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Physics Letters A www.elsevier.com/locate/pla

Genesis of quantum nonlocality L. de la Peña a , A. Valdés-Hernández a,∗ , A.M. Cetto a , H.M. França b a b

Instituto de Física, Universidad Nacional Autónoma de México, A.P. 20-364, México D.F., Mexico Instituto de Física, Universidade de São Paulo, CP 66318, São Paulo, Brazil

a r t i c l e

i n f o

Article history: Received 7 November 2010 Received in revised form 10 February 2011 Accepted 10 February 2011 Available online 12 February 2011 Communicated by P.R. Holland Keywords: Foundations of quantum mechanics Quantum nonlocality Quantum ﬂuctuations Phase-space quantum mechanics Stochastic processes

a b s t r a c t We revisit the problem of an otherwise classical particle immersed in the zero-point radiation ﬁeld, with the purpose of tracing the origin of the nonlocality characteristic of Schrödinger’s equation. The Fokker–Planck-type equation in the particle’s phase-space leads to an inﬁnite hierarchy of equations in conﬁguration space. In the radiationless limit the ﬁrst two equations decouple from the rest. The ﬁrst is the continuity equation; the second one, for the particle ﬂux, contains a nonlocal term due to the momentum ﬂuctuations impressed by the ﬁeld. These equations are shown to lead to Schrödinger’s equation. Nonlocality (obtained here for the one-particle system) appears thus as a property of the description, not of Nature. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Quantum nonlocality constitutes one of the most elusive properties of quantum systems, and one major conceptual diﬃculty raised by quantum mechanics. It is right at the root of quantum theory, ever since any (even a single-particle) state described by a Schrödinger wave implies nonlocal properties. Providing a wellgrounded physical explanation for such nonlocality is a fundamental task for a better realistic understanding of quantum theory. The present Letter intends to shed light on the physical origin and meaning of quantum nonlocality in the context of the one-body problem. We accomplish this within a theoretical framework that identiﬁes the presence of the stochastic zero-point radiation ﬁeld (zpf) in interaction with matter as the entity ultimately responsible for the ﬂuctuating properties of the momentum of the particle, which from the present perspective are at the root of both the nonlocal and the indeterministic nature of the quantum description.1

*

Corresponding author. E-mail addresses: [email protected]ﬁsica.unam.mx (L. de la Peña), [email protected]ﬁsica.unam.mx (A. Valdés-Hernández), [email protected]ﬁsica.unam.mx (A.M. Cetto), [email protected] (H.M. França). 1 A word of caution about our terminology is in order here. It is usual nowadays to speak of nonlocality in the context of several-particle systems, as, for example, when discussing Bell’s inequalities. In the concluding section we refer to such nonlocalities. However, in the one-particle case dealt with in this Letter a quantity (such as the quantum potential) whose value at point x depends on the particle distribution ρ is considered by deﬁnition nonlocal. This is consistent with a mechanical point of view. When the wave function is interpreted instead as a real ﬁeld (as e.g. 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.02.024

We start by considering the complete (inﬁnite dimensional) phase-space description of a system composed of an otherwise classical charged particle interacting with the zpf, and reduce the description (by successive averagings) to the conﬁguration space of the particle. In doing so, a term that originates in the ﬂuctuations of the momentum impressed by the zpf on the particle appears in the set of equations that govern the evolution of the mechanical system in the radiationless approximation. This term — by which the behavior of the system departs from any classical analog — is identiﬁed as the one that introduces the nonlocal character into the resulting formulation. It is found that the statistical equations that control the dynamics of the particle are completely equivalent to the Schrödinger equation. Our results reveal the genesis of quantum ﬂuctuations and of the characteristic quantum nonlocality, shown to be ascribable to the use of the reduced description rather than to any action at a distance. The present results also indicate that the quantum phenomenon is not intrinsic to matter alone — nor to the radiation ﬁeld alone — but an emergent phenomenon, generated by the permanent matter–ﬁeld interaction [2–4]. Altogether, these results show that a careful study of (otherwise classical) material systems in interaction with the zpf allows to achieve a better understanding of the meaning and richness of the quantum description.

in Bohm’s theory), the one-particle quantum potential has a local meaning, according to a ﬁeld-theoretical point of view. A nice discussion of quantum nonlocalities from the latter viewpoint is contained in Ref. [1], particularly Ch. 11.

L. de la Peña et al. / Physics Letters A 375 (2011) 1720–1723

means of the characteristic (moment generating) function Q˜ , obtained from the density Q by means of the relation

2. The reduced Liouville equation The physical system under study is a (single) particle of mass m and electric charge e embedded in the stochastic zero-point radiation ﬁeld. The complete system is classical except for the contribution of the (continuous) zero-point ﬁeld, whose energy per mode of frequency ω is E0 = h¯ ω/2 (this is the entry point for Planck’s constant into the picture). Let R (xs , p s , t ) be the density of points in the phase space of the complete system (xs and p s stand for the whole set of degrees of freedom: the position and momentum of the particle, x and p, and the quadratures xzpf , p zpf of the zpf). Our starting point is the stochastic Liouville equation for R (xs , p s , t ),

∂ ∂ ∂ x˙ s R + p˙ s R = 0. R+ ∂t ∂ xs ∂ ps

... p˙ = f (x) + mτ x + e E (t ),

(2)

where E (t ) stands for the electric component of the zpf in the ... long-wavelength approximation and mτ x stands for the radiation 2 3 reaction force, with τ = 2e /3mc (≈ 10−23 s for the electron). We will further approximate this term in the form (although this ... will be of little importance in what follows) mτ x τ x˙ f (x), where the prime denotes derivative with respect to x. Eq. (1) is a quite complicated equation for the stochastic density R. However, since we are interested only in the mean behavior of an ensemble of similar systems, we average R (xs , p s , t ) over the realizations of the stochastic ﬁeld and call Q (x, p , t ) the resulting density in the phase space of the particle. We do it using the projector technique [5,6], which leads after a somewhat lengthy calculation to the following Fokker–Planck-type equation for Q (x, p , t )

∂ ∂ 1 ∂ Q + pQ + ∂t m ∂x ∂p ∂ ˆ (t ) Q , D = e2 ∂p

f (x) +

τ m

f (x) p Q

(3)

ˆ (t ) is a complicated integro-differential operator such that where D

2k+1 ∞ ∂ ˆ ∂ ˆ ˆ ∂ e2 D (t ) Q = e PE eG (1 − Pˆ ) E Q. ∂p ∂p ∂p

(4)

k =0

∂ + Lˆ ∂t

t =

ˆ

−1 A (x, p , t )

e − L (t −t ) A x, p , t dt .

Q (x, p , t )e ipz dp .

(6)

(A parallel process can be followed to reduce the description to momentum space, by performing the Fourier transformation in conﬁguration space instead of momentum space.) From Eq. (6) it follows that the local probability density ρ (x, t ) is

ρ (x, t ) =

Q (x, p , t ) dp

= lim

z→0

Q (x, p , t )e ipz dp = Q˜ (x, 0, t ),

(7a)

so that the local (partially averaged) moments of the momentum p can be written as

p (x) ≡ pn x = n

1

ρ (x)

n

n

p Q dp = (−i )

. n ˜ ∂ z Q z =0 1 ∂ n Q˜

(7b)

By taking the Fourier transform of Eq. (3) one gets (we assume that all surface terms vanish at inﬁnity)

1 ∂ 2 Q˜ ∂ Q˜ τ ∂ Q˜ ˆ Q ). −i − izf (x) Q˜ − f z = −ie 2 z( D ∂t m ∂ x∂ z m ∂z

(8)

To reduce this to a description in conﬁguration space, we expand Eq. (8) into a power series around z = 0 and separate the coeﬃcients of zk (k = 0, 1, 2, . . .), thus obtaining for the ﬁrst two equations

∂ρ 1 ∂

p x ρ = 0, + ∂t m ∂x 1 ∂ 2 ∂

τ p x ρ + p xρ − f ρ − f p x ρ ∂t m ∂x m ˆ Q )|z=0 . = −e 2 ( D

(9a)

(9b)

The subsequent equations (corresponding to higher powers of z) are connected to the above couple by the same elements ρ , p x , p 2 x and further moments pn x , in addition to contributions de-

ˆ Q ). The resulting set constitutes an inﬁriving from the term z( D nite hierarchy of coupled differential equations. 4. Entry point for nonlocalities

Here Pˆ stands for the smoothing operator that averages over the ¯ E ) and Gˆ represents the realizations of the random ﬁeld E ( Pˆ A = A 1 ∂ ∂ p inverse Liouville operator (with Lˆ = m p ∂ x + ∂ p ( f + τ f m ))

Gˆ A (x, p , t ) =

Q˜ (x, z, t ) =

(1)

The motion of the mechanical (one-dimensional) subsystem is governed in the nonrelativistic limit by

x˙ = p /m,

1721

(5)

0

ˆ (t ) in the present work, We will not require the explicit form of D so that Eqs. (4) and (5) are included here solely for reference purposes. The detailed derivation can be seen in [6]. 3. Description in conﬁguration space We will now investigate the connection of Eq. (3) with quantum mechanics. Since the latter is usually described in conﬁguration space, we will restrict ourselves to the reduced description in the (x, t ) space. The transition can be systematically accomplished by

The ﬁrst member of the hierarchy, Eq. (9a), is the continuity equation (for the transfer of matter) in conﬁguration space, with the local (partially averaged) velocity v (x) = p x /m. The second equation describes the transfer of momentum and contains, in addition to ρ and v, the second moment p 2 x , whose value is determined in its turn by a third equation describing the transfer of kinetic energy. This coupling between successive equations creates a highly diﬃcult mathematical problem. However, in the radiationless approximation a crucial decoupling takes place, as we shall now see. First we introduce the change of variables

z+ = x + β z,

z− = x − β z,

(10)

with still undetermined parameter β that has the dimensions of an action. In terms of these new variables we write Q˜ in the general form

Q˜ (x, z, t ) = q+ ( z+ , t )q− ( z− , t )χ ( z+ , z− , t ). Thus using Eq. (7b) we obtain

(11)

1722

L. de la Peña et al. / Physics Letters A 375 (2011) 1720–1723

∂ ln Q˜ = −i β(∂+ ln Q˜ − ∂− ln Q˜ )|z=0 ∂ z z =0 = −i β ∂+ ln q+ − ∂− ln q− + (∂+ − ∂− ) ln χ z=0 , (12)

p x = mv x = −i

and

p2

2 ∂ 2 ˜ − p

= − ln Q x x 2 ∂z z =0 2 ∂ = −β 2 ln ρ + 4β 2 (∂+ ∂− ln χ )|z=0 ∂ x2 = −β 2

∂2 ln ρ + Σ, ∂ x2

with Σ = 4β 2 (∂+ ∂− ln χ )|z=0 . We now insert the above results into Eqs. (9) and extend the resulting equations to the 3-dimensional case by introducing the required indices (a sum over repeated indices is assumed), thus obtaining

m

(14a)

β2 ∂ ∂ ∂ ∂2 (v i ρ ) + m (v i v j ρ ) − ρ ln ρ − f i ρ ∂t ∂xj m ∂xj ∂ xi ∂ x j ∂ fi 1 ∂ ˆ Q )i . = τ vi ρ− Σi j ρ − e 2 ( D (14b) ∂x m ∂x j

j

z =0

The ﬁrst term on the second line of Eq. (14b), proportional to τ ∝ e 2 , is due to radiation reaction; the third one (containing inﬁnite memory), explicitly proportional to e 2 , is due to the action of the zpf. When detailed balance is attained between the mean power absorbed by the particle from the ﬁeld (due basically to the ﬂuctuating motion) and the mean radiated power (basically along the orbital motion), these terms become merely radiative corrections; hence in the radiationless approximation (i.e., when only the main effects of the ﬁeld on the particle are considered) they can be neglected and we are left with the term containing Σi j , which is a contribution from the (local) ﬂuctuations of the momentum, as follows from Eq. (13). However, by making a Taylor series expansion of the left-hand side of Eq. (8), it can be shown that in the radiationless approximation and for small values of z (which are the values of interest here, as seen from Eq. (7a)), the function χ becomes a constant (which we shall take equal to 1), hence Σi j |z=0 = 0. Taken altogether, these results mean that the righthand side of Eq. (14b) reduces to zero in the radiationless limit. Further, the function Q˜ factorizes as Q˜ (x, z, t ) = q+ ( z+ , t )q− ( z− , t ) (see Eq. (11)) and Eq. (12) reduces to

p x = mv x = −i β(∂+ ln q+ − ∂− ln q− )|z=0 .

(15)

Detailed derivations of these results are presented in an extended version of this Letter, to be published. The original problem of solving Eqs. (9) involved three unknowns: ρ , p x , and p 2 x , and thus we introduced three independent functions q+ , q− , and χ to determine them. But when Σ = 0 only two independent functions remain, since now Eq. (13) determines p 2 x in terms of ρ and p x . Hence, in this approximation two functions (q+ and q− ) are suﬃcient to determine the solution of Eqs. (9). With the above results Eqs. (14a), (14b) reduce to

∂ ∂ρ + ( v j ρ ) = 0, ∂t ∂xj m

σ p2 (x) = p 2 x − v 2 (x) = −β 2 (13)

∂ ∂ρ + ( v j ρ ) = 0, ∂t ∂xj

Since these equations do not contain second or higher order moments of p, they decouple from the rest of the hierarchy. Eqs. (16) control the evolution of the particle once the radiationless approximation is taken and the statistical description is reduced to conﬁguration space. The only explicit remnant of the momentum space in these expressions is the term containing ln ρ . As follows from Eq. (13), this contribution comes from the momentum ﬂuctuations, given now by

(16a)

β2 ∂ ∂ ∂ ∂2 (v i ρ ) + m (v i v j ρ ) − ρ ln ρ − f i ρ = 0. ∂t ∂xj m ∂xj ∂ xi ∂ x j (16b)

∂2 ln ρ . ∂ x2

(17)

σ p2 (x) over conﬁguration space we arrive at 2 2 1 ∂ρ . σ p (x) = σ p2 (x)ρ (x) dx = β 2 ρ ∂x Averaging

(18)

This formula (when multiplied by 1/2m) gives an extra contribution to the mean kinetic energy of the particle, due to the ﬂuctuations in momentum space. It is crucial to recognize that the contribution σ p2 (x) at a given point x bears information of what happens everywhere in space due to its dependence on the probability density ρ (x), and thus introduces an apparently nonlocal ingredient into Eq. (16b), as was already observed by Wallstrom [7]. 5. The Schrödinger equation With z = 0 the variables z± = x ± β z reduce both to x so that Eqs. (11) and (7a) give (generalizing to the 3-dimensional case)

ρ (x, t ) = Q˜ (x, 0, t ) = q− (x, t )q+ (x, t ) = ψ ∗ (x, t )ψ(x, t ),

(19)

where we have put q− = ψ ∗ , q+

= ψ , taking into account that ρ (x) is a real non-negative function (a formal justiﬁcation of the result q− = q∗+ appears in the extended version of this Letter). Now we introduce the change of variables

ψ(x, t ) = v(x) =

1 m

√

ρ e i S (x,t ) ,

p x = i

β m

(20a)

∗

∇ ln ψ − ln ψ =

2β m

∇ S,

(20b)

where for the last equation we resorted to Eq. (15) (applied to each component of p). With this change of variables Eq. (16b) gives after integration (we absorb the integration constant in S),

2β

√ ∂ S 2β 2 2β 2 ∇ 2 ρ + (∇ S )2 − + V = 0, √ ∂t m 2m ρ

(21a)

where V is the potential associated to the external force f(x). √ Except for the term proportional to ∇ 2 ρ that accounts for the local ﬂuctuations in momentum space, this is the Hamilton–Jacobi equation of classical mechanics. The entire statistical and nonlocal nature of the description is encapsulated in this term alone, which thus becomes the main source of the apparent nonlocal contributions in the resulting description, as stated above. Further, Eqs. (20) allow us to rewrite the continuity equation (16a) as

m

∂ρ + 2β ∇ 2 S ρ + 2β∇ S · ∇ ρ = 0. ∂t

(21b)

These last two coupled nonlinear equations can be separated and linearized by a standard procedure [6] that leads to

2i β

∂ψ 2β 2 2 =− ∇ ψ + Vψ ∂t m

(22)

and its complex conjugate. To determine the value of the constant β we observe that according to Eq. (18), β 2 is proportional to the intensity of the ﬂuctuations σ p2 (x). As these are generated by the action of the zpf,

L. de la Peña et al. / Physics Letters A 375 (2011) 1720–1723

they depend directly on the intensity of the ﬂuctuations of the latter, which is measured by Planck’s constant h¯ ; hence β (having the dimensions of action) must be of the form

β = ηh¯ ,

(23)

with a numerical universal constant η , since it is deﬁned by a universal change of variables. The simplest way to determine η is by considering the particular case of a harmonic oscillator of frequency ω . Since the material oscillator reaches equilibrium with the modes of the zpf having frequency ω and mean energy h¯ ω/2, this must also be the average energy of the particle. A direct calculation of the mean energy of the oscillator ﬁxes η = 1/2, so that

1

β = h¯ . 2

(24)

This result introduced into Eq. (22) conﬁrms the equivalence of Eqs. (16) to the Schrödinger equation and its complex conjugate. Together with the discussion that follows Eq. (18), this identiﬁcation shows that it is precisely the term containing ln ρ in Eq. (16b) what generates both the quantum ﬂuctuations and the quantum nonlocalities. This term is known in Bohm’s theory as the quantum potential, although it is of kinematic origin [1]. From the present perspective the source of the momentum ﬂuctuations is the zpf, an observation that ascribes a causal gist to such ﬂuctuations. 6. Concluding remarks Our results show that the quantum-mechanical description provided by the Schrödinger equation emerges naturally from a Fokker–Planck-type equation (of inﬁnite order) in phase space, when the presence of the zpf in interaction with matter is recognized. The effects of the ﬂuctuations in momentum space impressed by such ﬁeld are transferred to conﬁguration space, giving rise to an apparent nonlocality. The intrinsic nonlocality of the Schrödinger equation is thus shown to be a manifestation (on the mechanical subsystem) of the stochastic nature of the ﬁeld and a result of reducing the description to the conﬁguration space. These observations — which serve also to disclose the physical cause behind the dispersive nature of the quantum systems — mean that nonlocality is not a property of Nature, in the sense of being inherent to the physical system, but rather a hallmark of the quantum description. The above discussion has been restricted to the one-particle problem. For a problem of two or more particles, instead of Eq. (2) we would have a system of equations coupled through the respective radiation reaction terms [8]. This leads generally to a nonseparable phase-space distribution function, and eventually to a solution ψ that is not factorizable. When this occurs the probability density (in conﬁguration space) contains several terms, including those describing interferences among single-particle states, which generate additional nonlocalities. This phenomenon has become paradigmatic with the extensive studies prompted originally by Bell’s inequalities [9]. A detailed study on the relevance of the zpf for the emergence of entanglement between two (non-interacting) particles can be seen in [10]. It is appropriate to bear in mind that although the zpf could appear as a set of hidden variables introduced with the aim of completing the quantum description, this is not the case, since the zpf is not an ingredient added on top of the quantum-mechanical formalism to make it deterministic. Quite the contrary: nothing is here added to quantum mechanics, but rather quantum mechanics emerges from a more basic theory that embodies the zero-point ﬁeld. The ensuing description is then naturally indeterministic, since in every case the speciﬁc realization of the ﬁeld is unknown,

1723

the description requiring a statistical approach right from the outset. The present results explain the success of some theoretical works [11], and also of the numerical calculations of Cole et al. [12] leading to a correct prediction of the ground state orbit for the H-atom, starting from an otherwise classical atom immersed in the zero-point ﬁeld. The fact that the present treatment departs from a complete phase-space description deserves some comments. In particular, it is important to note that an inverse Fourier transformation of Eq. (6), with Q˜ given by Eq. (11), leads in principle to a true Kolmogorovian phase-space density for the material system. However, as soon as the radiationless approximation is made (and χ is taken as 1) one can no longer guarantee that the ensuing density is a true Kolmogorovian probability. With χ = 1 in Eq. (11) one gets precisely the Wigner function, which explains why this function — and more generally the phase-space probability functions employed in quantum mechanics — are often not true probability densities. Some time ago Wallstrom [7] criticized theories trying to recover the Schrödinger equation in which the probability density ρ is conveniently expressed as a product of two complex amplitudes in the form of Eq. (19) and a condition such as the second one in Eq. (20b) is used. He contended that the procedure can give rise to unwarranted quantized solutions that are foreign to a description in terms of the probability density alone. This argument certainly does not apply to the derivation offered here, where the factorization of the probability density ρ = ψ ∗ ψ is derived from the theory itself (the detailed derivation is presented in the extended version). Furthermore, the fact that the balance condition discussed below Eq. (14b) holds only for a selected (generally discrete) set of motions (see for example Ref. [2]), is a natural source of quantization, so this latter has a deep physical root within the present approach, and cannot be contended to be an artifact of the derivation. Acknowledgements A.V.-H. acknowledges ﬁnancial support from the Consejo Nacional de Ciencia y Tecnología, México. The authors wish to acknowledge the referees for valuable comments, and the Editor for a pertinent observation. References [1] P.R. Holland, The Quantum Theory of Motion, Cambridge Univ. Press, Cambridge, 1993. [2] L. de la Peña, A. Valdés-Hernández, A.M. Cetto, Found. Phys. 39 (2009) 1240; L. de la Peña, A.M. Cetto, in: T.M. Nieuwenhuizen, et al. (Eds.), Beyond the Quantum, World Scientiﬁc, Singapore, 2007. [3] L. de la Peña, A. Valdés-Hernández, A.M. Cetto, Am. J. Phys. 76 (2008) 947. [4] A. Valdés-Hernández, L. de la Peña, A.M. Cetto, in: A. Macías, L. Dagdug (Eds.), New Trends in Statistical Physics: Festschrift in Honor of Leopoldo GarcíaColín’s 80th Birthday, World Scientiﬁc, Singapore, 2010. [5] U. Frisch, in: A.T. Bharucha-Reid (Ed.), Probabilistic Methods in Applied Mathematics, vol. I, Academic Press, New York, 1968. [6] L. de la Peña, A.M. Cetto, J. Math. Phys. 18 (1977) 1612. [7] T.C. Wallstrom, Phys. Rev. A 49 (1994) 1613. [8] L. Landau, E. Lifshitz, The Classical Theory of Fields, Addison–Wesley, Cambridge, MA, 1951. [9] J.S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge Univ. Press, UK, 1987. [10] A. Valdés-Hernández, L. de la Peña, A.M. Cetto, Bipartite entanglement induced by a common background (zero-point) radiation ﬁeld, Found. Phys. (2011), DOI: 10.1007/s10701-010-9527-y. [11] H.M. França, H. Franco, C.P. Malta, Eur. J. Phys. 18 (1997) 343. [12] D.C. Cole, Y. Zou, Phys. Lett. A 317 (1–2) (2003) 14, quant-ph/0307154; D.C. Cole, Y. Zou, Phys. Rev. E 69 (2004) 016601; D.C. Cole, Y. Zou, J. Sci. Comput. 20 (2004) 43; D.C. Cole, Y. Zou, J. Sci. Comput. 20 (2004) 379; D.C. Cole, Y. Zou, J. Sci. Comput. 21 (2004) 145.

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