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Noncommutative geometry and stochastic processes

Marco Frasca∗

Via Erasmo Gattamelata, 3

00176 Roma (Italy)

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 reproduce in a proper way a stochastic process on a

noncommutative geometry. 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. In four

dimensions, the full set of Dirac matrices is needed and the corresponding stochastic process in a

noncommutative geometry is easily recovered as is the Dirac equation in the Klein–Gordon form

being it the Fokker–Planck equation of the process.

∗marcofrasca@mclink.it

1

I. INTRODUCTION

A comprehension of the link between stochastic processes and quantum mechanics can

provide a better understanding of the role of space–time at a quantum gravity level. Indeed,

noncommutative geometry, in the way Connes, Chamseddine and Mukhanov provided re-

cently [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 impor-

tant 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 applying 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[11]. This arises naturally by noting the apparent

formal similarity between the heat equation and the Schr¨odinger equation. But this formal

analogy is somewhat diﬃcult to understand due to the factor ientering into the Schr¨odinger

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 [4] and in

the subsequent deep analysis by Francesco Guerra and his group [5]. They dubbed this

reformulation of quantum mechanics as “stochastic mechanics”. This approach matches

directly a Wiener process to the Schr¨odinger equation passing through a Bohm-like set of

hydrodynamic 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” [6] showing contradiction with

predictions of quantum mechanics. Subsequent attempts to partially or fully recover this

view were proposed with non-Markovian processes [7] or repeated measurements [8–10].

In this paper we will show that a new set of stochastic processes can be devised that can

elucidate such a connection [11, 12]. We show their existence [13] and we will determine

how spin is needed also in the non-relativistic limit. Dirac equation for a free particle is also

obtained. These processes are characterized by the presence of a Bernoulli process yielding

2

the values 1 and i, exactly as expected in the volume quantization in noncommutative

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. For our aims we will use the simplest one: The

Euler–Maruyama technique. A numerical test will yield the proof of existence for this class

of stochastic processes. Also, the kernel of the Schr¨odinger equation is numerically obtained

through an ordinary Brownian motion.

The paper is so strctured. In Sec. II we discuss noncommutative geometry in informal

way, providing a general formula for a stochastic process on a quantized Riemannian man-

ifold. In Sec. III, we introduce the fractional powers of a Wiener process and we solve

the corresponding stochastic equation recovering the Wiener process we started from after

squaring its square root. In Sec. IV, we derive the formula for the square root of a Wiener

process expressing it through more elementary processes: This shows the need for a Cliﬀord

algebra and the Fokker–Planck equation is obtained for a free particle. In Sec. V, we show

numerically how the kernel of the Schr¨odinger equation is recovered by an ordinary Brownian

motion just with the extraction of its square root. In Sec. VI, we derive the Fokker–Planck

equation in presence of a potential and specialize to the case of a harmonic oscillator. In

Sec. VII, we show how to recover a stochastic process on a noncommutative geometry taking

the square root of more Wiener processes and using the algebra of the Dirac matrices. In

Sec. VIII, we recover the Dirac equation as the Fokker–Planck equation for a square root

process. Finally, in Sec. IX conclusions are presented.

II. NONCOMMUTATIVE GEOMETRY AND QUANTIZATION OF VOLUME

A. Deﬁnition of a noncommutative geometry

Common wisdom on geometry implies that one has to cope with points and minimum

paths between them. Indeed, the idea of geometry can be extended without the central

concept of points but rather functions and introducing a redeﬁnition of the concept of

distance beside the well-known one from a Riemannian geometry. This reformulation is

due to Alain Connes [14]. Essentially, one introduces a triple composed by an algebra

of functions Awith an involution operator like complex conjugation, playing the role of

3

coordinates, a Hilbert space L2, that we take the space of the square-summable spinors, and

a Dirac operator D=i(γ·∂+ωµ), being ωµa spin connection, representing momenta. The

algebra of functions has support on a Riemann manifold. When we change the algebra of

functions with a noncommutative algebra of operators acting on the given Hilbert space, in

the same way one quantize a classical theory, one gets a noncommutative geometry. So, a

geometry is identiﬁed by the triple (A,L2,D). A function fbelonging to Ashould satisfy

the Lipschitz condition on the Riemann manifold given by

Lip(f) : |f(x)−f(y)| ≤ L·dR(x, y) (1)

provided the constant Lexists and

dR(x, y) =inf

γZγ

ds(2)

is the usual (geodesic) distance on a Riemann manifold that coincides with the well-known

variational principle of minimum distance between two points. This grants some regularity

properties of the functions in Aand their gradient that is bounded. In this way, we can

introduce a new deﬁnition of distance dependent just on the algebra of functions Aand the

Dirac operator. This is given by

d(x, y) =sup

f

(|f(x)−f(y)|:f∈ A,kDf k<1) (3)

where the condition on the Dirac operator plays a crucial role. In this way, one recovers the

ordinary Riemann distance between points [15]. Indeed, one has for a spinor ψ∈L2

[D, f ]ψ=iγ ·∂f ψ. (4)

then we need

k[D, f ]k=kp∂µf∂µfk ≤ 1.(5)

This is nothing else than asking the boundedness of the gradient of f. We know that fis

Lipshitz on the manifold and so, we can apply the Cauchy mean value theorem implying

that

k[D, f ]k ≤ |f(x)−f(y)|

dR(x, y)(6)

because a constant Lexists that can limit the derivatives on the manifold. Now, this implies,

due to the condition kDf k<1, that

|f(x)−f(y)| ≤ dR(x, y) (7)

4

and this means that dR(x, y) is the upper extreme as required by our deﬁnition of distance.

The main conclusion is that the Dirac operator plays the role of the inverse of the distance

D∼ds−1.

B. Quantization of volume

A noncommutative geometry implies that the volume 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 states in the Hilbert space and so, we need to associate a

charge conjugation operator Jto our triple (A, H, D). To complete our characterization of

our geometry, we recall that the algebra of Dirac matrices implies a γ5, the chirality matrix.

For an ordinary Riemann manifold, the algebra Ais 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] = γ1γ2γ3γ4=−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]).(8)

So, in four dimension this gives

γ=−iJ ·γ5=−i·det(e)γ5(9)

being Jthe Jacobian, ea

µthe vierbein for the Riemann manifold and γ5=iγ1γ2γ3γ4for

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 proportional 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 Ybelonging to an operator algebra A′. These operators have the

properties

Y2=κI Y †=κY. (10)

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

5

conjugation of charge operator Cto 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 Yis by using an algebra of Dirac

matrices ΓAsuch that

{ΓA,ΓB}= 2δAB,(ΓA)∗=κΓA(11)

with A, B = 1 . . . d + 1, then

Y= ΓAYA.(12)

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= 2EC EC −1−I(13)

where E= (1 + Y+)/2 + (1 + iY−)/2 will project one or the other coordinate. We recognize

that the spectrum of Zis in (1, i) given eq.(10). 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. (14)

where we have introduced the average h...ithat, 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.(14) one has

VM=ZM

1

n!hZ[D, Z]...[D, Z]id3x(15)

and doing the traces one has

VM=ZM1

2ǫµν ǫABC YA

+∂µYB

+∂νYC

++1

2ǫµν ǫABC YA

−∂µYB

−∂νYC

−d3x. (16)

It is easy to see that this will yield[1, 2]

det(ea

µ) = 1

2ǫµν ǫABC YA

+∂µYB

+∂νYC

++1

2ǫµν ǫABC YA

−∂µYB

−∂νYC

−.(17)

The coordinates Y+and Y−belongs to unitary spheres and the Dirac operator has a discrete

spectrum, 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].

6

C. Stochastic processes on a quantized manifold

Diﬀerently from an ordinary stochastic process, a Wiener process on a quantized manifold

will yield the projection of the spectrum (1, i) of the coordinates on the two kind 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 kind of unitary 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+i1−B

2(18)

with Ba Bernoulli process producing the value ±1 depending on the unitary volume hit

by the particle such that B2=I, a deterministic process giving always 1, 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 Ais implied)

dY = ΓA·(κA+ξAdXA·BA+ζAdt +iηAγ5)·ΦA(19)

being κA, ξA, ζA, ηAarbitrary coeﬃcients of this linear combination. The Bernoulli pro-

cesses BAand the Wiener process dXAcannot be independent. Rather, the sign arising from

the Bernoulli process is the same of that of the corresponding Wiener process. This equation

provides the equivalent of the eq.(10) 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 kind of quanta. Underlying quantum

mechanics there appears to be a noncommutative geometry.

III. POWERS OF STOCHASTIC PROCESSES

We consider an ordinary Wiener Wprocess describing a Brownian motion and deﬁne the

α-th power of it. We do a proof of existence by construction using a numerical integration

technique of a stochastic diﬀerential equations (SDE) [13]. We will have the process (given

α∈R+) with deﬁnition

dX = (dW )α.(20)

7

We build it through the Euler–Maruyama deﬁnition of a stochastic process [16] at discrete

times

Xi=Xi−1+ (Wi−Wi−1)α.(21)

This is equivalent to take the power and then a (cumulative) sum exactly as done in simu-

lating a Wiener process when α= 1.

For our numerical test we consider the square root process with α= 1/2 as it is the one

of interest for quantum mechanics. In this case the Wiener process has two components:

one real and another imaginary. We just compare the original Brownian motion with the

square of its square root given by numerically solving eq.(21). The result is displayed in

Fig. 1. The results are perfectly identical and our deﬁnition by Euler–Maruyama technique

0 0.2 0.4 0.6 0.8 1

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10−3

BM

SR

Figure 1. Comparison between the square of the square root process and the original Brownian

motion. These coincide perfectly as expected and the square root process exists.

just works. The square root process is so shown to exist by construction. We note that

the need for a complex valued stochastic process is essential if we aim to recover quantum

mechanics. On the other side, taking the square root of values that can have both positive

8

and negative values entails coping with complex numbers. One can always take the power

of whatever sequence of numbers as that of a Wiener process.

IV. SQUARE ROOT FORMULA AND FOKKER–PLANCK EQUATION

Using It¯o calculus to express the square root process with more elementary stochastic

processes [17], (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(22)

being µ06= 0 an arbitrary scale factor and

Φ1

2=1−i

2sgn(dW ) + 1 + i

2(23)

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 iand 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. We have introduced the process sgn(dW ) that yields just the

signs of the corresponding Wiener process. Eq.(22) is unsatisfactory for a reason, taking the

square yields

(dX)2=µ2

0sgn(dW ) + dW (24)

and the original Wiener process is not exactly recovered. We ﬁnd added a process that has

the eﬀect to change the scale of 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 i6=ksuch that {σi, σk}= 0. We can rewrite the above identity as

I·dX =I·(dW )1

2=σiµ0+1

2µ0

dW ·sgn(dW )−1

8µ3

0

dt·Φ1

2+iσkµ0·Φ1

2(25)

and so, (dX)2=dW as it should, after removing the identity matrix on both sides. This

idea generalizes easily to higher dimensions using γmatrices. In the following we will omit

the contribution due to the Pauli matrices but it will be implied to remove the unwanted

scale changing process.

Now, let us consider a more general square root process where we assume also a term

proportional to dt. This forces to take µ0= 1/2 when the square is taken, to recover the

9

original stochastic process, and one has

dX(t) = [dW (t)+βdt]1

2=1

2+dW (t)·sgn(dW (t)) + (−1 + βsgn(dW (t)))dtΦ1

2(t).(26)

From the Bernoulli process Φ 1

2(t) we can derive

µ=−1 + i

2+β1−i

2σ2= 2D=−i

2.(27)

Then, we get a double Fokker–Planck equation for a free particle, being the distribution

function ˆ

ψcomplex valued,

∂ˆ

ψ

∂t =−1 + i

4+β1−i

2∂ˆ

ψ

∂X −i

4

∂2ˆ

ψ

∂X 2.(28)

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

Schr¨odinger equation for a complex distribution function. We can ask at this point if indeed

are recovering quantum mechanics. In the following section we will perform a numerical

check of this hypothesis.

V. RECOVERING THE KERNEL OF THE SCHR¨

ODINGER EQUATION

If really the square root process diﬀuses as a solution of the Schr¨odinger equation we

should be able to recover the corresponding solution for the kernel

ˆ

ψ= (4πit)−1

2exp ix2/4t(29)

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

2exp −x2/4t.(30)

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 gener-

ated 10000 paths of Brownian motion and extracted its square root in the way devised in

Sec. III. We have evaluated the corresponding distribution after Wick rotating the results

for the square root. The Wick rotation generates real results as it should be expected and

a comparison can be performed. The result is given in Fig. 2 The quality of the ﬁt can be

10

−4 −3 −2 −1 0 1 2 3 4

0

1

2

3x 105Brownian motion

−5 0 5

0

5

10

15 x 104Square root

Figure 2. Comparison between the distributions of the Brownian motion and its square root after

a Wick rotation.

evaluated being ˆµ= 0.007347 with conﬁdence interval [0.005916,0.008778], ˆσ= 0.730221

with conﬁdence interval [0.729210,0.731234] for the heat kernel while one has ˆµ= 0.000178

with conﬁdence interval [−0.002833,0.003189] and ˆσ= 1.536228 with conﬁdence interval

[1.534102,1.538360] for the Schr¨odinger kernel. Both are centered around 0 and there is a

factor ∼2 between standard deviations as expected from eq. (28). Both the ﬁts are exceed-

ingly good. Having recovered the Schr¨odinger kernel from Brownian motion with the proper

scaling factors in mean and standard deviation, we can conclude that we are doing quantum

mechanics: Square root of a Brownian motion describes the motion of a quantum particle.

Need for Pauli matrices, as shown in the preceding section, implies that spin cannot be

neglected.

11

VI. PARTICLE IN A POTENTIAL

In order to understand how to introduce a potential within this approach we use the

following mapping theorem between the Fokker–Planck and the Schr¨odinger equation [18,

19]:

Theorem 1. The Fokker–Planck operator for a gradient ﬂow can be written in the self-

adjoint form

∂ˆ

ψ

∂t =D∇ · e−U

D∇eU

Dˆ

ψ.(31)

Deﬁne now ψ(x, t) = eU

2Dˆ

ψ(x, t). Then ψsolves the PDE

∂ψ

∂t =D∆2ψ−V(x)ψ, V (x) := |∇U|2

4D−∆2U

2.(32)

On the basis of the given theorem, we can immediately generalize our formulation to the

case of a potential. We will have

dX(t) = [dW (t)+U(X, t)dt]1

2=1

2+dW ·sgn(dW (t)) + (−1 + U(X, t) sgn(dW (t)))dtΦ1

2(t).

(33)

The corresponding Fokker–Planck equation will be

∂ˆ

ψ

∂t =∂

∂X −1 + i

4+1−i

4U(X, t)ˆ

ψ−i

4

∂2ˆ

ψ

∂X 2.(34)

As an example we consider a harmonic oscillator with U(X) = kX2/2

dX(t) = dW (t) + k

2X2dt

1

2

=1

2+dW ·sgn(dW (t)) + −1 + k

2X2sgn(dW (t))dtΦ1

2(t).

(35)

Here kis an arbitrary constant and the quantum potential is V(X) = |k|2X2−k

2, using

the mapping between the Fokker–Planck and the Schr¨odinger equations. The corresponding

Schr¨odinger equation will be

−i∂ψ∗

∂t =−1

4

∂2ψ∗

∂X 2+|k|2X2−k

2ψ∗(36)

with the introduction of ψ∗as we get what is conventionally a time-reversed quantum evo-

lution.

12

VII. SQUARE ROOT AND NONCOMMUTATIVE GEOMETRY

We have seen that, in order to extract the 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. 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=−σkσii6=k. (37)

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 γmatrices {γk∈Cℓ1,3(C), k =

0,1,2,3}such that

γ2

0=I γ2

1=γ2

2=γ2

3=−I γiγk+γkγi= 2ηik (38)

being ηik the Minkowski metric. In this way one can introduce three diﬀerent Brownian

motions for each spatial coordinates and three diﬀerent Bernoulli processes for each of them.

The deﬁnition is now

dE =

3

X

k=1

iγkµk+1

2µk|dWk| − 1

8µ3

k

dt·Φ(k)

1

2

+

3

X

k=1

iγ0γkµkΦ(k)

1

2

(39)

It is now easy to check that

(dE)2=I·(dW1+dW2+dW3).(40)

The Fokker-Planck equations have a solution with 4 components, as now the distribution

functions are Dirac spinors. These are given by

∂ˆ

Ψ

∂t =

3

X

k=1

∂

∂Xkµkˆ

Ψ−i

4∆2ˆ

Ψ (41)

being µk=−1+i

4+βk1−i

2. This implies that, the general formula for the square root pro-

cess implies immediately spin and antimatter for quantum mechanics that now come out

naturally. But this appears just like the non-relativistic limit of the Dirac equation and so,

having already introduced the γmatrices at this stage, it should be natural to get a fully

covariant Dirac equation. In the next section we will show that this indeed the case so that,

the metric element of a noncommutative geometry arise naturally as the Fokker–Planck

equation of a stochastic process.

13

VIII. DIRAC EQUATION

Dirac equation works on a 4-dimensional manifold and so, we will need four Wiener

processes to derive it. This assures full Lorentz invariance but, on the other side, time

should be treated as any other space variable. We need a further time variable, a ﬁctitious

one (as happens in stochastic quantization), to get Fokker–Planck equations in this case.

To accomplish this one has to introduce the γ5matrix, as already seen in noncommutative

geometry, in the following way

dE =

3

X

k=0

iγkµk+1

2µk|dWk| − 1

8µ3

k

dt·Φ(k)

1

2

+

3

X

k=0

iγ5γkµkΦ(k)

1

2

.(42)

Now one has a ﬁctitious time variable τbut we have a full family of solutions to the Fokker-

Planck equations parametrized by τ. Only the ﬁxed point solutions, the eigenstate with zero

eigenvalue, reproduce the Klein-Gordon equation for a free massless particle with a Dirac

spinor.

∂ˆ

Ψ

∂τ =∂·µˆ

Ψ−i

4∂2ˆ

Ψ.(43)

This recovers completely Dirac theory for a free particle from Brownian motions. We recog-

nize in eq.(42) the same stochastic process arising in noncommutative geometry in eq.(19).

IX. CONCLUSIONS

We have shown the existence of a class of stochastic processes that can support quantum

behavior. A typical one is the square root of a Brownian motion from which the Schr¨odinger

equation comes out naturally. The case with a potential was also discussed and applied to

the harmonic oscillator. Finally, we have derived the Dirac equation while spin and anti-

matter are naturally introduced by a stochastic 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 formulated by Alain Connes through the more recent proposal of space quantization by

Connes himself, Chamseddine and Mukhanov. This quantization of volume entails two kind

of quanta implying naturally the unity (1, i) that arises in the square root of a Wiener pro-

cess. 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

14

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.

ACKNOWLEDGMENTS

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, Amsterdam, 1994).

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