Quantum fluctuation

In quantum physics, a quantum fluctuation is the temporary change in the amount of energy in a point in space, arising from Werner Heisenberg's uncertainty principle.

According to one formulation of the principle, energy and time can be related by the relation

: $Delta\; E\; Delta\; t\; approx\; \{h\; over\; 2\; pi\}$

That means that conservation of energy can appear to be violated, but only for small times. This allows the creation of particle-antiparticle pairs of virtual particles.The effects of these particles are measurable, for example, in the effective charge of the electron, different from its "naked" charge.

In the modern view, energy is always conserved, but the eigenstates of the Hamiltonian (energy observable) aren't the same as (e.g.the Hamiltonian doesn't commute with) the particle number operators.

Quantum fluctuations may have been very important in the origin of the structure of the universe: according to the model of inflation the ones that existed when inflation began were amplified and formed the seed of all current observed structure.

Quantum fluctuations of a field

A reasonably odd distinction can be made between quantum fluctuations and thermal fluctuations of a quantum field (at least for a free field; for interacting fields, renormalization complicates matters a lot). For the quantized Klein–Gordon field, we can calculate the probability density that we would observe a configuration $\{displaystylevarphi\_t(x)\}$ at a time $t$ in terms of its fourier transform $\{displaystyle\; ildevarphi\_t(k)\}$ to be:$ho\_0\; [varphi\_t]\; =\; exp\{left\; [-frac\{1\}\{hbar\}\; intfrac\{d^3k\}\{(2pi)^3\}\; ildevarphi\_t^*(k)sqrt\{|k|^2+m^2\};\; ilde\; varphi\_t(k)\; ight]\; \}.$In contrast, for the classical Klein–Gordon field at non-zero temperature, the Gibbs probability density that we would observe a configuration $\{displaystylevarphi\_t(x)\}$ at a time $t$ is:$ho\_E\; [varphi\_t]\; =\; exp\{\; [-H\; [varphi\_t]\; /kT]\; \}=exp\{left\; [-frac\{1\}\{kT\}\; intfrac\{d^3k\}\{(2pi)^3\}\; ildevarphi\_t^*(k)\{scriptstylefrac\{1\}\{2(|k|^2+m^2);\; ilde\; varphi\_t(k)\; ight]\; \}.$The amplitude of quantum fluctuations is controlled by the amplitude of Planck's constant $hbar$, just as the amplitude of thermal fluctuations is controlled by $kT$. Note that the following three points are closely related:
(1) Planck's constant has units of action instead of units of energy,
(2) the quantum kernel is $sqrt\{|k|^2+m^2\}$ instead of $\{scriptstylefrac\{1\}\{2(|k|^2+m^2)$ (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 modelled by classical continuous random fields.

See also

* Casimir effect
* Virtual particle
*Quantum annealing