Author’s postprint: International Journal of Theoretical Physics, 47, 1195-1205 (2008)

Winterberg’s conjectured breaking of the superluminal quantum correlations over large distances

Eleftherios Gkioulekas∗

Department of Mathematics, University of Central Florida, Orlando, FL, United States

We elaborate further on a hypothesis by Winterberg that turbulent fluctuations of the zero point field may lead to a breakdown of the superluminal quantum correlations over very large distances. A phenomenological model that was proposed by Winterberg to estimate the transition scale of the conjectured breakdown, does not lead to a distance that is large enough to be agreeable with recent experiments. We consider, but rule out, the possibility of a steeper slope in the energy spectrum of the turbulent fluctuations, due to compressibility, as a possible mechanism that may lead to an increased lower-bound for the transition scale. Instead, we argue that Winterberg overestimated the intensity of the ZPF turbulent fluctuations. We calculate a very generous corrected lower bound for the transition distance which is consistent with current experiments.

I. INTRODUCTION of light, but nonetheless finite. If it is true that quantum correlations break down beyond a certain “transition” length scale, then we are lead to the A compelling paradox in our current understanding of na- following implications: First, when we switch into a differ- ture is the fundamental inconsistency between the non-locality ent inertial reference frame, the transition length scale should of quantum mechanics and the Lorentz invariance demanded be expected to contract over certain directions in accordance by the theory of relativity. This inconsistency can be con- with the Lorentz transformation. Consequently, it becomes cealed to a large extent because it is possible to formulate rel- possible to establish a unique reference frame “at rest”, as the ativistic quantum theories that predict Lorentz-invariant sta- reference frame where the transition length scale is constant tistical behaviour. Thus, from a strictly empiricist standpoint, in all directions. This then, makes the idea of the original all appears to be well since non-locality cannot be exploited pre-Einstein theory of relativity by Lorentz and Poincare com- to transmit information. The inconsistency becomes more ob- pelling once again [21]. Furthermore, if the propagation speed vious when one attempts to formulate a Bohmian interpreta- of the superluminal interactions responsible for the collapse of tion of quantum mechanics [1–6]. Then it becomes necessary the wavefunction is indeed finite, that would then indicate the to introduce the quantum potential interactions which violate existence of a medium through which these interactions are Lorentz invariance. There is also the problem of making the transmitted. ρ |ψ|2 equilibrium condition = Lorentz invariant [7]. Though Winterberg proposed the hypothesis that this medium is the resolving this latter problem is possible for the case of non- field of zero-point vacuum energy, also known as the ZPF field interacting but entangled particles [8], the non-local nature of [20, 22]. The physical reality of the ZPF field has been estab- is the quantum potential interaction unavoidable by any in- lished experimentally by the Casimir effect [23]. However terpretation that agrees with experiment [9]. Even when one the underlying physical principles that govern the ZPF field decides to avoid the problem of interpretation altogether (by at the Planck scale are poorly understood. According to Win- ignoring it) one still has to contend with the presumably in- terberg, the simplest possible model for the ZPF field is as a stantaneous entanglement between distant quanta in situations “Planck-mass plasma” of two coupled superfluids, one with such as the well-known EPR two-photon experiment [10–13]. positive mass particles and one with negative mass particles The original Aspect experiment [14] confirmed the exis- [24–28]. Under small perturbations, the Winterberg Planck- tence of quantum entanglement over a range as large as 10 mass plasma is analogous to a compressible fluid with p ∝ ρ2 meters by confirming the violation of Bell’s inequality. More [24]. Winterberg [22] conjectured that over large length scales recent experiments have extended the range, initially to 4 km this medium undergoes turbulent fluctuations which disrupt [15], and subsequently to 11 km [16–18] and even 50 km [19]. the quantum correlations at length scales where the turbulence Winterberg [20] observed that we cannot presume, on the ba- energy spectrum exceeds the ZPF energy spectrum. sis of these experiments, that quantum correlations will indef- Although the physics of the ZPF field at the Planck scale is initely continue to hold over larger distances. For example, currently unknown to us, if it is assumed that the ZPF field is it would be a bold extrapolation from a 11 km experiment to Lorentz invariant, then it must follow the Boyer energy spec- presume that quantum correlations persist over interstellar and trum intergalactic distances. He also noted that the notion that the collapse of the wavefunction occurs instantaneously is proba- E(k) = ~ck3, (1) bly an unjustified idealization. It may be more reasonable to expect that quantum correlations propagate at a finite superlu- which can be shown [29, 30] to be the only Lorenz invari- minal speed, which may be significantly larger than the speed ant spectrum as long as it is infinitely extended to arbitrarily large wavenumbers k. More realistically, k should be cut off at wavenumbers corresponding to the Planck length scale rp, in which case Lorentz invariance would be broken only when ∗Electronic address: [email protected] the Planck length scale is approached. The cutoff wavenum- 2 ber defines a length, which in turn can define a rest reference II. THE SPECTRUM OF ZPF TURBULENCE frame as the one where the cut-off length is isotropic. The truncated spectrum The underlying assumption of Kolmogorov’s theory of tur- ( bulence [33–35] is that energy is injected into the system at ~ck3, if k ≤ 2π/r E(k) = p (2) small wavenumbers by random forcing, which is then cas- 0, if k > 2π/rp caded to larger wavenumbers where it is being dissipated. Be- tween the forcing range of wavenumbers where energy comes is still Lorentz invariant in terms of slope and numerical co- in, and the dissipation range of wavenumbers were energy efficient [31]. However, the cutoff scale contracts under a comes out, it is assumed that there is a so-called “inertial Lorentz transformation like any other length. It remains a range” of wavenumbers through which the energy is trans- point of controversy whether or not the actual Planck length ferred to small scales via nonlinear interactions. This theory scale is the same or different for different observers. Accord- has been confirmed both experimentally [37, 38], and repeat- ing to the recently proposed “double special relativity theory” edly with numerical simulations [39]. There has also been [32] (DSR), the Planck length scale should be the same for all considerable progress towards formulating a statistical theory observers. Under DSR, Eq. (2) would also be invariant for of the energy cascade [40–42]. On the other hand, it is im- all inertial frames of reference. However, that would still not portant to note that the theory has only been investigated ex- necessarily apply to the quantum correlations breakdown tran- tensively in the context of incompressible turbulence, and it is sition scale, which would still define a rest reference frame. not obvious that it is applicable for compressible turbulence. Knowing the energy spectrum of the ZPF field makes it pos- For the simplest case of adiabatic compressible turbulence sible to estimate the transition length scale λ0 where quantum where the pressure is dependent only on the density via the correlations break down by finding the transition wavenum- relation p ∝ ργ with γ the adiabatic constant, the Moisseev- ber k0 where the Boyer energy spectrum is comparable with Shivamoggi prediction [43, 44] is that the energy spectrum of the energy spectrum of the ZPF turbulent fluctuations. The the energy cascade is steeper and has slope k−a with a given salient feature of this approach is that it is reliant only on by our assumptions concerning energy spectra, and does not in- volve further speculative modelling assumptions. Winterberg 5γ − 1 a = . (5) [22] used the Kolmogorov-Batchelor [33–35] energy spec- 3γ − 1 trum E(k) ∝ k−5/3 and showed that The steeper spectral slope arises because of a distributed −3/14 λ0 = 12κ m, (3) energy dissipation throughout the inertial range by acoustic waves. The Moisseev-Shivamoggi spectrum reads: where κ represents the degree of turbulent fluctuations and ranges as 0 < κ < 1. For κ = 1, we get the lower-bound E(k) = C[ργ−1ε2γc−2k−(5γ−1)]1/(3γ−1) . (6) length scale of 12m which is too small to be consistent with the experiments of Gisin [16–19]. Winterberg [22] proposed Here, C is a universal constant, ρ the density, ε the rate of en- −5 that one should use instead the evaluation κ = 10 , which ergy injection per unit time and volume, and k the wavenum- is equal to the spatial temperature variation observed in the ber. The spectrum approaches the Kolmogorov spectrum cosmic microwave radiation. This choice was thought to be plausible because “the microwaves are refracted by the fluc- E(k) = C ρ1/3ε2/3k−5/3, (7) tuating gravitational field of the eddies” [36]. Nonetheless, a simple calculation shows that, contrary to Winterberg’s claim, when γ → ∞ and the Kadomtsev-Petviashvili spectrum this only gives us one more order of magnitude: E(k) = Cεc−1k−2, (8) −5 −3/14 λ0 = 12(10 ) m ≈ 120m. (4) when γ = 1 [45]. Note that arbitrarily steeper slopes are pos- This is still too small to agree with experiments. sible, in principle, if 1/3 < γ < 1, but a spectrum steeper than In the present paper we will consider the ramifications of k−3 would violate the locality of the nonlinear interactions that weakening the assumption that the ZPF turbulent fluctuations drive the energy cascade, and wouldn’t really represent phys- are governed by the Kolmogorov scaling k−5/3. The idea is ically a state of fully developed turbulence. Considering that that a steeper slope will give a larger value for the transition for small fluctuations the Winterberg Planck-mass plasma be- length scale. We will show that even with steeper slopes we haves as a compressible fluid with γ = 2 [24], the energy still do not get a transition scale larger than 10 km. Instead, spectrum of compressible turbulence may be more relevant to we will argue that the choice κ = 10−5 is too large because it the problem at hand. corresponds to an unrealistically large energy dissipation rate. Let us now generalize Winterberg’s calculation of the en- The paper is organized as follows. The energy spectrum of ergy spectrum of ZPF turbulence for the general slope a. As- the turbulent fluctuations is discussed in section 2. We recal- sume that the energy spectrum of the turbulent fluctuations culate the transition length scale in section 3. The problem of reads guessing the parameter κ is discussed in section 4. Conclu- −a sions are given briefly in section 5. Eturb(k) = Ak , (9) 3 where A is constant and a the scaling exponent. To estimate Here we have disregarded the numerical coefficients. We see the constant A, we follow Winterberg [22], and we assume that we essentially recover Winterberg’s spectrum for the case that the total energy of the turbulent fluctuations is κ = 1. However, if instead of ρc2 we use κ ρc2 for the universe

2 energy density, we do not obtain exactly the same dependence E = κ ρc , (10) on κ as we would from the generalization of Winterberg’s ar- where ρ is the average density of the universe and κ is a con- gument. So there is some ambiguity on how one should define stant 0 < κ < 1 that measures the degree of turbulence. We the degree κ of the turbulent fluctuations. estimate the density ρ with the critical density ρcrit which sep- arates the open and closed universe scenaria: 3H2 III. ESTIMATING THE TRANSITION SCALE ρ ≈ ρ = . (11) crit 8πG Here H is the Hubble constant and G Newton’s gravity con- We will now show that a steeper slope for the energy spec- stant. We also assume that the largest eddies have length scale trum of the turbulent fluctuations gives a larger transition of the order of the world radius R ∼ c/H. It follows from length scale. However, we will see that there is no case that these considerations that the total energy E also satisfies gives a large enough length scale to be consistent with Gisin’s experiments [16–19], unless, a smaller value of κ is chosen. Z ∞ A 1 E = Ak−a dk = − , (12) The energy spectrum of the zero-point vacuum energy is 1−a 1/R 1 − a R given by and combining (10) and (12) we find A: 3 Evac(k) = ~ck . (20) A = (a − 1)κ ρc2 R1−a. (13) The transition wavenumber k0 where we may expect quan- A related question, which was not addressed by Winterberg, tum correlations to break can be estinated, in the sense of an is how the energy is injected and dissipated for the turbulent upper-bound, by matching the two energy spectra Evac(k) and fluctuations of the field. On a cosmological scale there is an E (k) = Ak−a, which yields amount of energy which is injected into the universe due to the turb extension of the event horizon. This rate of energy injection ! 1 represents the total available energy and it is, of course, an A 3+a k ≈ ≡ k . (21) upper bound to the actual rate of energy injection into the ZPF ~c 0 field’s turbulent fluctuations, which can be very much smaller. Following this idea, it is possible to arrive to an alternative Here we deviate slightly from Winterberg [22] who compared derivation of the energy spectrum of the turbulent fluctuations the cumulative spectra instead. The advantage of this ap- of the ZPF field. proach is that we don’t need to make the assumption that these 2 If R is the radius of the event horizon, and ρc the av- power laws hold at all scales. Instead, it is sufficient that they erage energy density of the universe, then the rate with hold around the intersection wavenumber. which the horizon expansion increases the total energy is Using the evaluation A = (a − 1)κ ρc2 R1−a the transition ρc2(4πR2)(dR/dt). To get the rate of energy injection ε we wavenumber k reads: must divide this quantity with the volume of the universe. 0 Thus, since the radius R expands with the speed of light (i.e., ! 1 dR/dt = c), it follows that (a − 1)κ ρc2 R1−a 3+a k = (22) 0 ~c 4πR2dR 1 ε = ρc2 (14) ! 1 dt (4/3)πR3 κc 3H2  c 1−a 3+a = (a − 1) (23) = 3ρc2 R−1c = 3ρc3 R−1. (15) " ~ 8πG H # ! a 1 3(a − 1)κc2 H H 3+a Substituting ε to the Moisseev-Shivamoggi spectrum, and us- = . (24) ing the speed of light for c, yields: " 8πG~ c #

γ−1 3 −1 2γ −2 −(5γ−1) 1/(3γ−1) E(k) = C[ρ (3ρc R ) c k ] (16) This result generalizes equation (10) in Winterberg [22]. The = C[ρ3γ−132γ R−2γc6γ−2k−(5γ−1)]1/(3γ−1) (17) corresponding cross-over length scale reads ≈ ρc2 R−2γ/(3γ−1) k−(5γ−1)/(3γ−1) . (18) 1 2π 8πG~  c  a 3+a It is easy to see that solving a = (5γ − 1)/(3γ − 1) for γ gives λ = = 2π . (25) 0 k − 2 H γ = (1−a)/(5−3a) and that it follows that 2γ/(3γ−1) = a−1. 0 " 3(a 1)κc H # Thus, we can write E(k) in terms of a as follows: On a decimal logarithmic scale, the order of magnitude of λ0 E(k) ≈ ρc2 R1−a k−a. (19) is 4

! 1 3c2 H H log λ0 = log(2π) − log + log κ + log(a − 1) + a log . (26) 3 + a " 8πG~ c #

Let σ~, σG, σc, and σH be the measurement errors of the corresponding constants ~, G, c, and H, and assume for simplicity that these errors are statistically uncorrelated. The propagated statistical uncertainty is given by:

2 2 2 2 2 ∂λ0 ∂λ0 ∂λ0 ∂λ0 σ = σG + σc + σ~ + σH (27) λ0 " ∂G # " ∂c # " ∂~ # " ∂H # 1 λ σ 2 a − 2 λ σ 2 1 λ σ 2 1 λ σ 2 = 0 G + 0 c + 0 ~ + 0 H . (28) " 3 + a G # " 3 + a c # " 3 + a ~ # " 3 + a H #

Consequently, in terms of the relative standard uncertainties ec = σc/c, eG = σG/G, e~ = σ~/~, eH = σH /H, and eκ = σκ/κ we obtain

!2 2 2 2 2 2 2 2 σλ e + (a − 2) ec + e + (a + 1) e + eκ 0 = G ~ H . (29) 2 λ0 (3 + a)

The 2006 CODATA recommended values for the universal H is eH = 0.15, which is by far the dominant contribution to −11 constants are: c = 2997924583 m/s, G = 6.67428 × 10 N · σλ0 . We may thus estimate σλ0 practically via: m2kg−2, ~ = 1.054571628 × 10−34 J · s. The correspond- −4 ! ing relative standard uncertainties are: ec = 0, eG = 10 , σλ0 (a + 1)eH −5 ≈ . (30) e~ = 5 · 10 . Furthermore, the recent measurement [46] λ0 (a + 3) of the Hubble constant by NASA’s Chandra X-ray Observa- tory gives H = 77 km s−1Mpc−1 (with 1Mpc = 3.26 · 106 From these evaluations we also obtain the following empirical −18 −1 lightyears, we get H = 2.49 · 10 s ). The uncertainty of formula for the transition wavelength λ0

98.045658 − 60.049519a − log κ − log(a − 1) log λ = log(2π) − . (31) 0 a + 3

The relevant slopes a are: Kolmogorov scaling with intermit- Here we have assumed that the Kolmogorov constant is unity, tency corrections (i.e. a = 1.7), compressible scaling with since we shall be doing only order of magnitude calculations. γ = 2 (i.e. a = 1.8). It is also worthwhile to consider the ex- Solving for ε we obtain treme case a = 2 which corresponds to the shock-dominated / γ / γ Kadomtsev-Petviashvili spectrum. For κ = 1, we get the fol- A3γ−1 1 2 [(a − 1)κ ρc2 R1−a]3γ−1 1 2 ε = = (33) lowing transition scales: a = 1.7 gives λ0 = (16 ± 1)m, a = " ργ−1c−2 # " ργ−1c−2 # 1.8 gives λ = (53 ± 4)m, and a = 2 gives λ = (517 ± 46)m. 0 0 a − κR1−a 3γ−1 ρ2γc6γ 1/2γ The transition scales are increased for κ = 10−5 (Winterberg’s = [[( 1) ] ] (34) 3 1−a (3γ−1)/(2γ) choice) as follows: a = 1.7 gives λ0 = (185 ± 15)m, a = 1.8 = ρc [(a − 1)κR ] . (35) gives λ0 = (587±51)m, and a = 2 gives λ0 = (5172±465)m. In all cases the transition scale is less than 11km. This sug- This expression can be simplified further because the identity gests that Winterberg’s choice for κ is probably too large. 2γ/(3γ − 1) = a − 1 implies that (1 − a)(3γ − 1) = −1. (36) IV. WHAT κ IS REASONABLE? 2γ Thus the energy dissipation rate can be rewritten as To get a good sense of what κ is reasonable, we shall cal- culate the energy dissipation rate ε in terms of κ. From the ε = ρc3 R−1[(a − 1)κ]1/(a−1) . (37) Moisseev-Shivamoggi spectrum we see that A reads: To get some physical understanding, compare the amount A = [ργ−1ε2γc−2]1/(3γ−1) . (32) of energy dissipated over the entire universe under the rate ε 5 during time t, with the energy that would be released by the underestimates for any reasonable evaluation of the transition annihilation of N solar masses M. Then, N Mc2 = εR3t, and scale and they are both agreeable with Gisin’s experiments it follows that [16–19]. The actual transition scale probably exceeds these by some orders of magnitude and could be as large as, for εR3t ρc3 R−1[(a − 1)κ]1/(a−1) R3t N = = (38) example, the Earth-Moon distance. Mc2 Mc2 2 ρcR t 1/(a−1) 1/(a−1) = [(a − 1)κ] = N0[(a − 1)κ] , (39) M V. CONCLUSION AND DISCUSSION

2 where N0 = ρcR t/M is a dimensionless constant. The re- verse relation between κ and N is: In the present paper I have shown that the model of the ZPF turbulent fluctuations proposed by Winterberg [22] to account ! a−1 1 N for the conjectured breakdown of the quantum correlations is κ = . (40) a − 1 N plausible in the sense that it does not disagree with current 0 experiments. However, it is still unsatisfactory to not have a Using M = 1.98 · 1030kg (the mass of our Sun) and t = 1day, clear understanding of where the turbulence gets the energy, 57 −5 how the energy gets dissipated, and where it ends up after it is yields N0 = 10 . The value κ = 10 that was proposed by Winterberg gives N ≈ 1049 for a = 1.7. To get some sense of dissipated. This aspect of the model requires further elabora- what this means, consider the rescaled count tion. The strange behavior of quantum-mechanical systems that !3 !3 ` `H involves entanglement over large distances is a very tantaliz- N = N = N , (41) s R c ing mystery. It has led most physicists to the very extreme position of denying the existence of an objective reality un- with ` = 8 lightminutes, which is approximately the Earth- derlying quantum mechanics (the Copenhagen interpretation). 5 Sun distance. We find Ns ≈ 10 . This means that every This prompted Einstein to comment in a memorable way on day, within the neighborhood of Earth’s planetary orbit, the the non-existence of God’s gambling addiction. A more mild amount of the energy dissipated should be equal to the energy position along the same lines, which is still nonetheless a par- that would be released by the complete annihilation of 105 so- tial denial of objective reality, is the “relational interpretation” 6 lar masses. For a = 1.8 this count increases to Ns = 10 [47–49]. The idea here is to deny only as much of objective solar masses per day. Such an extraordinary amount of en- reality as is necessary to make the existing problems go away. ergy should have been somewhat conspicuous to all life on Part of the mystery is that we don’t really understand what Earth. This is why I believe that κ should be chosen to be the wavefunction really is. Winterberg [20] assumes that the much smaller. wavefunction is a genuine physical field that really collapses. As we have mentioned previously, turbulence requires both In the same paper he also reviews the early literature on the a mechanism for injecting energy into the system, and a mech- subject. It is hard to accept this viewpoint and not expect the anism for dissipating the energy at large wavenumbers. The collapse to propagate at a finite speed, or to not be disrupted by question of finding a reasonable choice for the variable κ, is possible noise in the mechanism that propagates it. From the equivalent to the question of deciding on an upper bound for standpoint of the Bohmian interpretation, the wavefunction of the dissipation rate of the turbulent fluctuations of the ZPF the combined physical system and the measuring apparatus field. Naturally, the underlying problem is understanding the never really collapses. Nonetheless, even in the Bohmian in- physical mechanism responsible for dissipating the energy terpretation, one models (instead of deriving from first princi- cascade of ZPF turbulence in the first place. ples) the Hamiltonian governing the interaction between sys- Rather than speculate on the particulars of the dissipation tem and apparatus during measurement. Furthermore, one can mechanism, we can simply assume that the rate of energy out- expect a breakdown in quantum correlations if there is a small put from the Sun is an extremely generous upper bound for amount of noise, presumably from subquantum processes, in the rate of energy dissipation over a spherical volume around the guidance condition that determines the particle velocities the Sun that reaches the Earth. Since the Sun has an overall from the wave function. lifetime of approximately 10 billion years (which is approx- There is also some controversy over the experiments that imately 1012 days), we can further overestimate the rate of have convinced us of the reality of quantum correlations in the −12 energy output by choosing Ns = 10 solar masses per day. first place. It is believed by some that these experiments are The corresponding κ is a dependent, and it is given by susceptible to certain “loopholes” (the locality loophole and the detection loophole) that prevent them from being conclu- ! a−1 1 N  c 3 sive [50–53]. There has been some interest in developing ex- κ = s . (42) a − 1 N0 `H periments that tried to close the loopholes [54–60]. Nonethe- less, the controversy continues [61]. It may turn out that, over For a = 1.7 , this gives κ = 9 × 10−18 , from which we get for increasing distances, these experiments begin to fail gradu- the transition scale λ0 = 67 ± 5 km. The situation is improved ally. The “loopholes” could then represent modes of failure if we choose a = 1.8 . Then we get κ = 2 × 10−20 which gives that are “irrelevant” at short distances but become increasingly λ0 = 630 ± 55 km. These numbers are extremely generous relevant over larger distances. Even if the particulars of the 6 model proposed by Winterberg turn out to be wrong, this un- derlying issue of understanding the possible role of distance with respect to entanglement remains.

Acknowledgments

The author would like to thank Dr. Friedwardt Winterberg for his e-mail correspondance.

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