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theorems:elitzur_s_theorem

# Elitzur's Theorem

## Intuitive

Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.

## Concrete

To quote Seiberg:

"a gauge symmetry is so big, it's too big to fail."

A local symmetry means that we have an independent copy of the symmetry (of the group $G$) at each spacetime point. A spacetime point is zero-dimensional. Hence we are dealing with symmetries of zero-dimensional subsystems and

A theorem due to Mermin and Wagner states that a continuous symmetry can only be spontaneously broken in a dimension larger than two. For a discrete symmetry this lower critical dimensionality is one. This is, in fact, well known since in quantum mechanics with finitely many degrees of freedom (corresponding to one-dimensional field theory) tunneling between degenerate classical minima allows for a unique symmetric ground state. […] The Mermin-Wagner theorem has been restated by Coleman in the framework of field theory. page 525 in Quantum Field Theory by Claude Itzykson, ‎Jean-Bernard Zuber

Therefore, combining the informations that

• for a local/gauge symmetry, we have a copy of the group at each spacetime point,
• a spacetime point is zero-dimensional and
• the statement of the Mermin-Wagner theorem that symmetry breaking of a continuous symmetry in less than two dimensions is impossible,

leads to the conclusion that the breaking of a local symmetry is impossible.

A global symmetry can be broken in a system with infinite degrees of freedom, because it takes an infinite amount of energy to switch to another ground state. However, for local symmetry there is no energy barrier, because all the degenerate vacua at each point are connected by a gauge transformation and therefore, there is no energy penalty.

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)

## Abstract

. 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

As a matter of fact, a non-perturbative analysis of the possible existence of a symmetry breaking order parameter, by using the euclidean functional inte- gral approach defined by the Lagrangean (19.1), gives symmetric correlation functions and in particular < φ >= 0 (Elithur-De Angelis-De Falco-Guerra (EDDG) theorem). 193 This means that the mean field ansatz is incompatible with the non-perturbative quantum effects and the approximation leading to (19.3) is not correct.

See S. Elitzur, Phys. Rev. D 12, 3978 (1975); G.F. De Angelis, D. De Falco and F. Guerra, Phys. Rev. D 17, 1624 (1978). The crux of the argument is that gauge invariance decouples the transformations of the fields inside a volume V (in a euclidean functional integral approach) from the transformation of the boundary, so that the boundary conditions are ineffective and cannot trigger non symmetric correlation functions. For a simple account of the argument, see e.g. F. Strocchi, Elements of Quantum Mechanics of Infinite Systems, World Scientific 1985, Part C, Sect. 2.5.

In order to avoid the vanishing of a symmetry breaking order parameter, one must reconsider the problem by adding to the Lagrangean (19.1) a gauge fixing LGF which breaks local gauge invariance. Then, the discussion of the Higgs mechanism necessarily becomes gauge fixing dependent; this should not appear strange, since the vacuum expectation of φ is a gauge dependent quantity.194

[…]

Since the gauge fixing breaks local gauge invariance, but not the invariance under the global group transformations, the EDDG theorem does not apply and one may consider the possibility of a symmetry breaking order parameter < φ ≯= 0. Now, another conceptual problem arises: the starting Lagrangean L is invariant under the U(1) global group and its breaking with a mass gap seems incompatible with the Goldstone theorem. As an explanation of such an apparent conflict, one finds in the literature the statement that the Gold- stone theorem does not apply if the two point function < j0(x)φ(y) > is not Lorentz covariant as it happens in the physical gauges, like the Coulomb gauge. As a matter of fact, the Goldstone-Salam-Weinberg proof of the Gold- stone theorem crucially uses Lorentz covariance; however, the more general proof discussed in Chapter 17 does not assume it, so that the quest of a better explanation remains.

P. 195 in Symmetry Breaking by Strocchi </blockquote>

## Why is it interesting?

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