Why is it interesting?

Elitzur demonstrated that a spontaneous breaking of a local symmetry is not possible.

https://journals.aps.org/prb/pdf/10.1103/PhysRevB.72.045137

Layman

Student

The phenomenon of spontaneously broken symmetries happens in large macroscopic systems where the breaking of a global symmetry involves a macroscopic number of degrees of freedom. This is not the case for a local gauge symmetry. The quantum fluctuations tend to smear the ground-state wave function of the system homogeneously over the whole orbit under the group. This results in

Theorem 13.1 (Elitzur) A local gauge symmetry cannot break spontaneously. The expectation value of any gauge non-invariant local observable must vanish.

Elitzur’s original proof in [38] applies to Abelian gauge theories but was later extended to non-Abelian models [40]. The proofs of Elitzur’s theorem are all based on the fact that inequalities which hold for any field configuration continue to hold after integrating with respect to a positive measure. In fact, positivity of the measure and gauge invariance are sufficient to prove the theorem. The theorem means that there is no analog of a magnetization: expectation values of a spin or link variables are zero, even if we introduce an external field (which explicitly breaks gauge invariance) and then carefully take first the infinite volume limit, and then the h → 0 limit. We must look, instead, to gauge-invariant observables which are unaffected by gauge transformations. These can be constructed by taking parallel transporters around closed loops, known as Wilson loops.

Elitzur’s theorem raises the question of whether the Higgs mechanism, which gives masses to the fermions and gauge bosons of the standard model, may perhaps not work. As demonstrated in [23] such fears are ungrounded, since the physical phenomena which are associated with the Higgs mechanism can be recovered in an approach that uses gauge-invariant fields only. The masses are extracted from expectation values of gauge-invariant combinations of the Higgs and gauge fields, without any need of introducing a non-zero expectation value of the Higgs field. In particular the electroweak phase transition can be described in purely gauge-invariant terms. For example, the expectation value of (φ,φ) exhibits a “jump” along the phase transition line in parameter space where the electroweak phase transition occurs.

Statistical Approach to Quantum Field Theoryby Andreas Wipf

Ref [23] is J. Fröhlich, G. Morchio, F. Strocchi, Higgs phenomenon without symmetry breaking order parameter. Nucl. Phys. B 190, 553 (1981)

Researcher

See: https://journals.aps.org/rmp/pdf/10.1103/RevModPhys.51.659

. In particular, zero- or one-dimensional theories with short-range interactions cannot exhibit a phase transition at any finite temperature. Additionally, the Mermin-Wagner theorem1 states that a continuous symmetry cannot be spontaneously broken at any finite temperature for two-dimensional theories with finite range interactions. On the other hand, Elitzur2 demonstrated that a spontaneous breaking of a local symmetry is not possible. Below, we will show that Elitzur’s theorem is a consequence of a reduction to zero of the effective dimension of the gauge invariant theory. Moreover, we will show that from the point of view of the noninvariant gauge fields, the presence of a “d-dimensional gauge
or gaugelike symmetry”
see definition below reduces the effective dimension of the theory from D to d. The dimension d is intermediate between local symmetries
d=0 and global symmetries
d=D. Ho https://journals.aps.org/prb/pdf/10.1103/PhysRevB.72.045137

Common Question 1

Common Question 2

Examples

Example1

Example2:

History