# Quantum Non-Locality and the CMB: What Experiments Say

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## Abstract

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## 1. Introduction

## 2. Vacuum State and Its Lorentz Invariance

## 3. The Basics of the Ether Drift Experiments

#### 3.1. Which Preferred Frame?

#### 3.2. The Old Experiments in Gaseous Media

#### 3.3. The Modern Experiments in Vacuum and Solid Dielectrics

## 4. Summary and Outlook

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**For ${\u03f5}_{v}$ as in Equation (59) and $\chi =2$, we report in units ${10}^{-15}$ two typical sets of 45 s for the two functions $2C\left(t\right)$ and $2S\left(t\right)$ of Equation (A2). The two sets belong to the same random sequence and refer to two sidereal times that differ by 6 h. The boundaries of the stochastic velocity components Equations (A11) and (A12) are controlled by ${(V,\alpha ,\gamma )}_{\mathrm{CMB}}$ through Equations (A5) and (A15). For a laser frequency of $2.8\xb7{10}^{14}$ Hz, the range $\pm 3.5\xb7{10}^{-15}$ corresponds to a typical frequency shift $\Delta \nu $ in the range $\pm 1$ Hz, as in our Figure 5.

## Notes

1 | According to Weinberg, “It is a bad sign that those physicists today who are most comfortable with quantum mechanics do not agree with one another about what it all means” [1]; or, according to Blanchard, Frohlich and Schubnel, “Given that quantum mechanics was discovered ninety years ago, the present rather low level of understanding of its deeper meaning may be seen to represent some kind of intellectual scandal” [2]. |

2 | “The impossibility of sending messages is sometimes taken to mean that there is nothing non-local going on. But non-locality refers here to causal interactions as described (in principle) by physical theories. Messages are far more anthropocentric than that, and require that humans be able to control these interactions in order to communicate. As remarked by Maudlin [8], the Big Bang and earthquakes cannot be used to send messages, but they have causal effects nevertheless” [7]. |

3 | The system where the CMB Kinematic Dipole [18] vanishes describes a motion of the solar system with average velocity ${V}_{\mathrm{CMB}}\sim 370$ km/s, right ascension ${\alpha}_{\mathrm{CMB}}\sim {168}^{o}$ and declination ${\gamma}_{\mathrm{CMB}}\sim -{7}^{o}$, approximately pointing toward the constellation Leo. |

4 | This reductive interpretation of Bell’s work is contested by Bricmont [25]. Spelling out precisely the meaning of Bell’s theorem, he is very explicit on this point: “Bell’s result, combined with the EPR argument, is rather that there are non-local physical effects (and not just correlations between distant events) in Nature”. |

5 | |

6 | In connection with the idea of ether, it should be better underlined that Einstein’s original point of view had been later reconsidered with the transition from Special Relativity to General Relativity [28]. Most probably, he realized that Riemannian geometry is also the natural framework to describe the dynamics of elastic media; see, e.g., [29]. |

7 | After these arguments, Chiao immediately adds the usual remark about the impossibility of information propagating at superluminal speed: “Relativistic causality forbids only the front velocity, i.e., the velocity of discontinuities, which connects causes to their effects, from exceeding the speed of light, but does not forbid a wave packet group velocity from being superluminal” [30]. |

8 | The presence of the cubic $g\langle \mathsf{\Phi}\rangle {h}^{3}$ interaction should not be overlooked. In fact, in the infrared region, it induces a strong coupling between bare ${b}^{\u2020}|{\mathsf{\Psi}}_{+}\rangle $ and ${b}^{\u2020}{b}^{\u2020}|{\mathsf{\Psi}}_{+}\rangle $ components in the Fock space of the broken-symmetry phase [50]. The net result is that, in the $\mathbf{k}\to 0$ limit, the effective 1-(quasi)particle spectrum deviates sizeably from the spectrum of the bare ${b}^{\u2020}|{\mathsf{\Psi}}_{+}\rangle $ states. |

9 | One may object that this conflict is merely a consequence of describing SSB as a (weak) first-order phase transition. Apparently, in the standard second-order picture, where no meaningful quantization in the symmetric phase is possible, there would be no such problem. However, this is illusory. In fact, as previously recalled, the same (weak) first-order scenario is found, within the conventional loop expansion [37], when studying SSB in the more realistic case of complex scalar fields interacting with gauge bosons. This first-order scenario is at the base of ’t Hooft’s description [26] of the physical vacuum as a Bose condensate of real, physical Higgs quanta (not of tachions with imaginary mass). Thus, the problem goes beyond the simplest ${\mathsf{\Phi}}^{4}$ model considered here and reflects the general constraint imposed by SSB on the renormalized mass parameter of the scalar fields in the symmetric phase. |

10 | This contrasts with the approach based on an energy-momentum tensor of the form [51,52]
$$\langle {\mathsf{\Psi}}_{\pm}^{(\Sigma )}|{W}_{\mu \nu}|{\mathsf{\Psi}}_{\pm}^{(\Sigma )}\rangle ={\rho}_{v}\text{}{\eta}_{\mu \nu}$$
${\rho}_{v}$ being a space–time-independent constant. In fact, from
$$\langle {\mathsf{\Psi}}_{\pm}^{\prime}|{W}_{\mu \nu}|{\mathsf{\Psi}}_{\pm}^{\prime}\rangle ={\mathsf{\Lambda}}^{\sigma}{}_{\mu}{\mathsf{\Lambda}}^{\rho}{}_{\nu}\text{}\langle {\mathsf{\Psi}}_{\pm}^{(\Sigma )}|{W}_{\sigma \rho}|{\mathsf{\Psi}}_{\pm}^{(\Sigma )}\rangle $$
$${L}_{0i}=-\int {d}^{3}x\text{}({x}_{i}{W}_{00}-{x}_{0}{W}_{0i})$$
If $|{\mathsf{\Psi}}_{\pm}^{(\Sigma )}\rangle $ is an eigenstate of the Hamiltonian, then an eigenvalue ${E}_{0}=0$ is needed to obtain ${L}_{0i}|{\mathsf{\Psi}}_{\pm}^{(\Sigma )}\rangle =0$. Instead, Equation (40) amounts to $\langle {\mathsf{\Psi}}_{\pm}^{(\Sigma )}\left|{L}_{0i}\right|{\mathsf{\Psi}}_{\pm}^{(\Sigma )}\rangle =0$. Thus, it is no surprise that one can run into contradictory statements. |

11 | By extending the Poincaré algebra, a remarkable case, which fulfills the zero-energy condition exactly, is that of an unbroken supersymmetric theory. This is because the Hamiltonian $H\sim {\overline{Q}}^{\alpha}{Q}^{\alpha}$ is bilinear in the supersymmetry generators ${Q}^{\alpha}$. Therefore, an exact supersymmetric state, for which ${Q}^{\alpha}|\mathsf{\Psi}\rangle =0$, has automatically zero energy. At present, however, an unbroken supersymmetry is not phenomenologically acceptable. |

12 | |

13 | A conceptual detail concerns the relation of the gas refractive index $\mathcal{N}$, as introduced in Equation (1), to the experimental quantity ${\mathcal{N}}_{\mathrm{exp}}$, which is extracted from measurements of the two-way velocity in the Earth laboratory. By assuming a $\theta -$dependent refractive index as in Equation (48), one should thus define ${\mathcal{N}}_{\mathrm{exp}}$ by an angular average, i.e., $\frac{c}{{\mathcal{N}}_{\mathrm{exp}}}\equiv {\langle \frac{c}{\overline{\mathcal{N}}\left(\theta \right)}\rangle}_{\theta}=\frac{c}{\mathcal{N}}\text{}\left[1-\frac{3}{2}(\mathcal{N}-1){\beta}^{2}\right]$. One can then determine the unknown value $\mathcal{N}\equiv \mathcal{N}(\Sigma )$ (as if the container of the gas were at rest in $\Sigma $), in terms of the experimentally known quantity ${\mathcal{N}}_{\mathrm{exp}}\equiv \mathcal{N}\left(\mathrm{Earth}\right)$ and of v. As discussed in refs. [32,33,34,35], for $v\sim 370$ km/s, the difference in the two quantities is well below the experimental accuracy and, for all practical purposes, can be neglected. |

14 | The idea of the physical vacuum as an underlying stochastic medium, similar to a turbulent fluid, is deeply rooted in basic foundational aspects of both quantum physics and relativity. For instance, at the end of the XIX century, the last model of the ether was a fluid full of very small whirlpools (a “vortex-sponge”) [72]. The hydrodynamics of this medium was accounting for Maxwell’s equations and thus providing a model of Lorentz symmetry as emerging from a system whose elementary constituents are governed by Newtonian dynamics. More recently, the turbulent ether model has been re-formulated by Troshkin [73] (see also [74,75]) within the Navier–Stokes equation, by Saul [76] by starting from Boltzmann’s transport equation and in [77] within Landau’s hydrodynamics. The same picture of the vacuum (or ether) as a turbulent fluid was Nelson’s [78] starting point. In particular, the zero-viscosity limit gave him the motivation to expect that “the Brownian motion in the ether will not be smooth” and, therefore, to conceive the particular form of kinematics at the base of his stochastic derivation of the Schrodinger equation. A qualitatively similar picture is also obtained by representing relativistic particle propagation from the superposition, at short time scales, of non-relativistic particle paths with different Newtonian mass [79]. In this formulation, particles randomly propagate (as in a Brownian motion) in an underlying granular medium, which replaces the trivial empty vacuum [80]. |

15 | In Ref. [34], a numerical simulation of the Piccard–Stahel experiment [63] is reported, for both the individual sets of 10 rotations of the interferometer and the experimental sessions (12 sets, each set consisting of 10 rotations). Our analysis confirms their idea that the optical path was much shorter than the instruments in the United States but their measurements were more precise because spurious disturbances were less important. |

16 | Joos’ optical system was enclosed in a hermetic housing and, as reported by Miller [59,88], measurements were performed in a partial vacuum. In his article, however, Joos is not clear on this particular aspect. Only when describing his device for electromagnetic fine movements of the mirrors, he refers to the condition of an evacuated apparatus [67]. Instead, Swenson [89,90] declares that Joos’ fringe shifts were finally recorded with optical paths placed in a helium bath. Therefore, we have followed Swenson’s explicit statements and assumed the presence of gaseous helium at atmospheric pressure. |

17 | Interestingly, a century after these old experiments, in a room-temperature environment, the fraction of millikelvin is still state of the art when measuring temperature differences; see [93,94,95]. This supports our idea that $\Delta {T}^{\mathrm{gas}}\left(\theta \right)$ is a non-local effect, which places a fundamental limit. |

18 | The RAV gives the variation of a function $f=f\left(t\right)$ sampled over steps of time $\tau $. By defining
$$\overline{f}({t}_{i};\tau )=\frac{1}{\tau}{\int}_{{t}_{i}}^{{t}_{i}+\tau}dt\text{}f\left(t\right)\equiv {\overline{f}}_{i}$$
$${\sigma}_{A}(f,\tau )=\sqrt{{\sigma}_{A}^{2}(f,\tau )}$$
$${\sigma}_{A}^{2}(f,\tau )=\frac{1}{2(M-1)}\sum _{i=1}^{M-1}{\left({\overline{f}}_{i}-{\overline{f}}_{i+1}\right)}^{2}$$
The integration time $\tau $ is given in seconds and the factor of 2 is introduced to obtain the same standard variance for uncorrelated data as for a white-noise signal with uniform spectral amplitude at all frequencies. |

19 | Numerical simulations indicate that our vacuum signal has the same characteristics of universal white noise. Thus, strictly speaking, it should be compared with the frequency shift of two optical resonators at the largest integration time ${\tau}_{0}$, where the pure white-noise branch is as small as possible but other types of noise are not yet important. In the experiments, we are presently considering this ${\tau}_{0}$ as typically 1 s. However, in principle, ${\tau}_{0}$ could also be considerably larger than 1 s, as, for instance, in the cryogenic experiment of Ref. [97]. There, the RAV at 1 s was around 10 times larger than the range of Equation (63) but, in the quiet phases between two refills of the refrigerator, ${\sigma}_{A}(\Delta \nu /{\nu}_{0},\tau )$ was monotonically following the white-noise trend ${\tau}^{-1/2}$ up to ${\tau}_{0}\sim 240$ s, where it reached its minimum value ${\sigma}_{A}(\Delta \nu /{\nu}_{0},{\tau}_{0})\sim 5.3\xb7{10}^{-16}$. Remarkably, for $\chi =2$, this is still consistent with the theoretical range of Equation (63). |

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**Figure 2.**A schematic illustration of the Michelson interferometer. Note that, by computing the transit times and the resulting fringe shifts via Equation (50), we are assuming the validity of Lorentz transformations so that the length of a rod does not depend on its orientation, in the frame ${S}^{\prime}$, where it is at rest.

**Figure 4.**The scheme of a modern ether drift experiment. The light frequencies are first stabilized by coupling the lasers to Fabry–Perot optical resonators. The frequencies ${\nu}_{1}$ and ${\nu}_{2}$ of the resonators are then compared in the beat note detector, which provides the frequency shift $\Delta \nu \left(\theta \right)={\nu}_{1}(\pi /2+\theta )-{\nu}_{2}\left(\theta \right)$. For a review, see, e.g., [96].

**Figure 5.**The experimental frequency shift reported in Figure 9a of Ref. [98] (courtesy Optics Communications). The black dots give the instantaneous signal, the red dots give the signal averaged over 1640 sequences. For a laser frequency ${\nu}_{0}=2.8\xb7{10}^{14}$ Hz, a $\Delta \nu =\pm 1$ Hz corresponds to a fractional value $\Delta \nu /{\nu}_{0}$ of approximately $\pm 3.5\xb7{10}^{-15}$.

**Figure 6.**A heavy mass M is carried on board of a freely falling system, case (

**b**). With respect to the ideal case (

**a**), the mass M modifies the local space–time units and could introduce a vacuum refractive index ${\mathcal{N}}_{v}\ne 1$ so that now ${c}_{\gamma}\ne c$. With a preferred frame, one would then expect off-diagonal elements ${g}_{0i}\sim 2({\mathcal{N}}_{v}-1)({v}_{i}/c)$ in the effective metric, which describes light propagation for the (

**b**) reference system.

**Figure 7.**We report two typical sets of 2000 s for our basic white-noise (WN) signal and its colored version, obtained by Fourier transforming the spectral amplitude of Ref. [57]. The boundaries of the random velocity components in Equations (A11) and (A12) were defined by Equation (A15) by inserting into Equation (A5) the CMB kinematical parameters, for a sidereal time $t=4000-6000$ s and for the latitude of Berlin–Düsseldorf; see Appendix A. The figure is taken from Ref. [35].

**Figure 8.**We report the Allan variance for the fractional frequency shift obtained from many simulations of sequences of 2000 s for our basic white-noise (WN) signal and for its colored version; see Figure 7. The direct experimental results of Ref. [57], for the non-rotating setup, are also shown. The figure is taken from Ref. [35].

**Table 1.**The average second harmonic amplitudes of classical ether drift experiments. These were extracted from the original papers by averaging the amplitudes of the individual observations and assuming the direction of the local drift to be completely random (i.e., no vector averaging of different sessions). These experimental values are then compared with the full statistical average Equation (57) for a projection 250 km/s $\lesssim V|sinz(t\left)\right|\lesssim $ 370 km/s of the Earth’s motion in the CMB and refractivities $\u03f5=2.8\xb7{10}^{-4}$ for air and $\u03f5=3.3\xb7{10}^{-5}$ for gaseous helium. The experimental value for the Morley–Miller experiment is taken from the observed velocities reported in Miller’s Figure 4, and our Figure 3. The experimental value for the Michelson–Pease–Pearson experiment refers to the only known session for which the fringe shifts are reported explicitly [66] and where the optical path was still fifty-five feet. The symbol $\pm \dots $ means that the experimental uncertainty cannot be determined from the available information. The table is taken from Ref. [35].

Experiment | Gas | ${\mathit{A}}_{2}^{\mathbf{EXP}}$ | $\frac{2\mathit{D}}{\mathit{\lambda}}$ | ${\langle {\mathit{A}}_{2}\left(\mathit{t}\right)\rangle}_{\mathbf{stat}}$ |
---|---|---|---|---|

Michelson (1881) | air | $(7.8\pm \dots )\xb7{10}^{-3}$ | $4\xb7{10}^{6}$ | $(0.7\pm 0.2)\xb7{10}^{-3}$ |

Michelson–Morley (1887) | air | $(1.6\pm 0.6)\xb7{10}^{-2}$ | $4\xb7{10}^{7}$ | $(0.7\pm 0.2)\xb7{10}^{-2}$ |

Morley–Miller (1902–1905) | air | $(4.0\pm 2.0)\xb7{10}^{-2}$ | $1.12\xb7{10}^{8}$ | $(2.0\pm 0.7)\xb7{10}^{-2}$ |

Miller (1921–1926) | air | $(4.4\pm 2.2)\xb7{10}^{-2}$ | $1.12\xb7{10}^{8}$ | $(2.0\pm 0.7)\xb7{10}^{-2}$ |

Tomaschek (1924) | air | $(1.0\pm 0.6)\xb7{10}^{-2}$ | $3\xb7{10}^{7}$ | $(0.5\pm 0.2)\xb7{10}^{-2}$ |

Kennedy (1926) | helium | $<0.002$ | $7\xb7{10}^{6}$ | $(1.4\pm 0.5)\xb7{10}^{-4}$ |

Illingworth (1927) | helium | $(2.2\pm 1.7)\xb7{10}^{-4}$ | $7\xb7{10}^{6}$ | $(1.4\pm 0.5)\xb7{10}^{-4}$ |

Piccard–Stahel (1928) | air | $(2.8\pm 1.5)\xb7{10}^{-3}$ | $1.28\xb7{10}^{7}$ | $(2.2\pm 0.8)\xb7{10}^{-3}$ |

Mich.–Pease–Pearson (1929) | air | $(0.6\pm \dots )\xb7{10}^{-2}$ | $5.8\xb7{10}^{7}$ | $(1.0\pm 0.4)\xb7{10}^{-2}$ |

Joos (1930) | helium | $(1.4\pm 0.8)\xb7{10}^{-3}$ | $7.5\xb7{10}^{7}$ | $(1.5\pm 0.6)\xb7{10}^{-3}$ |

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Consoli, M.; Pluchino, A.; Zizzi, P.
Quantum Non-Locality and the CMB: What Experiments Say. *Universe* **2022**, *8*, 481.
https://doi.org/10.3390/universe8090481

**AMA Style**

Consoli M, Pluchino A, Zizzi P.
Quantum Non-Locality and the CMB: What Experiments Say. *Universe*. 2022; 8(9):481.
https://doi.org/10.3390/universe8090481

**Chicago/Turabian Style**

Consoli, Maurizio, Alessandro Pluchino, and Paola Zizzi.
2022. "Quantum Non-Locality and the CMB: What Experiments Say" *Universe* 8, no. 9: 481.
https://doi.org/10.3390/universe8090481