Stochastic Mechanics Redux

Stochastic Mechanics Redux * Mark Davidson  October 28, 2025 (PREPRINT version 2) Abstract The intended purpose of this paper is to revive interest in stochastic mechanics among physicists, mathematicians, and philosophers. Edward Nelson, the principal advocate for it, lost faith in his theory mainly because of the difference between the multitime autocorrelation expectations of quantum mechanics and those of stochastic mechanics. In particular, he showed that although quantum mechanics decouples the dynamics of quantum systems which are separated by a large distance, stochastic mechanics does not. Some authors have argued that with collapse of the wave function incorporated into stochastic mechanics this deficiency can be mitigated. In this paper we show a way to avoid collapse of the wave function altogether and still get the equivalence of stochastic mechanics and quantum mechanics for multitime products, and to achieve dynamic separability. This is achieved by using generalized stochastic mechanics where the diffusion constant is an arbitrary constant, and by choosing a particular imaginary value for the diffusion constant. This value is chosen to reproduce the Heisenberg commutation rules for the non-commuting operators of the stochastic process by a technique that was introduced some years ago. It turns out that this implies that the stochastic process is happening in complex space, and with a relativistic generalization in mind, it implies motion in complex spacetime. The purpose of this paper is to prove that this procedure resolves the multitime dilemma facing stochastic mechanics. Other related topics discussed include Wallstrom’s insight that the wave function is not necessarily single-valued; the general objection to Hidden Variables from various no-go theorems, including Bell tests; the interpretation of complex spacetime; the role of non-Markovian processes; The possible origin of quantum mechanics from chaos theory; and why complex spacetime is an interesting possibility that might underlie quantum physics. 1 Introduction Stochastic mechanics (SM) provides a stochastic interpretation of quantum mechanics. Originally proposed by Imre F´enyes [1,2,3,4], it was later greatly elucidated and made * version 2  Affiliation: Independent Scholar, Spectel Research Corp, Palo Alto, CA; Email: [email protected] 1 more rigorous by Nelson [5,6], and by many other contributors. Nelson eventually came to believe that stochastic mechanics was flawed because of non-local effects [7,5,8]. It was consequently suggested that by including wave function collapse as a necessary feature of stochastic mechanics, that it could be rescued [9,10,11]. As far as I know, Nelson was not convinced by these arguments. The purpose of the present paper is to examine this topic within the framework of generalized stochastic mechanics (GSM) [12,13] where the diffusion constant can take on any real (or complex) value without affecting the experimental predictions, and in particular to examine this issue for the case of an imaginary diffusion constant νcequal to −iℏ/2m. This is just −itimes Nelson’s diffusion constant νN=ℏ/2m. The theory of stochastic mechanics with this imaginary diffusion constant has been analyzed by Wang [14,15], Rosenbrock [16,17,18,19], Kuipers [20], and Yang [21]. We show here that these theories apparently solve the problems that Nelson was concerned about. They are based on the same kind of Markovian stochastic differential equation (SDE) as Nelson’s stochastic mechanics, except that they have a different, and imaginary diffusion constant. It was proposed long ago that the non-commutative algebra of Heisenberg could be derived in the framework of generalized stochastic mechanics if one analytically continued the diffusion constant to ±iℏ/2m[22,23], with the minus sign being the one that is used in the standard treatment of quantum mechanics by convention. The other sign would also be possible in principle. The strategy of this paper is to show by means of a specific analytic continuation that the quantum expectations of multi-time correlations are the same for quantum mechanics and for stochastic mechanics with this imaginary diffusion constant. 2 Nelson’s disillusionment with Stochastic Mechanics After some years of championing it, Nelson lost faith in SM. His problem was its failure to yield dynamical separability. This is a different form of nonlocality than in EPR and Bell’s theorem. It’s a brilliantly honest problem that he found and illustrated in [7]. He starts his argument with the following observation. Suppose we have two distantly separated quantum systems described by two Hamiltonians H1and H2, that act on two independent Hilbert spaces H1and H2. He joins these together as a Cartesian product to form the full Hilbert space H=H1⊗H2(1) and the Hamiltonian for the system is simply H=H1⊗1+1⊗H2(2) and we assume that [H1, H2] = 0 (3) 2 Now a general state vector for this quantum system will be a sum of the form (in Dirac’s bra-ket notation) |Ψ⟩= imax X i=1 |φi⟩⊗|χi⟩(4) Now consider any Hermitian operator A1which acts only on the H1Hilbert space. In the Heisenberg picture we can propagate A1in time using the unitary time evolution operator to obtain A1(t)⊗1 = eitHA1(0) ⊗1e−itH =eitH1A1(0)e−itH1⊗1 (5) and we can calculate the expectation value of this operator in the following way ⟨Ψ|A1(t)⊗1|Ψ⟩=⟨Ψ|eitH1A1e−itH1⊗1|Ψ⟩(6) It is completely obvious that this expectation value does not depend on any of the parameters involved in defining H2. Moreover, products of the following form will also not depend on H2: ⟨Ψ|A1(0)A1(t)⊗1|Ψ⟩=⟨Ψ|A1eitH1A1e−itH1⊗1|Ψ⟩(7) This is what Nelson called dynamic separability, and he argued that since these two systems can be very widely separated, any violation of this principle would be a serious violation of locality, more serious than the violation of Bell’s inequalities by quantum mechanics. He then shows, for a particular simple case, that stochastic mechanics does not satisfy this separability property, and is therefore terminally unacceptable in his opinion. It can be argued that by thoughtfully introducing wave function collapse on measurements that one can avoid this problem [9,10,11], and certainly in condensed matter environments where separability is not to be expected, this would not be a problem. But Nelson felt that for well isolated systems it was unacceptable even if there were no way to send messages faster than light with it, and that it may even enable one to send such signals. I think that Nelson had a valid and important point. This same problem presumably plagues Bohmian mechanics as well, although I am not aware of a proof of this. 3 Generalized Stochastic Mechanics It was recognized long ago that the diffusion constant of F´enyes and Nelson (ℏ/2mfor Schr¨odinger’s equation) was not unique, and that it could be any positive number or even any complex number if the space was allowed to be complex [22,13,12]. In all these cases, the diffusion leads to the Born probability distribution on the real spatial axis ψ∗(x)ψ(x), and where ψ(x) doesn’t depend on the diffusion constant. In this paper we shall exploit this non-uniqueness by choosing the value −iℏ/2mfor it as this achieves two things. First it reproduces the Heisenberg operator algebra and second it solves the dynamic separability problem that so worried Nelson. This was proposed long ago as 3 an analytic continuation [22,23] in the diffusion constant. The detailed treatment of the complex diffusion with this value of the diffusion constant has now been more fully developed [17,19,20,15,21,14]. Nelson and his colleagues did not embrace generalized stochastic mechanics although they accepted its validity, but considered it as perhaps something to maybe take up in the future [5]. The action principle of Yasue seemed to reinforce this position [24], suggesting that Nelson’s value of the diffusion constant was unique. We shall show that diffusion in complex space with this special imaginary value of the diffusion constant leads to generalized stochastic mechanics being dynamically separable as Nelson required. In 1D (for simplicity) following Nelson [6] in the framework of Itˆo calculus, we postulate a forward SDE for the position dx(t) = b(x, t)dt +dw(t, ν) (8) along with a backwards equation dx∗(t) = b∗(x, t)dt +dw∗(t, ν) (9) b(x, t) = lim ϵ↓0 1 ϵE[x(t+ϵ)−x(t)|x(t) = x] (10) b∗(x, t) = lim ϵ↓0 1 ϵE[x(t)−x(t−ϵ)|x(t) = x] (11) The time truns forward for both of these equations, but in general b=b∗. The diffusion constant is defined by1 ν=1 2E[(dw(t, ν))2]/dt (12) w(t, ν) and w∗(t, ν) are both Wiener processes with diffusion constant ν. w(t, ν) = √2νw(t) (13) where w(t) is a standard Wiener process, The physical implications of these SDEs are contained in the Markov transition density functions, the Fokker-Planck equation, the initial probability density, and the various properties and partial differential equations for these. So whether we think of the process in terms of the Itˆo or Stratonovich framework makes no difference. The following nonlinear gauge invariance is a property of Schr¨odinger’s equation, as shown for real or complex-valued constant zin [12] −ℏ2 2m△+VeR+iSQ=iℏ∂ ∂teR+iSQ(14) −(zℏ)2 2m△+V+ℏ2 2m(z2−1)△√ρ √ρeR+iSQ/z =i(zℏ)∂ ∂teR+iSQ/z (15) 1was defined differently, by a factor of 2, in [22]. Sorry about that! 4 where ν=zℏ 2m(16) The special value of ν=−iℏ/2mthat sets z=−iprovides the most complete emulation of quantum mechanics. If the Schr¨odinger wave function is ψ=eR+iSQ(17) then in 1 dimension we have in GSM the relation that yields the Schr¨odinger equation is b(x, t, ν)=2ν∇(R(x, t) + SQ(x, t)/z) = ∇2νR(x, t) + ℏ mSQ(x, t)(18) Using the notation of [25] we define a function SN(x, t, ν) by the relations b(x, t)=2ν∇(R+SN) (19) b∗(x, t) = 2ν∇(−R+SN) (20) And in the case of GSM one finds SN(x, t, ν) = 1 ν ℏ 2mSQ(x, t) (21) Substituting into this equation the value ν=−iℏ/2mand z=−iwe obtain b(x, t) = −iℏ/m∇(R(x, t) + iSQ(x, t)) = ℏ/m∇(SQ(x, t)−iR(x, t)) (22) which simplifies to b(x, t) = −iℏ/m∇ln(ψ) (23) Many Body theory and spin If we have multiple particles interacting with different masses, it is easy to apply GSM to this case as well. Suppose we have a Hamiltonian of the for H= N X n=1 ℏ2∇2 xn 2mn +U(x1, ..., xN) (24) We can scale the x coordinates of the particles with different masses by the following formula Xn=xnrm0 mn (25) where m0is some arbitrary mass constant. So that the Hamiltonian in these scaled coordinates takes the form H=−ℏ2 2m0 N X n=1 ∇2 Xn+U(X1, ..., XN) (26) 5 and in this form we can treat the system as if it were Nparticles of equal mass and we can apply the equivalence of 14 and 15. In this way GSM is mathematically like a single particle moving in a 3N dimensional space, and the stochastic mechanics can be seen to be applicable quite easily. Hamiltonians with vector potentials and on Riemannian manifolds can be handled in GSM as well [13]. Spin was included in the framework of a Riemannian manifold by Dankel [26] where the charged particle was taken to be a spinning ball with charge on its surface. It is unlikely that this model can correctly describe the electromagnetic properties of an electron with a g factor of approximately 2. A purely electromagnetic model for the mass, angular momentum, and magnetic moment of an electron with a g factor of 2 was presented in [27], and I hope that this can be incorporated into the framework of generalized stochastic mechanics. One problem with this model though is that it predicts a nonzero value for the quadrupole moment of the electron. 4 Rosenbrock’s analysis of a complex diffusion model Rosenbrock has analyzed diffusion with an imaginary diffusion constant corresponding to ν=−iℏ/2min GSM [17]. We review his analysis here. Consider a one dimensional case for simplicity. Using the SDE dz(t) = b(z, t)dt +p−iℏ/mdw(t) (27) where w(t) is a standard Wiener process. Integrating this equation will generate trajectories in the complex zplane. It is assumed that b(z, t) is an analytic function of z except perhaps at isolated points. Let us write z=x+iy (28) Following Rosenbrock [19], we define X=x+yand Y=x−yso that an interesting property of the SDE can be revealed. From 22 we find: b(z, t) = ℏ/m∇z(SQ(z, t)−iR(z, t)) (29) where SQ(z, t) and R(z, t) are analytic continuations to complex zfrom their values on the real axis. Let bRand bIbe the real and imaginary parts of b: bR(x, y, t) = Real(b(x+iy, t)) (30) bI(x, y, t) = Imag(b(x+iy, t)) (31) We find the SDE for X=x+yand Y=x−yto be dX(t)=(BR(X, Y, t) + BI(X, Y, t)) dt (32) dY (t)=(BR(X, Y, t)−BI(X, Y, t)) dt +pℏ/mdw(t) (33) 6 where the functions BRand BIare the functions bRand bIreexpressed as functions of X and Y. We see that the SDE for X(t) has no diffusion term. Therefore, over very short time intervals, on the average we have |dX(t)|<< |dY (t)|. The Fokker Planck equation for the probability density ρ(x, y, t) is found by Rosenbrock to be ∂ρ ∂t +∂ ∂x(ρbR) + ∂ ∂y (ρbI)−ℏ 8m∂ ∂x −∂ ∂y 2 ρ= 0 (34) This is subject to the constraint on the real axis that the Born rule is satisfied ρ(x, 0, t) = e2R(x,t)=ψ∗(x, t)ψ(x, t) (35) The question is whether the multitime product expectation: E(z(t1)z(t2)) = Zρ(x1, y1, t1)(x1+iy1)(x2+iy2)PT(x1, y1, t1;x2, y2, t2)dx1dy1dx2dy2 (36) is the same as the quantum expectation with appropriate time ordering of the noncommuting quantum operators x(t1) and x(t2) ⟨ψ|x(t1)x(t2)|ψ⟩(37) If we can show that for complex stochastic mechanics this holds true in general, then it would resolve the separability problem. Our strategy will be to utilize a non-commuting operator approach for the stochastic process and show that the resulting expectations match those of quantum mechanics for ν=−iℏ/2m. 5 Non-commuting operators from stochastic processes There is a way to introduce non-commuting operators into the framework of stochastic processes that mimics Feynman’s time ordering technique [22,23,25]. Consider the following limiting procedure with t>s: Commutator = lim t↓s∂ ∂t −∂ ∂sE(x(t)x(s)) (38) The expectation E() here is the classical expectation for a stochastic process x(t). It can be evaluated by means of the forward and backward derivatives to give [25]: Commutator = E[(b∗(x, s)−b(x, s))x(s)] (39) Using the standard SM relation b∗(x, s)−b(x, s) = −2ν∇ρ(x, s)/ρ(x, s), and integrating by parts, we find: Commutator = −Z2ν∇ρ(x, s) ρ(x, s)xρ(x, s)dx = 2ν(40) 7 This simple result suggests that we can use the microscopic time ordering of expectations to define a non-commuting operator algebra. To get the usual Heisenberg commutation relations, we must set ν=−iℏ 2m(41) We introduce a Hilbert space Hswith the inner product: (f, g)s=E(f∗(x(s), s)g(x(s), s)) = Zρ(x, s)f∗(x, s)g(x, s)dx (42) This type of Hilbert space is called a weighted L2space and is often denoted as L2(Rn, ρ(x)dx). The density function ρ(x, s) must be smooth in both xand sto ensure that the inner product is finite and well-defined. This construction creates a Hilbert space that includes the functions of the standard L2space, but with a different inner product and norm. The reason for starting with this inner product is that it allows us to utilize the machinery of the stochastic process to generate non commuting operators for position and momentum very conveniently. Later we will apply a similarity transformation 52 to obtain the standard L2space. We define the position operator ˆxby simple multiplication. Next we define a velocity operator ˆ ˙xthat acts on elements of Hsas the following limit of a derivative of a conditional expectation: ˆ ˙xf(x, s) = lim t↓ulim u↓s ∂ ∂uE(x(u)f(x(t), t)|x(s) = x), t > u > s, f ∈ Hs(43) This can be written in terms of the Markov transition function densities P(xf, tf;xi, ti). We can write the probability density for three times, where t1< t2< t3, as ρ(x3, t3;x2, t2;x1, t1) = P(x3, t3;x2, t2)P(x2, t2;x1, t1)ρ(x1, t1) (44) To get the conditional expectation we simply drop the term ρ(x1, t1) to get ˆ ˙xf(x, s) = lim t↓ulim u↓s ∂ ∂u Zf(xt, t)xuP(xt, t;xu, u)P(xu, u;x, s)dxtdxu(45) Assuming uniform convergence to bring the derivative inside the integral and using the forward and backward Kolmogorov equations, this expression can be evaluated. The derivation is detailed in [25,22,23]. The result is: ˆ ˙x=b(x, s)+2ν∇x(46) It follows then that a commutator relation exists: [ˆ ˙x(s),ˆx(s)] = 2ν(47) which is the Heisenberg commutation rule if we set ν=−iℏ/2mand define momentum as mˆ ˙x. Note that ˆ ˙x(s) is non-Hermitian in general, unless νis imaginary, and unless 8 the wave function is single valued if in higher dimensions, and requiring a similarity transformation described below 52. Next, we define an acceleration operator from the second derivative of the conditional expectation: ˆ ¨xf(x, s)≡lim t↓ulim u↓s ∂2 ∂u2E(x(u)f(x(t), t)|x(s) = x) (48) The derivation of this is lengthy, but the result is [25,23]: ˆ ¨x=∂b(x, s) ∂s +ν(∆xb(x, s)) + 1 2∇xb2(x, s) (49) By comparing this operator with Nelson’s mean acceleration aNfrom [6] and using the equations of motion for GSM, one can show that this operator equation holds: mˆ ¨x=−∇xV(x, t)−2mν2+ℏ2 2m ∆√ρ √ρ(50) This is a generalized Ehrenfest theorem. The simplest possibility occurs when ν= ±iℏ/2m, for then the second term in the parenthesis vanishes, and we get the Heisenberg formula of quantum mechanics: mˆ ¨x=−∇xV(x, t),if ν=±iℏ/2m(51) This result can be put into canonical form by a similarity transformation that does not affect the commutation rules. Using 23 we obtain: ˆ ˙xcanonical =e−R−iSQˆ ˙xeR+iSQ=−iℏ m∇x(52) So we can have the exact Heisenberg momentum operator at an imaginary value for the diffusion constant: ˆ PHeisenberg =mˆ ˙xcanonical =−iℏ∇x,if ν=−iℏ/2m(53) with the canonical commutation relation [ˆx, ˆ PHeisenberg] = iℏ. Since ˆx and ˆ PHeisenberg are both self-adjoint operators on L2as is required from quantum mechanics, and so having established the commutation rules using the weighted Hilbert space, we can now introduce the standard Hilbert space of quantum mechanics. We can also form higher derivative operators ˆxmf(x, s)≡lim t↓ulim u↓s ∂m ∂umE(x(u)f(x(t), t)|x(s) = x) (54) These operators satisfy the following recursion formula ([23], eq. (29)). ˆxm+1 = [(b(x, t) + ν∇)·∇,ˆxm] + ∂ ∂t ˆxm(55) 9 A Sufficient conditions for analyticity of expectation values The argument for extending results from real to complex diffusion constants relies on the analyticity of multitime expectations as a function of ν. Here we state sufficient conditions for this analyticity, developed with use of gemini AI. Statement and Proof Let Z(t)∈CNsolve the complex SDE, for a parameter v∈ U⊂CN: dZj(t) = bj(Z(t); v)dt +σj(vj)dBj(t), j = 1, . . . , N (67) where {Bj}are independent standard real Brownian motions. Assumptions 1. (Analyticity, Lipschitz, growth) For any compact K⊂U:  The map v7→ b(z;v) is holomorphic on Ufor fixed z.  The map z7→ b(z;v) is entire for fixed v.  b(z;v) satisfies global Lipschitz and linear growth conditions in z, uniformly in v∈K.  Each σj(vj) is holomorphic on its domain. 2. (Initial moments) The initial condition Z(0) has a probability density p0on CN with sufficient finite moments. Theorem Suppose the assumptions hold. Let F: (CN)n→Cbe an entire observable with polynomial growth. Then for fixed times 0 < t1<··· < tn≤T, the expectation value u(v) = E[F(Zv(t1), . . . , Zv(tn))] (68) is a holomorphic function of von U. Sketch of proof The proof proceeds in four steps. First, standard SDE theory guarantees well-posedness and uniform moment bounds for the solution Zv(t) under the given assumptions. Second, for a fixed initial condition z0, parameter-analyticity results for SDEs show that the expectation is holomorphic in v. Third, the uniform moment bounds allow one to find a dominating function GK(z0) for the expectation that is independent of v∈Kand is integrable with respect to the initial density p0(z0). Finally, Morera’s theorem or differentiation under the integral sign proves that the full expectation u(v) is holomorphic on any compact K⊂U, and thus on all of U. 16 References [1] I. F´enyes. A deduction of schr¨odinger equation. Acta Bolyaiana, 1, 1946. 1 [2] I. F´enyes. Zur wellenmechanischen herleitung des statistischen atommodells. Zeitschrift f¨ur Physik, 125(5-7):336–338, 1948. 1 [3] Imre F´enyes. Eine wahrscheinlichkeitstheoretische begr¨undung und interpretation der quantenmechanik. Zeitschrift f¨ur Physik, 132(1):81–106, 1952. 1 [4] I. F´enyes. Stochastischer abh¨angigkeitscharakter der heisenbergschen ungenauigkeitsrelation. Die Naturwissenschaften, 39(21):500–500, 1952. 1 [5] Edward Nelson. Quantum Fluctuations. Princeton University Press, 1985. 1,3,11, 13 [6] Edward Nelson. 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