quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...

, the vacuum expectation value (VEV) of an operator is its average or expectation value in the

vacuum A vacuum (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressur ...

. The vacuum expectation value of an operator O 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 the

Casimir effect In quantum field theory, the Casimir effect (or Casimir force) is a physical force (physics), force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of a field (physics), field. The term Casim ...

. This concept is important for working with correlation functions in

. In the context of

spontaneous symmetry breaking Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion o ...

, an operator that has a vanishing expectation value due to symmetry can acquire a nonzero vacuum expectation value during a

phase transition In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic Sta ...

. Examples are: *The

Higgs field The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the excited state, quantum excitation of the Higgs field, one of the field (physics), fields in particl ...

has a vacuum expectation value of 246 GeV. This nonzero value underlies the

Higgs mechanism In the Standard Model of particle physics, the Higgs mechanism is essential to explain the Mass generation, generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles ...

of the

Standard Model The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...

. This value is given by

v = 1/\sqrt = 2M_W/g \approx 246.22\, \rm

, where ''M_W'' is the mass of the W Boson,

G_F^0

the reduced Fermi constant, and the weak isospin coupling, in natural units. It is also near the limit of the most massive nuclei, at v = 264.3 Da. *The chiral condensate in

quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the study of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of ...

, about a factor of a thousand smaller than the above, gives a large effective mass to

quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nucleus, atomic nuclei ...

s, and distinguishes between phases of quark matter. This underlies the bulk of the mass of most hadrons. *The gluon condensate in

may also be partly responsible for masses of hadrons. The observed

Lorentz invariance In a relativistic theory of physics, a Lorentz scalar is a scalar expression whose value is invariant under any Lorentz transformation. A Lorentz scalar may be generated from, e.g., the scalar product of vectors, or by contracting tensors. While ...

of space-time allows only the formation of condensates which are

Lorentz scalar In a relativistic theory of physics, a Lorentz scalar is a scalar expression whose value is invariant under any Lorentz transformation. A Lorentz scalar may be generated from, e.g., the scalar product of vectors, or by contracting tensors. Whil ...

s and have vanishing charge. Thus,

fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...

condensates must be of the form

\langle\overline\psi\psi\rangle

, where ψ is the fermion field. Similarly a

tensor field In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, ...

, G_μν, can only have a scalar expectation value such as

\langle G_G^\rangle

. In some vacua of

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...

, however, non-scalar condensates are found. If these describe our

universe The universe is all of space and time and their contents. It comprises all of existence, any fundamental interaction, physical process and physical constant, and therefore all forms of matter and energy, and the structures they form, from s ...

, then Lorentz symmetry violation may be observable.

References

External links

* Quantum field theory Standard Model {{Quantum-stub

See also

References

External links