Abstract
This chapter is about what at the turn of the 19th to the 20th century confused the fundamental physics community, namely the more or less obvious contradiction between classical mechanics and electrodynamics. The solution of this conceptual clash gave birth to special relativity. The Lorentz transformations are derived from the principle of relativity and the assumption that space-time is homogeneous and space is isotropic. The four-vector notions of Minkowski space is introduced, and the basic concept of relativistic field theories are described and explained on the example of Maxwell electrodynamics. Noether’s first theorem for global symmetries leading to conserved Noether currents and Noether charges is derived and explained on the example of energy-momentum conservation (canonical and Belinfante energy-momentum tensor). The second Noether theorem applying to the case of local, i.e. spacetime-dependent symmetry transformations, is shown to give rise to a chain of Klein-Noether identities with the consequence that field equations cease to be independent. A further section deals with the properties of the Poincaré group and its Lie algebra, and a discussion of all the kinematical symmetry groups allowed by the postulates of special relativity and their mutual contractions from the de Sitter group/algebra. The de Sitter group and the conformal group are treated in more detail, both being generalizations of the Poincaré group. Finally some current ideas how Poincaré invariance could be manifest as a broken symmetry are presented.
War es ein Gott der diese Zeichen schrieb?
Ludwig Boltzmann, who himself contributed to formulating the Maxwell equations in the familiar form (3.1), used this sentence from J. W. von Goethe’s “Faust” in order to praise their beauty and symmetry.
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Notes
- 1.
See the appendix “Units and Dimensions” in [298].
- 2.
This is not the full truth: If one allows for a specific non-local dependence of the transformed fields on the original fields, Maxwell’s equations can be made invariant under Galilei boosts; see [207].
- 3.
“Not only in mechanics but also in electrodynamics the phenomena have no properties corresponding to the concept of absolute rest.”
- 4.
“Light always propagates in empty space with a definite velocity \(c\), independent of the state of motion of the emitting body.”
- 5.
“Die Lektüre setzt etwa Maturitätsbildung ...voraus”.
- 6.
In the rest of this section I follow [337, 342, 367]; for similar approaches see [220, 472, 508].
- 7.
Only a few decades ago you could judge from a publication whether the author was educated in general relativity or in particle physics just by his or her use of the metric. Since these disciplines nowadays have converged to a new discipline called “cosmoparticle physics”, one should be able to read expressions in both metric conventions and to translate them. I allow myself to be inconsistent in using the “mostly plus” metric in some contexts, but tried to emphasize this deviation properly.
- 8.
In fact, Einstein attended lectures by Minkowski as a student.
- 9.
“The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. Henceforth space by itself, and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”
- 10.
“Since the mathematicians pounced on the relativity theory I no longer understand it myself.”; taken from the Einstein biography of C. Seelig from 1960.
- 11.
Wilhelm Killing (1847-1923) was a professor of mathematics in Münster, a town in Westfalia/Germany. His name is less well known than for instance those of S. Lie and E. Cartan, although results going under their names were found by him independently.
- 12.
In the rest of this chapter, Lorentz tensors will simply be called tensors, since no confusion with Riemann tensors–the basic entities in general relativity–can arise.
- 13.
I’m aware that J. Barbour [25] would object to this statement.
- 14.
If the integral \(I\) is invariant with respect to a \(\left[ \mathrm {group}\right] \) \(\mathcal {G}_\rho \), then \(\rho \) linearly independent combinations of the Lagrange expressions become divergences - and from this, conversely, invariance of I with respect to a \(\left[ \mathrm {group}\right] \) \(\mathcal {G}_\rho \), will follow. The theorem holds good even in the limiting case of infinitely many parameters.
- 15.
If the integral I is invariant with respect to a \(\mathcal {G}_{\infty \rho }\) in which the arbitrary functions occur up to the \(\sigma \)-th derivative, then there exist \(\rho \) identity relationships between the Lagrange expressions and their derivatives up to the \(\sigma \)-th order. In this case also, the converse holds.
- 16.
These transformations could be called ’fibre preserving’ because the new coordinates \(\hat{x}\) only depend on the old coordinates and not on the fields \(Q\).
- 17.
This is not the full truth: The Aharanov-Bohm effect reveals traces of the gauge potentials in the form of holonomies.
- 18.
Nearly all expressions in this book can be stated in an arbitrary number of dimensions. This may sound stupendous, since “obviously” our world is 4-dimensional. But, who knows: there are Kaluza-Klein theories, string models, brane worlds, ...
- 19.
The definition of \(M_{\mu \nu }\) in (3.93) is made use of by all those authors who use the metric convention chosen in this book; this definition has as a consequence that the Lorentz generators appear with a minus-sign in the group element.
- 20.
You certainly know that the unit of length is no longer derived from the circumference of the earth’s equator and represented in the physical meter prototype preserved in Paris (as I learned in school), nor is it any longer defined in terms of the wavelength of light emitted by a krypton isotope (as I learned at the university), but–since 1983–in terms of the velocity of light. The International Bureau of Weights and Measures (BIPM) states: “The meter is the length of the path travelled by light in vacuum during a time interval of 1/299792458 of a second.” This, by the way, answers the question about how precisely \(\,c\,\) is measured.
- 21.
In this subsection I use the notation of Bacry and Lévy-Leblond. You can recover the notation of this book by making the replacements \(\,J_k \rightarrow iJ^k\), etc. I also denote the translation operator as \(P\) although, strictly speaking this is the conserved momentum associated to translation symmetry generator \(T\).
- 22.
In a later publication [19] the requirement on parity and time reversal was dropped, given that these are not symmetries of nature–at least for weak interaction processes.
- 23.
Lévy-Leblond realized that this kind of world was already described in the literature. Lewis Carroll [77]: “A slow sort of country,” \(\gg \) said the Queen \( \ll \). “Now, here, you see, it takes all the running you can do, to keep in the same place.”
- 24.
Named after the German city Erlangen, now in the Federal state of Bavaria.
- 25.
Also, the term “dilatations” is used; this refers to British English.
- 26.
More about this in Sect. 7.3.1.
- 27.
I’m not aware whether all the infinite conservation laws can be derived from Noether’s theorem, how general the symmetry transformations are to be, and which symmetry groups are at work. I’m also not aware of a physical interpretation of the additional symmetries. It seems in any case that most of these conservation laws do not carry over to the case of interacting fields.
- 28.
In the quantum case this algebra receives a further term proportional to a central charge. This depends on the dimension of spacetime and vanishes for \(D=26\).
- 29.
“...da \({\mathbf {G}}_{\mathbf {c}}\) mathematisch verständlicher ist als \({\mathbf {G}}_\infty \), hätte wohl ein Mathematiker in freier Phantasie auf den Gedanken verfallen können, da \(\beta \) am Ende die Naturerscheinungen tatsächlich eine Invarianz nicht bei der Gruppe \({\mathbf {G}}_\infty \), sondern vielmehr bei einer Gruppe \({\mathbf {G}}_{\mathbf {c}}\) mit bestimmtem endlichen .... \(c\) besitzen.”
- 30.
Named after the Dutch astronomer Willem de Sitter (1872–1934)
- 31.
It seems that this is one of the rare circumstances where in 2011, a Nobel prize was awarded for increasing our nescience. The discovery led to the term “dark energy”: At least 70 % of the matter-energy content of the universe cannot yet be explained.
- 32.
This is – by general relativists – known by the names of the authors J. Ehlers, F. A. E. Pirani and A. Schild as the EPS axiomatics.
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Sundermeyer, K. (2014). Electrodynamics and Special Relativity. In: Symmetries in Fundamental Physics. Fundamental Theories of Physics, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-319-06581-6_3
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