Movatterモバイル変換

[0]ホーム

Jump to content

Vacuum expectation value

Edit links

From Wikipedia, the free encyclopedia

Type of operator expectation value

For articles related to the vacuum expectation value, seeQuantum vacuum (disambiguation).

Quantum field theory
Feynman diagram
History
Background Field theory Electromagnetism Weak force Strong force Quantum mechanics Special relativity General relativity Gauge theory Yang–Mills theory
Symmetries Symmetry in quantum mechanics C-symmetry P-symmetry T-symmetry Lorentz symmetry Poincaré symmetry Gauge symmetry Explicit symmetry breaking Spontaneous symmetry breaking Noether charge Topological charge
Tools Anomaly Background field method BRST quantization Correlation function Crossing Effective action Effective field theory Expectation value Feynman diagram Lattice field theory LSZ reduction formula Partition function Path Integral Formulation Propagator Quantization Regularization Renormalization Vacuum state Wick's theorem Wightman axioms
Equations Dirac equation Klein–Gordon equation Proca equations Wheeler–DeWitt equation Bargmann–Wigner equations Schwinger-Dyson equation Renormalization group equation
Standard Model Quantum electrodynamics Electroweak interaction Quantum chromodynamics Higgs mechanism
Incomplete theories String theory Supersymmetry Technicolor Theory of everything Quantum gravity
Scientists Adler Anderson Anselm Bargmann Becchi Belavin Bell Berezin Bethe Bjorken Bleuer Bogoliubov Brodsky Brout Buchholz Cachazo Callan Cardy Coleman Connes Dashen DeWitt Dirac Doplicher Dyson Englert Faddeev Fadin Fayet Fermi Feynman Fierz Fock Frampton Fritzsch Fröhlich Fredenhagen Furry Glashow Gell-Mann Glimm Goldstone Gribov Gross Gupta Guralnik Haag Hagen Han Heisenberg Hepp Higgs 't Hooft Iliopoulos Ivanenko Jackiw Jaffe Jona-Lasinio Jordan Jost Källén Kendall Kinoshita Kim Klebanov Kontsevich Kreimer Kuraev Landau Lee Lee Lehmann Leutwyler Lipatov Łopuszański Low Lüders Maiani Majorana Maldacena Matsubara Migdal Mills Møller Naimark Nambu Neveu Nishijima Oehme Oppenheimer Osborn Osterwalder Parisi Pauli Peccei Peskin Plefka Polchinski Polyakov Pomeranchuk Popov Proca Quinn Rouet Rubakov Ruelle Sakurai Salam Schrader Schwarz Schwinger Segal Seiberg Semenoff Shifman Shirkov Skyrme Sommerfield Stora Stueckelberg Sudarshan Symanzik Takahashi Thirring Tomonaga Tyutin Vainshtein Veltman Veneziano Virasoro Ward Weinberg Weisskopf Wentzel Wess Wetterich Weyl Wick Wightman Wigner Wilczek Wilson Witten Yang Yukawa Zamolodchikov Zamolodchikov Zee Zimmermann Zinn-Justin Zuber Zumino
v t e

Inquantum field theory, thevacuum expectation value (VEV) of anoperator is its average orexpectation value in thevacuum. The vacuum expectation value of an operatorO is usually denoted by $\langle O\rangle .$ One of the most widely used examples of an observable physical effect that results from the vacuum expectation value of an operator is theCasimir effect.

This concept is important for working withcorrelation functions inquantum field theory. In the context ofspontaneous symmetry breaking, an operator that has a vanishing expectation value due to symmetry can acquire a nonzero vacuum expectation value during aphase transition. Examples are:

TheHiggs field has a vacuum expectation value of 246GeV.^[1] This nonzero value underlies theHiggs mechanism of theStandard Model. This value is given by $v=1/{\sqrt {{\sqrt {2}}G_{F}^{0}}}=2M_{W}/g\approx 246.22\,{\rm {GeV}}$ , whereM_W is the mass of the W Boson, $G_{F}^{0}$ the reducedFermi constant, andg the weak isospin coupling, in natural units. It is also near the limit of the most massive nuclei, at v = 264.3Da.
Thechiral condensate inquantum chromodynamics, about a factor of a thousand smaller than the above, gives a large effective mass toquarks, and distinguishes between phases ofquark matter. This underlies the bulk of the mass of most hadrons.
Thegluon condensate inquantum chromodynamics may also be partly responsible for masses of hadrons.

The observedLorentz invariance of space-time allows only the formation of condensates which areLorentz scalars and have vanishingcharge.^{[citation needed]} Thus,fermion condensates must be of the form $\langle {\overline {\psi }}\psi \rangle$ , whereψ is the fermion field. Similarly atensor field,G_μν, can only have a scalar expectation value such as $\langle G_{\mu \nu }G^{\mu \nu }\rangle$ .

In somevacua ofstring theory, however, non-scalar condensates are found.^[which?] If these describe ouruniverse, thenLorentz symmetry violation may be observable.