Parcels of Universe or why Schr"odinger and Fourier are so relatives?

> REPLACE THIS LINE W ITH YOUR P APER IDENTIFICAT ION NUMBER (DOUB LE -CLICK HERE T O EDIT) < 1  This p aper is about the s urprising connection between the Fourier heat equation and the Schrödinger wave equation. In fact, if t he independent “time” variable in the heat equation is replaced by the time variable multiplied by     , the heat equation becomes t he Schrö dinger equation. T wo quite different phys ical phe nomena ar e put in clo se connection: the heat diff usion in a material and the probability a mplitude o f particles in a n atom. It is a fact of life that t he movements of a small particle f loating randoml y in a f luid, th e w ell- known Brownian motion, is regulated by the Fourier equation while the probabilistic b ehavior of the matter ar ound us, the quantum world, is dr iven b y the Schröd inger equation b ut no known stochastic proce ss seems at work here . The appar ent simplicity of the formal connection by a “time - rotation”, a Wick rotation as it is co mmonly kno wn, seems to po int otherwise. Why t his connection? I s there any physical intuitive explanation? Is there an y practical v alue ? I n this pap er, the authors atte mpt to shed some light o n the above questio ns. T he recent concept of volume q uantization in nonco mmutative geometry, due to Connes, Cha mseddine and M ukhanov, points again to stochastic pro cesses also under lying the q uantum world making Fourier and Sch rödi nger strict relatives. I. I NT RODUCTI ON In reference [ 1], it is shown that the n umbers alon g a ro w of a Tartaglia-Pascal triangle goes to fit a Gau ssian function. In t he paper [2], the authors obtain a kind of quantum Tartaglia - Pascal triangle that appears to be the “square root” of the classical one. A square ro ot could entail also imaginary values and it is here where the deep connection bet ween B rownian motion and quantum world starts. The diffusion heat (Fourier) equation is generally written in the form           where  is the di ffusion coefficient and  the prob ability distribution of the B rownian motion 1 . The Schrodinger equation is w ritten in the form               Marco Frasca is currentl y wo rking in MBDA Italia S.p .A. at Seeker Division (Rome) . Alfonso Farina was Senior VP S elex-Sistemi I ntegrati (retired), now consultant of L and&Naval Div ision of L eonardo Company (Rome). 1 I n Ref.[1] th e hea t equatio n w as written in the form where  was equivalent to our  and represe nted th e distributio n of the temperature in a body a nd  was the diffusion constant in the given b ody . In this paper,  is given by the microscopic motion of atoms colliding with a Brownian particle as devised b y Einstein [4]. where  is the wave f unction ,  the Planck constant d ivided by  and  th e m ass of the par ticle. T hese equations are formally ver y similar a nd can be transfor med i nto each other provided w e ch ange the ti me variable to    , what physicists call a W ick rotation [3 ]. This apparently innocu ous modification is i ndeed a drastic change as no w  is a complex function while  is a well-behaving real probability distribution. So, a questio n naturally arises : Where does the imaginary time co me from? The so lution of t he Fo urier equation, describi ng a probabilistic effect as the scatter ing of a small p article by the molecules of a fluid 2 , has a t ypical Gaussian for m that ar ises naturall y from the asymptotic form of the binomial co efficients, t h e Tartaglia-Pascal trian gle. As sho wn in [2], a Gaussi an probability distribu tion in a quantu m world represents a well- localized p article that, evolving i n time, loses such a precise localization, but this Gaussian pdf is the square of the wave function. Therefore , we can connect the bino mials of the Fourier equation with the square roo t of the probability distribution of a m oving free particle in a quantum world. However, this entails complex values. T his mapping 3 was proved in [2] and is the startin g point for a new view on stochastic processes and a p arceled world. So, the rather stunning conclusion is that there exists a square root of the Tartaglia -Pascal triangle, a q uantum T artaglia - Pascal trian gle (QT PT), that represents a co mpletely new complex-valued four dimensional fig ure (x, y,z,t) that lives in the realm of quantum mecha nics 4 . Our universe, as we currently understand it, has a peculiar mathematical str ucture. It is a Rie mann manifold with the Hausdorff property [5] . One can always have open sets without inter section with points inside. No patholog y whatsoever. A Rie mann m anifold is dee ply con nected to stochastic processes. In t wo dimensions, it can be always reconstructed from B rownian motion [6] . However , there is a deeper connection as shown b y Alain Connes, Ali Chamseddine and Viatc heslav Mu khanov. In esse nce a 2 Note that t he effect arises b y the averaged squared velocity of the molecule s that gives the ove rall temper ature of the fluid itsel f. 3 The mapp ing we proved states: “ There exists a discrete mapping onto the wave function that solves the Schrödinger equatio n for a free p article via the Tartaglia-Pascal triangle. Such a mapping gives complex-valued probability amplitudes wh ose squares are t he binomial c oefficients .” 4 Th e correspondence, as given i n [ 2], is b etwe en the binomial coefficient 󰇡   󰇢 and the quantum analog  󰇡   󰇢      󰇭 󰇡   󰇢              󰇮 . Parcels of Universe or why Schrödinger and Fourier are so relatives? Marco Frasca, Al fonso Farina LFellow, IEEE > REPLACE THIS LINE W ITH YOUR P APER IDENTIFICAT ION NUMBER (DOUB LE -CLICK HERE T O EDIT) < 2 Riemann manifold ca n b e rec onstructed by two sets of small volumes [7 ]-[8]. So, motion between s uch par cels, by a particle th at is able to sense them , can be ass imilated to a Brownian motion. Where do suc h small volumes come from? T he idea comes out from the nonco mmutative geo metry uncovered by Alain Connes in the last cen tury [9]. The idea behind noncommutative geometr y can b e traced back to the de ep connection b etween algebr a and geometry as initially conceived by Descartes. It is well -known that whatever algebraic expression one consider s there is a correspo nding geometrical object of it. E.g. , the equation        represents a circle on a Cartesian plane. Neverthele ss, all this works fine because complex numbers have the proper ty to commute ea ch other. We lear ned over the last ce ntury t hat our world does not seem to agree with this pro perty at its foundations. Werner Heisenberg initially co nceived this and it is now our understanding o f the b ehavior of ele mentary particles. In th is case, we have what Paul D irac called q - numbers and m omentum  and position  do n ot comm ute yielding the famous equation      (otherwise stated 󰇟    󰇠   where 󰇟  󰇠 is the commutator) fro m which th e uncertainty principle derives. T he reason is that such q - numbers are matrices with an infinite number of eleme nts or, better stated, op erators acting on a Hilbert spac e. Why such no ncommutative b ehavior changes the behavior of particles? W e know that dyna mics can be r epresented on geometric object called phase space where both momenta a nd positions ar e taken into acco unt. Dynamics i s d escribed by a trajectory o n such a spa ce. In add ition, when energy is conserved, on a p hase space the motion happens on a given geometrical objec t like a torus or a sphere. T herefore, there is a d eep connection bet ween Newton mechanics a nd geometry in a phase space. W illiam Rowan Ha milton discovered this two ce nturies ago. Therefore, what does it make deter ministic the Newton mechanics? This comes out fro m t he commutativity of momenta and positions. Other wise , we ha ve quantum mechanics and the structure of the p hase space changes dra matically. Quantu m mechanics is the conseque nce of the noncommutative geometr y of the phase space and one has to cope with operators and Hilbert spaces that are n ot properly geometric obj ects as our intuitio n lear ned from ordinary experience. One w orks with small volumes in the quantum phase space. T he order of magnitude i s given by  . As oppo site to the co ntinuity in the classical Newtonian phase space. As sta ted above, a Riemann manifold is characterized by a metric t hat d etermines the geodesic curves 5 on it. It is a differentiable ma nifold. This me ans that the d erivative is well defined on fu nctions d efined on the ma nifold. We can make a correspondence between the points on a Riemann m anifold and an algebra of functions defined on it that w e can call maps. These fu nctions can re present the manifold itself as the geometric concept of a point d oes. Stated in a simpler way, we 5 A geode sic cu rve is a generalization of the concept of “straight line” of the Euclidean sp ace to curved s paces as the shortest path joining two points. are moving fro m a picto rial representation to its algebraic elements as for the phase spac e we move fro m the geometrical structure of the motion like an orbit to functions rep resenting it like coordinates and momenta. W hen one moves from such functions to oper ators (or q -numbers a la Dirac as Heisenberg did) one gets a nonco mmutative Riemann manifold. Connes, Cha mseddine and Mukhanov proved that such a noncommutative ma nifold comes out made by t wo kinds o f elementary volumes and it is from here that the d eep connection w ith stochastic proce sses starts. This m eans t hat the relation bet ween Fourier and Schr ödinger eq uations , formally given b y a Wick rotation, has a deep physical meaning. Mathematicall y, it entail s the introd uction of a new class of stoc hastic processes: the fractional powers of a Wiener process [10]. In summary, the relat ion bet ween t he Fourier and the Schrödinger equations, thro ugh a Wic k ro tation, is not merely a mathematical curiosit y but rather has deep physic al implications a s the w orld is struct ured as a nonco mmutative Riemann manifold on which the particles, able to feel its quantization, perfor m B rownian motion with a pro cess equation, which is exactly t he Schrödinger eq uation. T his means t hat the analo gy between these famous eq uations rel ies on the kind of atoms one co nsiders: For Fourier, it is the matter being parceled while for Schrödinger, it is the space being made by ele mentary volumes ac ting like t he atoms in the matter. Ho wever , geometry is con tinuous in the for mer and discretized in the latter . Conse quently, in a lattice world, if one moves randomly between the sites of the la ttice, one obtains the so lution to the Fourier equation. In a quantum wor ld, represented - for instance - by a quantum T artaglia -Pascal triangle [2] , one has that the space on which the particle moves is discretized and so, moving o n such a space yields t he solution to the Schrödinger e quation. The quantum Tartaglia triangle is, in a se nse, the square r oot of the classical o ne [2] and this is the root o f the “i” factor 4 . In the re maining part of the p aper, we will d iscuss this deep connection showin g how the emerg ence o f a quantum w orld comes out from a quantized space. II. T HE WO RLD IN PA RCELS As sho wn b y Cha mseddine i n [11], we ca n always define a set of functions           to cover a volume with the condition 󰇛  󰇜   󰇛  󰇜     󰇛  󰇜    . These r epresent spheres i n    dimensions. We c an co ver t he volume o f a given manifold with such spher es so that the y beco mes a wa y to “measure” t he manifold itself . Ho wever, unfortunatel y, we will find holes e verywhere in our covering notwithstanding the manifold w e started with was simply connected. Therefore, w e ca nnot cl aim we are able to fully recover the original manifold fro m smaller volumes. In o rder to overcome t his di fficulty, le t us return to the case of quantum mechanics. As we stated above, in this case , we are studying t he motion of a particle in a particular m anifo ld > REPLACE THIS LINE W ITH YOUR P APER IDENTIFICAT ION NUMBER (DOUB LE -CLICK HERE T O EDIT) < 3 represented through the coordinates 󰇛  󰇜 w ith  being the position and  the momentum and this ea sily generalize s to the  dim ensional case. This is wh at physicists call phase space . W illiam Rowan Ha milton intr oduced this co ncept earlier in the XIX century b y reformulating Ne wton eq uations of motion in this way [12]. Fig. 1 . A par celed manifold 6 . Furthermore, Heisenberg noticed that the coord inates in phase space do not co mmute at all and the reason of t he no n- commutation must be attr ibuted to the existence of the Planck constant  [1 3]-[14]. T his is generally stated in the form 󰇟    󰇠   . So, one can ask if there exists a solution to this equation in such a way to formulate t he d ynamics of a particle that moves on suc h noncom mutative pha se space. T oday, we know the answer is affirmative, provided  and  are not ordinary c(omplex) -numbers but rat her q (uantum) -numbers that is, sel f-adjoint operators that act on a Hilbert space o f functions. So, we ha ve a so- called “triple”, in the nomenclature due to Alain C onnes, for med by an al gebra of operators, a Hilbert space of functions on which they ac t upon and we ca n postulate a spectral (Dirac) operator to measure distances i n such a noncommutative phase space that acts o n the functions i n the Hilbert sp ace. T o define the Dirac operato r one could use e.g. the Schrödinger equation that determines the way a particle m oves ar ound se nsing dista nces. We see that we ha ve built a geometr y without recurring to any ordinary concept of p oint, geodesic and similar but our definition o f a nonco mmutative pha se sp ace, a geo metric concept anywa y, is purely a lgebraic i n a per fect Cartesian spirit. 6 This figure was inspired b y a talk given by Al ain Connes in Castiglioncell o (Italy) o n 2014. What are the co nsequence s of ha ving such a nonco mmutative structure for the phase space? So, let us consider a particle of unitary mass moving under the ef fect of an ela stic spri ng. It is everyday experience that such a particle undergoes oscilla tor motion. T his is true if t he p hase space is a standard geo metric object. In such a case, the manifold on which the particle moves in phase space ha s the form o f an el lipsoid , degenerating to a n e llipse in t w o dimensions, g eometricall y characterized by the ener gy available to the p article. The effect of the Heisenberg commutation relation changes this dramatically as we get a fully quantized m anifold in small volumes having a well-defined magnitude. This is standard material in quantu m mechanics textbooks (e. g. see [15] ). Specifically, if the a vailable ener gy is  , we will have a number of quanta  propor tional to it a nd eac h quantum is measured by the pro duct of  multiplied by the pulsation  of the oscillatio n of the clas sical particle. Then, we can e valuate the volume o f the quantized ellipsoid that in this simple case is just given b y  , being     and       its semi-axes. Now, using energy quantization, the volume of the noncommutative manifold will be 2  󰇛  󰇜 and the phase space is q uantized i n small volumes ea ch o ne o f dimension propor tional to the Planck constant. T urning b ack to the Chamsedd ine’s exa mple, one can always try to co ver a given manifo ld with small volumes usi ng spheres but t he covering will always be unsatis factory. However, looking at quantum mechanics, we see that t he Heisenberg quantization conditi on gra nts us that a p erfect covering of a noncommutative manifold can be o btained. So, Chamseddine’s example can be made at work moving t o a noncommutative manifold. But, wh at should the quantizati on condition b e i n t his ca se? To understand t his, let us co nsider the case of a circle of radius 1. In this case, we can cover the circle with phase changes     and we can choose an arbitrary angle to ob tain the co vering with even smaller angles. A dista nce could be given b y  󰇛    󰇜        󰆒     sin 󰇡  󰆒  󰇢 . Now, i f  is the derivative with re spect to the angle and promote the angle to a position op erator, it is not difficult to observe that   󰇟    󰇠   ap pears to be fully analogous to t he Hei senberg quantization condition u se d above. T he q uantum prob lem informs us t hat the solution has a spectrum o f integers for  . If we try to extend thi s to a generic one-di mensional Riemann mani fold  , for  unitar y as for t he circle, t he q uantization conditio n   󰇟    󰇠   will give a so lution for  provided the length of the manifold satisfies      , that is the Riema nn manifold is quantized exactly as happened for the oscillator manifold. We have again a triple made b y an operator algebra, a Hilbert sp ace and a “Dirac” oper ator describing it. No w, we are cover ing a noncommutative manifold and the problem with an o rdinary manifold applies no more. These ideas ca n be generalized to arbitrar y di mensions and what Connes, Chamsed dine and Muk hanov proved w as that the quantization co ndition that w e have introd uced w ith the example of the circle can be consistentl y ap plied in 4 > REPLACE THIS LINE W ITH YOUR P APER IDENTIFICAT ION NUMBER (DOUB LE -CLICK HERE T O EDIT) < 4 dimensions. This permits the introduction of a fundamental experimental fact ob served in nature: For each particle seen in nature, there is it s anti -particle [1 6]. T his fact is widely known as the matter-a ntimatter parad igm. However, for sure, the Universe would not exist the way we know and are ab le to observe without this double nature of the matter. T o descr ibe this, one h as to introduce complex numbers. The effect of complex conjugation and time reversal moves the descriptio n from matter to anti-matter. T his has a dee p geo metrical origin as shown in noncommutative geo metry, as we need to consider a complex Riemann manifold to fit m atter and anti - matter in place. T his implies t hat a noncommutative Riemann manifold splits up in two kind s of a large number of parcels making up its volume as dep icted in Fig. 1. We assu me that these can be distributed in a r andom way because all the configurations are ad missible for the build ing of the manifold yielding a n identical result for the volume . So, we will be a ble to o btain a nonco mmutative Riemann manifold b ack pro vided we co nsider small parcels with volu me 1 an d  , the two fundamental unities in the world of mathematics. III. M OVING AROUND T HE PARCELS We can i magine that, for a p article with a given ener gy, moving on a manifold like that in Fig.1 means perfor ming a kind o f B rownian motion. Indeed, we ca n get a set of equivalent m anifolds, each one with the same volumes but with the parcels differentl y dispo sed. Fig. 2. Brow nian motion on a parce led manifold. To understand what is going on, w e extend the case of t he circle in the pr eceding section to the sphere. I n this case we have to introduce two d ifferent op erators   and   in f ull analogy with the case o f the circle o f the p receding section . We would like also to have a manifold t hat co uld be quantized consistently. This is generall y p ossible only i f one makes use of a Clifford algebra [ 16] . A Clifford algebra is a trick y way to take t he square root of the wave equation. T his wa s uncovered by Paul Dirac that in this way obtai ned the correct equation to d escribe the electro n and a p rope r generalization of the Sc hrödinger equatio n that co uld accou nt for t he relativity d iscovered by Albert Einstein [17]. The Dirac equation is essential in noncommutative geometr y to measure distances. Consider the wave equatio n on the surface o f the sphere o f radius 1. T his can be written as:           󰇛 󰇜        . Taking t he square root means t hat we have to use the nab la o perator  and w e do this by i ntroducing three new o bjects   ,   and   to be defined below. Then, let us consider the ne w (Dirac) operato r built as             where we have set    󰇛     󰇜 . One gets, taking the square   , that t he wave equatio n is full y recovered provided σs have the properties              and            w hen    . T his is what we call a Clifford algebra and the lower non-t ri vial di mensionality for the σ are 2x2 matrices firstly i ntroduced by Wolfgang P auli to describ e the spin of particles [ 18]. G iven this C lifford algeb ra of σ matrices , we ca n build a quantized manifold by considering the map                  ob tained by a nalogy from the square ro ot of the wave eq uation and the case o f t he circle in the preceding sectio n . Here we have introd uced a new Clifford al gebra, the same as above, but indep endent of it as this applies now to the “coor dinates”   ,   and   . This is the rea son why we called t hese matrices γ rather than σ . T hen, as Connes, Mu khanov and Ch amseddine proved, a manifold gets q uantized with respect to the conjugate operator  provided the Heisenberg-like quantization conditi on  󰇛 󰇟  󰇠 󰇟   󰇠󰇟   󰇠󰇜 =   σ holds, b eing    and σ a 2x2 matrix [7] -[8]. T he trace is over the γ matrices . T his quantization condition , which we have seen to arise natura lly from the ca se of the circle of the p receding sect ion , represent s a generalization o f the well -known Heisenberg condition 󰇟    󰇠   we discussed in the introduction for quantu m mechanics and the phase sp ace. The triple product o f 󰇟   󰇠 is due to the d imensionality of the manifold we are now considering. What Connes, M ukhanov a nd Chamsedd ine sho w is that this quantizatio n condit ion admits a solution, i.e.   ,   and   exist, i f and only if the volume is quantized. This is true given t wo kinds of small volumes (parcels): One has volume 1 and the other volu me i . The question i s how will a particle move on such a manifold? What will its eq uation of motion be? T he answer was given in a recent paper by o ne of t he authors [19] . To understand this, it is i mportant to notice th at this n oncommutative man ifold will have randoml y d istributed parcels that make it as, whatever co nfiguration we will choo se, t he volu me of the manifold will remai n the sa me. This situation is reall y si milar to the Brownian motion o f a particle in a fluid wh ose theory was put forward by Albert Einstein i n 190 5 [ 4]. He showed that the motion of a p article, under the effect o f t he scattering of the molecule s co mposing t he fluid, is r uled by t he Fourier heat eq uation. T his means tha t this is a stocha stic process with > REPLACE THIS LINE W ITH YOUR P APER IDENTIFICAT ION NUMBER (DOUB LE -CLICK HERE T O EDIT) < 5 a given pro bability di stribution functio n. In o ur case, there is something different as the m otion of the particle, although random as well, can hit b oth rea l and complex valued volumes randomly. This i mplies that o ur stochastic process cannot be a real one but, rather, a co mplex valued stochastic pro cess. In order to have a stoc hastic pro cess producing t he values 1 and i , we start with a Bernoulli process B that describes the tossing o f a coin, yielding a rando m sequence with +1 and -1. From this we can introd uce the p rocess         that will yield a si milar sequence with 1 and i . It is not dif ficult to observe that     . Then, the par ticle will move arou nd the maps Y tha t m ake the ma nifold and these are me mbers o f a Clifford algebra as we have seen. T herefore, this is a Wiener process but also means that the Bernoulli process B is n ot independent but will originate from the signs of the s ingle Brownian steps. Using the typical notation of stochastic calculus, where steps are written like differe ntials, the case o f our sphere will take a form l ike (see [19 ])   󰇟  󰇛             󰇜     󰇛             󰇜      󰇛        󰇜 󰇠 where the condition     must be understood and a, b, c , e, f, g and k, m a re numbers . W e used index es on the various Berno ulli pro cesses to distinguish their contributions on t he differe nt maps as shown in [10] and [19]; we are j ust extracting the s quare root of a W iener p rocess. This entails t he use o f a Cliffor d algeb ra ( the Dirac’s trick) as happened for the wave eq uation. So, summing everything up, we have seen th at a Riemann manifold can be quantized if it is noncommutative and, to accommodate the matter-antimatter paradig m experimentally, this quantization has parcels of unitary volu mes 1 and i . Moving on it entails a complex stoc hastic process that i s nothing else than the square roo t of the well-kno wn Bro wnian motion (see Fig. 2) . Now, we should as k what is the equivalent o f the Fourier heat equation for t his case as t his i s a characteristic of whatever stochastic process. The s urprising answer is that now, for a co mplex stoc hastic process, we ha ve a complex Fourier equation. T he magic happened and we are back to Schrödinger. IV. T HE SURPRI SING RELATION BETW EEN THE S CHRÖDI NGER AND F OURIER EQUATI ONS From the discussion given above it appea rs that t he Schrödinger equatio n is a Fourier equation in disg uise. They both represent a stochastic process es but one has to move from real sto chastic p rocesses to complex ones. Anyhow, the relation is deeper as one ari ses ta king a formal square root of the oth er. In the end, one has a diffusion equation for the random proce ss but, in a q uantum world, the equi valent of the probability distribution function is not real and one has to struggle working with m odulus square of w hat is commonly called a wave function that is now the p robability distribution function to consider for nat ural proce sses. This can be put at test very ea sily with a numerical computation. We compute a Brownian m otion and take its square root. The former has a histogra m rec overing the Fourier kernel, a Gaussia n distribution. We can Wick rotate the numerical data of the sq uare roo t yielding a Ga ussian curve agai n, the Wic k-rotated Schrödinger kernel, as e xpected by our co nnection [ 19]. T he result i s given i n Fig. 3 and is in excellent agreement with expectations. Fig.3. Compariso n between the Fourier ker nel of a Bro wnian motion and the Wick-rotated Schrödinger kernel of its square root. So, the formal change of the ti me variab le    , introduced by W ick, hides a deep physical fact: The ordinar y B rownian motion described by the Fourier equation changes into the motion on a par celed universe, w here matter a nd antimat ter exist, particles spin and th e Sch rödinger equation describes what is going on. But now, w e are w orking with complex quantities w hose only the modulus square can m ake s ense. Fig. 4 depicts this rel ation b etween Fourier and Schrö dinger equations and related physical conseque nces. Fig. 4. How Fourier and Schrödinger speak each other and Wick invented a communication w ay . V. C ONCLUSION S We have seen how a simple formal r elation between two famous equations entails a deep m athematical concept with new stochastic p rocesses underl ying it. A new mathematical technique also forecasts a wealth of possible app lications. > REPLACE THIS LINE W ITH YOUR P APER IDENTIFICAT ION NUMBER (DOUB LE -CLICK HERE T O EDIT) < 6 Indeed, this paper may benefit modern ap plications of quantum mechanics to en gineering i n co mmunication a nd sensing. Namel y, quantum co mmunications thro ugh entangled states [2 0], that are solutio ns o f the Sc hrödinger eq uation we derived, f rom a new clas s of co mplex stochastic processes, and quantu m radars [21] that can b enefit fro m quantum illumination [22] -[23] . Entangled states are particular solution of the Schrödinger equation w ith two or more particles li ke photons. Represent ing them through stochast ic process es could m ake eas ier their use in designing devices that employ such states. Stochastic proce sses l ike the ones that are disc ussed here can be easily si mulated digitally or through so me analogic device using discrete components. I n this vie w, filtering can witness new avenues of applicatio ns. In general, there co uld be a lot of possible new applic ations wherever a new class o f stochastic processes is uncovered. This, in view, is our hope for the future for this exciting mathematical achieve ment we have got from a world so distant from our common sense. R EFERENCES [1] A. Farina, S. Giompapa, A. Graziano, A . Liburdi, M. Ravanelli, F. Zirilli, “Tartaglia and Pascal triangle: a historical perspective with ap plications; fro m probability to modern physics, signal pro cessing, a nd finance”, S ignal, Image an d Video Processing, vol. 7 , Issue 1, pp. 173- 188 , 2013. [2] A. Farina, M. Frasca & M. Sedehi, “ Solving Schrödinger equation via Tartaglia/Pascal triangle: a possible link between stochastic pro cessing and quantum mechanics”, Signal, Imag e and Video Processing, vol. 8 , pp. 27 - 37, 201 4. [3] G. C. W ick, "P roperties of B ethe-Salpeter Wave Functions", P hysical Review, vol. 96, pp. 11 24 – 1134, 1954 . [4] A. Einstein, "Über die von der molekularkinetischen Theorie d er Wärme geford erte B ewegung von in ruhenden Flüssigkeiten suspendierte n Teilchen", Annalen der Ph ysik (in Ger man), vol. 3 22, pp. 549 – 560, 1905. [5] S.W . Hawking, G.F.R. Elli s, “The la rge scale structu re of the univers e”, Cambridge University P ress,1975 . [6] M. Frasca, “Two -di mensional Ricci flow a s a sto chastic process”, arXiv:0901.4703 [ math - ph ], 2009 , unpublished. [7] A. H. Chamseddine, A. Connes and V. Mu khanov, “Quanta of Geo metry: noncommutative aspects”, Physical Review Letter s, vol. 1 14, 091302 , 2015. [8] A. H. Chamseddine, A. Connes and V. Mukhanov, “Geometry and the quantum: basics”, J. High Energ. Phys. 2014: 9 8, 2014. [9] A. Connes, “Noncommu tative geo metry”, Acade mic Press, 1994 . [10] M. Frasca, A. 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Pauli, “Zur Quanten mechanik des magnetischen Elektrons”, Zeitschri ft für Physik 4 3, 601 (1927). [19] M. Frasca, “Nonco mmutative geometry and stoc hastic processes” in F. Nielsen and F.Barb aresco (eds), Geometric Science o f I nformation. GSI 2017 . Lect ure Notes in Co mputer Science, vo l 10589. Springer Cham. [20] C.Q. Choi, “Unhackable qu antum networks take to space”, IEEE Sp ectrum 54, Issue 8, 12 (2017). [21] M. Lanzagorta, “Quantum Radar (Synthesis Lectures on Quantu m Computi ng)”, Morgan & Cla ypool Publishers, 2011. [22] S. L loyd, “ Enhanced Sensitiv ity of P hotodetection via Quantum Illumination ”, Science 321, 1463 (2008). [23] Si -Hui T an, B. I. Erk men, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. P irandola, and J . H. Sh apiro, “Quantum Ill umination with Gaussian Sta tes”, Phys. Rev. Le tt. 10 1, 253601 (2008). Marco Frasca is curr ently e mployed at MBDA Italia S.p.A. working on signal p rocessing and se nsing. He is a lso a theoretical p hysicist with more than 80 publicatio ns on refereed jo urnals. Alfonso Fa rina in 1974, h e j oined Selenia, then Selex ES, where he beca me Director of the Analysis of I ntegrated Systems Unit a nd subseq uently Director of Engineeri ng of t he Large B usiness S ystems Division. In 2 012, he was Senior VP and Chief T echnology O fficer o f the Compan y, reporting di rectly to the Pr esident. He retired in October 2014. From 1979 to 1985, he was also professor o f “Radar T echniques” at the University o f Naple s (I T). Today, He is a Visiting Professor at University College Lon don (UCL), De pt. Electronic and Electrical Engineering. He is a Disti nguished Lecturer of IEEE AESS and I EEE SPS Distinguished Indu stry Speaker. He is a consultant to Leonard o S.p.A. “Land & Naval Defence Electronics Division” ( Rome). He is a Me mber of the Editorial Bo ard of IEEE SP Magazine.