The Importance of Being Symmetrical

Dear Colleagues,

In Strena seu de nive sexangula (1611), J. Kepler poses a mathematical problem on the packaging of spheres in an Euclidean space, but also anticipates a very modern style of reasoning in physics. In fact, by starting from the extraordinary symmetries of the snowflake, Keplero hypothesizes something about its "particle" structure, while ignoring the concept of a molecule. Like much of Kepler's work, this reflection is also in the balance between an ancient concept of symmetry, linked to "beauty", and a "dynamic" concept, connected instead to the interplay of interactions in a system. It is to the last one that the first modern considerations on symmetry in physics, due to Pierre Curie, must be referred (Sur la symetrie dans les phenomenes physiques. Symetrie d’un champ   ́electrique et d’un champ magnetique, Journal de Physique 3: 393-415), as follows:

1) When certain causes produce certain effects, the symmetry elements of the causes must be found in their effects.

2) When certain effects show a certain dissymmetry, this dissymmetry must be found in the causes that gave rise to them.

3) In practice, the converses of these two propositions are not true, that is, the effects can be more symmetric than their causes.

4) A phenomenon may exist in a medium having the same characteristic symmetry or the symmetry of a subgroup of its characteristic symmetry. In other words, certain elements of symmetry can coexist with certain phenomena, but they are not necessary. What is necessary, is that certain elements of symmetry do not exist. Dissymmetry is what creates the phenomenon.

Even today, one can still admire this synthesis, which foreshadows the powerful notion of symmetry breaking, at the base of unification programs in cosmology and particle physics. It is a sign of the high level of coherence of our theories on the nature that even the GR, in spite of physical differences, were later revealed to belong to the same "mathematical key bunch" of gauge theories.

The reason symmetries (and their breaks) work so well was highlighted in an exemplary way by Emmy Noether in 1915, which clarified the close relationship between symmetries, invariances, and conservation laws, indicating the ideal formalism in group structures. Almost everything we know about nature can be expressed in terms of symmetry, as follows: The universe after the big bang was in a hot phase that then lost its symmetries, like a drop of water when it freezes into ice, through a sequence of breaks that generated families of elementary particles by pairing with Higgs particles. The deep symmetries of the laws of nature are thus transformed into the richness of the observed world.

In the end, it is correct to say “That symmetry dictates interactions, that geometry is at the heart of physics” (Chen Non Yang, Einstein's impact on theoretical physics, Contemporary Physics, 1980).

The formidable questions posed by quantum gravity re-proposes the question from new points of view. Beyond the recent question within the Anti-de Sitter/conformal field theory correspondence (Daniel Harlow, Hirosi Ooguri, Constraints on Symmetries from Holography, PRL, 191601 (2019)), there is a wider front that tends to resolve symmetries according to an emergentist perspective, whose tutelary deities are Robert Laughlin and, recently, Edward Witten (Symmetry and emergence, Nature Physics volume 14, pages 116–119 (2018). Again, there are recurring questions in the history of physics, namely: are symmetries "written" in a background at the bottom of the world, or emerging from the constraints on this background? The closest suggestion comes from condensed matter and the numerous points of contact with QFTs.

Beyond these foundational questions, symmetry continues to be a solid guide for the exploration of the physical world, as demonstrated by the recent results of the PT symmetry in the study of dissipative systems and quantum computing (M. Naghiloo, M. Abbasi, Yogesh N. Joglekar, and K. W. Murch, Quantum state tomography across the exceptional point in a single dissipative qubit, Nature Physics, 2019).

This Special Issue is conceived both as a place to discuss about “Quantum Symmetries” in their philosophical and foundational aspects, and in the rich panorama of their applications.

-  The role of symmetry in physics: historical and philosophical aspects

-   Local and global symmetries

-   Continuous and discrete symmetries

-   Lorentz and de Sitter group

-   Unitary group in QFT

-   CPT symmetry in particles physics and cosmology

-   PT symmetry, dissipative systems, and quantum computing

-   Renormalization group

-   Dynamical symmetries and chaotic behavior in physical systems

-   Symmetry Breaking in Condensed Matter, Particle Physics and Cosmology

-   Irreversibility and time-symmetry breaking

Prof. Dr. Ignazio Licata
Prof. Dr. Roberto Passante
Guest Editors

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• The role of symmetry in physics: historical and philosophical aspects
• Local and global symmetries
• Continuous and discrete symmetries
• Lorentz and de Sitter group
• Unitary group in QFT
• CPT symmetry in particles physics and cosmology
• PT symmetry, dissipative systems, and quantum computing
• Renormalization group
• Dynamical symmetries and chaotic behavior in physical systems
• Irreversibility and time-symmetry breaking