Appendix B
Breaking of the Galilei Group and Plasmon
Energy Spectrum
An interesting physical model which clearly displays the structure discussed in the
previous Appendix is the electron gas in a uniform background.
15 In particular, one
may explicitly check that the Goldstone energy spectrum associated with the breaking
of the Galilei group has a gap corresponding to the plasma frequency (evasion of the
Goldstone theorem, as a consequence of the long range Coulomb potential).
In this way, one does not only get non-perturbative information on a collective
phenomenon in the many-body theory, but also displays a mechanism with intriguing
analogies with the Higgs phenomenon (in the Coulomb gauge).
B.1 Electron Gas in a Uniform Background
The model is obtained by neglecting the ion dynamics and by approximating the
ions with a uniform background of charge density ρ B ; it may be regarded as the zero
order expansion of the full theory in the ratio m/M, with m, M the electron and the
ion masses, respectively.
i) Infrared regularization
The infrared regularized Hamiltonian is
H L =
1
2m
d
3 x|∇ψ(x)|
2
−
d
3 x d
3 y ψ
∗
(x) ψ(x)U L (x − y) ρ B +
1
2
d
3 xd
3 y ψ
∗
(x) ψ
∗
(y) U L (x − y) ψ(y) ψ(x),
(B.1)
15 G. Morchio and F. Strocchi, Ann. Phys. 170, 310 (1986).
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0
237