Abstract
The main aspects of variational symmetries are discussed: how symmetries are encoded in actions and how they restrict the form of action functionals, how quantum physics builds a bridge from symmetries to group representations. The three types of variational symmetries in fundamental physics are summarized, namely symmetries with respect to diffeomorphism, with respect to Lorentz transformations and with respect to “internal” group transformations. The history of symmetry considerations, starting from Galilei and ending with the recent discovery of a scalar boson at the LHC, is reflected. Some philosophical-inspired questions deal with the potential unification of physics by symmetries, the role of symmetries as basic “principle of physics” and the origin of symmetries. Finally, some thoughts are given to physics beyond symmetries, and to manners of formulating physics beyond our present approaches.
\(\ldots \) the unreasonable effectiveness of symmetry in our understanding of Nature.
This citation from A. Zee [578] alludes to the title of an article by E. Wigner on the “Unreasonable Effectiveness of Mathematics in the Natural Sciences”
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Notes
- 1.
Observe that one speaks of the standard “model” of particle physics and the “theory” of general relativity. This is perhaps unreflected by many, but it reveals that GR, simply due its elegance–derived from one of the greatest symmetries imaginable–is far more respected as an ultimate theory than the standard model, because as discussed above, the latter has many unexplained features and more than a dozen parameters of unknown origin.
- 2.
W. Pauli was known to be harsh, so this sounds like Pauli. In any case, at an International Congress of Philosophers, held in Zürich in 1954, he stated: “It seems likely to me, that the reach of the mathematical group concept in physics is not yet fully exploited.” The essential role which “symmetries” were playing in the work of Pauli is investigated in [222].
- 3.
http://www.aip.org/history/ohilist/4965.html
- 4.
Perhaps I can be allowed to make a personal remark: Between leaving the Gymnasium in the early 1960’s and starting to study physics, I was stationed with the German army near Göttingen. In an antiquarian bookshop, I found a small brochure about group theory, which in its introduction remarked that group theory is of a certain importance to physics. I became curious, and I’m still curious...
- 5.
“Every urchin in our mathematical Göttingen knows more about four-dimensional geometry than Einstein. Nevertheless, it was Einstein who did the work and not the great mathematicians.
- 6.
“... daß das elektrische Feld ein notwendiges Begleitphänomen ... des materiellen durch \(\psi \) dargestellten Wellenfeldes ist.”
- 7.
In this philosophical context I will write Nature, World, Universe with capitals.
- 8.
Despite the discussion in the literature of the philosophy of science, I do not distinguish between laws of Nature and laws of physics.
- 9.
I, for instance, do not even understand the abstract of the PhD thesis of my niece Vera, which somehow deals with microelectronics–but I’m not sure because I don’t understand her!
- 10.
“Daneben bestand aber auch von Anfang an das Bestreben, eine alle diese Einzelwissenschaften vereinigende theoretische Basis aufzufinden, bestehend aus einem Minimum von Begriffen und Grundrelationen, aus denen alle Begriffe der Einzelwissenschaften sich auf rein logischem Wege sollen ableiten lassen. Es ist das Suchen nach einem Fundament der ganzen Physik. Das Vertrauen in die Erreichbarkeit dieses höchsten Zieles ist eine Hauptquelle der leidenschaftlichen Hingabe, welche die Forscher von jeher beseelt hat.”
- 11.
Your T-shirt could have the patch \(\,\delta S_{\mathcal {W}}=0\).
- 12.
The axiomatization doctrine received a setback by Gödel’s incompleteness theorem from 1931. It is not clear whether this also applies at the same time to physics, because the ‘axiomatization’ schemes may differ from those in mathematics.
- 13.
Another axiomatization could be based on the ‘relativity without light’ derivation of Lorentz transformations as given in Subsect. (3.2.1).
- 14.
The physicist and philosopher C.F. von Weizsäcker defined the end of physics as that state of affairs where all questions that one directs to Nature are outside of physics [543].
- 15.
One should keep in mind that many predictions of the standard model are not derived completely analytically, but are very often approximate results derived perturbatively, partly based on computer simulations within lattice gauge theories.
- 16.
The interested reader is referred to an article by D. Deutsch [117] on an examination of the slogan “law without law” brought into the world by J. A. Wheeler.
- 17.
There could be another more general principle as a guiding theme in the fundamental interactions: Physics should not depend on how we describe it.
- 18.
It really should be told in their dialect–as I experienced it from my high school teacher, Dr. Szillis, who is responsibly for my own fascination with mathematical physics.
- 19.
It also blends in well with the relational interpretation of spacetime in the sense of Leibniz and Mach.
- 20.
G. Ellis is a highly respected relativist and cosmologist, who later in his career also worked on complex systems. I mention this because to me this gives him more reputation to discuss emergence and self-organization than many philosophers of science who deal with this topic.
- 21.
The German title of this book is “Abschied von der Weltformel”–something like “Farewell to the World Formula”; and this reveals the contrasting aspect of Laughlin’s book to those bestsellers of B. Green and S. Hawking.
- 22.
There is a website: www.nbi.dk/ kleppe/random/qa/qa.html.
- 23.
In [65], the consequences of mostly slight changes of the parameters of the Standard Model on everyday life are elucidated.
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Sundermeyer, K. (2014). Conclusion. In: Symmetries in Fundamental Physics. Fundamental Theories of Physics, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-319-06581-6_9
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