In quantum field theory, the Casimir effect (or Casimir force) is a physical force that acts on large surfaces inside a confined space. This force happens because of tiny changes in a field caused by quantum mechanics. Sometimes, the force is described as "Casimir pressure" when measured as force spread over an area. The effect is named after the Dutch physicist Hendrik Casimir, who first predicted it for electromagnetic systems in 1948.
In the same year, Casimir and Dirk Polder described a similar force acting on a neutral atom near a large surface. This force is called the Casimir–Polder force. Their work connects to the London–van der Waals force, which describes attraction between atoms, and includes effects from the time it takes light to travel, since light moves at a finite speed. These forces share the same basic scientific principles.
In 1997, an experiment by Steven K. Lamoreaux measured the Casimir force directly and found it to be within 5% of the value predicted by theory.
The Casimir effect happens because materials like electrical conductors and dielectrics change the average energy level of the electromagnetic field in empty space. Since this energy depends on the shape and position of the materials, the effect creates a force between objects.
Any medium that allows waves or vibrations to move can have a version of the Casimir effect. For example, beads on a string or plates in moving water or gas show similar forces.
In modern physics, the Casimir effect is important in models of particles like protons. In applied physics, it influences developments in microtechnology and nanotechnology.
Physical properties
A common example involves two uncharged conductive plates placed in a vacuum and separated by a few nanometers. In a classical explanation, the absence of an external field means no field exists between the plates, and no force acts between them. However, when studied using quantum electrodynamic vacuum theory, the plates influence the virtual photons that make up the field, creating a net force—either attraction or repulsion, depending on the plates' arrangement. Although the Casimir effect can be described in terms of virtual particles interacting with objects, it is most accurately explained and calculated using the zero-point energy of a quantized field in the space between the objects. This force has been measured and is a clear example of an effect described by second quantization.
The way boundary conditions are treated in these calculations is debated. Casimir's original goal was to calculate the van der Waals force between polarizable molecules in the conductive plates. This means the effect can be understood without referencing the zero-point energy (vacuum energy) of quantum fields.
The force weakens quickly as the distance between objects increases, making it measurable only at very small distances. At these scales, the Casimir effect becomes the main force between uncharged conductors. For example, at distances of 10 nanometers—about 100 times the size of an atom—the Casimir effect creates pressure similar to 1 atmosphere (the exact value depends on the shape of the surfaces and other factors).
History
In 1947, Dutch scientists Hendrik Casimir and Dirk Polder at Philips Research Labs suggested that a force exists between two atoms that can be influenced by electric fields and between such an atom and a conducting plate. This specific type of force is known as the Casimir–Polder force. After discussing the idea with physicist Niels Bohr, who mentioned it might be related to zero-point energy, Casimir developed a theory in 1948 that predicted a force between neutral conducting plates. This phenomenon is called the Casimir effect.
Later, predictions about the force were expanded to include materials that do not conduct electricity perfectly, such as certain metals and dielectrics. Scientists also studied the force in more complex shapes. Before 1997, experiments showed the force in a general way, and indirect evidence supporting the Casimir energy was found by measuring the thickness of liquid helium films. In 1997, an experiment by Lamoreaux directly measured the force with accuracy within 5% of the theoretical prediction. Later experiments improved this accuracy to a few percent.
Possible causes
The Casimir effect is explained by quantum field theory, which states that all fundamental fields, such as the electromagnetic field, must be divided into small, discrete parts at every point in space. In a simple way, a "field" can be thought of as space filled with interconnected vibrating balls and springs. The strength of the field is shown by how far a ball moves from its resting position. Vibrations in the field move according to specific rules that describe wave motion. Quantum field theory requires that each of these ball-spring systems be divided into small, discrete parts, meaning the strength of the field at every point in space is also divided into small parts. At the most basic level, the field at each point in space behaves like a simple harmonic oscillator, and its division into small parts creates a quantum harmonic oscillator at each point. Excitations in the field represent the elementary particles studied in particle physics. However, even the vacuum, or empty space, has a complex structure, so all calculations in quantum field theory must consider this model of the vacuum.
The vacuum has properties that particles may have, such as spin, polarization (for light), and energy. On average, most of these properties cancel out, making the vacuum "empty" in this sense. One exception is the vacuum energy, or the average energy of the vacuum. The division of a simple harmonic oscillator into small parts shows that the lowest possible energy, called zero-point energy, is a specific value. Adding up this energy for all possible oscillators at every point in space results in an infinite value. However, only differences in energy are physically measurable (except for gravity, which is not fully explained by quantum field theory). This infinity is a feature of the math used, not the actual physics. This idea is the basis for the theory of renormalization, a method developed in the 1970s to handle infinities in calculations.
When considering gravity, the infinite vacuum energy remains a challenge. Scientists have not yet found a clear explanation for why this energy does not produce a cosmological constant much larger than what is observed. Since a complete theory of quantum gravity is still missing, there is no clear reason why the observed value of the cosmological constant exists.
The Casimir effect for fermions can be described as the imbalance in the properties of the fermion operator, known as the Witten index.
A 2005 paper by Robert Jaffe from MIT explains that the Casimir effect and the forces it creates can be calculated without using zero-point energy. These forces are quantum forces between charges and currents. The Casimir force between parallel plates decreases as the fine structure constant (a measure of electromagnetic interactions) approaches zero, and the standard result, which seems unrelated to this constant, applies when the constant is very large. The Casimir force is also described as a type of van der Waals force between metal plates. Casimir and Polder originally used this method to calculate the Casimir–Polder force. In 1978, Schwinger, DeRadd, and Milton used a similar approach to explain the Casimir effect between two parallel plates. More recently, Nikolic showed from the basic principles of quantum electrodynamics that the Casimir force does not come from the vacuum energy of the electromagnetic field. Instead, the force's microscopic origin is linked to van der Waals forces.
Effects
Casimir observed that the quantum electromagnetic field, when interacting with solid objects like metals or dielectrics, must follow the same boundary conditions as the classical electromagnetic field. This rule is important for calculating the vacuum energy near conductors or dielectrics.
For example, consider calculating the average energy of the electromagnetic field inside a metal cavity, such as a radar cavity or a microwave waveguide. To find the zero-point energy, scientists calculate the total energy of all possible standing waves inside the cavity. Each standing wave has a specific energy, labeled as Eₙ. The average energy of the field is then calculated as:
⟨E⟩ = (1/2) × (sum of all Eₙ)
This means the average energy is half the total energy of all standing waves. The factor of 1/2 comes from the zero-point energy of each wave, which is half the energy of the wave itself. This sum is very large, but it can still be used to create useful, finite results in calculations.
The dependence of zero-point energy on the shape of the cavity is important. Each energy level Eₙ depends on the cavity’s shape, so it is written as Eₙ(s). The average energy is then written as ⟨E(s)⟩. A key point is that the force on a part of the cavity wall depends on how the vacuum energy changes if the wall’s shape is slightly altered. This relationship is expressed as:
F(p) = – (change in ⟨E(s)⟩ / change in shape s) at point p
This value is often finite in practical calculations.
The attraction between plates can be understood by looking at a simple one-dimensional case. Imagine a movable conductive plate placed very close to one of two plates that are far apart. When the plates are very close, the space between them is narrow, and the energy levels of the electromagnetic field are far apart. In the larger space between the two plates, there are many energy levels that are closely spaced.
If the narrow distance between the plates decreases slightly (da), the energy of the wave in the narrow space increases by an amount proportional to -da/a. At the same time, the energy of the many waves in the larger space decreases by an amount proportional to -da/l. These two effects mostly cancel each other, but the energy in the larger space is slightly higher than the energy in the narrow space. This leads to a net decrease in energy, causing the plates to move closer together. The result is an attractive force between the plates.
Derivation of Casimir effect assuming zeta-regularization
In Casimir's original calculation, he studied the space between two parallel conducting metal plates separated by a distance "a." In this situation, the standing waves are easier to calculate because the electric field and magnetic field must meet specific conditions at the surface of the conductor. Assuming the plates are parallel to the xy-plane, the standing waves can be described as:
ψₙ(x, y, z; t) = e⁻ⁱωₙᵗ eⁱᵏₓˣ⁺ⁱᵏᵧʸ sin(kₙz),
where ψ represents the electric component of the electromagnetic field, and the magnetic component is not included here for simplicity. The variables kₓ and kᵧ are the wavenumbers in directions parallel to the plates, and kₙ = nπ/a is the wavenumber perpendicular to the plates. Here, "n" is an integer that ensures the wave vanishes on the metal plates. The frequency of the wave is:
ωₙ = c√(kₓ² + kᵧ² + (n²π²)/a²),
where "c" is the speed of light. The vacuum energy is calculated by summing over all possible wave modes. Since the plates are large, the sum is simplified by integrating over two dimensions in k-space. Using periodic boundary conditions, the average energy becomes:
⟨E⟩ = (ℏ/2) × 2 × ∫ [A dkₓ dkᵧ / (2π)²] × ∑ₙ=₁^∞ ωₙ,
where "A" is the area of the plates, and a factor of 2 accounts for the two possible polarizations of the wave. This expression is infinite, so a regulator is introduced to make it finite. The regulator is a mathematical tool used to handle the infinite sum, and it is removed after the calculation.
The zeta-regulated energy per unit area is:
⟨E(s)⟩/A = ℏ × ∫ [dkₓ dkᵧ / (2π)²] × ∑ₙ=₁^∞ ωₙ |ωₙ|⁻ˢ,
where "s" is a complex number. For real values of "s" greater than 3, the expression is finite. However, the sum has a pole (a point where the value becomes infinite) at s = 3, but it can be extended to s = 0, where the result is finite. After simplifying the expression using polar coordinates (q = √(kₓ² + kᵧ²)), the energy becomes:
⟨E(s)⟩/A = (ℏc¹⁻ˢ / 4π²) × ∑ₙ ∫₀^∞ 2πq dq |q² + (π²n²)/a²|⁰·⁵⁻⁰·⁵ˢ.
This integral converges when the real part of "s" is greater than 3. The final result, after taking the limit as s approaches 0, is:
⟨E⟩/A = −(ℏcπ² / 720a³).
This result shows that the vacuum energy between the plates is negative, which leads to an attractive force between the plates. The Casimir force per unit area is:
F_c/A = −(ℏcπ² / 240a⁴),
where:
– ℏ is the reduced Planck constant,
– c is the speed of light,
– a is the distance between the plates.
The negative sign indicates the force is attractive, meaning the energy decreases as the plates are brought closer together. The presence of ℏ confirms the force has a quantum-mechanical origin.
The energy required to separate the plates to infinity can be calculated by integrating the force:
U_E(a) = ∫ F(a) da = (ℏcπ² A) / (720a³).
Casimir's original work considered a movable plate placed near one of two plates separated by a large distance "L." He calculated the zero-point energy on both sides of the plate. Instead of using the zeta-regulation method, he used Euler-Maclaurin summation to handle the infinite sums.
Later, Evgeny Lifshitz extended Casimir's results to real materials, such as dielectrics and metals. Lifshitz's theory accounts for the material properties of the plates, such as finite conductivity, and can be used to calculate the Casimir force numerically. For large separations (much greater than the skin depth of the material), Lifshitz's theory reduces to Casimir's idealized 1/a force law. For small separations, it matches the London dispersion force (1/a law with a Hamaker constant).
Lifshitz's approach was later generalized to multilayer materials, anisotropic materials, and magnetic materials. For non-planar geometries, calculations were limited to a few idealized cases until the 2010s, when numerical methods from computational electromagnetics were adapted to compute Casimir forces for arbitrary shapes and materials.
Measurement
In 1958, Marcus Sparnaay at Philips in Eindhoven, Netherlands, performed one of the first experiments to test the Casimir effect. His experiment used parallel plates and produced results that did not conflict with Casimir theory, but the results had large errors due to the difficulty of the experiment.
In 1997, Steve K. Lamoreaux from Los Alamos National Laboratory and Umar Mohideen and Anushree Roy from the University of California, Riverside, measured the Casimir effect more accurately. Instead of using two parallel plates, which require extremely precise alignment, the experiments used a flat plate and a curved plate with a very large radius.
In 2001, a group of scientists at the University of Padua in Italy, including Giacomo Bressi, Gianni Carugno, Roberto Onofrio, and Giuseppe Ruoso, successfully measured the Casimir force between parallel plates using microresonators. These experiments were later summarized in a 2009 review by Klimchitskaya.
In 2013, a group of scientists from Hong Kong University of Science and Technology, University of Florida, Harvard University, Massachusetts Institute of Technology, and Oak Ridge National Laboratory created a compact silicon chip to measure the Casimir force. The chip was designed using electron-beam lithography and did not require additional alignment, making it useful for studying the Casimir force in complex shapes. In 2017 and 2021, the same group used this chip to demonstrate non-monotonic Casimir force and distance-independent Casimir force, respectively.
Regularization
To handle complex calculations, scientists often use a tool called a regulator. This tool helps control sums that would otherwise grow too large to manage. After using the regulator, scientists take a limit to remove it and return to the original problem.
One type of regulator is the heat kernel, which is written as ⟨E(t)⟩ = 1/2 × sum of ℏ|ωₙ| × e^(-t|ωₙ|). Here, t is a small number, and the final result is found by taking the limit as t approaches zero. For three-dimensional spaces, this sum often includes a term that grows as 1/t³, which represents an infinite part. This infinite part depends only on the size of the space, not its shape. The remaining finite part depends on the shape of the space.
Another regulator is the Gaussian type, written as ⟨E(t)⟩ = 1/2 × sum of ℏ|ωₙ| × e^(-t²|ωₙ|²). This form is easier for numerical calculations because it converges quickly, but it is harder to use in theoretical work. Other smooth regulators can also be used.
The zeta function regulator is written as ⟨E(s)⟩ = 1/2 × sum of ℏ|ωₙ| × |ωₙ|^{-s}. This method is not useful for numerical calculations but is helpful in theory. Divergences in this case appear as points where the function becomes undefined (poles) in the complex s plane, such as at s = 4. These poles can be removed through a process called analytic continuation, allowing the sum to be evaluated at s = 0.
Not all cavity shapes produce a finite part or shape-independent infinite parts. When this happens, additional physical factors must be considered. For example, at very high frequencies (above the plasma frequency), metals become transparent to light like X-rays, and dielectrics limit frequencies. These natural limits act as regulators. Similar effects in solid-state physics help keep calculations finite, as described in detailed studies like Landau and Lifshitz's "Theory of Continuous Media."
Generalities
The Casimir effect can also be studied using advanced math tools from quantum field theory, such as functional integrals. However, these methods are very abstract and hard to understand. They can only be used for the simplest shapes. Still, quantum field theory shows that calculations involving the vacuum expectation value are similar to adding up the effects of so-called "virtual particles."
It is also important to understand that the total energy of standing waves can be thought of as the sum of energy levels from a Hamiltonian. This idea helps explain atomic and molecular forces, like the Van der Waals force, as a type of Casimir effect. In this view, the Hamiltonian of a system depends on how objects, such as atoms, are arranged in space. Changes in the zero-point energy caused by these arrangements lead to forces between the objects.
In the chiral bag model of the nucleon, the Casimir energy is key to showing that the nucleon's mass does not depend on the size of the bag. Also, the imbalance in energy levels (spectral asymmetry) is linked to a non-zero average value of the baryon number, which cancels out the topological winding number of the pion field around the nucleon.
A "pseudo-Casimir" effect occurs in liquid crystal systems. Here, fixed walls that set the direction of the liquid crystal molecules create a long-range force, similar to the force between conducting plates.
Dynamical Casimir effect
The dynamical Casimir effect is the creation of particles and energy from a mirror that moves rapidly. This phenomenon was first predicted by mathematical solutions to quantum mechanics equations developed in the 1970s. In May 2011, scientists at Chalmers University of Technology in Gothenburg, Sweden, announced they had detected the dynamical Casimir effect. In their experiment, they produced microwave photons from empty space inside a superconducting microwave resonator. To achieve this, they used a modified SQUID to change the resonator’s length over time, simulating a mirror moving at a speed close to the speed of light. If confirmed, this would be the first experimental proof of the dynamical Casimir effect. In March 2013, a study published in the PNAS journal described an experiment that demonstrated the dynamical Casimir effect using a Josephson metamaterial. In July 2019, another study detailed an experiment showing evidence of the optical dynamical Casimir effect in a dispersion-oscillating fiber. In 2020, Frank Wilczek and others proposed a solution to the information loss paradox linked to the moving mirror model of the dynamical Casimir effect. This effect, studied within the framework of quantum field theory in curved spacetime, has helped scientists better understand the Unruh effect.
Repulsive forces
In some cases, the Casimir effect can create repulsive forces between objects that are not charged. Evgeny Lifshitz, a scientist, showed through theory that repulsive forces can occur under specific conditions, especially with liquids. This discovery has led to interest in using the Casimir effect for creating levitating devices. An experiment by Munday et al. confirmed Lifshitz's prediction, calling it "quantum levitation." Some scientists suggest using gain media for levitation, but this is debated because these materials might break basic rules about cause and effect and thermodynamic balance. Casimir and Casimir-Polder repulsion can occur for objects with highly directional electrical properties. For more information on this topic, refer to Milton et al. Recent research shows that chiral materials, like a special type of lubricant, can create repulsive, adjustable Casimir forces. Scientists Q.-D. Jiang and Frank Wilczek have studied this. In 1968, Timothy Boyer found that a spherical conductor can show repulsive forces, and the size doesn't affect the result. Later studies show that carefully selected dielectric materials can produce repulsive forces.
Speculative applications
The Casimir forces may be used in nanotechnology, especially in silicon-based micro- and nanoelectromechanical systems (MEMS) and devices called Casimir oscillators.
In 1995 and 1998, Maclay and others created the first models of a MEMS that included Casimir forces. These models did not use the Casimir force to perform work, but they showed that the Casimir effect must be considered when designing MEMS in the future. The Casimir effect could be a major cause of a problem called stiction failure, where parts of a MEMS stick together and stop working properly.
In 2001, Capasso and others demonstrated how the Casimir force can control the movement of a MEMS device. They attached a thin polysilicon plate to a small twisting bar. When they placed a metal-coated ball near the plate, the Casimir force pulled the plate, causing it to rotate. They also made the plate move back and forth, and found that the Casimir force slowed the movement and caused unusual behaviors, such as hysteresis and bistability. These results matched predictions made by scientists using math and theory.
The Casimir effect shows that quantum field theory allows energy in very small spaces to be lower than normal vacuum energy. However, this energy cannot be extremely low because the theory stops working at atomic distances. Scientists like Stephen Hawking and Kip Thorne have suggested that these effects might help stabilize a type of wormhole that could be used for travel.