Quantum fluctuation

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]

${\displaystyle \Delta E\Delta t\geq {h \over 4\pi }}$

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 because the particle number operator does not commute with a field's Hamiltonian or energy operator, the field's lowest-energy or ground state, often called the vacuum state, is not, as one might expect from that name, a state with no particles, but rather a quantum superposition of particle number eigenstates with 0, 1, 2...etc. particles.

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.

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). 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 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 (joule-seconds) instead of units of energy (joules),
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.

In the 1930s, Pascual Jordan knew that a star could equal zero energy because its matter energy was positive and its gravitational energy was negative and they cancelled each other out. And this led him to speculate what would prevent a quantum transition from creating a new star. And he had this idea because he was trying to figure out where matter might come from if we existed in an always-here universe.^[3]

In December, 1973, the British scientific journal Nature published an article by Edward P. Tryon titled "Is the Universe a Vacuum Fluctuation?" In this paper Tryon said our universe may have originated as a quantum fluctuation of the vacuum.^[3] Yet, the idea of our universe coming from a quantum fluctuation or quantum process was not taken seriously until inflationary theory came and was able to explain how our universe could inflate from a tiny particle.^[4]

See also

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

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).
3. Reynosa, Peter. "Why Isn't Edward P. Tryon A World-famous Physicist?". Huffington Post. Retrieved March 22, 2016.
4. Reynosa, Peter. "Some of the Changes Lawrence M. Krauss Should Make to the Second Edition of "A Universe from Nothing"". Huffington Post. Retrieved April 13, 2016.

External links

• Quantum Fluctuation at universe-review.ca