View source for Vacuum expectation value

{{Short description|Type of operator expectation value}}
{{For|articles related to the vacuum expectation value|Quantum vacuum (disambiguation)}}

{{Quantum field theory|cTopic=Tools}}
In [[quantum field theory]], the '''vacuum expectation value''' ('''VEV''') of an [[Operator (physics)|operator]] is its average or [[Expectation value (quantum mechanics)|expectation value]] in the [[Vacuum state|vacuum]]. The vacuum expectation value of an operator <var>O</var> is usually denoted by <math>\langle O\rangle.</math> 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]].

This concept is important for working with [[Correlation function (quantum field theory)|correlation functions]] in [[quantum field theory]]. In the context of [[spontaneous symmetry breaking]], an operator that has a vanishing expectation value due to symmetry can acquire a nonzero vacuum expectation value during a [[phase transition]]. Examples are:
*The [[Higgs field]] has a vacuum expectation value of 246 [[GeV]].<ref>{{Cite journal | last1 = Amsler | first1 = C. | last2 = Doser | first2 = M. | last3 = Antonelli | first3 = M. | last4 = Asner | first4 = D. | last5 = Babu | first5 = K. | last6 = Baer | first6 = H. | last7 = Band | first7 = H. | last8 = Barnett | first8 = R. | last9 = Bergren | first9 = E. | last10 = Beringer | first10 = J. | last11 = Bernardi | first11 = G. | last12 = Bertl | first12 = W. | last13 = Bichsel | first13 = H. | last14 = Biebel | first14 = O. | last15 = Bloch | first15 = P. | last16 = Blucher | first16 = E. | last17 = Blusk | first17 = S. | last18 = Cahn | first18 = R. N. | last19 = Carena | first19 = M. | last20 = Caso | first20 = C. | last21 = Ceccucci | first21 = A. | last22 = Chakraborty | first22 = D. | last23 = Chen | first23 = M. -C. | last24 = Chivukula | first24 = R. S. | last25 = Cowan | first25 = G. | last26 = Dahl | first26 = O. | last27 = d'Ambrosio | first27 = G. | last28 = Damour | first28 = T. | last29 = De Gouvêa | first29 = A. | last30 = Degrand | first30 = T. | display-authors = 1 | title = Review of Particle Physics⁎ | doi = 10.1016/j.physletb.2008.07.018 | journal = Physics Letters B | volume = 667 | pages = 1–6 | year = 2008 | issue = 1–5 | url = http://pdglive.lbl.gov/Rsummary.brl?nodein=S044&fsizein=1 | bibcode = 2008PhLB..667....1A | access-date = 2015-09-04 | archive-url = https://archive.today/20120712165412/http://pdglive.lbl.gov/Rsummary.brl?nodein=S044&fsizein=1 | archive-date = 2012-07-12 | url-status = dead | hdl = 1854/LU-685594 | s2cid = 227119789 | hdl-access = free }}</ref>  This nonzero value underlies the [[Higgs mechanism]] of the [[Standard Model]]. This value is given by <math>v = 1/\sqrt{\sqrt 2 G_F^0} = 2M_W/g \approx 246.22\, \rm{GeV}</math>, where   ''M<sub>W</sub>'' is the mass of the W Boson,  <math>G_F^0</math> the reduced [[Fermi constant]], and {{mvar|g}} the weak isospin coupling, in natural units. It is also near the limit of the most massive nuclei, at v = 264.3 [[Dalton (unit)|Da]].
*The [[chiral condensate]] in [[quantum chromodynamics]], about a factor of a thousand smaller than the above, gives a large effective mass to [[quark]]s, 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 scalar]]s and have vanishing [[charge (physics)|charge]].{{citation needed|date=April 2013}} Thus, [[fermion]] condensates must be of the form <math>\langle\overline\psi\psi\rangle</math>, where <var>ψ</var> is the fermion field. Similarly a [[tensor field]], <var>G</var><sub><var>μν</var></sub>, can only have a scalar expectation value such as <math>\langle G_{\mu\nu}G^{\mu\nu}\rangle</math>.

In some [[Vacuum#The quantum-mechanical vacuum|vacua]] of [[string theory]], however, non-scalar condensates are found.{{which|date=April 2013}} If these describe our [[universe]], then [[Lorentz symmetry#Lorentz violation|Lorentz symmetry violation]] may be observable.

==See also==
* [[Correlation function (quantum field theory)]]
* [[Dark energy]]
* [[Spontaneous symmetry breaking]]
* [[Vacuum energy]]
* [[Wightman axioms]]

== References ==
{{Reflist}}

== External links ==
* {{wikiquote-inline}}

{{DEFAULTSORT:Vacuum Expectation Value}}
[[Category:Quantum field theory]]
[[Category:Standard Model]]