Abstract
The reader probably followed a course on analytical mechanics during his/her university studies. If not, there are a great number of textbooks where its principles are explained in a detailed pedagogical way. I recommend two of them.
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Notes
- 1.
A functional is a function whose argument is itself a function.
- 2.
There are some reasons to believe that the Theory of Everything — a Holy Grail unified theory of all interactions including gravity — involves higher derivatives.
- 3.
We keep using the Einstein summation convention even though the set \(q_i\) is by no means a vector.
- 4.
Emmy Noether was a remarkable mathematician. Arguably, the most important woman in the history of mathematics. She worked in Germany in the beginning of the 20th century. In 1919 she got a professorship position at the University of Göttingen. But that was not easy for her. At that time, there were no women at the universities, and her colleagues hesitated to create a precedent. One of the faculty members protested, “What will our soldiers think when they return to the university and find that they are required to learn at the feet of a woman?” The controversy was settled by the intervention of David Hilbert, the leading, most respected German mathematician at that time. “Aber, meine Herren”, said he, “the university is still not a bath house!”.
- 5.
Even Feynman did not do so in his Lectures. Unfortunately, the courses of analytical mechanics and electromagnetism in the standard university curriculum do not usually “commute”.
- 6.
One easily derives it using \({\varvec{v}} = \dot{{\varvec{x}}_0}\).
- 7.
People do so to define and derive quite rigorously scattering amplitudes in quantum field theory. Such a derivation is beyond the scope of our book.
- 8.
Anticipating applications to high-energy physics, we have written this equation in relativistic notation using the 4-vector \(\partial _\mu \). But one can also write the equations of motion for a nonrelativistic field theory by replacing
$$\partial _\mu \left( \frac{\delta \mathcal{L}}{\delta (\partial _\mu \phi )} \right) \ \rightarrow \ \frac{\partial }{\partial t} \left( \frac{\delta \mathcal{L}}{\delta \dot{\Phi }_i } \right) + \frac{\partial }{\partial {\varvec{x}}} \left( \frac{\delta \mathcal{L}}{\delta ({\varvec{\nabla }}{\Phi }_i)} \right) \, , $$where \(\Phi _i\) is a relevant set of nonrelativistic fields. For example, the Eq. (4.27) could be derived in this way.
- 9.
Well, the complex conjugate of that equation, but that does not matter.
- 10.
- 11.
It represents the component \(T_{00}\) of a certain tensor \(T_{\mu \nu }\) called the energy-momentum tensor. We will meet it again in Chap. 15.
- 12.
The Hamiltonian in a free theory with \(\lambda =0\) and the potential \(V = - m^2 \phi ^* \phi \) also has no bottom. In this exotic tachyonic case, the field \(\phi \) does not oscillate, but grows exponentially with time. However, in free tachyonic theory (the presence of interactions change that), it never runs into a singularity; there is no collapse and there are no internal inconsistencies...
Well, this footnote was actually addressed not to our target reader, not to Sophie, but to an expert who may happen to see it. We will rediscuss these nontrivial issues in the last chapter.
- 13.
When acting on the Lagrangian density \(\mathcal{L}[\phi ({\varvec{x}}), \phi ^*({\varvec{x}});\, \partial _\mu \phi ({\varvec{x}}), \partial _\mu \phi ^*({\varvec{x}})]\), the symbol \(\delta /\delta \phi ({\varvec{x}})\) had the meaning of a usual partial derivative. But now it is a functional derivative acting on the wave functional (4.34).
- 14.
The reader is welcome to verify that
by trading the sum over the modes for the momentum integral:
$$ \sum _{{\varvec{n}}} \ \rightarrow \ \frac{V}{(2\pi )^3} \int d^3{\varvec{p}} \, . $$.
- 15.
Indeed, one can easily measure \({\varvec{E}}\) and \({\varvec{B}}\), but not \(\varphi \) and \({\varvec{A}}\).
- 16.
The Lagrangian density is defined up to a total divergence. The latter does not contribute to the action (7.35).
- 17.
- 18.
Incidentally, this explains our sign choice in (7.56) — we wanted to obtain an equation with oscillatory and not exponentially growing solutions — cf. the footnote on p. 117.
- 19.
The experimental limit for the photon mass is \(m_\gamma < 3\cdot 10^{-27}\) eV. This follows from the fact that the electromagnetic fields are long-range. We are sure that such long-range magnetic fields exist in our Galaxy because the light of the stars that we observe exhibits a small circular polarization, which can only be explained by interaction of the starlight with the interstellar magnetic fields. Hence the Compton wavelength of the photon cannot be much less than the size of the Galaxy.
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Smilga, A. (2017). Lagrangians and Hamiltonians. In: Digestible Quantum Field Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-59922-9_7
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DOI: https://doi.org/10.1007/978-3-319-59922-9_7
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