Abstract
As we have already noted in Chap. 1, Maxwell’s electromagnetic theory is by definition a relativistic theory, since it implies in particular the constancy of the speed of light in every RF. As such it must be covariant under the group of Lorentz transformations, or, using the terminology of the previous chapter, covariant under the group \(\mathrm{SO}(1,3)\).
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Notes
- 1.
We are using the so called Heaviside-Lorentz (HL) system of units, the most useful for theoretical considerations. It amounts to considering the electric charge as a quantity whose physical dimensions are derived from the basic dimensional quantities [M, L, T] (the corresponding units being [kilogram, meter, second]), by writing Coulomb’s law without additional physical constants, namely in the following form:
In this way the electric charge has the physical dimensions \([M^{\frac{1}{2}}L^{\frac{3}{2}}T^{-1}]\), and the electric field \([M^{\frac{1}{2}}L^{-\frac{1}{2}}T^{-1}]\). (Note that the presence of the factor \(\frac{1}{4\pi }\) in Coulomb’s law means that the HL system is rationalized, that is there are no factors \(4\pi \) explicitly appearing in Maxwell’s equations). Moreover the electric and magnetic fields are defined so as to have the same dimension, so that the Lorentz force reads:
The quickest way to translate formulae written in the international system of units (S.I.) into the HL one, is to redefine the electric charge as follows: Let \(\tilde{e}\) and e be the measures of the electric charge in the S.I. and the HL systems respectively. We then have:
$$ e=\frac{\tilde{e}}{\sqrt{\varepsilon _0}}\Rightarrow \rho =\frac{\tilde{\rho }}{\sqrt{\varepsilon _0}},\quad \mathbf{j}=\frac{\tilde{\mathbf{\jmath }}}{\sqrt{\varepsilon _0}}. $$Moreover the electric and magnetic fields \({\tilde{\mathbf {E}}}\), \(\tilde{\mathbf {B}}\) in the SI system are related to the analogous quantities \(\mathbf {E}\) e \(\mathbf {B}\) in the HL system as follows: \(\sqrt{\varepsilon _0}\,{\tilde{\mathbf{{E}}}}=\mathbf{E}\); \(\mathbf{B}=\frac{1}{\sqrt{\mu }_0}{\tilde{\mathbf{{B}}}}\).
For example, the energy density takes the form: \({\rho _E}=\frac{1}{2}\left( \varepsilon _0 |\tilde{\mathbf{E}}|^2 +\frac{ |\tilde{\mathbf{B}}|^2}{\mu _0}\right) = \frac{1}{2}\left( |\mathbf{E}|^2+|\mathbf{B}|^2\right) \).
- 2.
Actually, since Eq. (5.20) contains no free indices, it is a scalar equation, namely \(\partial _\mu J^\mu (x)= \partial _{\mu }' J^{\mu \prime }(x')\).
- 3.
Note that \(\mathbf {x}_k (t)\) is a kinematical variable referred to the kth particle, while \((x^\mu )=(c\,t,\mathbf{x})\) are space-time labels.
- 4.
In Cartesian coordinates, if \(\mathbf{x}=(x,y,z)\) and \(\mathbf{x}_k(t)= (x_k(t),y_k(t),z_k(t))\), the three-dimensional Dirac delta function reads: \(\delta ^3(\mathbf {x}-\mathbf {x}_k)=\delta (x-x_k(t))\,\delta (y-y_k(t))\,\delta (z-z_k(t))\).
- 5.
We used \(\delta (\alpha x)= \delta (x)/\alpha \), a particular case of the incoming formula (5.56).
- 6.
This property is easily proven on test functions \(f(x)=f(x^\mu )\). Indeed we can write \(\int d^4 x\,\delta ^4(\varvec{\varLambda }\cdot x)\,f(x)=\int d^4 x\,\delta ^4(x')\,f(\varvec{\varLambda }^{-1}\cdot x')= \int \frac{d^4 x'}{|\mathrm{det}(\varvec{\varLambda })|}\,\delta ^4(x')\,f(\varvec{\varLambda }^{-1}\cdot x')=\frac{f(0)}{|\mathrm{det}(\varvec{\varLambda })|}\), where we have changed the integration variable from \(x\equiv (x^\mu )\) to \(x'\equiv \varvec{\varLambda }\cdot x\). Recalling that \(f(0)=\int d^4 x\,\delta ^4(x)\,f(x)\), and being f(x) generic, we conclude that \(\delta ^4(\varvec{\varLambda }\cdot x)=\frac{1}{|\mathrm{det}(\varvec{\varLambda })|}\,\delta ^4(x)\).
- 7.
Note that since \(T_{\mathrm{part.}}^{\mu \nu }\) is a tensor, and being the product \(p^\mu p^\nu \) a rank-two tensor as well, from Eq. (5.61) it follows that \(\delta ^3(\mathbf {x}-\mathbf {x}_k(t))/E_k\) transforms as a rank 0-tensor, that is a scalar quantity.
- 8.
Note that in our conventions all the components of \(T^{\mu \nu }_{em}\) have the physical dimensions of a momentum density.
- 9.
While in the classical theory the only measurable physical quantities are \(\mathbf {E}\) and \(\mathbf {B}\), so that the four-potential seems not necessary for a complete description of the electromagnetic field, in quantum-mechanics the Aharonov-Bohm effect shows that the \(\mathbf {E}\) and \(\mathbf {B}\) fields are not sufficient for describing the electromagnetic field in interaction with matter, and that for its full description the four-potential \( A_\mu (x)\) is necessary.
- 10.
This is not the only possible gauge condition. Several other choices are possible. In particular, when discussing the quantization of the electromagnetic field in Chap. 6, we shall use the more convenient Coulomb gauge \(\mathbf {\nabla }\cdot \mathbf {A}(x)=0\). The Lorentz gauge has the advantage of being Lorentz covariant. In Chap. 11, the same quantization will be performed using the covariant Lorentz gauge.
- 11.
This interpretation was already proposed by Einstein in 1905 for the description of the photoelectric effect.
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D’Auria, R., Trigiante, M. (2016). Maxwell Equations and Special Relativity. In: From Special Relativity to Feynman Diagrams. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-22014-7_5
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DOI: https://doi.org/10.1007/978-3-319-22014-7_5
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