Stochastic Mechanics Redux

Stochastic Mechanics Redux Mark Davidson Independent Scholar, Spectel Research Co., Palo Alto, CA, USA. Contributing authors: [email protected]; [Preprint version 3, 11/07/25] 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 dierence 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 deciency 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 diusion constant is an arbitrary constant, and by choosing a particular imaginary value for the diusion 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 1 Introduction Stochastic mechanics (SM) provides a stochastic interpretation of quantum mechanics. Originally proposed by Imre Fényes [1,2,3,4], it was later greatly elucidated and made more rigorous by Nelson [5,6], and by many other contributors. Nelson eventually came to believe that stochastic mechanics was awed because of non-local eects [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 diusion constant can take on any real (or complex) value without aecting the experimental predictions, and in particular to examine this issue for the case of an imaginary diusion constant νc equal to −iℏ/2m . This is just −i times Nelson's diusion constant νN=ℏ/2m . The theory of stochastic mechanics with this imaginary diusion 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 dierential equation (SDE) as Nelson's stochastic mechanics, except that they have a dierent, and imaginary diusion 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 diusion constant to ±iℏ/2m [22,23,24], 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 specic analytic continuation that the quantum expectations of multi-time correlations are the same for quantum mechanics and for stochastic mechanics with this imaginary diusion 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 dierent 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 H1 and H2 , that act on two independent Hilbert spaces H1 and 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 consequently [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 A1 which acts only on the H1 Hilbert space. In the Heisenberg picture we can propagate A1 in 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 dening 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 diusion constant of Fényes and Nelson ( ℏ/2m for Schrödinger'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 diusion leads to the Born probability distribution on the real spatial axis ψ∗(x)ψ(x) , and where ψ(x) doesn't depend on the diusion constant. In this paper we shall exploit this non-uniqueness by choosing the value −iℏ/2m for 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 an analytic continuation [22,23] in the diusion constant. The 3 detailed treatment of the complex diusion with this value of the diusion 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 [25], suggesting that Nelson's value of the diusion constant was unique. We shall show that diusion in complex space with this special imaginary value of the diusion constant leads to generalized stochastic mechanics being dynamically separable as Nelson required. In 1D (for simplicity) following Nelson [6] in the framework of Itô 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 t runs forward for both of these equations, but in general b=b∗ . The diusion constant is dened by 1 ν=1 2E[(dw(t, ν))2]/dt (12) w(t, ν) and w∗(t, ν ) are both Wiener processes with diusion 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 dierential equations for these. So whether we think of the process in terms of the Itô or Stratonovich framework makes no dierence. The following nonlinear gauge invariance is a property of Schrödinger's equation, as shown for real or complex-valued constant z in [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) 1 ν was dened dierently, by a factor of 2, in Davidson [22]. 4 where ν=zℏ 2m (16) The special value of ν=−iℏ/2m that sets z=−i provides the most complete emulation of quantum mechanics. If the Schrödinger wave function is ψ=eR+iSQ (17) then in 1 dimension we have in GSM the relation that yields the Schrödinger 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 [24] we dene 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 nds SN(x, t, ν) = 1 ν ℏ 2mSQ(x, t) (21) Substituting into this equation the value ν=−iℏ/2m and z=−i we obtain b(x, t) = −iℏ/m∇(R(x, t) + iSQ(x, t)) = ℏ/m∇(SQ(x, t)−iR(x, t)) (22) which simplies to b(x, t) = −iℏ/m∇ln(ψ) (23) Many Body theory and spin If we have multiple particles interacting with dierent 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 dierent masses by the following formula Xn=xnrm0 mn (25) where m0 is 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) and in this form we can treat the system as if it were N particles of equal mass and we can apply the equivalence of (14) and (15). In this way GSM is mathematically 5 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 diusion model Rosenbrock has analyzed diusion with an imaginary diusion constant corresponding to ν=−iℏ/2m in 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 z plane. 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 dene X=x+y and Y=x−y so that an interesting property of the SDE can be revealed. From (22) we nd: 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 z from their values on the real axis. Let bR and bI be 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 nd the SDE for X=x+y and Y=x−y to 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) where the functions BR and BI are the functions bR and bI reexpressed as functions of X and Y. We see that the SDE for X(t) has no diusion term. Therefore, over very 6 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 satised 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)) = ˆρ(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 noncommuting 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 [22,23,24]. 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 [24]: 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 nd: Commutator =−ˆ2ν∇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 dene a non-commuting operator algebra. To get the usual Heisenberg commutation relations, we must set ν=−iℏ 2m (41) We introduce a Hilbert space Hs with the inner product: (f, g)s=E(f∗(x(s), s)g(x(s), s)) = ˆρ(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 x and s to ensure that the inner product is nite and well-dened. This construction creates a Hilbert space that includes the functions of the standard L2 space, but with a dierent 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 L2 space. We dene the position operator ˆ x by simple multiplication. Next we dene a velocity operator ˆ ˙ x that acts on elements of Hs as 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 ˆf(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 [24,22,23]. The result is: ˆ ˙ x=b(x, s)+2ν∇x (46) It follows then that a commutator relation exists: [ˆ ˙ x(s),ˆ x(s)] = 2ν (47) 8 which is the Heisenberg commutation rule if we set ν=−iℏ/2m and dene momentum as mˆ ˙ x . Note that ˆ ˙ x(s) is non-Hermitian in general, unless ν is imaginary, and unless the wave function is single valued if in higher dimensions, and requiring a similarity transformation described below (52). Next, we dene 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 [24,23]: 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 aN from [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 aect the commutation rules. Using (23) we obtain: ˆ ˙ x canonical =e−R−iSQˆ ˙ xeR+iSQ=−iℏ m∇x (52) So we can have the exact Heisenberg momentum operator at an imaginary value for the diusion constant: ˆ P Heisenberg =mˆ ˙ x canonical =−iℏ∇x, if ν=−iℏ/2m (53) with the canonical commutation relation [ˆ x, ˆ P Heisenberg ] = iℏ . Since ˆ x and ˆ P Heisenberg are both self-adjoint operators on L2 as 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 ˆ xmf(x, s)≡lim t↓ulim u↓s ∂m ∂umE(x(u)f(x(t), t)|x(s) = x) (54) 9 Acknowledgements I acknowledge valuable discussions with Tuck Choy and Hajo Lechske during the course of this work. Appendix Analyticity of Multi-Time Expectations (Prepared by Math Proof GPT for this paper) A Statement and Proof of Analyticity We consider a family of complex stochastic dierential equations depending on a complex parameter v∈U⊂Cp . Let Zv(t)∈CN solve the Itô SDE on a xed interval t∈[0, T] : dZv j(t) = bj(Zv(t); v)dt +σj(v)dwj(t), j = 1, . . . , N, driven by independent standard real Wiener processes w1, . . . , wN . The initial law at t= 0 has density p0 on CN . Assumptions Theorem A.1. Assumption (Analyticity and growth)  For each xed z∈CN , the map v7→ b(z;v) is holomorphic on U .  For each xed v∈U , the map z7→ b(z;v) is entire on CN .  Each σj:U→C is holomorphic.  For every compact K⋐U there exist constants LK, CK>0 such that ∥b(z;v)−b(z′;v)∥ ≤ LK∥z−z′∥,∥b(z;v)∥ ≤ CK(1 + ∥z∥), for all v∈K and all z, z′∈CN . Theorem A.2. Assumption (Initial moments) The initial distribution has nite polynomial moments: there exists m≥1 with ´CN∥z∥mp0(dz)<∞ . Let 0< t1<··· < tn≤T be xed and let F: (CN)n→C be entire with polynomial growth of order ≤m : |F(z1, . . . , zn)| ≤ C1 + n X j=1 ∥zj∥m. 16 Theorem A.3. Theorem Under the above assumptions, the map u(v) := E[F(Zv(t1), . . . , Zv(tn)) ], v ∈U, is holomorphic on U . Proof Step 1 (Well-posedness and uniform moment bounds). By the global Lipschitz and linear growth bounds, for each xed v∈U the SDE admits a unique strong solution Zv(t) on [0, T ] . Moreover, for each compact K⋐U there exists CK,m,T such that sup 0≤t≤T E[∥Zv(t)∥m]≤CK,m,T 1 + E∥Z(0)∥m. Step 2 (Pathwise holomorphy for xed initial condition). Fix z0 . Each Picard iterate is constructed by integrating the holomorphic-inv function b(·;v) and adding the term σ(v)w(t) ; hence each iterate is holomorphic in v . The iteration converges uniformly on compacts, so Zv(t, ω) is holomorphic in v for each t and almost every ω . Step 3 (Domination and dierentiation under expectation). The polynomial growth bound on F and the uniform moment estimates imply the existence of an integrable dominating random variable G(ω) such that sup v∈K |F(Zv(t1),...,Zv(tn))| ≤ G(ω),E[G]<∞. Hence one can interchange expectation and dierentiation, so u(v) is complex dierentiable in every direction. Step 4 (Holomorphy on U ). By Morera's theorem, for any closed contour Γ⊂U , ˛Γ u(v)dv =E h˛Γ F(Zv(t1),...,Zv(tn)) dvi= 0, and dominated convergence justies exchanging expectation and contour integration. Therefore, u is holomorphic on U . □ Remark A.4 . The assumptions can be relaxed (e.g. local Lipschitz with non-explosion via a localization/stopping argument). If the diusion coecients σ depend on z and v , the argument extends under the same holomorphy and growth assumptions. The above theorem provides the rigorous justication for analytic continuation of multi-time expectations in the diusion parameter ν used in the main text. References [1] Fényes, I.: A deduction of schrödinger equation. Acta Bolyaiana 1 (1946) [2] Fényes, I.: Zur wellenmechanischen herleitung des statistischen atommodells. Zeitschrift für Physik 125 (5-7), 336338 (1948) [3] Fényes, I.: Eine wahrscheinlichkeitstheoretische begründung und interpretation der quantenmechanik. 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Physical Review A 49 (3), 16131617 (1994) [32] Derakhshani, M.: A suggested answer to wallstrom's criticism: Zitterbewegung stochastic mechanics i (2015) arXiv:1510.06391 [quant-ph] [33] Davidson, M.: Multi-valued vortex solutions to the schrödinger equation and radiation. Annals of Physics 418 , 168196 (2020) [34] Davidson, M.: The Lorentz-Dirac equation in complex space-time. General Relativity and Gravitation 44 (11), 29392964 (2012) https://doi.org/10.1007/ s10714-012-1432-6 . arXiv: 1109.4923. Accessed 2014-07-02 [35] Davidson, M.P.: Quantum wave equations and non-radiating electromagnetic sources. Annals of Physics 322 (9), 21952210 (2007) https://doi.org/10.1016/j. aop.2006.10.005 . Accessed 2019-10-31 [36] Davidson, M.: A model for the stochastic origins of Schrödinger's equation. Journal of Mathematical Physics 20 (9), 18651869 (1979) https://doi.org/10.1063/1. 524304 . Accessed 2025-07-31 [37] De La Peña, L., Cetto, A.M., Valdés Hernández, A.: The Emerging Quantum: The Physics Behind Quantum Mechanics. Springer, Cham (2015). https://doi.org/10. 1007/978-3-319-07893-9 .https://link.springer.com/10.1007/978-3-319-07893-9 Accessed 2023-11-25 [38] Horwitz, L.P., Katz, N., Oron, O.: Could the classical relativistic electron be a strange attractor? Discrete Dynamics in Nature and Society 2004 (1), 179204 (2004) [39] Aharonovich, I., Horwitz, L.P.: Radiation-reaction in classical o-shell electrodynamics. I. The above mass-shell case. Journal of Mathematical Physics 53 (3), 032902 (2012) https://doi.org/10.1063/1.3694276 [40] Hooft, G.t.: Deterministic Quantum Mechanics: the Mathematical Equations. arXiv. arXiv:2005.06374 [hep-th, physics:quant-ph] (2020). https://doi.org/10. 48550/arXiv.2005.06374 .http://arxiv.org/abs/2005.06374 Accessed 2023-09-15 [41] Barandes, J.A.: Quantum Systems as Indivisible Stochastic Processes. arXiv. 19 arXiv:2507.21192 [quant-ph] (2025). https://doi.org/10.48550/arXiv.2507.21192 .http://arxiv.org/abs/2507.21192 Accessed 2025-09-29 20