# Quantum fluctuation

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The video of an experiment showing vacuum fluctuations (in the red ring) amplified by spontaneous parametric down-conversion.

In quantum physics, a quantum fluctuation (or quantum vacuum fluctuation or vacuum fluctuation) is the temporary change in the amount of energy in a point in space,[1] as explained in Werner Heisenberg's uncertainty principle.

According to one formulation of the principle, energy and time can be related by the relation[2]

$\Delta E \Delta t \approx {h \over 4 \pi}$

That means that conservation of energy can appear to be violated, but only to allow the creation of virtual particle-antiparticle pairs for exceedingly short period of time. 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) are not the same as (i.e. the Hamiltonian does not 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. Vacuum energy may also be responsible for the current accelerated expansion of the universe (cosmological constant).

## Quantum fluctuations of a field

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 $mc^2$). 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. This was proposed by scientist Heisenberg's study in 1916 at Harvard's Laboratory. Heisenberg's theory was later proven in the 1920s by Louis de Broglie and became a law of quantum physics.

In quantum field theory, fields undergo quantum fluctuations. A reasonably clear distinction can be made between quantum fluctuations and thermal fluctuations[how?] of a quantum field (at least for a free field; for interacting fields, renormalization substantially complicates matters). For the quantized Klein–Gordon field in the vacuum state, we can calculate the probability density that we would observe a configuration ${\displaystyle\varphi_t(x)}$ at a time $t$ in terms of its Fourier transform ${\displaystyle\tilde\varphi_t(k)}$ to be

$\rho_0[\varphi_t] = \exp{\left[-\frac{1}{\hbar} \int\frac{d^3k}{(2\pi)^3} \tilde\varphi_t^*(k)\sqrt{|k|^2+m^2}\;\tilde \varphi_t(k)\right]}.$

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

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

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 $k_\mathrm{B}T$, where $k_\mathrm{B}$ is Boltzmann's constant. 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 ${\scriptstyle\frac{1}{2}}(|k|^2+m^2)$ (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.

## References

1. ^ Browne, Malcolm W. (1990-08-21). "New Direction in Physics: Back in Time". The New York Times. Retrieved 2010-05-22. According to quantum theory, the vacuum contains neither matter nor energy, but it does contain fluctuations, transitions between something and nothing in which potential existence can be transformed into real existence by the addition of energy.(Energy and matter are equivalent, since all matter ultimately consists of packets of energy.) Thus, the vacuum's totally empty space is actually a seething turmoil of creation and annihilation, which to the ordinary world appears calm because the scale of fluctuations in the vacuum is tiny and the fluctuations tend to cancel each other out.
2. ^ Mandelshtam, Leonid; Tamm, Igor (1945), "The uncertainty relation between energy and time in nonrelativistic quantum mechanics", Izv. Akad. Nauk SSSR (ser. Fiz.) 9: 122–128. English translation: J. Phys. (USSR) 9, 249–254 (1945).