Gauge Symmetries, Symmetry Breaking, and Gauge-Invariant Approaches

Abstract

Gauge symmetries play a central role, both in the mathematical foundations as well as the conceptual construction of modern (particle) physics theories. However, it is yet unclear whether they form a necessary component of theories, or whether they can be eliminated. It is also unclear whether they are merely an auxiliary tool to simplify (and possibly localize) calculations or whether they contain independent information. Therefore their status, both in physics and philosophy of physics, remains to be fully clarified. In this overview we review the current state of affairs on both the philosophy and the physics side. In particular, we focus on the circumstances in which the restriction of gauge theories to gauge invariant information on an observable level is warranted, using the Brout-Englert-Higgs theory as an example of particular current importance. Finally, we determine a set of yet to be answered questions to clarify the status of gauge symmetries.

Geometric confinement in gauge theories

“…The map A → A φ is called the dressing transformation, the mass term (4.3) for A is the same as (4.10) for A φ , i.e. the group G G is an artificial gauge symmetry [10]. For the Abelian case G = U(1) this trading of degrees of freedom between A and scalar field φ ∈ U (1) was proposed by Stueckelberg [11].…”

“…For the Abelian case G = U(1) this trading of degrees of freedom between A and scalar field φ ∈ U (1) was proposed by Stueckelberg [11]. An overview of the dressing field method in gauge theories and many references can be found in [10].…”

Gauge Invariant Description of the $SU(2)$ Higgs model: Ward identities and Renormalization

“…where A a µ is the gauge field, h stands for the Higgs field, ρ a (a = 1, 2, 3) are the Goldstone bosons and v is the vacuum expectation value of the scalar complex field. These gauge invariant composite operators were first introduced by 't Hooft [3] and later on formalized by Fröhlich-Morchio-Strocchi(FMS) [4,5] in order to describe the Higgs phenomenon in a gauge invariant fashion, see [7,8,9,10,11] for recent accounts on the subject and [12,13] for a more historical account. In the U (1) case, a gauge invariant reformulation of the Higgs model was also proposed in [14], see also [12], but notice that the non-linear field redefinition invoked there misses a (non-trivial) Jacobian at the quantum level, see [22], which complicates matters.…”