HT k AT H HT k AT H We note that when we write C0 we have in mind that the operator C acts on the zero element of X. Now we will investigate the series gw = A n w n, 2. where w is a complex variable. If w > then we will make the change w = w and therefore w = w w > 0. Let Ω be the set of all w for which the series 2. is convergent. The set Ω is not empty because 0 Ω. Let w S w0 0. Then w < w 0 and A n w n = A n w0 n wn w n 0 = w n w 0 A n w0 n M w n w 0 and from here A n w n M w n <. w 0 If w < r A n w n = A n r n wn r n = r n A n w r n < w 0 n A n = A n w0 n M w r n. w r n w r n Consequently A n w n M w n <, r the series 2. is absolutely and uniformly convergent. 4

5 With R we will denotre the radius of convergence of 2.. From the definition of radius of convergence of power series we have Also, R = sup w. w Ω. If R = 0 then Ω = {0}. 2. If R = then the series 2. is convergent in all complex plane. 3. From Cauchy - Hadamard formula we have R =. lim n A n n Theorem 2.2. Let A, T, H LX, y, T > 0, H > 0. Then Proof. We have A n R > = i n n 2 T H. n AT H HT AT H n k k HT k n HT A T H n k n HT A T H n k n HT k AT H HT k AT H HT k AT H + T HA T n k H n k n H k T k A T H + T H A H n k T n k = 2 A T n H n n = 2 n A T n H n, from here therefore A n 2 n A T n H n, A n n 2 T H A n, n HT k AT H lim n A n n 2 T H lim n A n = 2 T H. HT AT H n k n HT A T H n k HT k AT H From here and Cauchy - Hadamard formulae we conclude that R = lim n A n n 2 T H. Theorem 2.3. If there exist positive constants M and l such that then R l. A n M l n 5

6 Proof. It is enough to be proved that the series 2. is uniformly bounded for w < l. Let w l = q <. Then A n w n = w n A n w n M l n = M q n, therefore Ak w k Ak w k M qk = M q <. Theorem 2.4. Let Then A n w n = Ã n w n in S R A n = Ãn for n N {0}. Proof. Since w = 0 Ω then, after we put w = 0 in 2.2 we get A 0 = Ã0. From here and 2.2 we obtain A k w k = Ã k w k in S R 0, k= from where k= k= A k w k = Ã k w k in S R k= We put w = 0 in the last equality and we have From here and 2.3 k=2 and etc. A = Ã. A k w k = Ã k w k in S R 0 k=2 Theorem 2.5. The function g is a continuous function in S R 0. Proof. Let ρ 0, R and w, w 0 S R 0. Then gw gw 0 n= An w n n= An w n 0 n= An w n w n 0 n= An w w 0 w n + w n 2 w w n 0, therefore gw gw 0 = n= An w w 0 w n + w n 2 w w n 0 n= An w w 0 w n + w n 2 w w n 0 n= An w w 0 w n + w n 2 w w n 0 n= An w w 0 w n + w n 2 w w 0 n n= An w w 0 ρ n + ρ n 2 ρ + + ρ n n= nρn A n, gw gw 0 nρ n A n w w n= Now we will prove that the series n= nρn A n is uniformly convergent for every ρ 0, R. Let ρ 0, R is arbitrary chosen and fixed. Let also ρ ρ, R. Then, since ρ < R then the series n= An ρ n is uniformly convergent, from where lim n A n ρ n = 0 and therefore the sequence { ρ n A n } n= is a bounded sequence. Consequently there exists a positive constant M 2 such that A n ρ n M 2 for n N. 6

7 From here n= n An ρ n = n n= n An ρ n ρ ρ ρ M2 ρ n= n ρ ρ n. Let q = ρ ρ. Then q < and n= nan ρ n n= nan ρ n n= n An ρ n M2 ρ = M2 ρ n= n ρ ρ n n= nqn. Let b k = kq k. Then lim k b k+ b k = lim k k+q k kq k = lim k k+ k q = q <. is conver- Consequently the series gent and then cρ := n= nqn n A n ρ n <. n= Since ρ S R 0 was arbitrary chosen then the series n= nan ρ n is uniformly convergent for every ρ S R 0. From 2.4 we obtain gw gw 0 cρ w w Since ɛ > 0 was arbitrary chosen and for it we find δ = δɛ > 0 such that form w w 0 < δ we have gw gw 0 < ɛ we conclude that g is a continuous function at w 0. Because w 0 S R 0 was arbitrary chosen then g is a continuous function in S R 0. Corollary 2.6. The series na n w n n= is a convergent series in S R 0. Proof. Let w < R and ρ w, R. Then w ρ < and from here and the proof of the previous Theorem we have n= nan w n n= nan w n n= n An w n n= n An ρ n w n ρ n n= n An ρ n <. Because w S R 0 was arbitrary chosen we conclude that n= nan w n is convergent in S R 0. Theorem 2.7. The function g is differentiable function in S R 0. Proof. For w S R 0 we define the function uw = na n w n. For w, w S R 0 we have gw gw uw nan w n Let ɛ > 0 be arbitrary chosen and fixed. Let also δ = ɛ +cρ. Then if w w 0 < δ, from 2.5, we get ɛ gw gw 0 cρ w w 0 < cρδ = cρ + cρ < ɛ. An w n w n = An w n An w n = An w n w n nan w n An w n w n gw gw w w uw = nan w n nw n, A n w n w n nw n. w w 2.6 7

8 We will note that w n w n Really, nw n = nn w w 0 θ θw + θw n 2 dθ. nn w w 2.7 for w, w S ρ 0 and from 2.8 we have gw gw uw = w w nn An 0 θ θw + θw n 2 dθ 0 θ θw + θw n 2 dθ = nn w w 0 θw + θw w n 2 dθ = nn 0 θw + θw w n 2 dw + θw w = n 0 θdw + θw w n = n θw + θw w n θ= +n 0 w + θw w n dθ = nw n θ=0 w w nn A n 0 θ θw + θw n 2 dθ = w w nn An 0 θ θw + θw n 2 dθ w w nn An 0 θ θw + θw n 2 dθ = w w nn An 0 θ θw + θw n 2 dθ w w nn An + n 0 w + θw w n dw + θw w = nw n + w + θw w n θ= θ=0 = wn w n nw n. Now we apply 2.7 in 2.6 and we obtain gw gw uw = w w n= nn An 0 θ θw + θw n 2 dθ. 2.8 Let ρ 0, R is arbitrary chosen and fixed. Then 8 0 θ θ w + θ w n 2 dθ w w nn An 0 θ θρ + θρn 2 dθ = w w n= nn An 0 θρn 2 dθ w w n= nn An ρ n 2, uw gw gw w w n= nn An ρ n Now we will prove that for every ρ 0, R the series nn An ρ n 2 is a convergent se-

9 ries. Really, let ρ 0, R is arbitrary chosen and fixed. and let also ρ ρ, R. Since 0 < ρ < R then the series An ρ n is a convergent series. Therefore lim n A n ρ n = 0 and from here the sequence {A n ρ n } n= is a convergent sequence. Consequently, there exists a positive constant M 3 so that A n ρ n M 3 for n N. Therefore n= nn An ρ n 2 We put nn An ρ n 2 nn An ρ n 2 = n 2 nn An ρ n ρ ρ ρ 2 M 3 R 2 nn ρ ρ n 2, nn An ρ n 2 M 3 R n 2. nn ρ ρ q 2 = ρ ρ. Then, using 2.0, q 2 < and nn An ρ n 2 Let Then M 3 R 2 nn qn 2 2. lim n d n+ d n d n = nn q n 2 2. n+nq = lim n 2 n nn q n 2 2 = q 2 lim n n+ n = q 2 <. 2. Consequently nn qn 2 is convergent and from 2.0 the series nn An ρ n 2 is convergent. Because ρ 0, R was arbitrary chosen then the series nn An ρ n 2 is convergent for every ρ 0, R. Therefore c ρ = nn A n ρ n 2 < for every ρ 0, R. From 2.9 follows that gw gw uw c ρ 2.2 w w for every ρ 0, R. Let ɛ > 0 be arbitrary ɛ chosen and fixed. Let also δ = +c. Then ρ from w w < δ we have gw gw uw c ρ w w < c ρδ = c ρ ɛ +c ρ < ɛ for every ρ 0, R. Because ɛ > 0 was arbitrary chosen and for it we found δ = δɛ > 0 so that from w w < δ we have gw gw uw < ɛ then the function g is a differentiable function at w and g w = uw. Since w S R 0 was arbitrary chosen then the function g is a differentiable function in S R 0 and for every w S R 0 we have g w = uw. Using induction one can prove Corollary 2.8. g C S R 0. Theorem 2.9. Let An be absolutely convergent series to 0. Then lim w,w: w w < gw = 0. Proof. Without loss of generality we will consider the case when w and 0 < w w <. Then w < and there exists a positive constant M 4 such that Let P n = 0 < w w M n A k, n = 0,, 2,.... 9

10 Then the sequence {P n } n= is a convergent sequence. A 0 = P 0, A k = P k P k, k =, 2,.... We put Then s n w = n A k w k. s n w = A 0 + A w + A 2 w A n w n = P 0 + P P 0 w + P 2 P w P n P n w n = P 0 w + P w w 2 + P 2 w 2 w P n w n w n 2 + P n w n = P 0 w + P w w + P 2 w 2 w + + P n w n w + P n w n, n s n w = w P k w k + P n w n. 2.4 Since An is an absolutely convergent series to 0 then for w < we have lim n P n w n = 0 and from 2.4 gw = lim s nw = w n P k w k. 2.5 Let ɛ > 0. Then there exists m N such that P n < ɛ for every n m. We choose w so that to satisfy 2.3 and w < ɛ. From here n=m P n w n n=m P n w n < ɛ 2.6 n=m w n = ɛ w m w. From 2.3 we have w w M 4 w, from where Since w < then w M 4 w +. w m w M 4 w + and from 2.6 we obtain n=m P n w n < ɛ M 4 w + = ɛm 4 + ɛ w. w From the last inequality we get w n=m P n w n w ɛm 4 + ɛ w = M 4 ɛ w + ɛ w w M 4 ɛ w + M 4 ɛ = M 4 ɛ + w and from 2.5 gw = w P n w n = w m P n w n + w n=m P 6nwn w m P n w n + w n=m P n w n w m P n w n +M 4 ɛ + w. Because ɛ > 0 was arbitrary chosen lim w,w:0< w w < gw = 0. 0

11 3 Acknowledgements This research has been supported in part by a grant from the R. M. Santilli Foundation. References [] S. Georgiev. Foundations of isodifferentail calculus. foundation.org/docs/isohandbook pdf [2] R. M. Santilli, On a possible Lieadmissible covering of Galilei s relativity in Newtonian mechanics for nonconservative and Galilei form-noninvariant systems, Hadronic J., , available in free pdf download from [3] R. M. Santilli, Rendiconti Circolo Matematico Palermo, Suppl. 42, , available as free download from [4] R. M. Santilli, Elements of Hadronic Mechanics, Vol. I 995 [5a], Vol. II 9995 [5b] Academy of Sciences, Kiev, available in free pdf downloads from foundation.org/docs/santilli- 300.pdf processesiclatip-3, Kathmandu University, Nepal, April20. [7] R. P. Feynman, Quantum ElectrodynamicsW. A. Benjamin, Inc, N. Y Svetlin Georgiev born on April 5th, 974 is a Bulgarian mathematician working in areas of ordinary differential equations, partial differential equations, stochastic differential equations, Clifford algebras, Clifford analysis, isomathematics. [5] R. M. Santilli, Hadronic Mathematics, Mechanics and Chemistry,, Vol. I [6a], II [6b], III [6c], IV [6d] and [6e], international academnioc press, 2008, available as free downlaods from [6] A. Animalu, R. M. Santilli, Nonunitary Lie - isotopic and Lie - admissible scattering theories of Hadronic Mechanics, II: Deformations - Isotopies of Lie s theory, Special Relativity and Mechanics, Proceeding of the third International Conference on Lie admissible treatment of irreversible