Quantum field theory for condensation of bosons and fermions

Giuseppe Vitiello

H. Chu, Y. C. Chang, Hiroomi Umezawa +1 more

Colin John Lambert

We examine in detail the microscopic theories of Beliaev and Hugenholtz and Pines as applied to an interacting system of bosons in its ground state and demonstrate that these theories are strictly applicable to a zero density gas only. After providing a re-normalization of the time dependent perturbation theory of a many fermion system in its ground state, we reformulate the corresponding boson theory without ab initio approximations concerning the k=0 mode.The resulting boson perturbation theory has a diagrammatic structure which is topologically identical with the corresponding fermion theory and provided the implicit assumptions concerning convergence are valid, we conclude that a finite fraction of the particles in an arbitrary interacting Bose gas at T=ok will not condense into the k=0 mode.

A. S. Alexandrov

Xiao-Gang Wen

This book has three types of theories: boson, fermion, and gauge theories. It asks which theories describe the elementary particles in our universe. Surprisingly, elementary particles are described by the two more complicated and non-local theories: fermion theory and gauge theory. In the last chapter and in this chapter it is demonstrated that fermion and gauge theory are not really needed. Boson theory may be able to explain everything, since both fermion theory and gauge theory can emerge from local bosonic models (or qbit models), if bosons (qbits) are organized to have a string-net condensed order. This chapter studies soluble quantum spin models to explain string-net condensation. It finds that gauge bosons are simply collective excitations associated with string density wave (with the electric field corresponding to the string density). The fermons can appear as the ends of open strings. Thus the string-net order explains the emergence of gauge bosons and fermions. It unifies gauge interaction and Fermi statistics. The chapter shows that non-orientable strings give rise to Z_2 gauge theory while the orientable string give rise to U(1) gauge theory.

Giuseppe Mussardo

Abstract In this chapter we discuss two explicit examples of conformal field theories. We start our analysis with the free massless bosonic theory that we have already seen in the previous chapter. After, we discuss the conformal field theory of a complex fermion operator (a Dirac fermion) using its decomposition in the real Majorana components. The central charge of both bosonic and fermion theories is c = 1 and this suggests the existence of an equivalence between them. The transformation that maps a bosonic into a fermion theory and vice versa is known as bosonization: it provides a useful tool both for the comprehension of the conformal theories and for a wide range of applications, in particular in low-dimensional condensed matter systems.