Abstract
Consider a charged particle of mass \(m_0\) and charge e, moving in the electromagnetic field \({\mathbf {E}},\,{\mathbf {B}}\), defined in terms of the usual electromagnetic potentials \(V,\,{\mathbf {A}}\).
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Notes
- 1.
There exists also the possibility to define a Lorentz-invariant Lagrangian whose integral with respect to an invariant parameter leads to the action of the relativistic particle in the covariant formalism. The procedure is thoroughly and transparently presented in the book of H. Goldstein, Classical Mechanics (2nd ed.), Addison-Wesley, 1980.
- 2.
Since \(A_\nu \) is non-zero, this means that the matrix \(S - \lambda I\) is singular, which in turn means that its determinant is 0 (non-invertible). Thus, the roots of the function \(\det \,(S - \lambda I)\) are the eigenvalues of S, so it is clear that this determinant is a polynomial in \(\lambda \).
- 3.
A hypersurface is a generalization of an ordinary two-dimensional surface embedded in three-dimensional space, to an \((n-1)\)-dimensional surface embedded in n-dimensional space; in our case, \(n = 4\).
- 4.
In comparing the formulas with those in Sect. 4.9, note that the symbol k there means \(|\mathbf {k}|\), while in this section it signifies the four-vector \(k^\mu \).
- 5.
The reason is that the translation group is a real continuous Abelian group (it is obvious that any two translations commute among themselves) and as such it admits only scalar, or trivial, irreducible representation. Proving this statement is beyond the scope of this book.
- 6.
In the static limit the d’Alembertian operator becomes the Laplacian operator.
- 7.
This is a potential used in particle and atomic physics and it is also called a screened Coulomb potential. The name of this potential comes from the Japanese theoretical physicist (and the first Japanese Nobel laureate) Hideki Yukawa (1907–1981).
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Chaichian, M., Merches, I., Radu, D., Tureanu, A. (2016). Relativistic Formulation of Electrodynamics in Minkowski Space. In: Electrodynamics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17381-3_8
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DOI: https://doi.org/10.1007/978-3-642-17381-3_8
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