Elitzur's theorem
Encyclopedia
Elitzur's theorem is a theorem
in quantum
and statistical field theory
stating that local gauge symmetries
cannot be spontaneously broken
. The theorem was proposed in 1975 by S. Elitzur, who proved it for Abelian
gauge fields on a lattice
. It is nonetheless possible to spontaneously break a global symmetry within a theory that has a local gauge symmetry, as in the Higgs mechanism
.
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
in quantum
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
and statistical field theory
Statistical field theory
A statistical field theory is any model in statistical mechanics where the degrees of freedom comprise a field or fields. In other words, the microstates of the system are expressed through field configurations...
stating that local gauge symmetries
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
cannot be spontaneously broken
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
. The theorem was proposed in 1975 by S. Elitzur, who proved it for Abelian
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
gauge fields on a lattice
Lattice gauge theory
In physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice. Gauge theories are important in particle physics, and include the prevailing theories of elementary particles: quantum electrodynamics, quantum chromodynamics and the Standard...
. It is nonetheless possible to spontaneously break a global symmetry within a theory that has a local gauge symmetry, as in the Higgs mechanism
Higgs mechanism
In particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking....
.