Vacuum expectation value

In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by ${\displaystyle \langle O\rangle .}$ One of the most widely used examples of an observable physical effect that results from the vacuum expectation value of an operator is the Casimir effect.

Quantum field theory
Feynman diagram
History
Background Casimir effect Field theory Electromagnetism Weak force Strong force Quartic interaction Quantum mechanics Special relativity General relativity Gauge theory
Symmetries Symmetry in quantum mechanics C-symmetry P-symmetry T-symmetry Space translation symmetry Time translation symmetry Rotation symmetry Lorentz symmetry Poincaré symmetry Gauge symmetry Explicit symmetry breaking Spontaneous symmetry breaking Yang–Mills theory Noether charge Topological charge
Tools Anomaly Crossing Effective field theory Expectation value Ghost fields Faddeev–Popov ghosts Feynman diagram Lattice gauge theory LSZ reduction formula Partition function Propagator Quantization Regularization Renormalization Vacuum state Wick's theorem Wightman axioms Feynman parametrization
Equations Dirac equation Klein–Gordon equation Proca equations Wheeler–DeWitt equation Bargmann–Wigner equations Joos–Weinberg equation Weyl equation
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This concept is important for working with correlation functions in quantum field theory. It is also important in spontaneous symmetry breaking. Examples are:

• The Higgs field has a vacuum expectation value of 246 GeV [1] This nonzero value underlies the Higgs mechanism of the Standard Model. This value is given by ${\displaystyle v=1/{\sqrt {{\sqrt {2}}G_{F}^{0}}}=2M_{W}/g\approx 246.22\,{\rm {GeV}}}$, where M_W is the mass of the W Boson, ${\displaystyle G_{F}^{0}}$ the reduced Fermi constant, and g the weak isospin coupling, in natural units.

• The chiral condensate in Quantum chromodynamics, about a factor of a thousand smaller than the above, gives a large effective mass to quarks, and distinguishes between phases of quark matter. This underlies the bulk of the mass of most hadrons.
• The gluon condensate in Quantum chromodynamics may also be partly responsible for masses of hadrons.

The observed Lorentz invariance of space-time allows only the formation of condensates which are Lorentz scalars and have vanishing charge. Thus fermion condensates must be of the form ${\displaystyle \langle {\overline {\psi }}\psi \rangle }$, where ψ is the fermion field. Similarly a tensor field, G_μν, can only have a scalar expectation value such as ${\displaystyle \langle G_{\mu \nu }G^{\mu \nu }\rangle }$.

In some vacua of string theory, however, non-scalar condensates are found. If these describe our universe, then Lorentz symmetry violation may be observable.