Casimir effect

Cause

The fluctuations Quantique S of the Vide are indeed present in all Quantum theory of the fields. The Casimir effect due to the fluctuations of the electromagnetic Champ, is described by the theory of the quantum electrodynamic .

Indeed, the energy of the “vacuum” between two plates is calculated by holding account only Photon S (and some other strange particles - virtual photons) of which the wavelengths are such as an integer of those represents the distance between the two plates ( $\ scriptstyle N \, \ lambda=L$ , where N is a positive entirety, λ the wavelength of a photon, and L the distance enters the two plates). What implies that the density of energy of the vacuum (between these two plates) is function of the number of photons which can exist between these two plates. The closer the plates are, the less there are photons obeying the rule $\ scriptstyle N \, \ lambda=L$ , because are excluded the photons of which the wavelength is higher than L. There is thus less energy. The force between these two plates (being the derivative of energy compared to L) is thus gravitational.

Energy of the vide

The Casimir effect is the result of the quantum theory of the fields, which indicates that all the fundamental fields, like the electromagnetic field, must be quantum in each point of space. In a very simple way, a physical field can be seen as if space were filled of balls and springs vibrating and inter-connected, and forces it field can be visualized like the displacement of a ball since a position at rest. The vibrations in this field are propagated and are controls by the equation of suitable wave for the particular field in question. The second quantum electromagnetic field of the quantum theory of the fields requires that each combination ball spring is quantum, i.e., that the force of the field will be quantum in each point of space. The field in each point of space is a simple harmonic oscillator. The excitations of the field correspond to elementary particles of particle physics. However, the vacuum has a complex structure. All calculations of the quantum theory of the fields must be made relative to this model of vacuum.

The vacuum has, implicitly, all the properties which a particle can have: the spin, polarization in the case of the light, energy etc On average, all these properties disappear: the vacuum is after all, “vacuum” in this direction. An important exception is the energy of the vacuum or the value hoped of the energy of the vacuum. The quantification of a simple harmonic oscillator indicates that the lowest possible energy or the energy of item zero that such an oscillator can have is

{E} = \ begin {matrix} \ frac {1} {2} \ end {matrix} \ hbar \ Omega \.

The office plurality of all the oscillators in all the points of space gives an infinite quantity. To remove this infinite, one can say that only the differences in energy are physically measurable; this principle is the base of the theory of Re-standardization In all practical calculations, it is always the manner that one treats the infinite one. In a major direction, however, Re-standardization is not satisfactory, and to eliminate this infinite is one of the challenges of the theory of the whole. There is currently no valid explanation saying to us, how this infinite must be treated, like primarily zero; a value different from zero is primarily the cosmological constant and all great value causes a problem in cosmology.

Expression of the force per unit of area

(Except remark, the effects edge are always neglected)

Dimensional analysis

Are two large metal plates plane of surface $S$ , parallels between them, and separated by a distance $L$ . It is supposed that, if the plates are rectangular with $\ scriptstyle S=D \ cdot H \,$ , spacing $L$ between the two parallel plates is small compared to the lengths $D$ and $H$ . One can then calculate a force per unit of area by neglecting the effects edges.

One supposes moreover than the plates are conducting perfect of electric Conductivité infinite, and than they are not charged. The relativistic effect being of quantum origin and , one expects that the force per unit of area of Casimir depends on the two fundamental constants $c$ (speed of light in the vacuum) and $\ hbar$ (quantum of action). Moreover, it is more than probable that the effect also depends on the distance $L$ between the plates. One thus postulates that the force per unit of area is written:

where $k$ is a pure number, without dimensions, and $\ alpha, \ beta, \ gamma$ three numbers to be determined. The dimensional Analyze gives the system of equations:

whose single solution is: $\ beta = \ gamma = 1$ and $\ alpha = - \, 4$ , are:

Exact result of Casimir

The exact calculation, made by Casimir in 1948, supposes a identically null Température thermodynamic: $T = 0$ K. It gives a nonnull value negative of the constant $k$ :

The minus sign indicates that this force is gravitational! The reader interested by this calculation will find it detailed in the article of review of Duplantier, corresponding if $\ alpha \ gg 1$ . Within this limit, one obtains:

the first in 1958 by Spaarnay. This experiment showed only one gravitational attraction which “is not in contradiction with the theoretical prediction of Casimir”. One can allot to this first experiment a bar of error of 100%.

the first experiment with the nonambiguous result goes back to 1978, and was carried out by van Blokland and Overbeeck. One can allot to this experiment a precision of about 25%.

the best current experiments check the theoretical prediction of Casimir with a precision of about 1%. On this level of precision, effects of imperfect reflection of the mirrors must be included in theoretical calculation.

NB: With the increase in the precision, one expects in the future to also observe effects due to a nonnull real temperature.

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