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Quantum fluctuation
Quantum fluctuation
Quantum fluctuation
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3D visualization of quantum fluctuations of the quantum chromodynamics (QCD) vacuum[1]

In quantum physics, a quantum fluctuation (also known as a vacuum state fluctuation or vacuum fluctuation) is the temporary random change in the amount of energy in a point in space,[2] as prescribed by Werner Heisenberg's uncertainty principle. They are minute random fluctuations in the values of the fields which represent elementary particles, such as electric and magnetic fields which represent the electromagnetic force carried by photons, W and Z fields which carry the weak force, and gluon fields which carry the strong force.[3]

The uncertainty principle states the uncertainty in energy and time can be related by[4] , where ⁠1/2⁠ħ ≈ 5.27286×10−35 J⋅s. This means that pairs of virtual particles with energy and lifetime shorter than are continually created and annihilated in empty space. Although the particles are not directly detectable, the cumulative effects of these particles are measurable. For example, without quantum fluctuations, the "bare" mass and charge of elementary particles would be infinite; from renormalization theory the shielding effect of the cloud of virtual particles is responsible for the finite mass and charge of elementary particles.

Another consequence is the Casimir effect. One of the first observations which was evidence for vacuum fluctuations was the Lamb shift in hydrogen. In July 2020, scientists reported that quantum vacuum fluctuations can influence the motion of macroscopic, human-scale objects by measuring correlations below the standard quantum limit between the position/momentum uncertainty of the mirrors of LIGO and the photon number/phase uncertainty of light that they reflect.[5][6][7]

Field fluctuations

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In quantum field theory, fields undergo quantum fluctuations. A reasonably clear distinction can be made between quantum fluctuations and thermal fluctuations of a quantum field (at least for a free field; for interacting fields, renormalization substantially complicates matters). An illustration of this distinction can be seen by considering relativistic and non-relativistic Klein–Gordon fields:[8] For the relativistic Klein–Gordon field in the vacuum state, we can calculate the propagator that we would observe a configuration at a time t in terms of its Fourier transform to be

In contrast, for the non-relativistic Klein–Gordon field at non-zero temperature, the Gibbs probability density that we would observe a configuration at a time is

These probability distributions illustrate that every possible configuration of the field is possible, with the amplitude of quantum fluctuations controlled by the Planck constant , just as the amplitude of thermal fluctuations is controlled by , where kB is the Boltzmann constant. Note that the following three points are closely related:

  1. the Planck constant has units of action (joule-seconds) instead of units of energy (joules),
  2. the quantum kernel is instead of (the relativistic quantum kernel is nonlocal differently from the non-relativistic classical heat kernel, but it is causal),[citation needed]
  3. the quantum vacuum state is Lorentz-invariant (although not manifestly in the above), whereas the classical thermal state is not (both the non-relativistic dynamics and the Gibbs probability density initial condition are not Lorentz-invariant).

A classical continuous random field can be constructed that has the same probability density as the quantum vacuum state, so that the principal difference from quantum field theory is the measurement theory (measurement in quantum theory is different from measurement for a classical continuous random field, in that classical measurements are always mutually compatible – in quantum-mechanical terms they always commute).

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Quantum fluctuation

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Quantum fluctuation is a fundamental phenomenon in and (QFT), arising from the , which prohibits the simultaneous precise knowledge of a system's and the time over which it is measured, or its position and momentum. This inherent uncertainty results in temporary, random variations in at any point in empty space, even in the vacuum's , manifesting as the brief creation and annihilation of virtual particle-antiparticle pairs. These fluctuations underscore that the quantum vacuum is not truly empty but a dynamic of potentiality, with non-zero that permeates all space. However, despite the presence of this zero-point energy, vacuum fluctuations cannot be harnessed to produce net extractable work for energy production, as any apparent energy gain is balanced by input energy, consistent with conservation laws. In QFT, quantum fluctuations are intrinsic to every quantum field, where the of the field is zero, but the variance—measuring fluctuation amplitude—is non-zero, leading to effects like and the divergent vacuum energy problem. Observable consequences include the , first predicted by Hendrik Casimir in 1948, in which two closely spaced, uncharged conducting plates experience an attractive force due to the modification of vacuum fluctuations between them, restricting the wavelengths of virtual photons compared to outside the plates. This force has been experimentally verified and scales inversely with the fourth power of the plate separation, providing direct evidence of quantum vacuum dynamics. Other manifestations appear in , such as the in energy levels, caused by interactions with vacuum fluctuations. Quantum fluctuations play a pivotal role in cosmology, particularly during the inflationary epoch of the early , where microscopic fluctuations in the are amplified by rapid exponential expansion to macroscopic scales, seeding the perturbations that evolve into galaxies and large-scale cosmic . Pioneering calculations by Viatcheslav Mukhanov and Gennady Chibisov in 1981 demonstrated how these quantum-origin perturbations produce a nearly scale-invariant power spectrum, consistent with observations. In 2015, experiments at NIST using superconducting circuits measured fluctuations in a mechanical cooled to its , revealing half a quantum of motion and bridging quantum theory with macroscopic reality. More recent work, such as University's 2025 experiments harnessing vacuum fluctuations in cavities to engineer , continues to explore these effects on mesoscopic scales.

Fundamental Principles

Definition and Origin

Quantum fluctuations represent temporary, random variations in the energy at a point in space, stemming from the fundamentally probabilistic interpretation of . These fluctuations arise because quantum systems cannot have precisely defined values for both energy and time simultaneously, as dictated by the Heisenberg uncertainty principle. This principle, formulated by in 1927, implies that empty space itself is not static but subject to inherent instabilities on microscopic scales. The concept originated in the 1920s amid the development of , particularly through Heisenberg's , which emphasized non-commuting observables and laid the groundwork for understanding quantum indeterminacy. It was further refined with Heisenberg's explicit statement of the in 1927, highlighting how such indeterminacy leads to unavoidable fluctuations in physical quantities. By the 1940s, these ideas were integrated into (QFT), where fluctuations were formalized as properties of the quantum vacuum, building on the quantization of fields pioneered by in the late 1920s and advanced through renormalization techniques by , , and Sin-Itiro Tomonaga. In contrast to classical fluctuations, such as thermal noise from particle agitation in gases or liquids, quantum fluctuations are not driven by temperature or statistical ensembles but persist even at and in the absence of matter. Classical average to zero as temperature approaches zero , whereas quantum ones remain due to the of quantum fields, ensuring perpetual activity in the . A key manifestation is the creation of short-lived virtual particles, which briefly "borrow" energy from the vacuum in violation of classical energy conservation, only to annihilate and repay it within the brief timescale permitted by the energy-time uncertainty relation.

Relation to Uncertainty Principle

The energy-time uncertainty relation, expressed as ΔEΔt≥ℏ/2\Delta E \Delta t \geq \hbar / 2, where ΔE\Delta E represents the in and Δt\Delta t the in time, fundamentally permits brief deviations from in . This relation allows to "borrow" for short durations Δt\Delta t, enabling transient processes that would be forbidden in , such as the appearance of virtual excitations. The energy-time uncertainty relation arises from the dynamics of and is often interpreted as relating the in to the timescale over which the system changes appreciably, such as the lifetime of an unstable state or the duration of a measurement. This principle is in nature for many applications, including the existence of virtual particles, and can be formally derived in specific contexts using or by considering the spread in measurement times. A concrete illustration is the creation of a virtual particle-antiparticle pair, where the pair's rest energy provides ΔE≈2[m](/page/Mass)c2\Delta E \approx 2 [m](/page/Mass) c^2 (with mm the particle and cc the ). Such a pair can then persist for a time Δt≈ℏ/(2ΔE)\Delta t \approx \hbar / (2 \Delta E), after which it must annihilate to restore energy balance on average. Proposed by in 1927, this principle demonstrates why quantum fluctuations are intrinsic and unavoidable, in stark contrast to the strict, deterministic in .

Theoretical Description

Vacuum Fluctuations in QFT

In (QFT), the vacuum state is defined as the unique state |0⟩ that is annihilated by all annihilation operators â_k, satisfying â_k |0⟩ = 0 for every mode k, representing the absence of real particles. Despite this, the vacuum possesses a non-zero arising from the sum of ground-state contributions (1/2)ℏω_k over all possible field modes, reflecting inherent quantum fluctuations. This concept of the vacuum as a fluctuating entity was formalized in the late through the by in 1927, who introduced the idea of field oscillators with zero-point energies. Concurrently, , along with and , developed the matrix mechanics framework for quantized fields in 1925–1926, while Jordan and established key commutation relations for field operators in 1928, laying the groundwork for relativistic QFT. The theory evolved significantly in the 1940s with the reformulation of (QED) by , , and Sin-Itiro Tomonaga, who addressed infinities in through techniques. Quantum fields in the vacuum state exhibit perpetual oscillations, characterized by fluctuations δφ(x,t) that denote deviations from the average field value ⟨φ(x,t)⟩ = 0. These fluctuations stem from the applied to field operators, allowing temporary energy borrowings that manifest as virtual excitations across all frequencies and momenta. As the of the QFT Hamiltonian, the vacuum's diverges due to the infinite number of modes in continuous , but physical predictions are rendered finite through , which subtracts unobservable infinities by redefining parameters like charge and mass. Unlike descriptions emphasizing transient particle-antiparticle pairs, vacuum fluctuations primarily describe the intrinsic quantum variability of the fields themselves, with particle-like behavior emerging as specific excitations of these fields.

Mathematical Formalism

In , the mathematical description of quantum fluctuations begins with the quantization of a free ϕ(x)\phi(x), which is expanded in terms of . For a real in , the field operator is expressed as ϕ(x)=∫d3k(2π)3ℏ2ωk[ake−iωkt+ik⋅x+ak†eiωkt−ik⋅x],\phi(x) = \int \frac{d^3 k}{(2\pi)^3} \sqrt{\frac{\hbar}{2 \omega_k}} \left[ a_{\mathbf{k}} e^{-i \omega_k t + i \mathbf{k} \cdot \mathbf{x}} + a_{\mathbf{k}}^\dagger e^{i \omega_k t - i \mathbf{k} \cdot \mathbf{x}} \right],
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