Probing the Vacuum of Particle Physics with Precise Laser Interferometry

Abstract

The discovery of the Higgs boson at LHC confirms that what we experience as empty space should actually be thought as a condensate of elementary quanta. This condensate characterizes the physically realized form of relativity and could play the role of preferred reference frame in a modern Lorentzian approach. This observation suggests a new interpretative scheme to understand the unexplained residuals in the old ether-drift experiments where light was still propagating in gaseous systems. Differently from present vacuum experiments, where anyhow deviations from Special Relativity are expected to be at the limit of visibility, these now acquire a crucial importance and become consistent with the Earth’s velocity of 370 km/s which characterizes the CMB anisotropy. In the same scheme, one can also understand the difference with the other experiments where light propagates in strongly bound systems such as solid or liquid transparent media. This non-trivial level of consistency motivates a new generation of precise laser interferometry experiments which explore the same particle physics vacuum and, in this sense, are complementary to those with high-energy accelerators.

Appendices

Appendix 1

Let us consider light propagation in a gaseous system of refractive index $\mathcal{N}$. By assuming isotropy, the time $t$ spent by refracted light to cover some given distance $L$ within the medium is $t=\mathcal{N}L/c$. This can be expressed as the sum of $t_0=L/c$ and $t_1=(\mathcal{N}-1)L/c$ where $t_0$ is the same time as in the vacuum and $t_1$ represents the additional, average time by which refracted light is slowed down by the presence of matter. If there are convective currents, due to the motion of the laboratory with respect to a preferred reference frame $\Sigma$, then $t_1$ will be different in different directions, and there will be an anisotropy of the velocity of light proportional to $(\mathcal{N}-1)$. In fact, let us consider light propagating in a 2-dimensional plane and express $t_1$ as

$$\begin{aligned} t_1={{L}\over {c}}f(\mathcal{N}, \theta , \beta ) \end{aligned}$$

(46)

with $\beta =V/c$, $V$ being (the projection on the considered plane of) the relevant velocity with respect to $\Sigma$ where the isotropic form

$$\begin{aligned} f(\mathcal{N}, \theta , 0)=\mathcal{N}-1 \end{aligned}$$

(47)

is assumed. By expanding around $\mathcal{N}=1$ where, whatever $\beta$, $f$ vanishes by definition, one finds for gaseous systems (where $\mathcal{N}-1 \ll 1$) the universal trend

$$\begin{aligned} f(\mathcal{N}, \theta ,\beta )\sim (\mathcal{N}-1)F(\theta ,\beta ) \end{aligned}$$

(48)

with

$$\begin{aligned} F(\theta ,\beta )\equiv (\partial f/\partial \mathcal{N})|_{ \mathcal{N}=1} \end{aligned}$$

(49)

and $F(\theta ,0)=1$. Therefore, by introducing the one-way velocity of light

$$\begin{aligned} t(\mathcal{N},\theta ,\beta )= {{L}\over {c_\gamma (\mathcal{N},\theta ,\beta )}}\sim {{L}\over {c}}+ {{L}\over {c}}(\mathcal{N}-1)~F(\theta ,\beta ) \end{aligned}$$

(50)

one gets

$$\begin{aligned} c_\gamma (\mathcal{N},\theta ,\beta )\sim {{c}\over { \mathcal{N} }}~ \left[ 1- (\mathcal{N}-1) ~(F(\theta ,\beta ) -1)\right] \end{aligned}$$

(51)

Analogous relations hold for the two-way velocity $\bar{c}_\gamma (\mathcal{N},\theta ,\beta )$

$$\begin{aligned} \bar{c}_\gamma (\mathcal{N},\theta ,\beta )&= {{2~c_\gamma (\mathcal{N},\theta ,\beta )c_\gamma (\mathcal{N},\pi +\theta ,\beta )}\over {c_\gamma (\mathcal{N},\theta ,\beta ) +c_\gamma (\mathcal{N},\pi +\theta ,\beta )}}\nonumber \\&\,\,\sim {{c}\over { \mathcal{N} }} \left[ 1- (\mathcal{N}-1) ~\left( {{F(\theta ,\beta ) + F(\pi +\theta ,\beta )}\over {2}} -1\right) \right] \end{aligned}$$

(52)

A more explicit expression can be obtained by exploring some general properties of the function $F(\theta ,\beta )$. By expanding in powers of $\beta$

$$\begin{aligned} F(\theta ,\beta )-1 = \beta F_1(\theta ) + \beta ^2 F_2(\theta )+ \cdots \end{aligned}$$

(53)

and taking into account that, by the very definition of two-way velocity, $\bar{c}_\gamma (\mathcal{N},\theta ,\beta )= \bar{c}_\gamma (\mathcal{N},\theta ,-\beta )$, it follows that $F_1(\theta )=-F_1(\pi + \theta )$. Therefore, to $\mathcal{O}(\beta ^2)$, by expressing the combination $F_2(\theta ) + F_2(\pi +\theta )$ as an infinite expansion of even-order Legendre polynomials, we get the general structure

$$\begin{aligned} \bar{c}_\gamma (\mathcal{N},\theta ,\beta ) \sim {{c}\over { \mathcal{N} }} \left[ 1- (\mathcal{N}-1)~\beta ^2 \sum ^\infty _{n=0}\zeta _{2n}P_{2n}(\cos \theta ) \right] \end{aligned}$$

(54)

which coincides with Eq. (17).

Appendix 2

To derive Eq. (18), let us first consider a dielectric medium of refractive index $\mathcal{N}$ whose container is at rest in $\Sigma$, the reference frame defined by imposing zero spatial momentum for the vacuum condensation phenomenon. For an observer at rest in this reference frame, light propagation within the medium is assumed to be isotropic and described by

$$\begin{aligned} \pi _\mu \pi _\nu \gamma ^{\mu \nu }= 0 \end{aligned}$$

(55)

where

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

(56)

and $\pi _\mu$ denotes the light 4-momentum vector for the $\Sigma$ observer. Let us now consider that the container of the medium is moving with some velocity $\mathbf{V}$ with respect to $\Sigma$ and is at rest in some other frame $S'$. By analogy, light propagation within the medium for the observer in $S'$ will be described by

$$\begin{aligned} p_\mu p_\nu g^{\mu \nu }= 0 , \end{aligned}$$

(57)

where $p_\mu \equiv (E/c,\mathbf{p})$ and $g^{\mu \nu }$ denote respectively the light 4-momentum and the effective metric for $S'$. On this basis, by introducing the $S'$ dimensionless velocity 4-vector $u^\mu \equiv (u^0,\mathbf{V}/c)$ (with $u_\mu u^\mu =1$), let us define $A^{\mu }{_\nu }=A^{\mu }{_\nu }(u_\mu ,\mathcal{N})$ to be the transformation matrix which produces $g^{\mu \nu }$ from the reference isotropic metric $\gamma ^{\mu \nu }$, i.e.

$$\begin{aligned} g^{\mu \nu }=A^{\mu }{_\sigma }A^{\nu }{_\rho }\gamma ^{\sigma \rho } \end{aligned}$$

(58)

In this context, requiring consistency between vacuum condensation and Special Relativity corresponds to place all reference frames on the same footing and assume $g^{\mu \nu }=\gamma ^{\mu \nu }$ or (SR=Special Relativity)

$$\begin{aligned} \left[ A^{\mu }{_\nu }(u_\mu ,\mathcal{N})\right] _\mathrm{SR}=\delta ^{\mu }_{\nu } \end{aligned}$$

(59)

In Special Relativity, where there is no preferred reference frame, this identification is independent of the physical nature of the medium, being valid for gaseous systems as well as for liquid or solid transparent media.

On the other hand, let us now consider the case where the medium inside the container becomes more and more rarefied, i.e. let us take the $\mathcal{N} \rightarrow 1$ limit. Here, the usual identification of the transformation matrix is instead

$$\begin{aligned} A^{\mu }{_\nu }(u_\mu ,\mathcal{N}=1)=\Lambda ^{\mu }_{\nu } \end{aligned}$$

(60)

$\Lambda ^{\mu }_{\nu }$ being the Lorentz transformation matrix from $\Sigma$ to $S'$. In fact, this is the standard choice to solve non-trivially the equation $g^{\mu \nu }=\gamma ^{\mu \nu }$ when light propagates in vacuum and $\gamma ^{\mu \nu }$ reduces to the Minkowski tensor $\eta ^{\mu \nu }$. But then, by continuity, it is conceivable that, to describe the case $\mathcal{N}=1+ \epsilon$ of gaseous media, the right transformation matrix is closer to Eq. (60) rather than to Eq. (59). Otherwise, it would be very hard to justify the sudden change from Eq. (60) to

$$\begin{aligned} \left[ A^{\mu }{_\nu }(u_\mu ,\mathcal{N}=1+\epsilon )\right] _\mathrm{SR}=\delta ^{\mu }_{\nu } \end{aligned}$$

(61)

for any $\epsilon > 0$ and for any value of the $S'$ velocity.

Notice however that no contradiction arises once $\mathcal{N}$ starts to differ substantially from unity. In this other limit, one can reconcile Eqs. (59) and (60). As discussed in Sects. 1 and 2, the possibility of two different regimes is in agreement with the idea of a tiny vacuum energy–momentum flow associated with a Lorentz non-invariant vacuum state. This flow, acting as an effective thermal gradient, could induce convective currents of the gas molecules along the light path and a non-zero anisotropy of refracted light. The point is that, differently from strongly bound systems, such as solid or liquid transparent media, where $\mathcal{N}$ differs substantially from unity, the elementary constituents of such weakly bound matter can be set in motion by an arbitrarily small thermal gradient. This picture also agrees with the phenomenological pattern of experiments [21, 22] which indicates the existence of two different regimes. A former region of gaseous systems where $\mathcal{N}\sim 1$ and a latter region where the difference of $\mathcal{N}$ from unity is substantial, e.g. $\mathcal{N}\sim 1.5$ as with perspex in the experiment by Shamir and Fox [23]. Although it would be difficult to describe in a fully quantitative way the transition between the two regimes, some simple arguments can be given along the lines suggested by de Abreu and Guerra (see pages 165–170 of Ref. [51]).

Thus we conclude that, for $\mathcal{N}=1+ \epsilon$ and with a Lorentz non-invariant vacuum, it makes more sense to compute $g^{\mu \nu }$ through the relation

$$\begin{aligned} g^{\mu \nu }=\Lambda ^{\mu }{_\sigma }\Lambda ^{\nu }{_\rho }\gamma ^{\sigma \rho } \end{aligned}$$

(62)

rather than setting $g^{\mu \nu }= \gamma ^{\mu \nu }$. This gives the effective metric for $S'$ ($\kappa =\mathcal{N}^2 - 1$)

$$\begin{aligned} g^{\mu \nu }= \eta ^{\mu \nu } + \kappa u^\mu u^\nu \end{aligned}$$

(63)

In this way, Eq. (57) gives a photon energy ($u^2_0=1 + \mathbf{V}^2/c^2$)

$$\begin{aligned} E(| {\mathbf {p}}| , \theta )= c~{{ -\kappa u_0 \zeta + \sqrt{ |{\mathbf {p}}|^2(1+\kappa u^2_0) - \kappa \zeta ^2 }}\over { 1 + \kappa u^2_0}} \end{aligned}$$

(64)

with

$$\begin{aligned} \zeta ={\mathbf {p}}\cdot {{{\mathbf {V}}}\over {c}}= |{\mathbf {p}}|\beta \cos \theta , \end{aligned}$$

(65)

where $\beta ={{|{\mathbf {V}}|}\over {c}}$ and $\theta \equiv \theta _\mathrm{lab}$ indicates the angle defined, in the laboratory $S'$ frame, between the photon momentum and ${\mathbf {V}}$. By using the above relation, one gets the one-way velocity of light

$$\begin{aligned}&{{E(| {\mathbf {p}}| , \theta )}\over {|{\mathbf {p}}|}}= c_\gamma (\theta )= c~{{ -\kappa \beta \sqrt{1+\beta ^2} \cos \theta + \sqrt{ 1+ \kappa + \kappa \beta ^2 \sin ^2\theta } } \over {1+ \kappa (1+\beta ^2)}} . \end{aligned}$$

(66)

or to $\mathcal{O}(\kappa )$ and $\mathcal{O}(\beta ^2)$

$$\begin{aligned} c_\gamma (\theta ) \sim {{c} \over {\mathcal{N}}}~\left[ 1- \kappa \beta \cos \theta - {{\kappa }\over {2}} \beta ^2(1+\cos ^2\theta )\right] \end{aligned}$$

(67)

From this one can compute the two-way velocity ($\kappa \sim 2(\mathcal{N} - 1) \equiv 2\epsilon$)

$$\begin{aligned} \bar{c}_\gamma (\theta )&= {{ 2 c_\gamma (\theta ) c_\gamma (\pi + \theta ) }\over { c_\gamma (\theta ) + c_\gamma (\pi + \theta ) }} \nonumber \\&\sim {{c} \over {\mathcal{N}}}~\left[ 1-\epsilon \beta ^2\left( 2 - \sin ^2\theta \right) \right] \end{aligned}$$

(68)

which, as anticipated, is a special form of the more general Eq. (17).⁷

One conceptual detail concerns the gas refractive index whose reported values are experimentally measured on Earth by two-way measurements. For instance, for air the most precise determinations are at the level $10^{-7}$, say $\mathcal{N}_\mathrm{air}=1.0002926..$ at STP (Standard Temperature and Pressure). By assuming a non-zero anisotropy in the Earth’s frame, one should interpret the isotropic value $c/\mathcal{N_\mathrm{air}}$ as an angular average of Eq.(68), i.e.

$$\begin{aligned} {{c}\over { \mathcal{N}_\mathrm{air} }}\equiv \langle \bar{c}_\gamma (\bar{\mathcal{N}}_\mathrm{air},\theta ,\beta )\rangle _\theta = {{c}\over { \bar{\mathcal{N}} _\mathrm{air} }} ~[1-{{3}\over {2}} (\bar{\mathcal{N}}_\mathrm{air} -1)\beta ^2] \end{aligned}$$

(69)

From this relation, one can determine in principle the unknown value $\bar{\mathcal{N}} _\mathrm{air} \equiv \mathcal{N}(\Sigma )$ (as if the gas were at rest in $\Sigma$), in terms of the experimentally known quantity $\mathcal{N}_\mathrm{air}\equiv \mathcal{N}(Earth)$ and of $V$. In practice, for the standard velocity values involved in most cosmic motions, say $V \sim$ 300 km/s, the difference between $\mathcal{N}(\Sigma )$ and $\mathcal{N}(Earth)$ is at the level $10^{-9}$ and thus completely negligible. The same holds true for the other gaseous systems at STP (say nitrogen, carbon dioxide, helium,..) for which the present experimental accuracy in the refractive index is, at best, at the level $10^{-6}$. Finally, the isotropic two-way speed of light is better determined in the low-pressure limit where $(\mathcal{N}-1)\rightarrow 0$. In the same limit, for any given value of $V$, the approximation $\mathcal{N}(\Sigma )=\mathcal{N}(Earth)$ becomes better and better.