Vacuum expectation value

In <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a> the vacuum expectation value (also called condensate or simply VEV) of an <a href="/facts/Operator_(physics)/rCJw8UFz">operator</a> is its average or <a href="/facts/Expectation_value_(quantum_mechanics)/szGNS7Dx">expectation value</a> in the <a href="/facts/Vacuum_state/dTvXYPad">vacuum</a>. The vacuum expectation value of an operator O is usually denoted by 
 
 
 
 ⟨
 O
 ⟩
 .
 
 
 {\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 <a href="/facts/Casimir_effect/xRhpDH6T">Casimir effect</a>.
This concept is important for working with <a href="/facts/Correlation_function_(quantum_field_theory)/zH7RInAw">correlation functions</a> in <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a>. It is also important in <a href="/facts/Spontaneous_symmetry_breaking/VmXqNLlo">spontaneous symmetry breaking</a>. Examples are:

<ul><li>The <a href="/facts/Higgs_field/3LAYXfft">Higgs field</a> has a vacuum expectation value of 246 <a href="/facts/GeV/YMT1fqX9">GeV</a>. This nonzero value underlies the <a href="/facts/Higgs_mechanism/Vs5Cbxyy">Higgs mechanism</a> of the <a href="/facts/Standard_Model/LhVGf9ow">Standard Model</a>. This value is given by 
 
 
 
 v
 =
 1
 
 /
 
 
 
 
 
 2
 
 
 
 G
 
 F
 
 
 0
 
 
 
 
 =
 2
 
 M
 
 W
 
 
 
 /
 
 g
 ≈
 246.22
 
 
 
 G
 e
 V
 
 
 
 
 {\displaystyle v=1/{\sqrt {{\sqrt {2}}G_{F}^{0}}}=2M_{W}/g\approx 246.22\,{\rm {GeV}}}
 
, where MW is the mass of the W Boson, 
 
 
 
 
 G
 
 F
 
 
 0
 
 
 
 
 {\displaystyle G_{F}^{0}}
 
 the reduced <a href="/facts/Fermi_constant/uuzvty7m">Fermi constant</a>, and g the weak isospin coupling, in natural units. It is also near the limit of the most massive nuclei, at v = 264.3 <a href="/facts/Dalton_(unit)/IMATU3vf">Da</a>.</li>
<li>The <a href="/facts/Chiral_condensate/WlqsAJtp">chiral condensate</a> in <a href="/facts/Quantum_chromodynamics/3cA2ctyb">quantum chromodynamics</a>, about a factor of a thousand smaller than the above, gives a large effective mass to <a href="/facts/Quark/hwuuVkV5">quarks</a>, and distinguishes between phases of <a href="/facts/Quark_matter/vbsPu9rL">quark matter</a>. This underlies the bulk of the mass of most hadrons.</li>
<li>The <a href="/facts/Gluon_condensate/N8M4sdy7">gluon condensate</a> in <a href="/facts/Quantum_chromodynamics/3cA2ctyb">quantum chromodynamics</a> may also be partly responsible for masses of hadrons.</li></ul>
The observed <a href="/facts/Lorentz_invariance/K5AEuGKg">Lorentz invariance</a> of space-time allows only the formation of condensates which are <a href="/facts/Lorentz_scalar/USPNCG1l">Lorentz scalars</a> and have vanishing <a href="/facts/Charge_(physics)/jnQEDBI3">charge</a>. Thus, <a href="/facts/Fermion/LEckdDsR">fermion</a> condensates must be of the form 
 
 
 
 ⟨
 
 
 ψ
 ¯
 
 
 ψ
 ⟩
 
 
 {\displaystyle \langle {\overline {\psi }}\psi \rangle }
 
, where ψ is the fermion field. Similarly a <a href="/facts/Tensor_field/cAC6F6rF">tensor field</a>, Gμν, can only have a scalar expectation value such as 
 
 
 
 ⟨
 
 G
 
 μ
 ν
 
 
 
 G
 
 μ
 ν
 
 
 ⟩
 
 
 {\displaystyle \langle G_{\mu \nu }G^{\mu \nu }\rangle }
 
.
In some <a href="/facts/Vacuum/BTk3goyV">vacua</a> of <a href="/facts/String_theory/HUJFIPYH">string theory</a>, however, non-scalar condensates are found.[which?] If these describe our <a href="/facts/Universe/XbfpnDmM">universe</a>, then <a href="/facts/Lorentz_symmetry/K5AEuGKg">Lorentz symmetry violation</a> may be observable.

Vacuum expectation value open-in-new

Vacuum expectation value