Quantum Non-Locality and the CMB: What Experiments Say

Abstract

“Non-locality is most naturally incorporated into a theory in which there is a special frame of reference. One possible candidate for this special frame of reference is the one in which the Cosmic Microwave Background (CMB) is isotropic. However, other than the fact that a realistic interpretation of quantum mechanics requires a preferred frame and the CMB provides us with one, there is no readily apparent reason why the two should be linked” (L. Hardy). Starting from this remark, we first argue that, given the present view of the vacuum, the basic tenets of Quantum Field Theory cannot guarantee that Einstein Special Relativity, with no preferred frame, is the physically realized version of relativity. Then, to try to understand the nature of the hypothetical preferred $\Sigma -$frame, we consider the so-called ether drift experiments, those precise optical measurements that try to detect, in the laboratory, a small angular dependence of the two-way velocity of light and then to correlate this angular dependence with the direct CMB observations with satellites in space. By considering all experiments performed so far, from Michelson–Morley to the present experiments with optical resonators, and analyzing the small observed residuals in a modern theoretical framework, the long-sought $\Sigma -$frame tight to the CMB naturally emerges. Finally, if quantum non-locality reflects some effect propagating at vastly superluminal speed $v_{QI}\to \infty$, its ultimate origin could be hidden somewhere in the infinite speed $c_s\to \infty$ of vacuum density fluctuations.

1. Introduction

In spite of its extraordinary success in the description of experiments, many conceptual aspects of Quantum Mechanics are still puzzling.1 In this paper, we will focus on a particular aspect that is perhaps the most controversial: the violation of Einstein locality and the conflict with (Einstein) relativity. This has been the subject of a long debate, which started with the seminal paper by Einstein–Podolski–Rosen (EPR) [3]; it was substantially influenced by the work of Bell [4], and continues unabated until today. To have an idea, among the various implications, one may conclude that “a free choice made by an experimenter in one space-time region can influence a second region that is space-like separated from the first” [5] (see however [6]).

For completeness, we observe that the problem dates back to the very early days of Quantum Mechanics, even before EPR. Indeed, the basic issue is already found in Heisenberg’s 1929 Chicago Lectures: “We imagine a photon represented by a wave packet… By reflection at a semi-transparent mirror, it is possible to decompose into a reflected and a transmitted packet… After a sufficient time the two parts will be separated by any distance desired; now if by experiment the photon is found, say, in the reflected part of the packet, then the probability of finding the photon in the other part of the packet immediately becomes zero. The experiment at the position of the reflected packet thus exerts a kind of action (reduction of the wave packet) at the distant point and one sees that this action is propagated with a velocity greater than that of light”.

Then, Heisenberg adds immediately the following remark: “However, it is also obvious that this kind of action can never be utilized for the transmission of signals so that it is not in conflict with the postulates of relativity”. However, one may ask, if there were really something that propagates nearly instantaneously, could such an extraordinary entity be so easily dismissed? Namely, could we ignore this “something” simply because it cannot be efficiently controlled to send “messages” [7]?2 After all, this explains why Dirac, more than forty years later, was still concluding that “The only theory which we can formulate at the present is a non-local one, and of course one is not satisfied with such a theory. I think one ought to say that the problem of reconciling quantum theory and relativity is not solved” [9].

Reduced to its essential terms, this locality problem concerns the time ordering of two events, A and B, along the world line of a hypothetical effect propagating with speed $>c$. This ordering would be different in different frames, because, in some frame $S^{\prime }$, one could find $t_A^{\prime }>t_B^{\prime }$, and in some other frame $S^{″}$, the opposite $t_B^{″}>t_A^{″}$. This causal paradox is the main reason that superluminal signals are not believed to exist. However, for instance, one cannot exclude superluminal sound, i.e., density fluctuations propagating with speed $c_s>c$. In fact, “it is an open question whether $c_s/c$ remains less than unity when non-electromagnetic forces are taken into account” [10]. Thus, superluminal sound cannot be excluded but is confined to very dense media and thus considered irrelevant for the vacuum. As we shall see at the end of Section 4, however, the physical vacuum is a peculiar medium and this conclusion may be too naive.

Therefore, in principle, one could also change perspective and try to dispose of the causality paradox if there were a preferred reference system $\Sigma$ where the superluminal effect propagates isotropically; see, e.g., [11,12,13,14,15,16]. More explicitly, in our case at hand of Quantum Mechanics, if we look at the “Quantum Information” as a transport phenomenon [17], which propagates in space with superluminal velocity $v_{QI}\gg c$, Quantum Mechanics corresponds to the $v_{QI}\to \infty$ limit. Equivalently, by comparison with experiments, one can set a lower limit on $v_{QI}$. Assuming that the preferred $\Sigma -$frame coincides with the reference system where the Cosmic Microwave Background (CMB) is exactly isotropic,3 present experimental determinations give vastly superluminal values whose lower limit has now increased from the original $v_{QI}>{10}^{4}c$ [19,20,21,22] up to the more recent determination $v_{QI}>{10}^{6}c$ [23].

A frequent objection to the idea of relativity with a preferred frame is that, after all, Quantum Mechanics is not a fundamental description of the world. One should instead start from a fundamental Quantum Field Theory (QFT), which incorporates the locality requirement. More precisely, there are violations of (micro)causality in QFT that originate from the lack of sharp localizability of relativistic quantum systems [24]. However, these violations are confined to such small scales so as to be completely irrelevant for the problem that we are considering. On this basis, some authors have concluded that Bell’s proof of non-locality is either wrong or can simply be used to rule out a particular class of hidden-variable theories.4

Therefore, if we adopt the perspective of an underlying, fundamental QFT, solving the problem of locality in Quantum Mechanics could be reduced to finding a particular, missing logical step, which prevents the deduction, from the basic tenets of QFT, that Einstein Special Relativity, with no preferred frame, is the physically realized version of relativity. This is the version that is always assumed when computing S-matrix elements for microscopic processes. However, what one is actually using is the machinery of Lorentz transformations, whose first, complete derivation dates back, ironically, to Larmor and Lorentz, who were assuming the existence of a fundamental state of rest (the ether). Our point here is that there is indeed a particular element that has been missed so far and depends on the nature of the vacuum state. Most likely, this is not Lorentz-invariant due to the phenomenon of vacuum condensation, i.e., due to the macroscopic occupation of the same quantum state.

In the physically relevant case of the Standard Model, the phenomenon of vacuum condensation can be summarized by saying that “What we experience as empty space is nothing but the configuration of the Higgs field that has the lowest possible energy. If we move from field jargon to particle jargon, this means that empty space is actually filled with Higgs particles. They have Bose condensed” [26].5 Clearly, this type of medium is not the ether of classical physics. However, it is also different from the “empty” space–time of Special Relativity that Einstein had in mind in 1905.6

To our knowledge, the idea that the phenomenon of vacuum condensation could produce “conceptual tensions” with the basic locality of both Special and General Relativity was first discussed by Chiao [30]: “The physical vacuum, an intrinsically non-local ground state of a relativistic quantum field theory, which possesses certain similarities to the ground state of a superconductor… This would produce an unusual ‘quantum rigidity’ of the system, associated with what London called the ‘rigidity of the macroscopic wave function’… The Meissner effect is closely analog to the Higgs mechanism in which the physical vacuum also spontaneously breaks local gauge invariance”.7 Therefore, it is not inconceivable that the macroscopic occupation of the same quantum state, say $\mathbf{k}=0$ in some reference system $\Sigma$, can represent the origin of the sought preferred frame. In particular, as we will discuss in Section 2, imposing that only local, scalar operators (such as the Higgs field, or the gluon condensate, or the chiral condensate, etc.) acquire a non-zero vacuum expectation value does not imply the much stronger requirement of an exact Lorentz-invariant vacuum state.

Since our arguments in Section 2 are rather formal and give no information on the nature of the preferred frame, we will then look for definite experimental indications. As anticipated, existing lower limits on $v_{QI}$ have assumed that $\Sigma$ is tight to the CMB. However, as remarked by Hardy [12], there is no readily apparent reason for this identification. Therefore, to find the link, we will consider the so-called ether drift experiments, where, by precise optical measurements, one tries (i) to detect in the laboratory a small angular dependence $\frac{\Delta {\overline{c}}_{\theta }}{c}\ne 0$ of the two-way velocity of light and then (ii) to correlate this angular dependence with the direct CMB observations with satellites in space.

Of course, experimental evidence for both the undulatory and corpuscular aspects of radiation has substantially modified the consideration of an underlying ethereal medium, as support of the electromagnetic waves, and its logical need for the physical theory. Nevertheless, by accepting the idea of a preferred frame, the final physical description could become qualitatively very similar. To this end, let us consider light propagating in a medium of refractive index $\mathcal{N}=1+ϵ$, with $0\le ϵ\ll 1$, and the effective space–time metric $g^{\mu \nu }=g^{\mu \nu }\left(\mathcal{N}\right)$, which should be replaced into the relation $g^{\mu \nu }{p}_{\mu }{p}_{\nu }=0$. At the quantum level, this metric was derived by Jauch and Watson [31] when quantizing the electromagnetic field in a dielectric. They observed that the formalism introduces a preferred reference system, where the photon energy does not depend on the direction of light propagation, and which “is usually taken as the system for which the medium is at rest”. This conclusion is obvious in Special Relativity, where there is no preferred system, but less obvious here, where an isotropic propagation is only assumed when both medium and observer are at rest in $\Sigma$.

To be more specific, let us place this medium in two identical optical resonators, namely resonator 1, which is at rest in $\Sigma$, and resonator 2, which is at rest in an arbitrary frame $S^{\prime }$. Let us also introduce ${\pi }_{\mu }\equiv (\frac{E_{\pi }}{c},\mathsf{ß})$, to indicate the light 4-momentum for $\Sigma$ in his cavity 1, and $p_{\mu }\equiv (\frac{E_p}{c},\mathbf{p})$, to indicate the analogous 4-momentum of light for $S^{\prime }$ in his cavity 2. Finally, let us define by $g^{\mu \nu }$ the space–time metric used by $S^{\prime }$ in the relation $g^{\mu \nu }{p}_{\mu }{p}_{\nu }=0$ and by

$${\gamma }^{\mu \nu }=\mathrm{diag}({\mathcal{N}}^2,-1,-1,-1)$$

(1)

the metric that $\Sigma$ adopts in the analogous relation ${\gamma }^{\mu \nu }{\pi }_{\mu }{\pi }_{\nu }=0$ and which produces the isotropic velocity $c_{\gamma }=E_{\pi }/\left|\mathsf{ß}\right|=\frac{c}{\mathcal{N}}$.

The peculiar view of Special Relativity is that no observable difference can exist between two reference systems that are in uniform translational motion. Instead, with a preferred frame $\Sigma$, as far as light propagation is concerned, this physical equivalence is only assumed in the ideal $\mathcal{N}=1$ limit. In fact, for $\mathcal{N}\ne 1$, where light is absorbed and then re-emitted, the fraction of refracted light could keep track of the particular motion of matter with respect to $\Sigma$ and produce, in a frame $S^{\prime }$ where matter is at rest, a $\Delta {\overline{c}}_{\theta }\ne 0$. Likewise, assuming that the solid parts of cavity 2 are at rest in the inertial frame $S^{\prime }$ no longer implies that the medium that stays inside, e.g., a gas, is in thermodynamic equilibrium. Thus, one should keep an open mind and exploit the implications of the basic condition

$$g^{\mu \nu }(\mathcal{N}=1)={\gamma }^{\mu \nu }(\mathcal{N}=1)={\eta }^{\mu \nu }$$

(2)

where ${\eta }^{\mu \nu }$ is the Minkowski tensor. This standard equality amounts to introducing a transformation matrix, say $A_{\nu }^{\mu }$, which produces $g^{\mu \nu }$ from the reference metric ${\gamma }^{\mu \nu }$ and such that

$$g^{\mu \nu }(\mathcal{N}=1)=A_{\rho }^{\mu }{A}_{\sigma }^{\nu }{\gamma }^{\rho \sigma }(\mathcal{N}=1)=A_{\rho }^{\mu }{A}_{\sigma }^{\nu }{\eta }^{\rho \sigma }={\eta }^{\mu \nu }$$

(3)

This relation is strictly valid for $\mathcal{N}=1$. However, by continuity, one is driven to conclude that an analogous relation between $g^{\mu \nu }$ and ${\gamma }^{\mu \nu }$ should also hold in the $ϵ\to 0$ limit. The crucial point is that the chain in (3) does not fix uniquely $A_{\nu }^{\mu }$. In fact, it is fulfilled either by choosing the identity matrix, i.e., $A_{\nu }^{\mu }={\delta }_{\nu }^{\mu }$, or by choosing a Lorentz transformation, i.e., $A_{\nu }^{\mu }={\mathsf{\Lambda }}_{\nu }^{\mu }$. It thus follows that $A_{\nu }^{\mu }$ is a two-valued function and there are two possible solutions [32,33,34,35] for the metric in $S^{\prime }$. In fact, when $A_{\nu }^{\mu }$ is the identity matrix, we find

$${\left[g^{\mu \nu }\left(\mathcal{N}\right)\right]}_1={\delta }_{\rho }^{\mu }{\delta }_{\sigma }^{\nu }{\gamma }^{\rho \sigma }={\gamma }^{\mu \nu }\sim {\eta }^{\mu \nu }+2ϵ{\delta }_0^{\mu }{\delta }_0^{\nu }$$

(4)

while, when $A_{\nu }^{\mu }$ is a Lorentz transformation, we obtain

$${\left[g^{\mu \nu }\left(\mathcal{N}\right)\right]}_2={\mathsf{\Lambda }}_{\rho }^{\mu }{\mathsf{\Lambda }}_{\sigma }^{\nu }{\gamma }^{\rho \sigma }\sim {\eta }^{\mu \nu }+2ϵv^{\mu }{v}^{\nu }$$

(5)

where $v_{\mu }$ is the $S^{\prime }$ 4-velocity, $v_{\mu }\equiv (v_0,\mathbf{v}/c)$ with $v_{\mu }{v}^{\mu }=1$. As a consequence, the equality ${\left[g^{\mu \nu }\left(\mathcal{N}\right)\right]}_1={\left[g^{\mu \nu }\left(\mathcal{N}\right)\right]}_2$ can only hold for $v^{\mu }={\delta }_0^{\mu }$, i.e., for $\mathbf{v}=0$ when $S^{\prime }\equiv \Sigma$. Notice that by choosing the first solution ${\left[g^{\mu \nu }\left(\mathcal{N}\right)\right]}_1$, which is implicitly assumed in Special Relativity to preserve isotropy in all reference frames also for $\mathcal{N}\ne 1$, we are considering a transformation matrix $A_{\nu }^{\mu }$, which is discontinuous for any $ϵ\ne 0$. In fact, all emphasis on Lorentz transformations depends on enforcing the last equality in (3) for $A_{\nu }^{\mu }={\mathsf{\Lambda }}_{\nu }^{\mu }$ so that ${\mathsf{\Lambda }}^{\mu \sigma }{\mathsf{\Lambda }}_{\sigma }^{\nu }={\eta }^{\mu \nu }$ and the Minkowski metric, if valid in one frame, applies to all equivalent frames.

In conclusion, with a preferred frame $\Sigma$, there may be non-zero $g_{0i}$ in $S^{\prime }$, which plays the role of a velocity field and produces a small anisotropy of the two-way velocity [32,33,34,35]

$$\begin{array}{ccc}\hfill {\overline{c}}_{\gamma }\left(\theta \right)& =& \frac{2c_{\gamma }\left(\theta \right)c_{\gamma }(\pi +\theta )}{c_{\gamma }\left(\theta \right)+c_{\gamma }(\pi +\theta )}\sim \frac{c}{\mathcal{N}}\left[1-ϵ{\beta }^2\left(1+{cos}^2\theta \right)\right]\equiv \frac{c}{\overline{\mathcal{N}}\left(\theta \right)}\hfill \end{array}$$

(6)

The only difference with respect to the old ether model is that the resulting fractional anisotropy $\frac{\Delta {\overline{c}}_{\theta }}{c}\sim ϵ(v^2/c^2)$ would be much smaller than the classical prediction $\frac{\Delta {\overline{c}}_{\theta }}{c}{|}_{\mathrm{class}}\sim (v^2/2c^2)$. However, this has only quantitative significance and has to be decided by experiments.

With this in mind, and addressing refs. [32,33,34,35,36] for more details, we will summarize in Section 3 the analysis of all data from Michelson–Morley to the present experiments with optical resonators. As a matter of fact, once the small residuals are analyzed in a modern theoretical framework, the $\Sigma -$frame tight to the CMB is naturally emerging. Section 4 will finally contain a summary and some general arguments that may indicate the existence of nearly instantaneous effects in the physical vacuum.

2. Vacuum State and Its Lorentz Invariance

The discovery of the Higgs boson at LHC has confirmed the basic idea of Spontaneous Symmetry Breaking (SSB), where particle masses originate from the particular structure of the vacuum. As anticipated in the Introduction by ’t Hooft’s words [26], this means that empty space is actually filled with the elementary quanta of the Higgs field, whose trivial, empty vacuum is not the lowest-energy state of the theory. In this section, we will summarize the basic picture of SSB, in the case of a one-component scalar field $\mathsf{\Phi }\left(x\right)$ with only a discrete reflection symmetry $\mathsf{\Phi }\to -\mathsf{\Phi }$, i.e., no Goldstone bosons. In spite of its simplicity, this system can already display the general aspects that are relevant for the problem of a Lorentz-invariant vacuum state.

Our analysis will be based on the condensation process of physical quanta and thus assumes a description of symmetry breaking as a (weak) first-order phase transition. Namely, SSB occurs when the renormalized mass squared $m_R^2$ of the quanta of the symmetric phase is extremely small but still in the physical region $m_R^2>0$. In the presence of gauge bosons, this was shown in the loop expansion by Coleman and Weinberg [37] long ago. Today, the same (weak) first-order scenario is now supported by the most recent lattice simulations [38,39,40] of a pure ${\mathsf{\Phi }}^4$ theory, which we adopt as our basic model.

To introduce the issue of Lorentz invariance, let us first recall that inertial transformations are represented in Hilbert space by unitary operators, which correspond to the Poincaré group. This means a representation of the 10 generators $P_{\mu }$ and $L_{\mu \nu }$ ($\mu$, $\nu$= 0, 1, 2, 3), where $P_{\mu }$ describe the space–time translations and $L_{\mu \nu }=-L_{\nu \mu }$ the space rotations and Lorentz boosts, with commutation relations

$$[P_{\mu },P_{\nu }]=0$$

(7)

$$[L_{\mu \nu },P_{\rho }]=i{\eta }_{\nu \rho }{P}_{\mu }-i{\eta }_{\mu \rho }{P}_{\nu }$$

(8)

$$[L_{\mu \nu },L_{\rho \sigma }]=-i{\eta }_{\mu \rho }{L}_{\nu \sigma }+i{\eta }_{\mu \sigma }{L}_{\nu \rho }-i{\eta }_{\nu \sigma }{L}_{\mu \rho }+i{\eta }_{\nu \rho }{L}_{\mu \sigma }$$

(9)

where again ${\eta }_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$ is the Minkowski tensor. An exact Lorentz-invariant vacuum has to be annihilated by all 10 generators (see, e.g., [41]).

These premises are well known but one should also be aware that, with the exception of low-dimensionality cases, a construction of the Poincaré algebra is only known for the free-field case. For the interacting theory, at present, one can only implement it perturbatively. This means that one should start with a definite free-field limit, and therefore with a unique vacuum, where the simplest prescription of the Wick, normal ordering allows for a consistent representation of the commutation relations.

Let us thus consider a system of free, spinless quanta with mass m and energy $E\left(k\right)=\sqrt{{\mathbf{k}}^2+m^2}$. In terms of annihilation and creation operators $a\left(\mathbf{k}\right)$ and $a^{†}\left(\mathbf{k}\right)$ of an empty vacuum $|\mathrm{o}\rangle$, with $a\left(\mathbf{k}\right)|\mathrm{o}\rangle =\langle \mathrm{o}|a^{†}\left(\mathbf{k}\right)=0$, and commutation relations $[a\left(\mathbf{k}\right),a^{†}\left({\mathbf{k}}^{\prime }\right)]={\delta }_{\mathbf{k}{\mathbf{k}}^{\prime }}$, the required representation of the generators is then

$$P_0\equiv H_2\left(m\right)=:\int d^{3}x\left[\frac{1}{2}\left({\mathsf{\Pi }}^2+{(\nabla \mathsf{\Phi })}^2+m^2{\mathsf{\Phi }}^2\right)\right]:=\sum _{\mathbf{k}}E\left(k\right)a^{†}(\mathbf{k})a\left(\mathbf{k}\right)$$

(10)

$$P_i=:\int d^{3}x\mathsf{\Pi }\left(x\right){\partial }_i\mathsf{\Phi }\left(x\right):=\sum _{\mathbf{k}}{k}_i{a}^{†}\left(\mathbf{k}\right)a\left(\mathbf{k}\right)$$

(11)

$$\begin{array}{c}\hfill L_{ij}=:\int d^{3}x[x_i\mathsf{\Pi }\left(x\right){\partial }_j\mathsf{\Phi }\left(x\right)-x_j\mathsf{\Pi }\left(x\right){\partial }_i\mathsf{\Phi }\left(x\right)]:=\\ \hfill =\frac{i}{2}\sum _{\mathbf{k}}[k_i{a}^{†}(\mathbf{k})\frac{\overleftrightarrow{\partial }}{\partial k_j}a\left(\mathbf{k}\right)-k_j{a}^{†}(\mathbf{k})\frac{\overleftrightarrow{\partial }}{\partial k_i}a\left(\mathbf{k}\right)]\end{array}$$

(12)

$$\begin{array}{c}\hfill L_{0i}=x_0{P}_i-\frac{1}{2}:\int d^{3}x{x}_i[{\mathsf{\Pi }}^2+{(\nabla \mathsf{\Phi })}^2+m^2{\mathsf{\Phi }}^2]:=\\ \hfill =\frac{i}{2}\sum _{\mathbf{k}}E\left(k\right)a^{†}(\mathbf{k})\frac{\overleftrightarrow{\partial }}{\partial k_i}a\left(\mathbf{k}\right)\end{array}$$

(13)

With the above expressions, the Poincaré algebra is reproduced and the empty vacuum $|\mathrm{o}\rangle$ is annihilated by all 10 generators. As such, the general requirements for an exact Lorentz-invariant vacuum are fulfilled. In particular, notice the crucial role of the zero-energy condition $P_0|\mathrm{o}\rangle =0$, which, from the commutation relations $[P_i,L_{0i}]=i{P}_0$ (no summation over i), is needed for consistency with $P_i|\mathrm{o}\rangle =0$ and $L_{0i}|\mathrm{o}\rangle =0$.

Let us now introduce the interaction and consider the limit of a very weakly coupled $g{\mathsf{\Phi }}^4$ theory, i.e., with a coupling g in the range $0<g\ll 1$. In this case, which is the typical example of nonlinear QFT with polynomial interaction $P\left(\mathsf{\Phi }\right)=g{\mathsf{\Phi }}^4$, there are two basically different options. In a first approach, one could try to introduce a suitable de-singularized operator, say $::P\left(\mathsf{\Phi }\right(x\left)\right)::$, which extends the standard normal ordering of the free-field case so that $\langle {\mathsf{\Psi }}_0|::P\left(\mathsf{\Phi }\left(x\right)\right)::|{\mathsf{\Psi }}_0\rangle =0$ in the true vacuum state $|{\mathsf{\Psi }}_0\rangle$. This type of approach, which has been followed by very few authors, was discussed by Segal [42]. His conclusion was that $::P\left(\mathsf{\Phi }\right(x\left)\right)::$ is not well defined until the physical vacuum is known, but, at the same time, the physical vacuum also depends on the definition given for $::P\left(\mathsf{\Phi }\right(x\left)\right)::$. From this type of circularity, Segal was deducing that, in general, in such a nonlinear QFT, the physical vacuum will not be invariant under the full Lorentz symmetry of the underlying Lagrangian density.

A second approach, followed nowadays by most authors, is instead to consider $g{\mathsf{\Phi }}^4$ theory in the framework of a perturbative Renormalization Group approach. In this case, $g\equiv g\left(\mu \right)$ should be understood as the running coupling constant at a variable mass scale $\mu$ and, in its value, would carry the information on the asymptotic pair $(g_0,{\mathsf{\Lambda }}_s)$, where $g_0$ is the bare coupling at some minimum locality scale fixed by the ultraviolet cutoff ${\mathsf{\Lambda }}_s$. Clearly, with a finite ${\mathsf{\Lambda }}_s$, one is explicitly breaking Lorentz symmetry. However, by the generally accepted “triviality” of $g{\mathsf{\Phi }}^4$ theory in four space–time dimensions, one finds that $g\left(\mu \right)\sim {ln}^{-1}({\mathsf{\Lambda }}_s/\mu )\to 0^{+}$ for ${\mathsf{\Lambda }}_s\to \infty$, whatever the value of $g_0$. By defining m the mass of the scalar quanta in the symmetric phase $\langle \mathsf{\Phi }\rangle =0$, one can then assume that, for $\mu \sim m$, g is so small (or equivalently the cutoff ${\mathsf{\Lambda }}_s$ is so large) that one can meaningfully obtain $|{\mathsf{\Psi }}_0\rangle$ by perturbing around the previous free-field vacuum $|\mathrm{o}\rangle$. Likewise, with a formally Lorentz-invariant interaction such as ${\mathsf{\Phi }}^4$, it should be possible to construct a representation of the Poincaré algebra in Equations (7)–(9), which holds to any finite order in g and goes smoothly into the previous free-field structure for ${\mathsf{\Lambda }}_s\to \infty$. Finally, as in the free-field case, since the true vacuum $|{\mathsf{\Psi }}_0\rangle$ is always assumed to have zero spatial momentum, from the commutator $[P_i,L_{0i}]=i{P}_0$, its invariance under boosts will also require one to implement the zero-energy condition in the perturbative expansion.

This general strategy is briefly sketched below with a Hamiltonian

$$P_0\equiv H=H_2\left(m\right)+H_I+\Delta E$$

(14)

which, beside the interaction

$$H_I=:\int d^{3}x\frac{g}{4!}{\mathsf{\Phi }}^4:$$

(15)

should also include an additive constant $\Delta E$ given as a power series in g

$$\Delta E=g\Delta E\left(1\right)+g^2\Delta E\left(2\right)+\dots$$

(16)

with coefficients $\Delta E\left(1\right)$, $\Delta E\left(2\right)$… determined by imposing that the true ground state

$$|{\mathsf{\Psi }}_0\rangle =|\mathrm{o}\rangle +g|\mathrm{o}\left(1\right)\rangle +g^2|\mathrm{o}\left(2\right)\rangle +\dots$$

(17)

has exactly zero energy to any finite order in g

$$P_0|{\mathsf{\Psi }}_0\rangle =0$$

(18)

Again, $|\mathrm{o}\rangle$ is the vacuum of the free-field $H_2\left(m\right)$, with $H_2\left(m\right)|\mathrm{o}\rangle =0$, so that, if, with a short-hand notation, we denote by $|\mathrm{n}\rangle$ its higher eigenstates, i.e., $H_2\left(m\right)|\mathrm{n}\rangle =E_n|\mathrm{n}\rangle$ with $E_n>0$, we find $\Delta E\left(1\right)=0$ and the first few relations

$$g|\mathrm{o}\left(1\right)\rangle =-\sum _{n\ne 0}\frac{|\mathrm{n}\rangle \langle \mathrm{n}|H_I|\mathrm{o}\rangle }{E_n}$$

(19)

$$g^2|\mathrm{o}\left(2\right)\rangle =\sum _{m\ne 0}\sum _{n\ne 0}\frac{|\mathrm{n}\rangle \langle \mathrm{n}|H_I|\mathrm{m}\rangle \langle \mathrm{m}|H_I|\mathrm{o}\rangle }{E_n{E}_m}$$

(20)

$$g^2\Delta E\left(2\right)=\sum _{n\ne 0}\frac{\langle \mathrm{o}|H_I|\mathrm{n}\rangle \langle \mathrm{n}|H_I|\mathrm{o}\rangle }{E_n}$$

(21)

The issue of Lorentz invariance would finally require the analysis of the boost generators $L_{0i}$, which should also be re-defined in perturbation theory [43,44] by starting from the free-field form $L_{0i}(g=0)$ in Equation (13), say

$$L_{0i}\left(g\right)=L_{0i}(g=0)+g\Delta L_{0i}\left(1\right)+g^2\Delta L_{0i}\left(2\right)+\dots$$

(22)

In this way, with a zero-energy vacuum state $|{\mathsf{\Psi }}_0\rangle$, and assuming that the Poincaré algebra Equations (7)–(9) and the condition $L_{0i}\left(g\right)|{\mathsf{\Psi }}_0\rangle =0$ can be fulfilled to any finite order $g^n$, i.e., up to $g^{n+1}$ terms, the Lorentz invariance of $|{\mathsf{\Psi }}_0\rangle$ can be considered exact.

Let us now consider the phenomenon of SSB. Here, for constant field configurations, the basic quantity is the effective potential $V_{\mathrm{eff}}\left(\varphi \right)$. Within the class of normalized quantum states $\langle \mathsf{\Psi }|\mathsf{\Psi }\rangle =1$, this is defined in the infinite volume limit $\mathsf{V}\to \infty$ as

$$V_{\mathrm{eff}}\left(\varphi \right)=\frac{1}{\mathsf{V}}·{\min }_{\mathsf{\Psi }}\langle \mathsf{\Psi }\left|H\right|\mathsf{\Psi }\rangle$$

(23)

with the condition

$$\langle \mathsf{\Psi }\left|\mathsf{\Phi }\right|\mathsf{\Psi }\rangle =\varphi$$

(24)

By expanding around $\varphi =0$, the quadratic shape of the effective potential gives the renormalized mass of the symmetric-phase quanta

$$m_R^2=V_{\mathrm{eff}}^{″}\left(\varphi \right){|}_{\varphi =0}$$

(25)

given as a power serious expansion

$$m_R^2=m^2+g^2\Delta m^2\left(2\right)+g^3\Delta m^2\left(3\right)+\dots$$

(26)

In turn, from the connection between the variational principle and the eigenvalue equation for the Hamiltonian, for a translational invariant field configuration, the previous zero-energy condition of the symmetric ground state at $\langle {\mathsf{\Psi }}_0\left|\mathsf{\Phi }\right|{\mathsf{\Psi }}_0\rangle =0$ implies

$$V_{\mathrm{eff}}(\varphi =0)=0$$

(27)

to any finite order in g.

The conventional second-order picture of SSB assumes that no non-trivial minima can exist if $m_R^2\ge 0$. Instead, as indicated by most recent lattice simulations [38,39,40] of a pure ${\mathsf{\Phi }}^4$ theory, we will adopt the view of SSB as a (weak) first-order phase transition. This means that, with the same Hamiltonian Equation (14), before reaching the massless $m_R^2=0$ limit, for some very small $m_R^2=m_c^2$, there will arise a first pair of vacua, say with $\langle \mathsf{\Phi }\rangle =\pm v_c$, which have the same zero energy as the empty vacuum at $\langle \mathsf{\Phi }\rangle =0$; see Figure 1. Therefore, the critical mass is defined as that value

$$m_c^2=V_{\mathrm{eff}}^{″}\left(\varphi \right){|}_{\varphi =0}$$

(28)

for which there are $three$ vacuum states with the same energy

$$0=V_{\mathrm{eff}}(\varphi =0)=V_{\mathrm{eff}}(\varphi =\pm v_c)$$

(29)

On the other hand, for $0\le m_R^2<m_c^2$, SSB takes place and the energy of the two degenerate vacua, say $|{\mathsf{\Psi }}_{\pm }\rangle$ with $\langle {\mathsf{\Psi }}_{\pm }\left|\mathsf{\Phi }\right|{\mathsf{\Psi }}_{\pm }\rangle =\pm v$, will definitely be lower than its value at $\langle \mathsf{\Phi }\rangle =0$

$$V_{\mathrm{eff}}(\varphi =\pm v)<V_{\mathrm{eff}}(\varphi =0)=0$$

(30)

Explicit calculations, either in the loop expansion [27] or variational [45], indicate that $V_{\mathrm{eff}}(\varphi =\pm v)$ has a non-analytic $exp(-1/g)$ behavior. This reflects the basic non-perturbative nature of SSB, which cannot be found to any finite order in g.

Here, we emphasize a few aspects:

(i) To understand intuitively the first-order nature of the phase transition, the crucial observation is that the quanta of the symmetric phase, besides the +$g{\delta }^{\left(3\right)}\left(\mathbf{r}\right)$ repulsion, also feel a $-g^2\frac{e^{-2mr}}{r^3}$ attraction [27], which shows up at the one-loop level and whose range becomes longer and longer in the $m\to 0$ limit. A calculation of the energy density in the dilute-gas approximation, which is equivalent to the one-loop effective potential, indicates that, for very small m, the attractive tail dominates. Higher-order corrections simply renormalize the strength of these two basic effects [27], whose interplay explains the instability of the symmetric phase producing a $\langle \mathsf{\Phi }\rangle \ne 0$

(ii) The field fluctuations around each of the two non-symmetric vacua are conveniently described by the shifted field $h\left(x\right)\equiv \mathsf{\Phi }\left(x\right)-\langle \mathsf{\Phi }\rangle$, with $\langle h\rangle =0$ by definition. In the realistic case of the Standard Model with a SU(2)xU(1) symmetry, the lowest $h-$field excitation has mass $m_h\sim$ 125 GeV as observed at the Large Hadron Collider of CERN. Notice that this $m_h$ is conceptually different from the $m_R$ discussed above, the two mass parameters describing the quadratic shape of the effective potential at two different values of $\varphi$, namely

$$m_h^2=V_{\mathrm{eff}}^{″}\left(\varphi \right){|}_{\varphi =\pm v}$$

(31)

(iii) Strictly speaking, Wightman’s axioms [41] require a unique vacuum state. The usual way out is that the two states $|{\mathsf{\Psi }}_{\pm }\rangle$, giving the absolute minima of the effective potential, have zero overlap $\langle {\mathsf{\Psi }}_{-}|{\mathsf{\Psi }}_{+}\rangle \to 0$ in the infinite-volume $\mathsf{V}\to \infty$ [46].

(iv) Due to the equivalence between the variational method and eigenvalue equation, the two spontaneously broken vacua $|{\mathsf{\Psi }}_{\pm }\rangle$, at the minima of the effective potential, are the lowest eigenstates of the Hamiltonian. However, by assuming Equation (18), their energy cannot be zero. Then, from $[P_i,L_{0i}]=i{P}_0$, $P_i|{\mathsf{\Psi }}_{\pm }\rangle =0$ and $P_0|{\mathsf{\Psi }}_{\pm }\rangle \ne 0$, it follows that $L_{0i}|{\mathsf{\Psi }}_{\pm }\rangle \ne 0$, so that the two states $|{\mathsf{\Psi }}_{\pm }\rangle$ cannot be Lorentz-invariant [47,48,49].

Of course, concerning (iv), we could define a different free-field limit to preserve the Lorentz invariance of the two $|{\mathsf{\Psi }}_{\pm }\rangle$. To this end, one should first express the field as $\mathsf{\Phi }\left(x\right)=h\left(x\right)+\langle \mathsf{\Phi }\rangle$ so that the original ${\mathsf{\Phi }}^4$ term will also produce a cubic term $g\langle \mathsf{\Phi }\rangle h^3$, which reflects the interaction of the h-field fluctuation with the vacuum condensate $\langle \mathsf{\Phi }\rangle$. Then, $h\left(x\right)$ should be quantized in terms of new annihilation and creation operators, say $b\left(\mathbf{k}\right)$ and $b^{†}\left(\mathbf{k}\right)$ for $|{\mathsf{\Psi }}_{+}\rangle$ and $c\left(\mathbf{k}\right)$ and $c^{†}\left(\mathbf{k}\right)$ for $|{\mathsf{\Psi }}_{-}\rangle$, so that $b\left(\mathbf{k}\right)|{\mathsf{\Psi }}_{+}\rangle =0$ and $c\left(\mathbf{k}\right)|{\mathsf{\Psi }}_{-}\rangle =0$. This means that one should select one of the two non-symmetric vacuum states, for instance $|{\mathsf{\Psi }}_{+}\rangle$, and define the normal ordering procedure, in terms of $b\left(\mathbf{k}\right)$ and $b^{†}\left(\mathbf{k}\right)$. After having re-defined the free-field limit, the perturbative expansion should also be re-formulated because, now, there is a new dimensionful coupling constant $g\langle \mathsf{\Phi }\rangle$.8 Finally, the additive constant $\Delta E$ should now be determined by requiring zero energy for $|{\mathsf{\Psi }}_{+}\rangle$ with a change in the Hamiltonian Equation (14). Correspondingly, the previous symmetric vacuum $|{\mathsf{\Psi }}_0\rangle$, would now be higher, of the same amount associated with $V_{\mathrm{eff}}(\varphi =\pm v)$.

With these two different procedures, one has to choose between the Lorentz invariance of the two degenerate minima $|{\mathsf{\Psi }}_{\pm }\rangle$ and the Lorentz invariance of the original $|{\mathsf{\Psi }}_0\rangle$. In particular, with the alternative re-arrangement described above, we are implicitly admitting that, even in the simplest case of a one-component, massive ${\mathsf{\Phi }}^4$ theory with $g\to 0^{+}$, there is no way to start from the free-field vacuum $|\mathrm{o}\rangle$ and preserve, in perturbation theory, the basic Lorentz symmetry embodied in the operatorial structure Equations (10)–(13).9 Perhaps, a way out could be to impose the condition $m_R^2=m_c^2$, which fixes the mass of the symmetric phase at the particular critical value where Equation (29) holds true and the three local minima have all the same zero energy (a choice that, however, would imply the meta-stability of the broken-symmetry phase).

By discarding the particular case $m_R^2=m_c^2$, we will then return to our original choice of a Lorentz-invariant $|{\mathsf{\Psi }}_0\rangle$. The point is that the resulting Lorentz-non-invariance of the two non-symmetric minima $|{\mathsf{\Psi }}_{\pm }\rangle$ has an intuitive physical meaning. Indeed, it reflects the macroscopic occupation of the same quantum state by the basic elementary quanta of ${\mathsf{\Phi }}^4$, those described by $a\left(\mathbf{k}\right)$ and $a^{†}\left(\mathbf{k}\right)$, and which behave as hard spheres at the asymptotic cutoff scale ${\mathsf{\Lambda }}_s$.

To this end, let us consider the field expansion in terms of $a\left(\mathbf{k}\right)$ and $a^{†}\left(\mathbf{k}\right)$

$$\mathsf{\Phi }\left(\mathbf{x}\right)=\sum _{\mathbf{k}}\frac{1}{\sqrt{2\mathsf{V}E\left(k\right)}}\left[a\left(\mathbf{k}\right)e^{i\mathbf{k}·\mathbf{x}}+a^{†}\left(\mathbf{k}\right)e^{-i\mathbf{k}·\mathbf{x}}\right]$$

(32)

and introduce the number of field quanta through the operator

$$\widehat{\mathsf{N}}=\sum _{\mathbf{k}}{a}_{\mathbf{k}}^{†}{a}_{\mathbf{k}}$$

(33)

For a temperature $T=0$, following’t Hooft [26]—see also [27]—SSB corresponds to a macroscopic number $\mathsf{N}$ of quanta in the state $\mathbf{k}=0$ of some reference frame $\Sigma$. In this case, where $\widehat{\mathsf{N}}\sim a_0^{†}{a}_0$, one can effectively consider $a_0$ as a c-number $\sqrt{\mathsf{N}}$ with

$$\varphi =\langle \mathsf{\Phi }\rangle \sim \frac{1}{\sqrt{2\mathsf{V}m}}2a_0\sim \sqrt{\frac{2\mathsf{N}}{\mathsf{V}m}}$$

(34)

a density $n\equiv \mathsf{N}/\mathsf{V}$ (with $\mathsf{N}\to \infty$, $\mathsf{V}\to \infty$ at fixed n) given by

$$n\sim \frac{1}{2}m{\varphi }^2$$

(35)

and mass density ${\rho }_m=(m\mathsf{N}/\mathsf{V})$ reproducing the quadratic term $\frac{1}{2}{m}^2{\varphi }^2$ in the potential.

In this formalism, it will be natural to adopt a particular notation for each of the two degenerate vacua $|{\mathsf{\Psi }}_{\pm }\rangle$, say $|{\mathsf{\Psi }}_{\pm }^{(\Sigma )}\rangle$, which defines the vacuum assignment for that observer which is at rest in $\Sigma$. As we have seen, if these states have a non-zero energy $E_0<0$, and thus are not Lorentz-invariant, boost operators $U^{\prime }$, $U^{″}$, … will transform non-trivially $|{\mathsf{\Psi }}_{\pm }^{(\Sigma )}\rangle$ into new states $|{\mathsf{\Psi }}_{\pm }^{\prime }\rangle$, $|{\mathsf{\Psi }}_{\pm }^{″}\rangle$, … appropriate to moving observers $S^{\prime }$, $S^{″}$, … For instance, by defining the boost operator $U^{\prime }=e^{i{\lambda }^{\prime }{L}_{01}}$, one finds

$$|{\mathsf{\Psi }}_{\pm }^{\prime }\rangle =e^{i{\lambda }^{\prime }{L}_{01}}|{\mathsf{\Psi }}_{\pm }^{(\Sigma )}\rangle$$

(36)

so that, by using the relations

$$e^{-i{\lambda }^{\prime }{L}_{01}}{P}_1{e}^{i{\lambda }^{\prime }{L}_{01}}=cosh{\lambda }^{\prime }{P}_1+sinh{\lambda }^{\prime }{P}_0$$

(37)

$$e^{-i{\lambda }^{\prime }{L}_{01}}{P}_0{e}^{i{\lambda }^{\prime }{L}_{01}}=sinh{\lambda }^{\prime }{P}_1+cosh{\lambda }^{\prime }{P}_0$$

(38)

we obtain

$$P_1|{\mathsf{\Psi }}_{\pm }^{\prime }\rangle =E_{0}sinh{\lambda }^{\prime }|{\mathsf{\Psi }}_{\pm }^{\prime }\rangle P_0|{\mathsf{\Psi }}_{\pm }^{\prime }\rangle =E_{0}cosh{\lambda }^{\prime }|{\mathsf{\Psi }}_{\pm }^{\prime }\rangle$$

(39)

Thus, the physical, realized form of relativity would now contain a preferred frame $\Sigma$ with zero spatial momentum. In a generic moving system $S^{\prime }$, one would instead find10

$$P_i|{\mathsf{\Psi }}_{\pm }^{\prime }\rangle =E_0{b}_i|{\mathsf{\Psi }}_{\pm }^{\prime }\rangle P_0|{\mathsf{\Psi }}_{\pm }^{\prime }\rangle =E_{0}a|{\mathsf{\Psi }}_{\pm }^{\prime }\rangle a^2-b_i{b}_i=1$$

(43)

We observe that, traditionally, with the exception of Chiao’s [30] mentioned ”conceptual tensions“, SSB was never believed to be in potential conflict with Einstein relativity, the motivation being, perhaps, that the mean properties of the condensed phase are summarized into the vacuum expectation value $\langle \mathsf{\Phi }\rangle$ of the Higgs field, which transforms as a world scalar under the Lorentz group. However, this does not imply that the vacuum state itself is Lorentz-invariant. Lorentz transformation operators $U^{\prime }$, $U^{″}$, … could transform non-trivially the reference vacuum states $|{\mathsf{\Psi }}_{\pm }^{(\Sigma )}\rangle$ and, yet, for any Lorentz scalar operator S, i.e., for which $S={\left(U^{\prime }\right)}^{†}S{U}^{\prime }={\left(U^{″}\right)}^{†}S{U}^{″}\dots$, one would find

$$\langle {\mathsf{\Psi }}_{\pm }^{(\Sigma )}\left|S\right|{\mathsf{\Psi }}_{\pm }^{(\Sigma )}\rangle =\langle {\mathsf{\Psi }}_{\pm }^{\prime }\left|S\right|{\mathsf{\Psi }}_{\pm }^{\prime }\rangle =\langle {\mathsf{\Psi }}_{\pm }^{″}\left|S\right|{\mathsf{\Psi }}_{\pm }^{″}\rangle \dots$$

(44)

Another aspect, which is always implicitly assumed but very seldom spelled out, concerns the condition $V_{\mathrm{eff}}(\varphi =0)=0$. This is usually interpreted as a matter of convention, as if one were actually computing $\Delta V_{\mathrm{eff}}\left(\varphi \right)\equiv V_{\mathrm{eff}}\left(\varphi \right)-V_{\mathrm{eff}}(\varphi =0)$. However, starting from this apparently innocent assumption, many authors have raised the problem of the non-zero cosmological constant in Einstein’s field equations, which is generated by SSB. This problem has only a definite meaning if the condition $V_{\mathrm{eff}}(\varphi =0)=0$ is not arbitrary but is actually assumed from the start for consistency. With our previous analysis of the symmetric vacuum, we can now understand the motivations of this implicit assumption. Starting from the free-field structure in Equations (10)–(13), the condition in Equation (27) expresses the requirement of having a zero-energy vacuum at $\varphi =0$ and, therefore, of preserving its Lorentz invariance in the interacting theory. Nonetheless, near the critical mass, where Equation (29) holds true, the induced cosmological constant could be made arbitrarily small.11

Truly enough, the previous arguments are rather formal and give no information on the preferred frame $\Sigma$ tight to the reference vacua $|{\mathsf{\Psi }}_{\pm }^{(\Sigma )}\rangle$. For this reason, in the following Section 3, we will turn our attention to experiments and try to understand if $\Sigma$ really exists and if, eventually, it is tight to the CMB, as assumed in refs. [19,20,21,22,23].

3. The Basics of the Ether Drift Experiments

3.1. Which Preferred Frame?

Looking for the preferred $\Sigma -$frame, the natural candidate is the reference system where the temperature of the CMB looks exactly isotropic or, more precisely, where the CMB Kinematic Dipole [18] vanishes. This dipole is in fact a consequence of the Doppler effect associated with the motion of the Earth ($\beta =V/c)$

$$T\left(\theta \right)=\frac{T_o\sqrt{1-{\beta }^2}}{1-\beta cos\theta }$$

(45)

Accurate observations with satellites in space [53] have shown that the measured temperature variations correspond to a motion of the solar system described by an average velocity $V\sim 370$ km/s, a right ascension $\alpha \sim {168}^o$ and a declination $\gamma \sim -7^o$, pointing approximately in the direction of the constellation Leo. This means that, if one sets $T_o\sim$ 2.725 K and $\beta \sim 0.00123$, there are angular variations of a few millikelvin

$$\Delta T^{\mathrm{CMB}}\left(\theta \right)\sim T_o\beta cos\theta \sim \pm 3.36\mathrm{mK}$$

(46)

which represent by far the largest contribution to the CMB anisotropy.

Therefore, one may ask, could the reference system with a vanishing CMB dipole represent a fundamental preferred frame for relativity, as in the original Lorentzian formulation? The standard answer is that one should not confuse these two concepts. The CMB is a definite medium and, as such, sets a rest frame where the dipole anisotropy is zero. Our motion with respect to this system has been detected but there is no contradiction with Special Relativity.

Nonetheless, to good approximation, this kinematic dipole arises by combining the various forms of peculiar motion that are involved (rotation of the solar system around the center of the Milky Way, motion of the Milky Way toward the center of the Local Group, motion of the Local Group of galaxies in the direction of the Great Attractor, etc.) [53]. Thus, if one could switch off the local inhomogeneities that produce these peculiar forms of motion, it is natural to imagine a global frame of rest associated with the Universe as a whole. A vanishing CMB dipole could then simply indicate the existence of this fundamental system $\Sigma$ that we may conventionally decide to call ”ether“, but the cosmic radiation itself would not coincide with this form of ether. Due to the group properties of Lorentz transformations, two observers, S′ and S″, moving individually with respect to $\Sigma$, would still be connected by a Lorentz transformation with the relative velocity parameter fixed by the standard relativistic composition rule.12 However, as anticipated in the Introduction, the ultimate consequences would be far-reaching.

The answer cannot be found on pure theoretical grounds and this is why one looks for a small anisotropy of the two-way velocity of light $\frac{\Delta {\overline{c}}_{\theta }}{c}\ne 0$ in the Earth laboratory. Here, the general consensus is that no genuine ether drift has ever been observed, all measurements (from Michelson–Morley to the most recent experiments with optical resonators) being seen as a long sequence of null results, i.e., typical instrumental effects in experiments with better and better systematics (see, e.g., Figure 1 of Ref. [57]).

However, this is not necessarily true. In the original measurements, light was propagating in gaseous systems (air or helium at atmospheric pressure), while now, in modern experiments, light propagates in a high vacuum or inside solid dielectrics. Therefore, in principle, the difference with the modern experiments might not depend on the technological progress only but also on the different media that are tested, thus preventing a straightforward comparison. This is even more true if one takes into account that, in the past, the greatest experts (such as Hicks and Miller) have seriously questioned the traditional null interpretation of the very early measurements. The observed ”fringe shifts“, although much smaller than the predictions of classical physics, were often non-negligible as compared to the extraordinary sensitivity of the interferometers. It is then conceivable that, in some alternative framework, the small residuals could acquire a physical meaning. As a definite example, in the following Section 3.2, we will summarize the theoretical scheme of refs. [32,33,34,35] starting with the old experiments. The modern experiments will be considered in Section 3.3.

3.2. The Old Experiments in Gaseous Media

In the old experiments in gases (Michelson–Morley, Miller, Tomaschek, Kennedy, Illingworth, Piccard–Stahel, Michelson–Pease–Pearson, Joos) [58,59,60,61,62,63,64,65,66,67], with refractive index $\mathcal{N}=1+ϵ$, the velocity of light in the interferometers, say $c_{\gamma }$, was not the same parameter c of Lorentz transformations. Hence, assuming their exact validity, deviations from isotropy could only be due to the small fraction of refracted light that keeps track of the velocity of matter with respect to $\Sigma$ and produces a direction-dependent refractive index. As anticipated in the Introduction, from symmetry arguments valid in the $ϵ\to 0$ limit [32,33,34,35], one would then expect a two-way velocity

$$\begin{array}{ccc}\hfill {\overline{c}}_{\gamma }\left(\theta \right)& =& \frac{2c_{\gamma }\left(\theta \right)c_{\gamma }(\pi +\theta )}{c_{\gamma }\left(\theta \right)+c_{\gamma }(\pi +\theta )}\sim \frac{c}{\mathcal{N}}\left[1-ϵ{\beta }^2\left(1+{cos}^2\theta \right)\right]\equiv \frac{c}{\overline{\mathcal{N}}\left(\theta \right)}\hfill \end{array}$$

(47)

with an effective $\theta -$dependent refractive index13

$$\overline{\mathcal{N}}\left(\theta \right)\sim \mathcal{N}[1+ϵ{\beta }^2(1+{cos}^2\theta )]$$

(48)

and a fractional anisotropy

$$\frac{\Delta {\overline{c}}_{\theta }}{c}=\frac{{\overline{c}}_{\gamma }(\pi /2+\theta )-{\overline{c}}_{\gamma }\left(\theta \right)}{c}\sim ϵ{\beta }^{2}cos2\theta$$

(49)

In the above relations, $\beta \equiv v/c$, with v and $\theta =0$ indicating, respectively, the magnitude and the direction of the drift in the plane of the interferometer.

By introducing the optical path D—see Figure 2—and the light wavelength $\lambda$, this would produce a fringe pattern

$$\frac{\Delta \lambda \left(\theta \right)}{\lambda }=\frac{2D}{\lambda }\frac{\Delta {\overline{c}}_{\theta }}{c}\sim \frac{2D}{\lambda }ϵ\frac{v^2}{c^2}cos2\theta$$

(50)

so that the dragging of light in the Earth frame is described as a pure second-harmonic effect that is periodic in the range $[0,\pi ]$, as in the classical theory (see, e.g., [68]), with the exception of its amplitude

$$A_2=\frac{2D}{\lambda }ϵ\frac{v^2}{c^2}$$

(51)

This is suppressed by the factor $2ϵ$ relatively to the classical amplitude for the orbital velocity of 30 km/s

$$A_2^{\mathrm{class}}=\frac{D}{\lambda }{\left(\frac{30\mathrm{km}/\mathrm{s}}{c}\right)}^2$$

(52)

This difference could then be re-absorbed into an observable velocity that depends on the gas refractive index

$$v_{\mathrm{obs}}^2\sim 2ϵv^2$$

(53)

and is the very small velocity 5–10 km/s traditionally extracted from the classical analysis of the early experiments through the relation

$$v_{\mathrm{obs}}\sim 30\mathrm{km}/\mathrm{s}\sqrt{\frac{A_2^{\mathrm{EXP}}}{A_2^{\mathrm{class}}}}$$

(54)

see, e.g., Figure 3.

Thus, it is no surprise that the resulting $\frac{|\Delta {\overline{c}}_{\theta }|}{c}\sim ϵ(v^2/c^2)$ was much smaller than the classical expectation. For instance, in the old experiments in air (at room temperature and atmospheric pressure, where $ϵ\sim 2.8·{10}^{-4}$), a typical value was $\frac{|\Delta {\overline{c}}_{\theta }{|}_{\exp }}{c}\sim 3·{10}^{-10}$. By assuming the classical relations (52), (54), this was originally interpreted as an observable velocity $v_{obs}\sim 7.3$ km/s, but, by using Equation (53), this observable velocity would now correspond to a true kinematic velocity $v\sim 310$ km/s. Analogously, in the old experiment in gaseous helium (at room temperature and atmospheric pressure, where $ϵ\sim 3.3·{10}^{-5}$), a typical value was $\frac{|\Delta {\overline{c}}_{\theta }{|}_{\exp }}{c}\sim 2.2·{10}^{-11}$. This was classically interpreted as an (observable) velocity of 2 km/s but would now correspond to a true kinematical value $v\sim 240$ km/s.

Another observation concerns the time dependence of the data and the precise definition of v and $\theta =0$ in the above relations. Traditionally, for short-time observations of a few days, where there can be no sizeable change in the orbital motion of the Earth, the genuine signal for a preferred frame consisted of the regular modulations induced by the Earth’s rotation. Instead, the data had an irregular behavior indicating sizeably different directions of the drift at the same hour on consecutive days. This was a strong argument to interpret the small residuals as typical instrumental artifacts. However, this conclusion derives from the traditional identification of the local velocity field, which describes the drift, say $v_{\mu }\left(t\right)$, with the corresponding projection of the global Earth motion, say ${\tilde{v}}_{\mu }\left(t\right)$. This identification is equivalent to a form of regular, laminar flow where global and local velocity fields coincide and, in principle, may be incorrect.

The model of ether drift adopted in refs. [32,33,34,35] starts from Maxwell’s original argument [69]. After having considered all known properties of light, he was driven to consider the idea of a substratum: “…We are therefore obliged to suppose that the medium through which light is propagated is something distinct from the transparent media known to us…”. He was calling this substratum “ether”, while, today, we prefer to call it a “physical vacuum”. However, this is irrelevant. The essential point for the propagation of light, e.g., inside an optical cavity, is that, differently from the solid parts of the apparatus, this physical vacuum is not totally entrained with the Earth’s motion. Therefore, to explain the irregular character of the data, the original idea of refs. [70,71] was to model this vacuum as a turbulent fluid or, more precisely, as a fluid in the limit of zero viscosity.14 Then, the simple picture of a laminar flow is no more obvious due to the subtlety of the infinite-Reynolds-number limit; see, e.g., Section 41.5 in Vol. II of Feynman’s lectures [81]. Namely, besides $v_{\mu }\left(t\right)={\tilde{v}}_{\mu }\left(t\right)$, there is also another solution where $v_{\mu }\left(t\right)$ is a continuous, nowhere differentiable velocity field [82,83]. This leads to the idea of a signal with a fundamental stochastic nature, as when turbulence, at small scales, becomes homogeneous and isotropic. One should thus first analyze the data for $\frac{\Delta {\overline{c}}_{\theta }\left(t\right)}{c}$ and extract the (second-harmonic) phase and amplitude $A_2\left(t\right)$ by concentrating on the latter, which is positive-definite and remains non-zero under any averaging procedure.

For a quantitative description, let us assume a set of kinematic parameters $(V,\alpha ,\gamma )$ for the Earth’s cosmic motion, a latitude $\varphi$ of the laboratory and a given sidereal time $\tau ={\omega }_{\mathrm{sid}}t$ (with ${\omega }_{\mathrm{sid}}\sim \frac{2\pi }{{23}^h{56}^{\prime }}$). Then, $\tilde{v}\left(t\right)=V|sinz\left(t\right)|$ is the magnitude of the projection in the plane of the interferometer, $sinz\left(t\right)$ being defined by [84]

$$cosz\left(t\right)=sin\gamma sin\varphi +cos\gamma cos\varphi cos(\tau -\alpha )$$

(55)

In this scheme, the amplitude ${\tilde{A}}_2\left(t\right)$ associated with the global motion is

$${\tilde{A}}_2\left(t\right)\sim \frac{2D}{\lambda }·ϵ·\frac{V^2{sin}^{2}z\left(t\right)}{c^2}$$

(56)

Although the local, irregular $v_{\mu }\left(t\right)$ is not a differentiable function, it could still be simulated in terms of random Fourier series [82,85,86] This method was adopted in refs. [32,33,34,35] in a simple uniform probability model, where the kinematic parameters of the global ${\tilde{v}}_{\mu }\left(t\right)$ are merely used to fix the boundaries for the local random $v_{\mu }\left(t\right)$. The essential components are summarized in Appendix A.

We emphasize that the instantaneous, irregular $A_2\left(t\right)$ is very different from the smooth ${\tilde{A}}_2\left(t\right)$. However, the relation with the statistical average ${\langle A_2\left(t\right)\rangle }_{\mathrm{stat}}$ is very simple

$${\langle A_2\left(t\right)\rangle }_{\mathrm{stat}}\sim \frac{{\pi }^2}{18}·{\tilde{A}}_2\left(t\right)\sim \frac{2D}{\lambda }·\frac{{\pi }^2}{18}ϵ\frac{V^2{sin}^{2}z\left(t\right)}{c^2}$$

(57)

Furthermore, by using Equation (55), if the amplitude is measured at various sidereal times, one can also obtain information on the angular parameters $\alpha$ and $\gamma$.

Altogether, these old measurements $\frac{|\Delta {\overline{c}}_{\theta }{|}_{\exp }}{c}\sim 3·{10}^{-10}$ and $\frac{|\Delta {\overline{c}}_{\theta }{|}_{\exp }}{c}\sim 2.2·{10}^{-11}$, respectively, for air or gaseous helium at atmospheric pressure, can thus be interpreted in three different ways: (a) as 7.3 and 2 km/s, in a classical picture; (b) as 310 and 240 km/s, in a modern scheme and in a smooth picture of the drift; (c) as 418 and 324 km/s, in a modern scheme but now allowing for irregular fluctuations of the signal. In this last case, in fact, by replacing Equation (56) with Equation (57), from the same data, one would now obtain kinematical velocities that are larger by a factor $\sqrt{18/{\pi }^2}\sim 1.35$. In this third interpretation, the average of the two values agrees very well with the CMB velocity of 370 km/s. A comparison with all classical experiments is shown in

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 $ϵ=2.8·{10}^{-4}$ for air and $ϵ=3.3·{10}^{-5}$ 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	$(7.8\pm \dots )·{10}^{-3}$	$4·{10}^6$	$(0.7\pm 0.2)·{10}^{-3}$
Michelson–Morley (1887)	air	$(1.6\pm 0.6)·{10}^{-2}$	$4·{10}^7$	$(0.7\pm 0.2)·{10}^{-2}$
Morley–Miller (1902–1905)	air	$(4.0\pm 2.0)·{10}^{-2}$	$1.12·{10}^8$	$(2.0\pm 0.7)·{10}^{-2}$
Miller (1921–1926)	air	$(4.4\pm 2.2)·{10}^{-2}$	$1.12·{10}^8$	$(2.0\pm 0.7)·{10}^{-2}$
Tomaschek (1924)	air	$(1.0\pm 0.6)·{10}^{-2}$	$3·{10}^7$	$(0.5\pm 0.2)·{10}^{-2}$
Kennedy (1926)	helium	$<0.002$	$7·{10}^6$	$(1.4\pm 0.5)·{10}^{-4}$
Illingworth (1927)	helium	$(2.2\pm 1.7)·{10}^{-4}$	$7·{10}^6$	$(1.4\pm 0.5)·{10}^{-4}$
Piccard–Stahel (1928)	air	$(2.8\pm 1.5)·{10}^{-3}$	$1.28·{10}^7$	$(2.2\pm 0.8)·{10}^{-3}$
Mich.–Pease–Pearson (1929)	air	$(0.6\pm \dots )·{10}^{-2}$	$5.8·{10}^7$	$(1.0\pm 0.4)·{10}^{-2}$
Joos (1930)	helium	$(1.4\pm 0.8)·{10}^{-3}$	$7.5·{10}^7$	$(1.5\pm 0.6)·{10}^{-3}$

.

Notice the substantial difference with the analogous summary Table I of Ref. [87], where those authors were comparing the much larger classical amplitudes of Equation (52) and emphasizing the much smaller magnitude of the experimental fringes. Here, it is just the opposite. In fact, our theoretical statistical averages are often smaller than the experimental results, indicating, most likely, the presence of systematic effects in the measurements.

At the same time, by adopting Equation (57), from the experiments in air, we find ${\tilde{v}}_{\mathrm{air}}\sim 418\pm 62$ km/s, and from the two experiments in gaseous helium, ${\tilde{v}}_{\mathrm{helium}}\sim 323\pm 70$ km/s, with a global average $\langle \tilde{v}\rangle \sim 376\pm 46$ km/s, which agrees well with the 370 km/s from the CMB observations. Moreover, from the two most precise experiments of Piccard–Stahel (Bruxelles and Mt. Rigi in Switzerland)15 and of Joos (Jena)16, we find, in our stochastic scheme, two determinations, $\tilde{v}={360}_{-110}^{+85}$ km/s and $\tilde{v}={305}_{-100}^{+85}$ km/s, respectively, whose average $\langle \tilde{v}\rangle \sim {332}_{-80}^{+60}$ km/s reproduces to high accuracy the projection of the CMB velocity at a typical Central European latitude. Finally, by using Equation (55) and fitting the amplitudes obtained from Joos’ observations (data collected at steps of 1 h to cover the sidereal day), one finds [32,34] $\alpha \left(\mathrm{fit}-\mathrm{Joos}\right)=(168\pm 30)$ degrees and $\gamma \left(\mathrm{fit}-\mathrm{Joos}\right)=(-13\pm 14)$ degrees, which are consistent with the present values $\alpha \left(\mathrm{CMB}\right)\sim$ 168 degrees and $\gamma \left(\mathrm{CMB}\right)\sim -$7 degrees.

As it often happens, symmetry arguments can successfully describe a phenomenon regardless of the physical mechanisms behind it. The same is true here with our relation $\frac{|\Delta {\overline{c}}_{\theta }|}{c}\sim ϵ(v^2/c^2)$. It gives a consistent description of the data but does not explain the ultimate origin of the tiny observed anisotropy in the gaseous systems. For instance, as a first mechanism, we considered the possibility of different polarizations in different directions in the dielectric, depending on its state of motion. However, if this works in weakly bound gaseous matter, the same mechanism should also work in a strongly bound solid dielectric, where the refractivity is $({\mathcal{N}}_{\mathrm{solid}}-1)=O\left(1\right)$, and thus produce a much larger $\frac{|\Delta {\overline{c}}_{\theta }|}{c}\sim ({\mathcal{N}}_{\mathrm{solid}}-1)(v^2/c^2)\sim {10}^{-6}$. This is in contrast with the Shamir–Fox [91] experiment in perspex, where the observed value was smaller by several orders of magnitude. We have thus reconsidered [33,34,36] the traditional thermal interpretation [87,92] of the observed residuals. The idea was that, in a weakly bound system such as a gas, a small temperature difference $\Delta T^{\mathrm{gas}}\left(\theta \right)$, of a millikelvin or so, in the air of the two optical arms could produce a difference in the refractive index and a light anisotropy proportional to ${ϵ}_{\mathrm{gas}}\Delta T^{\mathrm{gas}}\left(\theta \right)/T$, where $T\sim$ 300 K is the temperature of the laboratory. Miller was aware of this potentially large effect [59,88] and objected that casual changes in the ambient temperature would largely cancel out when averaging over many measurements. Only temperature effects with a definite angular periodicity would survive. The overall consistency, in our scheme, of different experiments would now indicate that such $\Delta T^{\mathrm{gas}}\left(\theta \right)$ must have a non-local origin. As anticipated, this could be due to the non-zero momentum flow Equation (43), or it could reflect the interactions with the background radiation, which transfer a part of $\Delta T^{\mathrm{CMB}}\left(\theta \right)$ in Equation (46) and thus bring the gas out of equilibrium. Those old estimates were, however, slightly too large. In fact, in the ideal gas approximation, from the Lorentz–Lorenz equation for the molecular polarizability, one actually finds [33,34,35] $\Delta T^{\mathrm{gas}}\left(\theta \right)=(0.2-0.3)$ mK.17 For the CMB case, this would mean that the interactions of the gas with the CMB photons are so weak that, on average, the induced temperature differences in the optical paths were only 1/10 of the $\Delta T^{\mathrm{CMB}}\left(\theta \right)$ in Equation (46). Nevertheless, whatever its precise origin, this typical magnitude can help intuition. In fact, it can explain the quantitative reduction in the effect in the vacuum limit where ${ϵ}_{\mathrm{gas}}\to 0$ and the qualitative difference with solid dielectrics, where such small temperature differences cannot produce any appreciable deviation from isotropy in the remaining frame of the medium.

Now, admittedly, the idea that small modifications of the gaseous matter, produced by the tiny CMB temperature variations, can be detected by precise optical measurements in a laboratory, while certainly unconventional, has not the same implications of a genuine preferred-frame effect due to the vacuum structure. Nonetheless, this thermal explanation of the small residuals in gases has important predictive power. In fact, it implies that if a tiny, but non-zero, fundamental signal were definitely detected in vacuum, then, with very precise measurements, the same universal signal should also show up in a solid dielectric, where temperature differences of a fraction of a millikelvin become irrelevant. Detecting such “non-thermal” light anisotropy, for the same cosmic motion indicated by the CMB observations, would finally confirm the idea of the $\Sigma -$frame assumed in refs. [19,20,21,22,23].

3.3. The Modern Experiments in Vacuum and Solid Dielectrics

This expectation of a “non-thermal” light anisotropy, which could be detected in vacuum and in solid dielectrics, was then compared with the modern experiments, where $\frac{\Delta {\overline{c}}_{\theta }}{c}\sim \frac{\Delta \nu \left(\theta \right)}{{\nu }_0}$ is now extracted from the frequency shift of two optical resonators; see Figure 4.

By starting with vacuum resonators, after averaging many observations, the present limit is a residual $\langle \frac{\Delta {\overline{c}}_{\theta }}{c}\rangle ={10}^{-18}-{10}^{-19}$. However, this only reflects the very irregular nature of the signal because its typical instantaneous magnitude $\frac{|\Delta {\overline{c}}_{\theta }{\left(t\right)|}_v}{c}\sim {10}^{-15}$ is approximately 1000 times larger; see Figure 5. This ${10}^{-15}$ signal is found with vacuum resonators [97,98,99,100,101,102] made of different materials, operating at room temperature and/or in the cryogenic regime. As such, it cannot be interpreted as a spurious effect, e.g., thermal noise [103].

In the same model discussed for the classical experiments, we are thus led to the concept of a refractive index ${\mathcal{N}}_v$ for the vacuum or, more precisely, for the physical vacuum that is established in an apparatus placed on the Earth’s surface. This ${\mathcal{N}}_v$ should differ from unity at the ${10}^{-9}$ level, in order to give $\frac{|\Delta {\overline{c}}_{\theta }{\left(t\right)|}_v}{c}\sim ({\mathcal{N}}_v-1)(v^2\left(t\right)/c^2)\sim {10}^{-15}$, and thus would fit with Ref. [104], where, for an apparatus placed on the Earth’s surface, a vacuum refractivity ${ϵ}_v\sim (2G_{N}M/c^{2}R)\sim 1.4·{10}^{-9}$ was considered, $G_N$ being the Newton constant and M and R the mass and radius of the Earth. The idea is that, if the curvature observed in a gravitational field reflects local deformations of the physical space–time units and of the velocity of light [105], for an apparatus on the Earth’s surface, there could be a tiny difference with that ideal free-fall environment, which, in the presence of gravitational effects, is always assumed to define operationally the limit where the velocity of light in a vacuum $c_{\gamma }$ coincides with the parameter c of Lorentz transformations. This would reflect the physical difference, which, indeed, exists [104] between an observer in a true free-falling elevator and the modified situation of an observer that is in free fall in the same external potential but is now carrying onboard a heavy mass M; see Figure 6.

Therefore, if $\delta U$ is the extra Newtonian potential produced by the heavy mass M at the experimental setup, the vacuum refractivity, for system in Figure 6b, can be expressed as

$${ϵ}_v={\mathcal{N}}_v-1\sim \frac{\chi }{2}\left(\frac{2\left|\delta U\right|}{c^2}\right)$$

(58)

In General Relativity, one assumes $\chi =0$, while the two non-zero values, $\chi =$ 1 or 2, account for the two alternatives traditionally reported in the literature for the effective refractive index in a gravitational potential. For $\chi =2$, the resulting refractivity is the same as reported by Eddington [106] to explain in flat space the observed deflection of light in a gravitational field. A difference is found with Landau’s and Lifshitz’ textbook [107], where the vacuum refractive index entering the constitutive relations is instead defined as ${\mathcal{N}}_v\sim 1+\frac{\left|\delta U\right|}{c^2}$. We refer to Broekaert’s article [105], and in particular to his footnote 3, for a more detailed discussion of the two choices of $\chi$.

In our case, of an observer on the Earth’s surface, by introducing the Newton constant, the radius R and the mass M of the Earth, so that $\delta U=\frac{G_{N}M}{R}$, we can express the refractivity as

$${ϵ}_v\sim \frac{\chi }{2}1.4·{10}^{-9}$$

(59)

Referring to Ref. [35], here, we only report the comparison with Ref. [102], which, at present, is the most precise experiment in a vacuum. We compared this with the average instantaneous variation of the frequency shift over 1 s; see their Figure 3, bottom part. This is defined by the Root Square of the Allan Variance (RAV)18 (for ${\tau }_0\sim$ 1 s)

$${\left[{\sigma }_A(\Delta \nu ,{\tau }_0)\right]}_{\exp }\sim 0.24\mathrm{Hz}$$

(60)

or, in units of the reference frequency ${\nu }_0=2.8·{10}^{14}$ Hz (for ${\tau }_0\sim$ 1 s) (Ref. [102]),

$${\left[{\sigma }_A(\frac{\Delta \nu }{{\nu }_0},{\tau }_0)\right]}_{\exp }\sim 8.5·{10}^{-16}$$

(61)

As discussed in Ref. [35], our instantaneous, stochastic signal is, to very good approximation, pure white noise, for which the RAV coincides with the standard variance. At the same time, for a very irregular signal where $\langle \Delta \nu \rangle =0$, the standard variance $\sigma (\Delta \nu )$ coincides with the average magnitude $\langle |\Delta \nu |\rangle$. Therefore, since, in our stochastic model, the average magnitude of the dimensionless frequency shift $\frac{\Delta \nu \left(\theta \right)}{{\nu }_0}\sim \frac{\Delta {\overline{c}}_{\theta }}{c}$ is given in Equation (57), we find (for ${\tau }_0\sim$ 1 s)

$${\left[{\sigma }_A(\frac{\Delta \nu }{{\nu }_0},{\tau }_0)\right]}_{\mathrm{theor}}\sim {\left[\frac{\langle |\Delta \nu |\rangle }{{\nu }_0}\right]}_{\mathrm{theor}}\sim \frac{{\pi }^2}{18}·{ϵ}_v·\frac{V^2}{c^2}{sin}^{2}z\left(t\right)$$

(62)

In this way, by replacing Equation (59), and for a projection 250 km/s $\lesssim Vsinz\left(t\right)\lesssim$ 370 km/s, for ${\tau }_0\sim$ 1 s, our prediction for the RAV can finally be expressed as

$${\left[{\sigma }_A(\frac{\Delta \nu }{{\nu }_0},{\tau }_0)\right]}_{\mathrm{theor}}\sim \frac{\chi }{2}·(8.5\pm 3.5)·{10}^{-16}$$

(63)

By comparison with Equation (61), we see that the data definitely favor $\chi =2$, which is the only free parameter of our scheme. Moreover, the very good agreement with our simulated value indicates that, at least for an integration time of 1 s, the corrections to our model should be negligible.19

Let us now compare the modern experiments in solid dielectrics, in particular with the very precise Ref. [57]. This is a cryogenic experiment, with microwaves of 12.97 GHz, where almost all electromagnetic energy propagates in a medium, sapphire, with a refractive index of around 3 (at microwave frequencies). As anticipated, with a thermal interpretation of the residuals in gaseous media, we expect that a fundamental ${10}^{-15}$ vacuum anisotropy could also become visible here.

Following refs. [33,34,35], we first observe that for ${\mathcal{N}}_v=1+{ϵ}_v$, there will be a very tiny difference between the refractive index defined relatively to the ideal vacuum value c and the refractive index relatively to the physical isotropic vacuum value $c/{\mathcal{N}}_v$ measured on the Earth’s surface. The relative difference between these two definitions is proportional to ${ϵ}_v\lesssim {10}^{-9}$ and, for all practical purposes, can be ignored. All materials would now exhibit, however, the same background vacuum anisotropy. To this end, let us replace the average isotropic value

$$\frac{c}{{\mathcal{N}}_{\mathrm{solid}}}\to \frac{c}{{\mathcal{N}}_v{\mathcal{N}}_{\mathrm{solid}}}$$

(64)

and then use Equation (48) to replace ${\mathcal{N}}_v$ in the denominator with its $\theta -$dependent value

$${\overline{\mathcal{N}}}_v\left(\theta \right)\sim 1+{ϵ}_v{\beta }^2(1+{cos}^2\theta )$$

(65)

This is equivalent to defining a $\theta -$dependent refractive index for the solid dielectric

$$\begin{array}{c}\hfill \frac{{\overline{\mathcal{N}}}_{\mathrm{solid}}\left(\theta \right)}{{\mathcal{N}}_{\mathrm{solid}}}\sim 1+{ϵ}_v{\beta }^2(1+{cos}^2\theta )\end{array}$$

(66)

so that

$${\left[{\overline{c}}_{\gamma }\left(\theta \right)\right]}_{\mathrm{solid}}=\frac{c}{{\overline{\mathcal{N}}}_{\mathrm{solid}}\left(\theta \right)}\sim \frac{c}{{\mathcal{N}}_{\mathrm{solid}}}\left[1-{ϵ}_v{\beta }^2(1+{cos}^2\theta )\right]$$

(67)

with an anisotropy

$$\frac{{\left[\Delta {\overline{c}}_{\theta }\right]}_{\mathrm{solid}}}{\left[c/{\mathcal{N}}_{\mathrm{solid}}\right]}\sim {ϵ}_v{\beta }^{2}cos2\theta$$

(68)

In this way, a genuine vacuum effect, if present, could also be detected in a solid dielectric.

In Ref. [35], a detailed comparison with [57] was presented. First, from Figure 3c of [57], it was seen that the spectral amplitude of this particular apparatus becomes flat at frequencies $\omega \ge 0.5$ Hz, indicating that the white-noise branch of the signal reaches its minimum value for an integration time ${\tau }_0\sim$ 1 s (at which the other spurious disturbances are still negligible). These data for the spectral amplitude were then fitted to an analytic, power-law form to describe the lower-frequency part 0.001 Hz $\le \omega \le 0.5$ Hz, which reflects apparatus-dependent disturbances. This fitted spectrum was then used to generate a signal by Fourier transform. Finally, very long sequences of this signal were stored to produce a “colored” version of our basic white-noise signal.

To obtain a qualitative impression of the effect, we report in Figure 7 a sequence of our basic white-noise signal and a sequence of its colored version. By averaging over many 2000 s sequences of this type, the corresponding RAVs for the two signals are then reported in Figure 8. The experimental RAV extracted from Figure 3b of Ref. [57] is also reported (for the non-rotating setup). At this stage, the agreement of our simulated, colored signal with the experimental data remains satisfactory only up $\tau =$ 50 s. Reproducing the signal at larger $\tau$s will require further efforts, but this is not relevant here, our scope being merely to understand the modifications of our stochastic signal near the 1 s scale.

As one can check from Figure 3b of Ref. [57], the value of the experimental RAV for the fractional frequency shift (at ${\tau }_0=1$ s) is (Ref. [57])

$${\sigma }_A{(\frac{\Delta \nu }{{\nu }_0},{\tau }_0)}_{\exp }\sim 8.5·{10}^{-16}$$

(69)

This is precisely the same value of Equation (61) that we extracted from Ref. [102] after normalizing their experimental result, ${\sigma }_A{(\Delta \nu ,{\tau }_0)}_{\exp }\sim$ 0.24 Hz, to their laser frequency ${\nu }_0=2.8·{10}^{14}$ Hz. At the same time, it also agrees with our Equation (63) ${\sigma }_A{(\frac{\Delta \nu }{{\nu }_0},{\tau }_0)}_{\mathrm{theor}}=(8.5\pm 3.5)·{10}^{-16}$ for $\chi =2$. Therefore, this excellent agreement, between Ref. [102] (a vacuum experiment at room temperature) and Ref. [57] (a cryogenic experiment in a solid dielectric), on the one hand, and with our Equation (63), on the other hand, confirms our interpretation of the experiments in terms of a stochastic signal associated with the Earth’s cosmic motion within the CMB.

Two ultimate experimental checks still remain. First, one should try to detect our predicted, daily variations in Equation (63) in the range $(5-12)·{10}^{-16}$ corresponding to 250 km/s $\lesssim Vsinz\left(t\right)\lesssim$ 370 km/s. Due to the excellent systematics, these should remain visible with both experimental setups. Second, one more complementary test should be performed by placing the vacuum (or solid dielectric) optical cavities onboard a satellite, as in the OPTIS proposal [108]. In this ideal free-fall environment, as in panel (a) of our Figure 6, the typical instantaneous frequency shift should be much smaller (by orders of magnitude) than the corresponding ${10}^{-15}$ value measured with the same interferometers on the Earth’s surface.

4. Summary and Outlook

In this paper, we have considered one of the most controversial aspects of Quantum Mechanics, namely the apparent violation of Einstein locality and the conflict with (Einstein) relativity. Since the original paper by Einstein–Podolski–Rosen (EPR) [3] and through the work of Bell [4], many authors have thus arrived at the conclusion that, to dispose of the causality paradox in a realistic interpretation of the theory, it may be natural to introduce a special frame of reference $\Sigma$ for relativity. Then, one can consider the idea that some “Quantum Information” propagates at a vastly superluminal speed $v_{QI}\gg c$, with standard Quantum Mechanics corresponding to the $v_{QI}\to \infty$ limit. In this way, by comparison with experiments [19,20,21,22,23], one finds the lower bounds $v_{QI}>{10}^4-{10}^{6}c$ if the preferred $\Sigma -$frame is identified with the reference system where the Cosmic Microwave Background (CMB) is seen as isotropic, namely that particular system where the observed CMB kinematic dipole [18] vanishes exactly.

A frequent objection to this idea of a preferred frame is that, after all, Quantum Mechanics is not a fundamental description of the world. One should instead start from a fundamental QFT that incorporates the locality requirement. For this reason, we have tried to understand whether, from the perspective of an underlying, fundamental QFT, there could be a missing logical step that prevents the deduction that Einstein Special Relativity, with no preferred frame, is the physically realized version of relativity. Einstein Relativity is always assumed when computing S-matrix elements for elementary particle processes, but what one is actually using is the machinery of Lorentz transformations, whose first, complete derivation dates back to Larmor and Lorentz, who were assuming the existence of a fundamental state of rest (the ether).

In particular, in our opinion, an element missed so far derives from the present view of the lowest energy state in elementary particle theory, which is called a “vacuum” but should actually be thought of as a condensate of elementary quanta. Thus far, one has always been imposing the traditional constraint that, as far as local operators are concerned, the only possible vacuum expectation values are those which transform as world scalars under the Lorentz group.

However, as discussed in Section 2, this does not imply the vacuum to be a Lorentz-invariant state. This would rather require the vacuum to be annihilated by the generators of the Lorentz boosts. In four space–time dimensions, this requirement is only fulfilled for the free-field case, which, by definition, has a unique vacuum and where the simplest prescription of the Wick, normal ordering allows for a consistent representation of the Poincaré algebra. Instead, in the interacting theory and in the presence of Spontaneous Symmetry Breaking, one can meaningfully argue that the physically realized form of relativity contains a preferred reference system $\Sigma$.

However, since our arguments have not the status of a theorem, to decide if $\Sigma$ in fact exists, we then looked for definite experimental indications from the ether drift experiments by summarizing in Section 3 the extensive work of refs. [32,33,34,35,36], where all data from Michelson–Morley to the present experiments with optical resonators were considered. Ours is not the only possible scheme to analyze the data but, still, in this theoretical framework, which assumes the validity of Lorentz transformations and allows for irregular fluctuations of the signal, the long-sought $\Sigma -$frame tight to the CMB is naturally emerging.

Finally, before closing our paper, we will return to our starting point: the idea that, eventually, the non-locality of Quantum Mechanics could be understood as the consequence of some “Quantum Information” that propagates at a vastly superluminal speed $v_{QI}\gg c$. This was, after all, Bell’s conviction, namely that his result, combined with the EPR argument, implies non-local physical effects, and not simply correlations between distant events [25]. More specifically, if, as we have argued, the physical vacuum is in fact the ultimate origin of the $\Sigma -$frame, the hypothetical superluminal effects could be hidden somewhere in the physical structure of the condensed vacuum. To exploit this possibility, we will assume that this physical vacuum, however different from ordinary matter, is nevertheless a medium with a certain degree of substantiality. As such, it should exhibit density fluctuations. In this case, these density fluctuations would propagate with a speed $c_s\gg c$. We believe that, in the present context, this can be a relevant issue, even without a definite model where the previous $v_{QI}\gg c$ is directly related to a $c_s\gg c$.

To this end, we first recall that, as anticipated in the Introduction,“it is an open question whether $c_s/c$ remains less than unity when non-electromagnetic forces are taken into account” [10]. This is why superluminal sound has been meaningfully considered by several authors; see, e.g., refs. [109,110,111,112,113]. The point is that the sought non-local effect may derive from two different space–time regions. The first region is universal and is associated with the localization of the interacting particles, i.e., their Compton wavelength. This type of effect remains confined to microscopic distances. The second region, on the other hand, depends on the basic interaction, which, dealing with non-electromagnetic interactions, as in our case of a hard-sphere cutoff ${\mathsf{\Phi }}^4$ theory, could be instantaneous. Therefore, if each successive event leads to a small violation of causality, with a sufficiently long chain of scattering events, the effect could be amplified to macroscopic distances. Notice that we are not speaking of scattering events with single-particle propagation over large distances. In Bose condensates, each particle moves slowly back and forth of a very small amount and it only scatters with those particles which are immediately nearby. It is the coherent effect of these local scattering processes which propagates at much higher speed [114] and could produce, in principle, a faster-than-light sound wave.

With this premise, in a purely hydrodynamic description, valid over length scales much larger than the mean free path of the elementary constituents, an argument for superluminal sound could be the following. Let us consider the basic relation

$$P=-\mathcal{E}+n\frac{d\mathcal{E}}{dn}$$

(70)

which relates the pressure P and the energy density $\mathcal{E}$ in a medium of density n. By expanding the energy density around some given value $n=n_0$, we find

$$\mathcal{E}\sim \mathcal{E}\left(n_0\right)+A·(n-n_0)+\frac{1}{2}B·{(n-n_0)}^2$$

(71)

$$P\sim -\mathcal{E}\left(n_0\right)+A·n_0+\frac{1}{2}B·(n^2-n_0^2)$$

(72)

so that, in units of $c=1$, the speed of sound is

$$c_s^2=\frac{\partial P}{\partial \mathcal{E}}{|}_{\mathrm{n}={\mathrm{n}}_0}=\frac{dP/dn}{d\mathcal{E}/dn}{|}_{\mathrm{n}={\mathrm{n}}_0}=\frac{B{n}_0}{A}$$

(73)

By following Stevenson [115], one can envisage two different regimes: (a) the “empty vacuum” and (b) the “condensed vacuum”. Case (a) corresponds to a very small density of particles near the trivial empty state $n_0=0$ and is dominated by the rest mass term $\mathcal{E}\left(n\right)=mn+O\left(n^2\right)$. This limit has a vanishingly small speed of sound

$$c_s^2=\frac{\partial P}{\partial \mathcal{E}}{|}_{\mathrm{n}=0}=0$$

(74)

The situation changes substantially in the condensed vacuum, where the effective potential $V_{\mathrm{eff}}\left(\varphi \right)$ achieves its minimum at some $\varphi =\pm v$. Then, due to Equation (35), the energy density $\mathcal{E}\left[n\left(\varphi \right)\right]=V_{\mathrm{eff}}\left(\varphi \right)$ has its minimum at $n(\varphi =\pm v)\equiv n_v$, where now $A=0$. Therefore, the speed of sound is formally infinite

$$c_s^2=\frac{\partial P}{\partial \mathcal{E}}{|}_{n=n_v}=\frac{B{n}_v}{A}{|}_{\mathrm{A}=0}=+\infty$$

(75)

After this, Stevenson’s analysis [115] goes actually much farther, touching upon other aspects (such as shock waves, post-hydrodynamic approximations, etc.), which reach beyond the scope of our paper. We thus refer to Ref. [115] and also to Ref. [116] for more details.

Of course, the above elementary analysis cannot help to deduce a (potentially) infinitely large $v_{QI}\to \infty$ from the (potentially) infinitely large $c_s\to \infty$ of Equation (75). It only shows that the physical vacuum medium is incompressible or, better, that it can support density fluctuations whose wavelengths $\lambda$ will become larger and larger in the $c_s\to \infty$ limit in order that $c_s/\lambda$ can remain finite. Thus, with sound waves of such long wavelengths, it would be challenging to produce the sharp wavefronts needed for transmitting information.

Nonetheless, the argument indicates that the present view of the vacuum is probably too narrow because, regardless of “messages”, it is far from obvious that c, the speed of light in this type of vacuum, is a limiting speed. In this sense, it adds up to our discussion in Section 2, indicating that the idea of a Lorentz-invariant vacuum, and therefore of the overall consistency with Einstein Special Relativity without a preferred frame, is far from obvious.

Additionally, it also adds up to our analysis of the ether drift experiments in Section 3, indicating that the standard null interpretation of the data is, again, far from obvious once one starts to understand the observed, irregular nature of the signal (compare, e.g., the data in Figure 5 with our simulations in Figure A1 of Appendix A). The required conceptual effort is modest, we believe, if compared with the implications of the new perspective. In fact, by conducting a laser interferometry experiment, indoors inside a laboratory, from the remarkable agreement of the experimental results of Equations (61) and (69) with the theoretical predictions of Equations (62), (63) and (68), it is possible to perceive the motion of the solar system, of our galaxy, within the background radiation. Independently of the interpretation of relativity, this possibility of perceiving reality in a global way (precisely in a completely non-local way) is something fascinating. Clearly, this is implicit in the idea of revealing our motion with respect to a privileged reference system and fits well with the quantum view of correlations over arbitrarily large distances. However, this global vision of reality perhaps goes beyond quantum correlations: it seems to have to do, in a sense, with the quantum holographic principle [117] that all quantum information is “globalized”. Nonetheless, perhaps, it also has to do with the vision of the internal observer, i.e., the observer inside [118] the quantum system, meaning that he is located in a quantum space that is in a one-to-one relationship with the quantum computational system under consideration.