Quantum fluctuation
Encyclopedia
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
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 particle
s.
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 doesn't commute with) the particle number operator
s.
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.
, 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
at a time 
in terms of its Fourier transform 
to be
In contrast, for the classical Klein–Gordon field at non-zero temperature, the Gibbs probability density
that we would observe a configuration
at a time 
is
The amplitude of quantum fluctuations is controlled by the amplitude of Planck's constant
, just as the amplitude of thermal fluctuations is controlled by 
. 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
instead of 
(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.
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
's uncertainty principle
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
.
According to one formulation of the principle,
energy and time can be related by the relation
That means that conservation of energy
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
can appear to be violated, but only for small times. This allows the creation of particle-antiparticle
Antiparticle
Corresponding to most kinds of particles, there is an associated antiparticle with the same mass and opposite electric charge. For example, the antiparticle of the electron is the positively charged antielectron, or positron, which is produced naturally in certain types of radioactive decay.The...
pairs of virtual particle
Virtual particle
In physics, a virtual particle is a particle that exists for a limited time and space. The energy and momentum of a virtual particle are uncertain according to the uncertainty principle...
s.
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
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
is always conserved
Conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
, but the eigenstates of the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
(energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
observable
Observable
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
) are not the same as (i.e. the Hamiltonian doesn't commute with) the particle number operator
Particle number operator
In quantum mechanics, for systems where the total number of particles may not be preserved, the number operator is the observable that counts the number of particles.The number operator acts on Fock space...
s.
Quantum fluctuations may have been very important in the origin of the structure of the universe: according to the model of inflation
Cosmic inflation
In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary epoch comprises the first part...
the ones that existed when inflation began were amplified and formed the seed of all current observed structure.
Quantum fluctuations of a field
In quantum field theoryQuantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, fields undergo quantum fluctuations. A reasonably clear distinction can be made between quantum fluctuations and thermal fluctuations
Thermal fluctuations
In statistical mechanics, thermal fluctuations are random deviations of a system from its equilibrium. All thermal fluctuations become larger and more frequent as the temperature increases, and likewise they disappear altogether as temperature approaches absolute zero.Thermal fluctuations are a...
of a quantum field (at least for a free field; for interacting fields, renormalization
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
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
at a time in terms of its Fourier transform to beIn contrast, for the classical Klein–Gordon field at non-zero temperature, the Gibbs probability density
Gibbs state
In probability theory and statistical mechanics, a Gibbs state is an equilibrium probability distribution which remains invariant under future evolution of the system...
that we would observe a configuration
at a time isThe amplitude of quantum fluctuations is controlled by the amplitude of Planck's constant
Planck constant
The Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...
, just as the amplitude of thermal fluctuations is controlled by . 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
instead of (the quantum kernel is nonlocal from a classical heat kernelHeat kernel
In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a particular domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum of the Laplace operator, and is thus of some...
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
Gibbs state
In probability theory and statistical mechanics, a Gibbs state is an equilibrium probability distribution which remains invariant under future evolution of the system...
is not a Lorentz invariant initial condition).
We can construct a classical continuous random field
Field (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
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.
See also
- Casimir effectCasimir effectIn quantum field theory, the Casimir effect and the Casimir–Polder force are physical forces arising from a quantized field. The typical example is of two uncharged metallic plates in a vacuum, like capacitors placed a few micrometers apart, without any external electromagnetic field...
- Quantum annealingQuantum annealingIn mathematics and applications, quantum annealing is a general method for finding the global minimum of a given objective function over a given set of candidate solutions , by a process analogous to quantum fluctuations...
- Quantum foamQuantum foamQuantum foam, also referred to as spacetime foam, is a concept in quantum mechanics, devised by John Wheeler in 1955. The foam is supposed to be the foundations of the fabric of the universe. Additionally, it can be used as a qualitative description of subatomic spacetime turbulence at extremely...
- Virtual particleVirtual particleIn physics, a virtual particle is a particle that exists for a limited time and space. The energy and momentum of a virtual particle are uncertain according to the uncertainty principle...