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Parcels of Universe Or Why Schrödinger and Fourier Are So Relatives? > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 1 Parcels of Universe or why Schrödinger and Fourier are so relatives? Marco Frasca, Alfonso Farina LFellow, IEEE where ψ is the wave function, ℏ the Planck constant divided This paper is about the surprising connection between the by 2휋 and 푚 the mass of the particle. These equations are Fourier heat equation and the Schrödinger wave equation. In formally very similar and can be transformed into each other fact, if the independent “time” variable in the heat equation is provided we change the time variable to 푡 → 푖푡, what replaced by the time variable multiplied by 풊 = √−ퟏ, the heat physicists call a Wick rotation [3]. This apparently innocuous equation becomes the Schrödinger equation. Two quite modification is indeed a drastic change as now ψ is a complex different physical phenomena are put in close connection: the function while 푃 is a well-behaving real probability heat diffusion in a material and the probability amplitude of distribution. So, a question naturally arises: Where does the particles in an atom. It is a fact of life that the movements of a imaginary time come from? small particle floating randomly in a fluid, the well-known Brownian motion, is regulated by the Fourier equation while The solution of the Fourier equation, describing a probabilistic the probabilistic behavior of the matter around us, the effect as the scattering of a small particle by the molecules of 2 quantum world, is driven by the Schrödinger equation but no a fluid , has a typical Gaussian form that arises naturally from known stochastic process seems at work here. The apparent the asymptotic form of the binomial coefficients, the simplicity of the formal connection by a “time-rotation”, a Tartaglia-Pascal triangle. As shown in [2], a Gaussian Wick rotation as it is commonly known, seems to point probability distribution in a quantum world represents a well- otherwise. Why this connection? Is there any physical intuitive localized particle that, evolving in time, loses such a precise explanation? Is there any practical value? In this paper, the localization, but this Gaussian pdf is the square of the wave authors attempt to shed some light on the above questions. The function. Therefore, we can connect the binomials of the recent concept of volume quantization in noncommutative Fourier equation with the square root of the probability geometry, due to Connes, Chamseddine and Mukhanov, points distribution of a moving free particle in a quantum world. 3 again to stochastic processes also underlying the quantum However, this entails complex values. This mapping was world making Fourier and Schrödinger strict relatives. proved in [2] and is the starting point for a new view on stochastic processes and a parceled world. I. INTRODUCTION So, the rather stunning conclusion is that there exists a square root of the Tartaglia-Pascal triangle, a quantum Tartaglia- In reference [1], it is shown that the numbers along a row of a Pascal triangle (QTPT), that represents a completely new Tartaglia-Pascal triangle goes to fit a Gaussian function. In the complex-valued four dimensional figure (x,y,z,t) that lives in paper [2], the authors obtain a kind of quantum Tartaglia- the realm of quantum mechanics4. Pascal triangle that appears to be the “square root” of the classical one. A square root could entail also imaginary values Our universe, as we currently understand it, has a peculiar and it is here where the deep connection between Brownian mathematical structure. It is a Riemann manifold with the motion and quantum world starts. Hausdorff property [5]. One can always have open sets without intersection with points inside. No pathology The diffusion heat (Fourier) equation is generally written in whatsoever. A Riemann manifold is deeply connected to 휕푃 휕2푃 the form = 퐷 where 퐷 is the diffusion coefficient and 푃 휕푡 휕푥2 stochastic processes. In two dimensions, it can be always the probability distribution of the Brownian motion1. The reconstructed from Brownian motion [6]. However, there is a 휕ψ ℏ2 휕2ψ deeper connection as shown by Alain Connes, Ali Schrodinger equation is written in the form 푖 = − 휕푡 2푚 휕푥2 Chamseddine and Viatcheslav Mukhanov. In essence a Marco Frasca is currently working in MBDA Italia S.p.A. at Seeker 2 Division (Rome). Note that the effect arises by the averaged squared velocity of the molecules that gives the overall temperature of the fluid itself. Alfonso Farina was Senior VP Selex-Sistemi Integrati (retired), now 3 consultant of Land&Naval Division of Leonardo Company (Rome). The mapping we proved states: “There exists a discrete mapping onto the 1 In Ref.[1] the heat equation was written in the form wave function that solves the Schrödinger equation for a free particle via the Tartaglia-Pascal triangle. Such a mapping gives complex-valued probability where 푢 was equivalent to our 푃 and amplitudes whose squares are the binomial coefficients.” represented the distribution of the temperature in a body and 훾 was the 4 The correspondence, as given in [2], is between the binomial coefficient 푛 2 diffusion constant in the given body. In this paper, 퐷 is given by the (푘− ) 푛 1 푛 푖( 2 √ −1− tan−1 √ −1) 푛 푛 4 2 4 microscopic motion of atoms colliding with a Brownian particle as devised by 푛 푛 − ( ) and the quantum analog √( ) 2 2 푒 . Einstein [4]. 푘 푘 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 2 Riemann manifold can be reconstructed by two sets of small are moving from a pictorial representation to its algebraic volumes [7]-[8]. So, motion between such parcels, by a elements as for the phase space we move from the geometrical particle that is able to sense them, can be assimilated to a structure of the motion like an orbit to functions representing Brownian motion. it like coordinates and momenta. When one moves from such functions to operators (or q-numbers a la Dirac as Heisenberg Where do such small volumes come from? The idea comes out did) one gets a noncommutative Riemann manifold. from the noncommutative geometry uncovered by Alain Connes in the last century [9]. The idea behind Connes, Chamseddine and Mukhanov proved that such a noncommutative geometry can be traced back to the deep noncommutative manifold comes out made by two kinds of connection between algebra and geometry as initially elementary volumes and it is from here that the deep conceived by Descartes. It is well-known that whatever connection with stochastic processes starts. This means that algebraic expression one considers there is a corresponding the relation between Fourier and Schrödinger equations, geometrical object of it. E.g., the equation 푥2 + 푦2 = 1 formally given by a Wick rotation, has a deep physical represents a circle on a Cartesian plane. Nevertheless, all this meaning. Mathematically, it entails the introduction of a new works fine because complex numbers have the property to class of stochastic processes: the fractional powers of a commute each other. We learned over the last century that our Wiener process [10]. world does not seem to agree with this property at its foundations. Werner Heisenberg initially conceived this and it In summary, the relation between the Fourier and the is now our understanding of the behavior of elementary Schrödinger equations, through a Wick rotation, is not merely particles. In this case, we have what Paul Dirac called q- a mathematical curiosity but rather has deep physical numbers and momentum 푝 and position 푞 do not commute implications as the world is structured as a noncommutative yielding the famous equation 푞푝 − 푝푞 = 푖ℏ (otherwise stated Riemann manifold on which the particles, able to feel its [푞, 푝] = 푖ℏ where [, ] is the commutator) from which the quantization, perform Brownian motion with a process uncertainty principle derives. The reason is that such q- equation, which is exactly the Schrödinger equation. This numbers are matrices with an infinite number of elements or, means that the analogy between these famous equations relies better stated, operators acting on a Hilbert space. on the kind of atoms one considers: For Fourier, it is the matter being parceled while for Schrödinger, it is the space Why such noncommutative behavior changes the behavior of being made by elementary volumes acting like the atoms in particles? We know that dynamics can be represented on the matter. However, geometry is continuous in the former and geometric object called phase space where both momenta and discretized in the latter. Consequently, in a lattice world, if one positions are taken into account. Dynamics is described by a moves randomly between the sites of the lattice, one obtains trajectory on such a space. In addition, when energy is the solution to the Fourier equation. In a quantum world, conserved, on a phase space the motion happens on a given represented - for instance - by a quantum Tartaglia-Pascal geometrical object like a torus or a sphere. Therefore, there is triangle [2], one has that the space on which the particle a deep connection between Newton mechanics and geometry moves is discretized and so, moving on such a space yields the in a phase space. William Rowan Hamilton discovered this solution to the Schrödinger equation. The quantum Tartaglia two centuries ago. Therefore, what does it make deterministic triangle is, in a sense, the square root of the classical one [2] the Newton mechanics? This comes out from the and this is the root of the “i” factor4. commutativity of momenta and positions. Otherwise, we have quantum mechanics and the structure of the phase space In the remaining part of the paper, we will discuss this deep changes dramatically.
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