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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Casimir effect | 2/6 | https://en.wikipedia.org/wiki/Casimir_effect | reference | science, encyclopedia | 2026-05-05T10:54:44.616020+00:00 | kb-cron |
The causes of the Casimir effect are described by 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 simplified view, 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. At the most basic level, 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 particles of particle physics. However, even the vacuum has a vastly complex structure, so 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: spin, polarization in the case of light, energy, and so on. On average, most of these properties cancel out: the vacuum is, after all, "empty" in this sense. One important exception is the vacuum energy 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
=
1
2
ℏ
ω
.
{\displaystyle {E}={\tfrac {1}{2}}\hbar \omega \,.}
Summing over all possible oscillators at all points in space gives an infinite quantity. Since only differences in energy are physically measurable (with the notable exception of gravitation, which remains beyond the scope of quantum field theory), this infinity may be considered a feature of the mathematics rather than of the physics. This argument is the underpinning of the theory of renormalization. Dealing with infinite quantities in this way was a cause of widespread unease among quantum field theorists before the development in the 1970s of the renormalization group, a mathematical formalism for scale transformations that provides a natural basis for the process. When the scope of the physics is widened to include gravity, the interpretation of this formally infinite quantity remains problematic. There is currently no compelling explanation as to why it should not result in a cosmological constant that is many orders of magnitude larger than observed. However, since we do not yet have any fully coherent quantum theory of gravity, there is likewise no compelling reason as to why it should instead actually result in the value of the cosmological constant that we observe. The Casimir effect for fermions can be understood as the spectral asymmetry of the fermion operator (−1)F, where it is known as the Witten index.
=== Relativistic van der Waals force === Alternatively, a 2005 paper by Robert Jaffe of MIT states that "Casimir effects can be formulated and Casimir forces can be computed without reference to zero-point energies. They are relativistic, quantum forces between charges and currents. The Casimir force (per unit area) between parallel plates vanishes as alpha, the fine structure constant, goes to zero, and the standard result, which appears to be independent of alpha, corresponds to the alpha approaching infinity limit", and that "The Casimir force is simply the (relativistic, retarded) van der Waals force between the metal plates." Casimir and Polder's original paper used this method to derive the Casimir–Polder force. In 1978, Schwinger, DeRadd, and Milton published a similar derivation for the Casimir effect between two parallel plates. More recently, Nikolic proved from first principles of quantum electrodynamics that the Casimir force does not originate from the vacuum energy of the electromagnetic field, and explained in simple terms why the fundamental microscopic origin of Casimir force lies in van der Waals forces.
== Effects == Casimir's observation was that the second-quantized quantum electromagnetic field, in the presence of bulk bodies such as metals or dielectrics, must obey the same boundary conditions 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 find the zero-point energy of the field is to sum the energies of the standing waves of the cavity. To each and every possible standing wave corresponds an energy; say the energy of the nth standing wave is En. The vacuum expectation value of the energy of the electromagnetic field in the cavity is then
⟨
E
⟩
=
1
2
∑
n
E
n
{\displaystyle \langle E\rangle ={\tfrac {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 is present because the zero-point energy of the nth mode is 1/2En, where En is the energy increment for the nth mode. (It is the same 1/2 as appears in the equation E = 1/2ħω.) 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 En depends on the shape, and so one should write En(s) for the energy level, and ⟨E(s)⟩ 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 δs, at p. That is, one has