Quantized Noncommutative Riemann Manifolds and Stochastic Processes: The theoretical foundations of the square root of Brownian motion
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Marco Frasca ∗ and Alfonso Farina † Via Erasmo Gattamelata, 300176 Roma (Italy) Via Helsinki, 1400144 Roma (Italy) (Dated: May 7, 2019)We consider Brownian motion on quantized non-commutative Riemannian manifolds and showhow a set of stochastic processes on sets of complex numbers can be devised. This class of stochasticprocesses are shown to yield at the outset a Chapman-Kolmogorov equation with a complex diffusioncoefficient that can be straightforwardly reduced to the Schr¨odinger equation. The existence of theseprocesses has been recently shown numerically. In this work we provide an analogous support forthe existence of the Chapman-Kolmogorov-Schr¨odinger equation for them. Besides, it is numericallyseen as a Wick rotation can change the heat kernel into the Schr¨odinger one. ∗ [email protected] † [email protected] I. INTRODUCTION
One of the hotly debated problems about quantum mechanics is if it could be derived from some stochastic process.One the most promising proposal was put forward by Nelson [1]. This idea was widely discussed [2–7] but it remainsan open question if it could be a solution to the problem. Quite recently, we proposed an approach, based on theextraction of the square root of a Wiener process [8, 9], that identifies a new class of stochastic processes based oncomplex evaluated random variables . These processes can have a Schr¨odinger equation as Kolmogorov-Chapmandiffusion equation. This proposal was put at test with a numerical study in [10] and we showed a theorem aboutfractional powers of Wiener processes. Using a simple integration technique, the Euler-Maruyama method, we solvedthe stochastic differential equations arising in the square root case proving the existence of such a process. Complexevaluated stochastic processes can have convergence problems when managed with the standard techniques of realevaluated stochastic processes. This is the main reason why we recurred to numerical methods. Particularly, in thispaper we will give numerical evidence of the existence of the corresponding diffusion process, given by the Schr¨odingerequation, for the square root case. This is especially interesting for the possible applications in physics where theSchr¨odinger equation is moved from the role of a postulate to that of a theorem.These processes arise naturally as a Brownian motion on a noncommutative Riemann manifold. Connes, Chamsed-dine and Mukhanov proved that such a noncommutative manifold is quantized and made by two kinds of elementaryvolumes [11, 12], identified by the unities (1 , i ), and it is from here that the deep connection with stochastic processesstarts. This means that the relation between an ordinary diffusion process a la Fourier and the Schr¨odinger equa-tions, formally given by a Wick rotation t → − it , has a deep physical meaning. Mathematically, as already said, itentails the introduction of a new class of stochastic processes: the fractional powers of a Wiener process [8–10]. Thisconnection with the noncommutative geometry is a natural one as a square root stochastic process can only be builtif a Clifford algebra [13] exists to support it. Otherwise, the ordinary Wiener process cannot be recovered by takingthe square because a spurious shifting term will appear.In this paper, we will show, by numerical evidence, that the square root process, proved recently to exist, givesrise to a diffusion process ruled by the Schr¨odinger equation for complex evaluated random variables. The workis structured as follows. In Sec. II, we give some elements of noncommutative geometry for a quantized Riemannmanifold and introduce the stochastic process on it. In Sec. III, we yields some results about fractional powers ofa Wiener process, specifically for the case 1 /
2. In Sec. IV, we present the results of our Monte Carlo study of thediffusion process for this class of stochastic processes. Finally, in Sec. V, conclusions are given.
II. QUANTIZED RIEMANN MANIFOLDSA. Noncommutative geometry
A noncommutative geometry is characterized by the triple ( A , H, D ) being A a set of operators forming a ∗ -algebra, H a Hilbert space and D a Dirac operator. This yields that the volume of the corresponding noncommutative Riemannmanifold is quantized with two distinct classes of unity of volume (1 , i ). A prove of this theorem was provided byConnes, Chamseddine and Mukhanov[11, 12]. The need of two kinds of elementary volumes arises from the factthat the Dirac operator should not be limited to Majorana (neutral) states in the Hilbert space but we have moregeneral states and we have to add a charge conjugation operator J to our triple. Finally, we recall that the Cliffordalgebra of Dirac matrices implies the existence of a γ matrix [14], the chirality matrix that changes the parity of thestates. For a commutative Riemann manifold, the algebra A is the Abelian algebra of smooth functions. One has[ D, a ] = iγ · ∂a , and noting that, in four dimensions, x , x , x , x are legal functions of A , we can generate γ as[ D, x ][ D, x ][ D, x ][ D, x ] = γ γ γ γ = − iγ . Similarly, for arbitrary functions in A , a , a , a , a , a , . . . a d ,summing over all the possible permutations one has a Jacobian. Then, we can define a more general chirality operator γ = X P ( a [ D, a ] . . . [ D, a d ]) , (1)that, in four dimension, gives γ = − iJ · γ = − i · det( e ) γ (2) These come out naturally trying to extend the Tartaglia-Pascal triangle to quantum mechanics. As shown in [8], the correspondence isbetween the binomial coefficient (cid:0) nk (cid:1) and the discrete quantum analog q(cid:0) nk (cid:1) − n e i (cid:20) ( k − n/ n √ n − − arctan √ n − (cid:21) . being J the Jacobian, e aµ the vierbein [13] for the Riemann manifold, characterizing the metric, and γ = iγ γ γ γ for d = 4, a well-known result. We used the fact that det( e ) = √ g , being g µν the metric tensor. So, our definition ofchirality operator is just proportional to the metric factor that yields the volume of a Riemannian orientable manifold.A Riemannian manifold can be properly quantized when, instead of functions, we consider operators Y belongingto an operator algebra A ′ . These operators have the properties Y = κI Y † = κY. (3)These are compact operators that have the role of coordinates as in the Heisenberg commutation relations. To accountfor the existence of the conjugation of charge operator C such that CAC − = Y † , we need two sets of coordinates, Y + and Y − as we expect a conjugation of charge operator C to exist such that CAC − = Y † . This is the analogousof complex conjugation for a function. Such coordinates appear naturally out of a Dirac algebra of gamma matrices.Indeed, a natural way to write down the operators Y is by using a Clifford algebra of Dirac matrices Γ A such that { Γ A , Γ B } = 2 δ AB , (Γ A ) ∗ = κ Γ A (4)with A, B = 1 . . . d + 1, so that Y = Γ A Y A . (5)We will need two different sets of gamma matrices for Y + and Y − having these independent traces. Using the chargeconjugation operator C , we can introduce a new coordinate Z = 2 ECEC − − I (6)where E = (1 + Y + ) / iY − ) / Z is in (1 , i ), given eq.(3). We can now generalize our definition of the chirality operator by taking the trace on Z s,properly normalized to the number of components. This yields1 n ! h Z [ D, Z ] . . . [ D, Z ] i = γ, (7)being the average h . . . i , in this case, just matrix traces. We can now see the quantization of the volume. Let usconsider a three dimensional manifold M and the sphere S . From eq.(7) one has V M = Z M n ! h Z [ D, Z ] . . . [ D, Z ] i d x. (8)By taking the traces we get V M = Z M (cid:18) ǫ µν ǫ ABC Y A + ∂ µ Y B + ∂ ν Y C + + 12 ǫ µν ǫ ABC Y A − ∂ µ Y B − ∂ ν Y C − (cid:19) d x. (9)It is not difficult to see that this will reduce to [11, 12]det( e aµ ) = 12 ǫ µν ǫ ABC Y A + ∂ µ Y B + ∂ ν Y C + + 12 ǫ µν ǫ ABC Y A − ∂ µ Y B − ∂ ν Y C − . (10)The coordinates Y + and Y − belong to unitary spheres while the Dirac operator has a discrete spectrum as it is definedon a compact manifold. This means that we are covering all the manifold with a large integer number of these spheres.Therefore, the volume is quantized as this is required by the above condition. An extension to four dimensions is alsopossible with some more work [11, 12]. B. Stochastic processes on a quantized manifold
We expect that a Wiener process on a quantized manifold will account for the spectrum (1 , i ) of the coordinates onthe two kinds of spheres Y + , Y − . Assuming a completely random distribution of the two kinds of spheres that makethe Riemann manifold, the result will depend on the motion of the particle on it. A process Φ can be defined suchthat, like for tossing of a coin, one gets either 1 or i as outcome, once we assume that the distribution of the unitaryvolumes is uniform. The definition of this process isΦ = 1 + B i − B B a Bernoulli process such that B = I that yields the value ± = B . For a Brownian motion of the particle on such a manifold, the possible outcomes will beeither Y + or Y − . For a given set of Γ matrices and chirality operator γ , one can write the most general form for sucha stochastic process as (summation on A is implied) dY = Γ A · ( κ A + ξ A dX A · B A + ζ A dt + iη A γ ) · Φ A (12)being κ A , ξ A , ζ A , η A arbitrary coefficients of this linear combination (a pictorial view is given in Fig. 1). FIG. 1. Pictorial representation of motion (green lines) of a particle on a noncommutative Riemann manifold.
The Bernoulli processes B A and the Wiener process dX A are not independent. We expect that the sign arising fromthe Bernoulli process should be the same of that of the corresponding Wiener process. This stochastic differentialequation is the equivalent of the eq.(3) for the coordinates on the manifold. As we will see below, this is the sameas the formula for the square root of a Wiener process. This represents the motion of a particle on a quantizednoncommutative Riemann manifold. In this way, the Schr¨odinger equation can be removed from the state of apostulate and, underlying quantum mechanics, we have a quantized manifold. III. FRACTIONAL POWERS OF WIENER PROCESSES
With It¯o calculus we can express the “square root” process through more elementary stochastic processes [18],( dW ) = dt , dW · dt = 0, ( dt ) = 0 and ( dW ) α = 0 for α >
2, we set dX = ( dW ) ? = (cid:18) µ + 12 µ dW · sgn( dW ) − µ dt (cid:19) · Φ (13)being µ = 0 an arbitrary scale factor and Φ = 1 − i dW ) + 1 + i ) = sgn( dW ). The possible outcomes forthis process are 1 and i and represent a particle executing Brownian motion scattering two different kinds of smallpieces of space, each one contributing either 1 or i to the process, randomly. We have already seen this process forthe noncommutative geometry in eq.(11). We have introduced the process sgn( dW ) that yields just the signs of thecorresponding Wiener process. But Eq.(13) is not satisfactory for, taking the square, yields( dX ) = µ sgn( dW ) + dW (15)and we do not exactly recover the original Wiener process. We see that we have added a process that has an overalleffect to shift upward the original Brownian motion even if its shape is preserved.This problem can be fixed by using the Clifford algebra formed by the Pauli matrices [14]. Taking two differentPauli matrices σ i , σ k with i = k such that { σ i , σ k } = 0 we can rewrite the above identity as I · dX = I · ( dW ) = σ i (cid:18) µ + 12 µ dW · sgn( dW ) − µ dt (cid:19) · Φ + iσ k µ · Φ (16)and so, ( dX ) = dW as it should. This idea can be easily generalized to higher dimensions using Dirac’s γ matrices.We see that we have recovered a similar stochastic process as in eq.(12).This view agrees very well with the recent results by Connes, Chamseddine and Mukhanov [11, 12] and yields ahint for the underlying possible quantization of space.For consistency reasons we also provide the operational definitions for the involved processes needed to completethe above derivation. These are sgn( dW ) = { sgn( W ) , sgn( W ) , sgn( W ) , . . . } (17)such that (sgn( dW )) = I , | dW | = {| W | , | W | , | W | , . . . } (18)and | dW | sgn( dW ) = {| W | · sgn( W ) , | W | · sgn( W ) , | W | · sgn( W ) , . . . } = dW. (19)These definitions are also used in the numerical evaluation for the proof by construction in Sec. IV.We can consider a more general “square root” process by adding a term proportional to dt . We take for grantedthat the Pauli matrices are used to remove the sgn so, we will do it by hand. Assuming for the sake of simplicity µ = 1 /
2, one has dX ( t ) = [ dW ( t ) + βdt ] = (cid:20)
12 + dW ( t ) · sgn( dW ( t )) + ( − β sgn( dW ( t ))) dt (cid:21) Φ ( t ) , (20)being β an arbitrary constant. From the Bernoulli process Φ ( t ) one gets µ = − i β − i σ = 2 D = − i . (21)Therefore, we have a double Fokker–Planck equation for a free particle, being the distribution function ˆ ψ complexvalued, ∂ ˆ ψ∂t = (cid:18) i − β − i (cid:19) ∂ ˆ ψ∂X − i ∂ ˆ ψ∂X . (22)This result is not unexpected as, having complex random variables, we should have a Fokker–Planck equation for thereal part and another for the imaginary part. The surprising result is that we get an equation strongly resemblingthe Schr¨odinger equation. We will see below that we are really recovering quantum mechanics, by recovering the heatkernel from the Monte Carlo simulation of the “square root” process, after a Wick rotation. IV. MONTE CARLO STUDY
A recent Monte Carlo study by the authors [10] has shown the existence of fractional Wiener processes, provideda proper definition of the involved random evaluated functions is given. In this way, a straightforward numericalimplementation is possible. Having this in mind, we use the same technique to perform a Monte Carlo evaluation ofthe diffusion process involved with our complex random processes and show that the so obtained mean, variance andprobability distribution agree fairly well with what we have obtained theoretically so far. We just note that meanand variance should be evaluated by dividing by µ and µ respectively. µ should be chosen greater than one. Inour case β , that appears in eq. (20), is assumed to be zero.We performed a Monte Carlo study where each Brownian path is evaluated for 1000 steps for 20000 runs . In thisway we were able to evaluate both the Wiener process, its square root and eq. (20) obtained by Euler-Maruyamamethod. We expect that the kernel is the standard heat kernel for the first case and a Schr¨odinger kernel otherwise.But this should be correlated by a Wick rotation. So, in order to perform a fit with a Gaussian distribution, weneed to be certain that the phases of the Schr¨odinger kernel, producing the imaginary part, are removed after a Wickrotation. This is indeed the case. Therefore, given the set of random complex numbers ψ obtained by numericallyevaluating the square of a Wiener path sample, we evaluate the module ρ and the phase θ for each one of them. Nowwe have for the Schr¨odinger kernelˆ ψ = (4 πit ) − exp (cid:0) ix / t (cid:1) = (4 πt ) − (cid:20) cos (cid:18) x t − π (cid:19) + i sin (cid:18) x t − π (cid:19)(cid:21) . (23)A Wick rotation, t → − it , turns it into a heat kernel giving immediately K = (4 πt ) − (cid:20) cos (cid:18) i x t − i π (cid:19) + i sin (cid:18) i x t − i π (cid:19)(cid:21) e π . (24)Given the phases and modules computed by our set of samples, this can be easily expressed using them. The result The code is available on request to M.F. is given in Fig. 2. -3 -2 -1 0 1 2 3 a) b) FIG. 2. Comparison between the heat kernel a) and the Schr¨odinger kernel b) obtained after a Wick rotation as given ineq.(24).
One sees that one gets a perfect normal distribution in both the cases as it should. We just note that, in our case,the Schr¨odinger kernel has its center shifted, in agreement with our expectations.
TABLE I. Means, variances and diffusion coefficients.Process Mean Variance Diffusion coefficientBrownian − . ± .
004 0 . ± . . ± . . ± . . ± . i ± · − − (0 . ± . i − (0 . ± . i In Fig. 3, we show the distributions of the averages and the variances of the square root of the Wiener process. a) b) c) -4 d) FIG. 3. Distributions of the means (real part a) and imaginary part b)) and variances (real part c) and imaginary part d)) ofthe square root process.
In table I, we report the values of their means, variances and diffusion coefficients. The agreement with ourtheoretical results, looking at eq. (22), is exceedingly good confirming that we are observing a diffusion process ruledby the Schr¨odinger equation arising from the square root of a Wiener process. Particularly, we notice the values(1 + i ) / β = 0), andthe variance being − i/ V. CONCLUSIONS
We have shown, through a Monte Carlo study, the existence of a diffusion process, described by a Schr¨odingerequation, arising by taking the square root of an ordinary Brownian motion. We have a complete agreement with thetheoretical expectations. As a concluding remark, we are pleased to note that our theory has recently been appliedin the field of stock exchange prediction as a refinement of the Black and Scholes equation [19]. [1] E. Nelson,
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