Quantum fluctuation

A quantum fluctuation is the temporary appearance of energetic particles out of empty space, as allowed by the uncertainty principle. The uncertainty principle states that for a pair of conjugate variables such as position/momentum or energy/time, it is impossible to have a precisely determined value of each member of the pair at the same time. For example, a particle pair can pop out of the vacuum during a very short time interval.

An extension is applicable to the "uncertainty in time" and "uncertainty in energy" (including the rest mass energy ${\displaystyle mc^{2}}$). When the mass is very large like a macroscopic object, the uncertainties and thus the quantum effect become very small, and classical physics is applicable.

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 quantum and classical Klein-Gordon fields: For the quantized Klein–Gordon field in the vacuum state, we can calculate the probability density that we would observe a configuration ${\displaystyle {\displaystyle \varphi _{t}(x)}}$ at a time ${\displaystyle t}$ in terms of its Fourier transform ${\displaystyle {\displaystyle {\tilde {\varphi }}_{t}(k)}}$ to be

${\displaystyle \rho _{0}[\varphi _{t}]=\exp {\left[-{\frac {1}{\hbar }}\int {\frac {d^{3}k}{(2\pi )^{3}}}{\tilde {\varphi }}_{t}^{*}(k){\sqrt {|k|^{2}+m^{2}}}\;{\tilde {\varphi }}_{t}(k)\right]}.}$

In contrast, for the classical Klein–Gordon field at non-zero temperature, the Gibbs probability density that we would observe a configuration ${\displaystyle {\displaystyle \varphi _{t}(x)}}$ at a time ${\displaystyle t}$ is

${\displaystyle \rho _{E}[\varphi _{t}]=\exp {[-H[\varphi _{t}]/k_{\mathrm {B} }T]}=\exp {\left[-{\frac {1}{k_{\mathrm {B} }T}}\int {\frac {d^{3}k}{(2\pi )^{3}}}{\tilde {\varphi }}_{t}^{*}(k){\scriptstyle {\frac {1}{2}}}(|k|^{2}+m^{2})\;{\tilde {\varphi }}_{t}(k)\right]}.}$

These probability distributions illustrate that every possible configuration of the field is possible, with the amplitude of quantum fluctuations controlled by Planck's constant ${\displaystyle \hbar }$, just as the amplitude of thermal fluctuations is controlled by ${\displaystyle k_{\mathrm {B} }T}$, where ${\displaystyle k_{\mathrm {B} }}$ is Boltzmann's constant. Note that the following three points are closely related:

1. Planck's constant has units of action (joule-seconds) instead of units of energy (joules),
2. the quantum kernel is ${\displaystyle {\sqrt {|k|^{2}+m^{2}}}}$ instead of ${\displaystyle {\scriptstyle {\frac {1}{2}}}(|k|^{2}+m^{2})}$ (the quantum kernel is nonlocal from a classical heat kernel viewpoint, but it is local in the sense that it does not allow signals to be transmitted),
3. the quantum vacuum state is Lorentz invariant (although not manifestly in the above), whereas the classical thermal state is not (the classical dynamics is Lorentz invariant, but the Gibbs probability density is not a Lorentz invariant initial condition).

We can construct a classical continuous random field 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). Quantum effects that are consequences only of quantum fluctuations, not of subtleties of measurement incompatibility, can alternatively be models of classical continuous random fields.

The success of quantum fluctuation theories have given way to metaphysical interpretations on the nature of reality and their potential role in the origin and structure of the universe:

• The fluctuations are a manifestation of the innate uncertainty on the quantum level[3]
• Fluctuations of the fields in each element of our universe's spacetime could be coherent throughout the universe by mesoscopic quantum entanglement.
• A fundamental particle arising out of its quantum field is always inescapably subject to this reality and is thus describable by an associated wave function.

The wave function of a quantum particle represents the reality of the innate quantum fluctuations at the core of the universe and bestows the particle its counterintuitive quantum behavior.

In the double slit experiment each particle makes an unpredictable choice between alternative possibilities, consistent with an interference pattern with the inherent fluctuations of the underlying quantum field rendering the electron to do so.[4]

Such an underlying immutable quantum field by which quantum fluctuations are correlated in a universal scale may explain the non-locality of quantum entanglement as a natural process.[5]

As it has been recently demonstrated, charged extended particles can experience self-oscillatory dynamics as a result of classical electrodynamic self-interactions.[6] This trembling motion has a frequency that is closely related to the zitterbewegung frequency appearing in Dirac's equation. The mechanism producing these fluctuations arises because some parts of an accelerated charged corpuscle emit electromagnetic perturbations that can affect another part of the body, producing self-forces.

Using the Liénard-Wiechert potential as solutions to Maxwell's equations with sources, it can be shown that these forces can be described in terms of state-dependent delay differential equations, which display limit cycle behavior. Therefore, the principle of inertia, as appearing in Newton's first law, would only hold in an average sense, since uniform motion is unstable. Consequently, pilot waves would be necessary attached to any electrodynamic body.

• Casimir effect
• Cosmic microwave background
• Quantum annealing
• Quantum foam
• Virtual particle
• Virtual black hole
• Stochastic interpretation
• Zitterbewegung