# What are the current experimental limits on the polarizability of the vacuum?

Many (all?) materials are polarizable at some level; meaning that when we apply an electric field across the material, there is an induced dipole moment $$\vec{p}=\alpha(\omega)\vec{E},$$ where $\alpha(\omega)$ is the polarizability of the material at frequency $\omega$, $\vec{E}$ is the applied electric field, and $\vec{p}$ is the induced polarization. If we assume that the electrons in the material are bound to their nuclei as simple harmonic oscillators (the standard physics assumption), then we find that the polarizability of the material is given by $$\alpha(\omega)=\frac{e^2/m}{\omega_0^2-\omega^2},$$ where $\omega_0$ contains the information about the stiffness of the binding. This leads to interesting optical phenomena in materials, one example of which is that it can be related to the index of refraction by $$n^2(\omega)=1+\frac{N}{\epsilon_0}\alpha(\omega),$$ where $N$ is the number density of the atoms and $\epsilon_0$ is the permittivity of free space. All of this is just background information.

My actual questions are: What are the experimental limits on the polarizibility of the vacuum, and which experiments set these limits? Is anyone actively trying to improve the limit? It is obviously expected to be identically zero, but we have been surprised by measurements in the past.

• lots of documents on PV at [calphysics] (calphysics.org/sci_articles.html) (I have a lot of links on PV but I dont have the time to search what is exp or theory), sorry – Helder Velez Feb 28 '14 at 0:13

The vacuum is polarizable. The polarization can be with respect to electric charge or color charge. In the presence of an electric field, virtual electron-positron pairs briefly exist (created from virtual photons of sufficient energy). The virtual pairs act as dipoles and orient with respect to the field. For example, near a proton, the virtual electron of such a pair will orient nearer the proton and the virtual positron further away.

The first experimental confirmation of vacuum polarization was through spectroscopy of the hydrogen atom. For the hydrogen atom, Dirac’s relativistic quantum mechanics predicted that the $2S_{\frac12}$ and $2P_{\frac12}$ energy levels should be equal. However, experimentally the difference corresponds to 1058MHz, first measured by Lamb in 1947. This energy level difference is the "Lamb Shift". The electron and proton of the hydrogen atom do not experience the classical Coulomb potential. The polarization screens points away from the proton from the full change that it would otherwise have. Because the S level has higher electron probability density at/near the proton, it experiences a less-screened proton charge. Vacuum polarization contributes -27Mhz to the Lamb shift.

As far as I know, electric polarization of the vacuum has not been experimentally measured across a range of applied frequencies point by point. However, through quantum electrodynamics, calculations of the Lamb Shift (including the vacuum polarization contribution) to increasingly high orders of virtual particle effects have been progressively made over the years, and compared to increasingly accurate experiments.

Current Theoretical Lamb Shift 1057.833 (4) MHz

Current Experimental Lamb Shift 1057.845 (3) MHz

Though vacuum polarization is a relatively small contribution to the Lamb Shift of ordinary hydrogen, it is the dominant contribution to the Lamb Shift in muonic hydrogen. Study of the Lamb Shift of muonic hydrogen is an active area of research.

Other experimental confirmations of vacuum polarization involve the anomalous electron magnetic moment (how much g deviates the value "2" predicted by Dirac theory) and Josephson junctions.

References:

http://isites.harvard.edu/fs/docs/icb.topic473482.files/16-vacuumpol.pdf

http://www.pha.jhu.edu/~rt19/hydro/node8.html

http://courses.washington.edu/phys432/lamb_shift/lamb_shift.pdf

http://arxiv.org/pdf/1208.2637v2.pdf

Heisenbeg and Euler (1936) http://arxiv.org/pdf/physics/0605038.pdf

Euwema and Wheeler (1956) http://journals.aps.org/pr/pdf/10.1103/PhysRev.103.803

• Interesting. My question was an experimental one though; has the effect been measured? What is the value of $\alpha$? – Chris Mueller Feb 26 '14 at 17:57
• My understanding is that experimental measurements of the Lamb Shift and anomalous magnetic moment of the electron are compared to predictions from quantum electrodynamics, and in this sense it is measured. Fig. 2 of Euwema and Wheeler (1956) plots vacuum polarization as a function of frequency. – DavePhD Feb 26 '14 at 18:58

The Casimir effect is used as experimental proof of the existence of the vacuum virtual exchanges.

The typical example is of two uncharged metallic plates in a vacuum, placed a few micrometers apart. In a classical description, the lack of an external field also means that there is no field between the plates, and no force would be measured between them. When this field is instead studied using the QED vacuum of quantum electrodynamics, it is seen that the plates do affect the virtual photons which constitute the field, and generate a net force2—either an attraction or a repulsion depending on the specific arrangement of the two plates. Although the Casimir effect can be expressed in terms of virtual particles interacting with the objects, it is best described and more easily calculated in terms of the zero-point energy of a quantized field in the intervening space between the objects. This force has been measured, and is a striking example of an effect captured formally by second quantization

italics mine.

Also the Casimir effect can be considered as polarization of the vacuum:

The Casimir effect is one of the most interesting manifestations of nontrivial prop- erties of the vacuum state in quantum field theory 1, 2. Since it's first prediction by Casimir in 1948 3 this effect is investigated for various cases of boundary geometries and various types of fields. The Casimir effect can be viewed as a polarization of vacuum by boundary conditions.

I found this review that refers also to the experiments done. It treats extensively polarization due to the effect .

And there seem to be proposals for checking the effect further experimentally . It seems to be important for nano technology devices and one cannot be more experimental than this.

So, since the Casimir effect is a measured effect and is described theoretically as a type of vacuum polarization there is not a limit, in the sense of a lower bound. For the numbers one would have to delve into the experimental papers .