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The Dressing Field Method of Gauge Symmetry Reduction, a Review with Examples

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Foundations of Mathematics and Physics One Century After Hilbert

Abstract

Gauge symmetries are a cornerstone of modern physics but they come with technical difficulties when it comes to quantization, to accurately describe particles phenomenology or to extract observables in general. These shortcomings must be met by essentially finding a way to effectively reduce gauge symmetries. We propose a review of a way to do so which we call the dressing field method. We show how the BRST algebra satisfied by gauge fields, encoding their gauge transformations, is modified. We outline noticeable applications of the method, such as the electroweak sector of the Standard Model and the local twistors of Penrose.

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Notes

  1. 1.

    See [45] for a critical discussion of its scope and limits.

  2. 2.

    Weyl topped this with an even stronger endorsement of the importance of symmetries in physics: “As far as I see, all a priori statements in physics have their origin in symmetry” [68].

  3. 3.

    See the nice short appendix by S. S. Chern of the book on differential geometry he co-authored [12].

  4. 4.

    In the general theory the group \(G'\) is replaced by a \(C^*\)-algebra A.

  5. 5.

    For instance, such a term is the one for a spontaneous symmetry breaking mechanism.

  6. 6.

    In fact, it could even be reduced to a technical step useful to perform the usual field quantization procedure, which relies heavily on the identification of propagators and mass terms in the Lagrangian.

  7. 7.

    While in the previous example we had \(G=K=SU(2)\subset H=U(1)\times SU(2)\).

  8. 8.

    Notice that from now on we shall make use of “\(\cdot \)” to denote Greek indices contractions, while Latin indices contraction is naturally understood from matrix multiplication.

  9. 9.

    Beware of the fact that in this index free notation a is the set of components of the 1-form a. This should be clear from the context.

  10. 10.

    Let us mention here how it is has been difficult, in several occasions, to convince some colleagues that these relations are not mathematically on the same footing.

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Authors and Affiliations

  1. Centre de Physique Théorique, Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

    J. Attard, S. Lazzarini & T. Masson

  2. Institut Élie Cartan de Lorraine, Université de Lorraine, UMR 7502, Vandœuvre-lès-Nancy, F-54506, France

    J. François

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  1. J. Attard
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Correspondence to T. Masson .

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  1. Nice and Sophia Antipolis University, Nice, France

    Joseph Kouneiher

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Attard, J., François, J., Lazzarini, S., Masson, T. (2018). The Dressing Field Method of Gauge Symmetry Reduction, a Review with Examples. In: Kouneiher, J. (eds) Foundations of Mathematics and Physics One Century After Hilbert. Springer, Cham. https://doi.org/10.1007/978-3-319-64813-2_13

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