Vacuum expectation value

Vacuum expectation value
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Vacuum_expectation_value".

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Quantum field theory

Feynman diagram

History of...

Background
Gauge theory Field theory Poincaré symmetry Quantum mechanics Spontaneous symmetry breaking

Symmetries
Crossing Charge conjugation Parity Time reversal

Tools
Anomaly Effective field theory Expectation value Faddeev–Popov ghosts Feynman diagram LSZ reduction formula Partition function Propagator Quantization Renormalization Vacuum state Wick's theorem Wightman axioms

Equations
Dirac equation Klein–Gordon equation Proca equations Wheeler–deWitt equation

Standard model
Electroweak interaction Higgs mechanism Quantum chromodynamics Quantum electrodynamics Yang–Mills theory

Incomplete theories
Quantum gravity String theory Supersymmetry Theory of everything

Scientists

Adler • Bethe • Bogoliubov • Callan • Coleman • DeWitt • Dyson • Fermi • Feynman • Fierz • Fröhlich • Gell-Mann • Goldstone • Gross • 't Hooft • Jackiw • Klein • Landau • Lee • Lehmann • Majorana • Nambu • Parisi • Polyakov • Salam • Schwinger • Skyrme • Stueckelberg •Symanzik • Tomonaga • Veltman • Weinberg • Weisskopf • Wilson • Witten • Yang • Yukawa • Zimmermann • Zinn-Justin

In quantum field theory the vacuum expectation value (also called condensate) of an operator is its average, expected value in the vacuum. The vacuum expectation value of an operator O is usually denoted by $\langle O\rangle$ . One of the best known examples of the vacuum expectation value of an operator leading to a physical effect is the Casimir effect.

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. This nonzero value allows the Higgs mechanism to work.
The chiral condensate in Quantum chromodynamics gives a large effective mass to quarks, and distinguishes between phases of quark matter.
The gluon condensate in Quantum chromodynamics may 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 $\langle\overline\psi\psi\rangle$ , where ψ is the fermion field. Similarly a tensor field, G_μν, can only have a scalar expectation value such as $\langle G_{\mu\nu}G^{\mu\nu}\rangle$ .

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

See also