By Cox R.T., Hubbard J.C.

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In a non-relativistic quantum theory the introduction of the multi-particle space is never mandatory, but is of convenience since it allows for an automatic way of respecting the quantum statistics of the particles even when interactions are present. It is also quite handy, but again not mandatory, when it comes to the description of symmetry broken states such as the cases of condensed states of fermions in superconductors and superﬂuid 3 He, and for describing Bose–Einstein condensates of bosons.

Show that [np , a†p ]s = a†p , [np , ap ]s = s ap . 4. 1 25 Phonons The bose ﬁeld does not occur only in connection with the elementary bosonic particles of the standard model, but can be useful in describing collective phenomena such as the long wave length oscillations of the ions in say a metal or a semiconductor, and we turn to see how this comes about. The Hamiltonian describing the ions of mass M and density ni in a crystal lattice is given by the kinetic energy term for the ions and an eﬀective ion–ion interaction determined by the screened Coulomb interaction.

These relations are therefore similar to those for bosons except that obnoxious sign factors occur owing to the Fermi statistics. The occupation number representation for fermions is therefore not attractive as the wedge is not explicit. 6 Summary In this chapter we have considered the quantum mechanical description of systems which can be in superposition of states with an arbitrary content of particles. To deal with such situations, endemic to relativistic quantum theory, quantum ﬁelds were introduced, describing the creation and annihilation of particles.