Casimir Effect

Dutch Physicist Hendrik B. G. Casimir first proposed the existence of the force, and he formulated an experiment to detect it in 1948 while participating in research at Philips Research Labs. The classic form of his experiment used a pair of Uncharged parallel metal plates in a vacuum, and successfully demonstrated the force to within 15% of the value he had predicted according to his theory. The Van Der Waals Force between a pair of neutral Atom s is a similar effect. In modern Theoretical Physics , the Casimir effect plays an important role in the Chiral Bag Model of the Nucleon , and in applied physics, it is becoming of increasing importance in development of the ever-smaller, miniaturised components of emerging Micro- and Nano- technologies. OVERVIEW The Casimir effect can be understood by the idea that the presence of conducting metals and Dielectric s alter the Vacuum Expectation Value of the energy of the Electromagnetic Field . Since the value of this energy depends on the shapes and positions of the conductors and dielectrics, the Casimir effect manifests itself as a force between such objects. VACUUM ENERGY The Casimir effect is an outcome of Quantum Field Theory , which states that all of the various fundamental Fields , such as the Electromagnetic Field , must be quantized at each and every point in space. In a naïve sense, a field in physics may be envisioned as if space were filled with interconnected vibrating balls and springs, and the strength of the field can be visualized as the displacement of a ball from its rest position. Vibrations in this field propagate, and are governed by the appropriate Wave Equation for the particular field in question. The Second Quantization of quantum field theory requires that each such ball-spring combination be quantized, that is, that the strength of the field be quantized at each point in space. Canonically, the field at each point in space is a Simple Harmonic Oscillator , and its quantization places a Quantum Harmonic Oscillator at each point. Excitations of the field correspond to the Elementary Particle s of Particle Physics . However, as this picture shows, even the Vacuum has a vastly complex structure. All calculations of quantum field theory must be made in relation to this model of the vacuum. The vacuum has, implicitly, all of the properties that a particle may have: or the Vacuum Expectation Value of the energy. The quantization of a simple harmonic oscillator states that the lowest possible energy or Zero-point Energy that such an oscillator may have is : ${E} = \begin{matrix} rac{1}{2} \end{matrix} \hbar \omega \ .$ Summing over all possible oscillators at all points in space gives an infinite quantity. To remove this infinity, one may argue that only differences in energy are physically measurable; this argument is the underpinning of the theory of Renormalization . In all practical calculations, this is how the infinity is always handled. In a deeper sense, however, renormalization is unsatisfying, and the removal of this infinity presents a challenge in the search for a Theory Of Everything . As Of 2006 , there is no compelling explanation for how this infinity should be treated as essentially zero; a non-zero value is essentially the Cosmological Constant and any large value causes trouble in Cosmology . THE CASIMIR EFFECT Casimir's observation was that the Second-quantized , quantum electromagnetic field, in the presence of bulk bodies such as metals or Dielectric s, must obey the same Boundary Condition s that the classical electromagnetic field must obey. In particular, this affects the calculation of the vacuum energy in the presence of a Conductor or dielectric. Consider, for example, the calculation of the vacuum expectation value of the electromagnetic field inside a metal cavity, such as, for example, a Radar Cavity or a Microwave Waveguide . In this case, the correct way to sum the zero point energy of the field is to sum the energies of the Standing Wave s of the cavity. To each and every possible standing wave corresponds an energy; say the energy of the ''n''th standing wave is $E_n$ . The vacuum expectation value of the electromagnetic field in the cavity is then : $\langle E angle = rac{1}{2} \sum_n E_n$ with the sum running over all possible values of ''n'' enumerating the standing waves. The factor of 1/2 corresponds to the fact that the zero-point energies are being summed (it is the same 1/2 as appears in the equation $E=\hbar \omega/2$ ). Written in this way, this sum is clearly divergent; however, it can be used to create finite expressions. In particular, one may ask how the zero point energy depends on the shape ''s'' of the cavity. Each energy level $E_n$ depends on the shape, and so one should write $E_n(s)$ for the energy level, and $\langle E(s) angle$ for the vacuum expectation value. At this point comes an important observation: the force at point ''p'' on the wall of the cavity is equal to the change in the vacuum energy if the shape ''s'' of the wall is perturbed a little bit, say by $\delta s$ , at point ''p''. That is, one has : $F(p) = - \left. rac{\delta \langle E(s) angle} {\delta s} ightert_p\,$ Amazingly, this value is finite in many practical calculations. CASIMIR'S CALCULATION In the original calculation done by Casimir, he considered the space between a pair of conducting metal plates a distance ''a'' apart. In this case, the standing waves are particularly easy to calculate, since the transverse component of the electric field and the normal component of the magnetic field must vanish on the surface of a conductor. Assuming the parallel plates lie in the x-y plane, the standing waves are : $\psi_n(x,y,z,t) = e^{-i\omega_nt} e^{ik_xx+ik_yy} \sin \left( k_n z ight)$ where $\psi$ stands for the electric component of the electromagnetic field, and, for brevity, the Polarization and the magnetic components are ignored here. Here, $k_x$ and $k_y$ are the Wave Vector s in directions parallel to the plates, and : $k_n = rac{n\pi}{a}$ is the wave-vector perpendicular to the plates. Here, ''n'' is an integer, resulting from the requirement that ψ vanish on the metal plates. The energy of this wave is : $\omega_n = c \sqrt$ where ''c'' is the Speed Of Light . The vacuum energy is then the sum over all possible excitation modes : $\langle E angle = rac{\hbar}{2} \cdot 2 \int rac{dk_x dk_y}{(2\pi)^2} \sum_{n=-\infty}^\infty A\omega_n$ where ''A'' is the area of the metal plates, and a factor of 2 is introduced for the two possible polarizations of the wave. This expression is clearly infinite, and to proceed with the calculation, it is convenient to introduce a Regulator (discussed in greater detail below). The regulator will serve to make the expression finite, and in the end will be removed. The Zeta-regulated version of the energy per unit-area of the plate is : $rac{\langle E(s) angle}{A} = \hbar \int rac{dk_x dk_y}{(2\pi)^2} \sum_{n=-\infty}^\infty \omega_n ert \omega_nert^{-s}$ In the end, the limit $s o 0$ is to be taken. Here ''s'' is just a Complex Number , not to be confused with the shape discussed previously. This integral/sum is finite for ''s'' Real and larger than 3. The sum has a Pole at ''s''=3, but may be Analytically Continued to ''s''=0, where the expression is finite. Expanding this, one gets : $rac{\langle E(s) angle}{A} = rac{\hbar c^{1-s}}{4\pi^2} \sum_n \int_0^\infty 2\pi qdq \left ert q^2 + rac{\pi^2 n^2}{a^2} ightert^{(1-s)/2}$ where Polar Coordinates $q^2 = k_x^2+k_y^2$ were introduced to turn the Double Integral into a single integral. The integral is easily performed, resulting in : $rac{\langle E(s) angle}{A} = -rac {\hbar c^{1-s} \pi^{2-s}}{2a^{3-s}} rac{1}{3-s} \sum_n ert nert ^{3-s}$ The sum may be understood to be the Riemann Zeta Function , and so one has : $rac{\langle E angle}{A} = \lim_{s o 0} rac{\langle E(s) angle}{A} = -rac {\hbar c \pi^{2}}{6a^{3}} \zeta (-3)$ But $\zeta(-3)=1/120$ and so one obtains : $rac{\langle E angle}{A} = rac {-\hbar c \pi^{2}}{3 \cdot 240 a^{3}}$ The Casimir force per unit area $F_c / A$ for idealized, perfectly conducting plates with vacuum between them is : ${F_c \over A} = - rac{d}{da} rac{\langle E angle}{A} = -rac {\hbar c \pi^2} {240 a^4}$ where : $\hbar$ (hbar, ℏ) is the Reduced Planck Constant , : $c$ is the Speed Of Light , : $a$ is the Distance between the two plates. The force is negative, indicating that the force is attractive: by moving the two plates closer together, the energy is lowered. The presence of $\hbar$ shows that the Casimir force per unit area $F_c / A$ is very small, and that furthermore, the force is inherently of quantum-mechanical origin. MEASUREMENT One of the first experimental tests was conducted by Marcus Spaarnay at Philips in Eindhoven, in 1958, in a delicate and difficult experiment with parallel plates, obtaining results not in contradiction with the Casimir theory, but with large experimental errors. The Casimir effect was measured more accurately in 1997 by Steve K. Lamoreaux of Los Alamos National Laboratory and by Umar Mohideen of the University Of California At Riverside and his colleague Anushree Roy . In practice, rather than using two parallel plates, which would require phenomenally accurate alignment to ensure they were parallel, the experiments use one plate that is flat and another plate that is a part of a Sphere with a large Radius Of Curvature . In 2001, a group at the University Of Padua finally succeeded in measuring the Casimir force between parallel plates using microresonators. Further research has shown that, with materials of certain Permittivity and Permeability , or with a certain configuration, the Casimir effect could be made repulsive instead of attractive, although there are no experimental demonstrations of these predictions. REGULARIZATION In order to be able to perform calculations in the general case, it is convenient to introduce a Regulator in the summations. This is an artificial device, used to make the sums finite so that they can be more easily manipulated, followed by the taking of a limit so as to remove the regulator. The Heat Kernel or Exponentially regulated sum is
	:<math>\langle E(t) Angle	rac{1}{2} \sum_n \hbar \omega_n
	:<math>\langle E(s) Angle	rac{1}{2} \sum_n \hbar \omega_n \omega_n^{-s}</math>

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Casimir Effect