Noncommutative geometry and stochastic processes
07/11/2017- Accès libre pour les ayants-droit
Résumé
These processes are characterized by producing complex values and so, the corresponding Fokker–Planck equation resembles the Schr¨odinger equation. Indeed, by a direct numerical check, one can recover the kernel
of the Schr¨odinger equation starting by an ordinary Brownian motion.
This class of stochastic processes needs a Clifford algebra to exist.
Collection
- Accès libre pour les ayants-droit
These processes are characterized by producing complex values and so, the corresponding Fokker–Planck equation resembles the Schr¨odinger equation. Indeed, by a direct numerical check, one can recover the kernel
of the Schr¨odinger equation starting by an ordinary Brownian motion.
This class of stochastic processes needs a Clifford algebra to exist.
Auteurs
Marco Frasca |
Média
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Noncommutative geometry and stochastic processes Marco Frasca MBDA Italy S.p.A., Via Monte Flavio, 45, 00131 Rome, Italy, marco.frasca@mbda.it, WWW home page: http://marcofrasca.wordpress.com Abstract. The recent analysis on noncommutative geometry, showing quantization of the volume for the Riemannian manifold entering the geometry, can support a view of quantum mechanics as arising by a stochastic process on it. A class of stochastic processes can be devised, arising as fractional powers of an ordinary Wiener process, that repro- duce in a proper way a stochastic process on a noncommutative geome- try. These processes are characterized by producing complex values and so, the corresponding Fokker–Planck equation resembles the Schrödinger equation. Indeed, by a direct numerical check, one can recover the kernel of the Schrödinger equation starting by an ordinary Brownian motion. This class of stochastic processes needs a Clifford algebra to exist. 1 Introduction A comprehension of the link between stochastic processes and quantum mechan- ics can provide a better understanding of the role of space–time at a quantum gravity level. Indeed, noncommutative geometry, in the way Connes, Chamsed- dine and Mukhanov provided recently [1, 2], seems to fit well the view that a quantized volume yields a link at a deeper level of the connection between stochastic processes and quantum mechanics. This is an important motivation as we could start from a reformulation of quantum mechanics to support or drop proposals to understand quantum gravity and the fabric of space-time. A deep connection exists between Brownian motion and binomial coefficients. This can be established by recovering the kernel of the heat equation from the binomial distribution for a random walk (Pascal–Tartaglia triangle) and apply- ing the theorem of central limit [3]. When an even smaller step in the random walk is taken a Wiener process is finally approached. So, it is a natural question to ask what would be the analogous of Pascal–Tartaglia triangle in quantum mechanics[4]. This arises naturally by noting the apparent formal similarity be- tween the heat equation and the Schrödinger equation. But this formal anal- ogy is somewhat difficult to understand due to the factor i entering into the Schrödinger equation. An answer to this question hinges on a deep problem not answered yet: Is there a connection between quantum mechanics and stochastic processes? The formal similarity has prompted attempts to answer as in the pioneering work of Edward Nelson [5] and in the subsequent deep analysis by Francesco Guerra and his group [6]. They dubbed this reformulation of quantum mechanics as “stochastic mechanics”. This approach matches directly a Wiener process to the Schrödinger equation passing through a Bohm-like set of hydrody- namic equations and so, it recovers all the drawbacks of Bohm formulation. This view met severe criticisms motivating some researchers to a substantial claim that “no classical stochastic process underlies quantum mechanics” [7], showing contradiction with predictions of quantum mechanics. Subsequent attempts to partially or fully recover this view were proposed with non–Markovian processes [8] or repeated measurements [9–11]. In this paper we will show that a new set of stochastic processes can be devised, starting from noncommutative geometry, that can elucidate such a con- nection [4, 12]. We show how spin is needed also in the non-relativistic limit. These processes are characterized by the presence of a Bernoulli process yielding the values 1 and i, exactly as expected in the volume quantization in noncom- mutative geometry. In this latter case, it appears that a stochastic process on a quantized manifold is well represented by a fractional power of an ordinary Wiener process when this is properly defined through a technique at discrete time [13]. The kernel of the Schrödinger equation is numerically evaluated through a Brownian motion. A similar idea to use noncommutative geometry in stochastic processes was proposed in [14] but there it was used to fix univocally the kinetic equation of a real stochastic process. 2 Noncommutative geometry and quantization of volume A noncommutative geometry, given by the triple (A, H, D) being A a set of operators belonging to a ∗ -algebra, H a Hilbert space and D a Dirac operator, implies that the volume of the corresponding Riemann manifold is quantized with two classes of unity of volume (1, i). This has been recently proved by Connes, Chamseddine and Mukhanov[1, 2]. The two classes of volume arise from the fact that the Dirac operator should not be limited to Majorana (neutral) states in the Hilbert space and so, we need to associate a charge conjugation operator J to our triple. To complete the characterization of our geometry, we recall that the algebra of Dirac matrices implies a γ5 , the chirality matrix that changes the parity of the states. For an ordinary Riemann manifold, the algebra A is that of functions and is commuting. Remembering that [D, a] = iγ ·∂a, and noting that, in four dimensions, x1, x2, x3, x4 are legal functions of A, it is [D, x1][D, x2][D, x3][D, x4] = γ1 γ2 γ3 γ4 = −iγ5 . For generally chosen functions in A, a0, a1, a2, a3, a4, . . . ad, summing over all the possible permutations one has a Jacobian, we can define the chirality operator γ = X P (a0[D, a1] . . . [D, ad]). (1) So, in four dimension this gives γ = −iJ · γ5 = −i · det(e)γ5 (2) being J the Jacobian, ea µ the vierbein for the Riemann manifold and γ5 = iγ1 γ2 γ3 γ4 for d = 4, a well-known result. We used the fact that det(e) = √ g, being gµν the metric tensor. So, the definition of the chirality operator is propor- tional to the factor determining the volume of a Riemannian orientable manifold. In order to see if a Riemannian manifold can be properly quantized, instead of functions we consider operators Y belonging to an operator algebra A′ . These operators have the properties Y 2 = κI Y † = κY. (3) This is a set of compact operators playing the role of coordinates as in the Heisenberg commutation relations. We have to consider two sets of them Y+ and Y− as we expect a conjugation of charge operator C to exist such that CAC−1 = Y † for a given operator or complex conjugation for a function. This appears naturally out of a Dirac algebra of gamma matrices. So, a natural way to write down the operators Y is by using an algebra of Dirac matrices ΓA such that {ΓA , ΓB } = 2δAB , (ΓA )∗ = κΓA (4) with A, B = 1 . . . d + 1, then Y = ΓA Y A . (5) We will have two different set of gamma matrices for Y+ and Y− that will have independent traces. Using the charge conjugation operator C, we can define a new coordinate Z = 2ECEC−1 − I (6) where E = (1 + Y+)/2 + (1 + iY−)/2 will project one or the other coordinate. We recognize that the spectrum of Z is in (1, i) given eq.(3). Now, we generalize our equation for the chirality operator imposing a trace on Γs both for Y+ and Y−, normalized to the number of components, and we will have 1 n! hZ[D, Z] . . . [D, Z]i = γ. (7) where we have introduced the average h. . .i that, in this case, reduces to matrix traces. In order to see the quantization of the volume, let us consider a three dimensional manifold and the sphere S2 . From eq.(7) one has VM = Z M 1 n! hZ[D, Z] . . . [D, Z]id3 x (8) and doing the traces one has VM = Z M 1 2 ǫµν ǫABCY A + ∂µY B + ∂νY C + + 1 2 ǫµν ǫABCY A − ∂µY B − ∂νY C − d3 x. (9) It is easy to see that this will yield[1, 2] det(ea µ) = 1 2 ǫµν ǫABCY A + ∂µY B + ∂νY C + + 1 2 ǫµν ǫABCY A − ∂µY B − ∂νY C − . (10) The coordinates Y+ and Y− belongs to unitary spheres and the Dirac opera- tor has a discrete spectrum being evaluated on a compact manifold, so we are covering all the manifold with a large integer number of these spheres. Thus, the volume is quantized as this condition requires. This can be extended to four dimensions with some more work [1, 2]. Differently from an ordinary stochastic process, a Wiener process on a quan- tized manifold will yield the projection of the spectrum (1, i) of the coordinates on the two kinds of spheres Y+, Y−. This will depend on the way a particle moves on the manifold taking into account that the distribution of the two kinds of uni- tary volumes is absolutely random. One can construct a process Φ such that, against a toss of a coin, one gets 1 or i as outcome, assuming the distribution of the unitary volumes is uniform. This can be written Φ = 1 + B 2 + i 1 − B 2 (11) with B a Bernoulli process such that B2 = I producing the value ±1 depending on the unitary volume hit by the particle and Φ2 = B. If we want to consider the Brownian motion of the particle on such a manifold we should expect the outcomes to be either Y+ or Y−. So, given the set of Γ matrices and the chirality operator γ, the most general form for a stochastic process on the manifold can be written down (summation on A is implied) dY = ΓA · (κA + ξAdXA · BA + ζAdt + iηAγ5 ) · ΦA (12) being κA, ξA, ζA, ηA arbitrary coefficients of this linear combination. The Bernoulli processes BA and the Wiener process dXA cannot be independent. Rather, the sign arising from the Bernoulli process is the same of that of the cor- responding Wiener process. This equation provides the equivalent of the eq.(3) for the coordinates on the manifold. This is exactly the formula we will obtain for the fractional powers of a Wiener process. It just represents the motion on a quantized Riemannian manifold with two kinds of quanta. Underlying quantum mechanics there appears to be a noncommutative geometry. 3 Powers of stochastic processes We consider an ordinary Wiener W process describing a Brownian motion and define the α-th power of it. We do a proof of existence by construction [13]. A Wiener process W is computed by the cumulative sum of the increments at discrete steps Wi − Wi−1. Similarly, we will have the process (given α ∈ R+ ) with the formal definition dX = (dW)α . (13) built through the Euler–Maruyama definition of a stochastic process [15] at discrete times obtained by the cumulative sum Xi = Xi−1 + (Wi − Wi−1)α . (14) as done in computing a Wiener process with α = 1. A complete proof of existence of these processes has been shown in [13]. 4 “Square root” formula and Fokker–Planck equation Using Itō calculus to express the “square root” process with more elementary stochastic processes [16], (dW)2 = dt, dW · dt = 0, (dt)2 = 0 and (dW)α = 0 for α > 2, we could tentatively set dX = (dW) 1 2 ? = µ0 + 1 2µ0 dW · sgn(dW) − 1 8µ3 0 dt · Φ1 2 (15) being µ0 6= 0 an arbitrary scale factor and Φ1 2 = 1 − i 2 sgn(dW) + 1 + i 2 (16) a Bernoulli process equivalent to a coin tossing that has the property (Φ1 2 )2 = sgn(dW). This process is characterized by the values 1 and i and it is like the Brownian motion went scattering with two different kinds of small pieces of space, each one contributing either 1 or i to the process, randomly. This is the same process seen for the noncommutative geometry in eq.(11). We have introduced the process sgn(dW) that yields just the signs of the corresponding Wiener process. Eq.(15) is unsatisfactory for a reason, taking the square yields (dX)2 = µ2 0 sgn(dW) + dW (17) and the original Wiener process is not exactly recovered. We find added a process that has the effect to shift upward the original Brownian motion while retaining the shape. We can fix this problem by using Pauli matrices. Let us consider two Pauli matrices σi, σk with i 6= k such that {σi, σk} = 0. We can rewrite the above identity as I·dX = I·(dW) 1 2 = σi µ0 + 1 2µ0 dW · sgn(dW) − 1 8µ3 0 dt ·Φ1 2 +iσkµ0·Φ1 2 (18) and so, (dX)2 = dW as it should, after removing the identity matrix on both sides. This idea generalizes easily to higher dimensions using γ matrices. We see that we have recovered a similar stochastic process as in eq.(12). This will be extended to four dimensions below. Now, let us consider a more general “square root” process where we assume also a term proportional to dt. We assume implicitly the Pauli matrices simply removing by hand the sgn process at the end of the computation. This forces to take µ0 = 1/2 when the square is taken, to recover the original stochastic process, and one has dX(t) = [dW(t)+βdt] 1 2 = 1 2 + dW(t) · sgn(dW(t)) + (−1 + β sgn(dW(t)))dt Φ1 2 (t). (19) From the Bernoulli process Φ1 2 (t) we can derive µ = − 1 + i 2 + β 1 − i 2 σ2 = 2D = − i 2 . (20) Then, we get a double Fokker–Planck equation for a free particle, being the distribution function ψ̂ complex valued, ∂ψ̂ ∂t = − 1 + i 4 + β 1 − i 2 ∂ψ̂ ∂X − i 4 ∂2 ψ̂ ∂X2 . (21) This should be expected as we have a complex stochastic process and then two Fokker–Planck equations are needed to describe it. We have obtained an equation strongly resembling the Schrödinger equation for a complex distribution func- tion. We can ask at this point if we indeed are recovering quantum mechanics. In the following section we will perform a numerical check of this hypothesis. 5 Recovering the kernel of the Schrödinger equation If really the “square root” process diffuses as a solution of the Schrödinger equa- tion we should be able to recover the corresponding solution for the kernel ψ̂ = (4πit)− 1 2 exp ix2 /4t (22) sampling the square root process. To see this we note that a Wick rotation, t → −it, turns it into a heat kernel as we get immediately K = (4πt)− 1 2 exp −x2 /4t . (23) A Montecarlo simulation can be easily executed extracting the square root of a Brownian motion and, after a Wick rotation, to show that a heat kernel is obtained. We have generated 10000 paths of Brownian motion and extracted its square root in the way devised in Sec. 3. We have evaluated the corresponding distribution after Wick rotating the results for the square root. The Wick ro- tation generates real results as it should be expected and a comparison can be performed. The result is given in Fig. 1 −4 −3 −2 −1 0 1 2 3 4 0 1 2 3 x 10 5 Brownian motion −5 0 5 0 5 10 15 x 10 4 Square root Fig. 1. Comparison between the distributions of the Brownian motion and its square root after a Wick rotation. The quality of the fit can be evaluated being µ̂ = 0.007347 with confidence in- terval [0.005916, 0.008778], σ̂ = 0.730221 with confidence interval [0.729210, 0.731234] for the heat kernel while one has µ̂ = 0.000178 with confidence interval [−0.002833, 0.003189] and σ̂ = 1.536228 with confidence interval [1.534102, 1.538360] for the Schrödinger kernel. Both are centered around 0 and there is a factor ∼ 2 between standard deviations as expected from eq. (21). Both the fits are exceedingly good. Having recovered the Schrödinger kernel from Brownian motion with the proper scaling factors in mean and standard deviation, we can conclude that we are doing quan- tum mechanics: The “square root” process describes the motion of a quantum particle. Need for Pauli matrices, as shown in the preceding section, implies that spin cannot be neglected. 6 Square root and noncommutative geometry We have seen that, in order to extract a sort of square root of a stochastic process, we needed Pauli matrices or, generally speaking, a Clifford algebra. This idea was initially put forward by Dirac to derive his relativistic equation for fermions and the corresponding algebra was proven to exist by construction as it also happens for Pauli matrices. The simplest and non-trivial choice is obtained, as said above, using Pauli matrices {σk ∈ Cℓ3(C), k = 1, 2, 3} that satisfy σ2 i = I σiσk = −σkσi i 6= k. (24) This proves to be insufficient to go to dimensions higher than 1+1 for Brownian motion. The more general solution is provided by a Dirac algebra of γ matrices {γk ∈ Cℓ1,3(C), k = 0, 1, 2, 3} such that γ2 0 = I γ2 1 = γ2 2 = γ2 3 = −I γiγk + γkγi = 2ηik (25) being ηik the Minkowski metric. In this way one can introduce three different Brownian motions for each spatial coordinates and three different Bernoulli pro- cesses for each of them. The definition is now dE = 3 X k=1 iγk µk + 1 2µk |dWk| − 1 8µ3 k dt · Φ (k) 1 2 + 3 X k=1 iγ0γkµkΦ (k) 1 2 (26) It is now easy to check that (dE)2 = I · (dW1 + dW2 + dW3). (27) The Fokker-Planck equations have a solution with 4 components, as now the distribution functions are Dirac spinors. These are given by ∂Ψ̂ ∂t = 3 X k=1 ∂ ∂Xk µkΨ̂ − i 4 ∆2Ψ̂ (28) being µk = −1+i 4 + βk 1−i 2 . This implies that, the general formula for the square root process implies immediately spin and antimatter for quantum mechanics that now come out naturally. 7 Conclusions We have shown the existence of a class of stochastic processes that can support quantum behavior. This formalism could entail a new understanding of quantum mechanics and give serious hints on the properties of space-time for quantum gravity. This yields a deep connection with noncommutative geometry as formu- lated by Alain Connes through the more recent proposal of space quantization by Connes himself, Chamseddine and Mukhanov. This quantization of volume entails two kinds of quanta implying naturally the unity (1, i) that arises in the “square root” of a Wiener process. Indeed, a general stochastic process for a particle moving on such a quantized volume corresponds to our formula of the “square root” of a stochastic process on a 4-dimensional manifold. Spin appears to be an essential ingredient, already at a formal level, to treat such fractional powers of Brownian motion. Finally, it should be interesting, and rather straightforward, to generalize this approach to a Dirac equation on a generic manifold. The idea would be to recover also Einstein equations as a fixed point solution to the Fokker-Planck equations as already happens in string theory. Then they would appear as a the result of a thermodynamic system at the equilibrium based on noncommutative geometry. This is left for further study. I would like to thank Alfonso Farina for giving me the chance to unveil some original points of view on this dusty corner of quantum physics. References 1. A. H. Chamseddine, A. Connes and V. Mukhanov, Phys. Rev. Lett. 114, no. 9, 091302 (2015) [arXiv:1409.2471 [hep-th]]. 2. A. H. Chamseddine, A. Connes and V. Mukhanov, JHEP 1412, 098 (2014) [arXiv:1411.0977 [hep-th]]. 3. G. Weiss, Aspects and Applications of the Random Walk (North-Holland, Amster- dam, 1994). 4. A. Farina, M. Frasca, M. Sedehi, SIViP (2014) 8: 27. doi:10.1007/s11760-013-0473-y. 5. E. Nelson, Dynamical Theories of Brownian Motion (Princeton University Press, Princeton, 1967). 6. F. Guerra Phys. Rept. 77, 263 (1981). 7. H. Grabert, P. Hänggi, P. Talkner, Phys. Rev. A 19, 2440 (1979). 8. G. A. Skorobogatov, S. I. Svertilov, Phys. Rev. A 58, 3426 (1998). 9. P. Blanchard, S. 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