In
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
the
vacuum expectation value (also called
condensate) of an
operatorIn physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
is its average,
expected valueIn probability theory and statistics, the expected value of a random variable is the integral of the random variable with respect to its probability measure....
in the vacuum. The vacuum expectation value of an operator
O is usually denoted by . One of the best known examples of the vacuum expectation value of an operator leading to a physical effect is the
Casimir 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, placed a few micrometers apart, without any external electromagnetic field...
.
This concept is important for working with
correlation functionsIn quantum field theory, correlation functions generalize the concept of correlation functions in statistics. In the quantum mechanical context they are computed as the matrix element of a product of operators inserted between two vectors, usually the vacuum states.Sometimes, the time-ordering...
in
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
. It is also important in
spontaneous symmetry breakingIn physics, spontaneous symmetry breaking occurs when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. When that happens, the system no longer appears to behave in a symmetric manner...
.
In
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
the
vacuum expectation value (also called
condensate) of an
operatorIn physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
is its average,
expected valueIn probability theory and statistics, the expected value of a random variable is the integral of the random variable with respect to its probability measure....
in the vacuum. The vacuum expectation value of an operator
O is usually denoted by . One of the best known examples of the vacuum expectation value of an operator leading to a physical effect is the
Casimir 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, placed a few micrometers apart, without any external electromagnetic field...
.
This concept is important for working with
correlation functionsIn quantum field theory, correlation functions generalize the concept of correlation functions in statistics. In the quantum mechanical context they are computed as the matrix element of a product of operators inserted between two vectors, usually the vacuum states.Sometimes, the time-ordering...
in
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
. It is also important in
spontaneous symmetry breakingIn physics, spontaneous symmetry breaking occurs when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. When that happens, the system no longer appears to behave in a symmetric manner...
. Examples are:
- The Higgs field has a vacuum expectation value of 246 GeV
GEV or GeV may stand for:*Generalized extreme value distribution*GeV or gigaelectronvolt, a unit of energy equal to billion electron volts*Ground effect vehicle*G.E.V., a tabletop game by Steve Jackson games, based on OGRE...
. This nonzero value allows the Higgs mechanismIn the standard model of particle physics, the Higgs mechanism is a theoretical framework which explains how the masses of the W and Z bosons arise as a result of electroweak symmetry breaking....
to work.
- The chiral condensate in Quantum chromodynamics
In theoretical physics, Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
gives a large effective mass to quarkA quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never found in...
s, and distinguishes between phases of quark matter.
- The gluon condensate
In Quantum chromodynamics , the gluon condensate is a non-perturbative property of the QCD vacuum which could be partly responsible for giving masses to certain hadrons.If the gluon field tensor is represented as Gμν, then...
in Quantum chromodynamicsIn theoretical physics, Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
may be partly responsible for masses of hadrons.
The observed Lorentz invariance of space-time allows only the formation of condensates which are
Lorentz scalarIn physics a Lorentz scalar is a scalar which is invariant under a Lorentz transformation. A Lorentz scalar is generated from vectors and tensors...
s and have vanishing
chargeIn physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges are associated with conserved quantum numbers.-Formal definition:...
. Thus
fermionIn particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle. Thus, if more than one...
condensates must be of the form , where
ψ is the fermion field. Similarly a tensor field,
Gμν, can only have a scalar expectation value such as .
In some vacua of
string theoryString theory is a developing branch of theoretical physics that combines quantum mechanics and general relativity into a quantum theory of gravity...
, however, non-scalar condensates are found. If these describe our
universeThe Universe comprises everything that physically exists, the entirety of space and time, all forms of matter and energy, and the physical laws and constants that govern them...
, then Lorentz symmetry violation may be observable.
See also
- Wightman axioms
In physics the Wightman axioms are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the early 1950s but they were first published only in 1964, after Haag-Ruelle scattering theory affirmed their significance.The axioms exist in...
and Correlation function (quantum field theory)In quantum field theory, correlation functions generalize the concept of correlation functions in statistics. In the quantum mechanical context they are computed as the matrix element of a product of operators inserted between two vectors, usually the vacuum states.Sometimes, the time-ordering...
- vacuum energy
Vacuum energy is an underlying background energy that exists in space even when devoid of matter . The vacuum energy is deduced from the concept of virtual particles, which are themselves derived from the energy-time uncertainty principle...
or dark energyIn physical cosmology, astronomy and celestial mechanics, dark energy is a hypothetical form of energy that permeates all of space and tends to increase the rate of expansion of the universe. Dark energy is the most popular way to explain recent observations that the universe appears to be...
- Spontaneous symmetry breaking
In physics, spontaneous symmetry breaking occurs when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. When that happens, the system no longer appears to behave in a symmetric manner...