In this paper, it has been shown that the usual demonstrations of the phenomenon of Bose-Einstein condensation are invalid. Exact formal solutions have been derived for the partition functions and the mean occupation numbers in classical and Bose statistics, and the theory of fluctuations in the case of Bose statistics has been given. In a second paper, convenient approximations will be derived for the mean occupation numbers in Bose statistics, and it will be shown that for temperatures well above the critical temperature of the supposed Bose-Einstein condensation the well-known result for the mean occupation numbers is valid, whilst for temperatures rather lower than this critical value it will be shown that particles will condense into the lowest available state. The critical temperatureT0is given by where the summation is over all energy levels except the lowest.
The order-by-order renormalization of the self-consistent mean-field potential in many-body perturbation theory for normal Fermi systems is investigated in detail. Building on previous work mainly by Balian and de Dominicis, as a key result we derive a thermodynamic perturbation series that manifests the consistency of the adiabatic zero-temperature formalism with perturbative statistical mechanics---for both isotropic and anisotropic systems---and satisfies at each order and for all temperatures the thermodynamic relations associated with Fermi-liquid theory. These properties are proved to all orders.
On the basis of an exact perturbational expression for the interacting one-particle Green function
G corresponding to bosons / fermions in terms of the bare interaction potential
v and permanents / determinants of the non-interacting one-particle Green function
G0, we deduce four recursive perturbation expansions for the self-energy
Σ. With
W denoting the dynamic screened interaction potential, these perturbation expansions are identical to those of
Σ in terms of (i) self-energy diagrams and
(v,G0), (ii)
G-skeleton self-energy diagrams and
(v,G), (iii)
W-skeleton self-energy diagrams and
(W,G0), and (iv)
G- and
W-skeleton self-energy diagrams and
(W,G). For the calculation of
W, we rely on a similar exact perturbational expression for the interacting two-particle Green function
G2 as for
G. From this expression, we deduce four recursive perturbation expansions for the polarization function
P, necessary for the calculation of
W, that are similar to those for
Σ specified above. The correlation functions considered in this paper may be corresponding to ground states, and thermal ensemble of states. For thermal ensemble of states, we consider both the imaginary-time formalism of Matsubara, and the real-time formalism of thermo-field dynamics (TFD). The latter is advantageous for the direct calculation of dynamic correlation functions. In an appendix, we apply the formalisms presented in this paper to the Hubbard Hamiltonian for spin-
21 fermions on a lattice in arbitrary
d spatial dimensions. In two further appendices, we present methods and short programs for determining the
νth-order diagrams corresponding to the perturbation expansions of
G in terms of
(v,G0), and
Σ in terms of
(v,G0) and
(v,G) on the basis of the cycle decompositions of the elements of the symmetric group
S2ν.
It is shown in this article that the statistical character of the nonrelativistic, one particle quantum probabilities can be inferred from the function which results from the Wigner transformation on the field operator ψ̂†(x⃗) which, in turn, must be expressed in the occupation number representation. The function which results from the Wigner transformation is shown to be the state function of a single particle which is defined in the conventional quantum mechanics. The integral of the norm squared of that function is proven to be unity which is the step that proves the Max Born hypothesis. The compatibility of the expression for the field amplitude, obtained here, with the measurability of the Bosonic and the Fermionic fields is explained.
We consider a dilute homogeneous mixture of bosons and spin-polarized fermions at zero temperature. We first construct the formal scheme for carrying out systematic perturbation theory in terms of single particle Green's functions. We especially focus on the description of the boson-fermion interaction. To do so we need to introduce the renormalized boson-fermion
T matrix, which we determine to second order in the boson-fermion
s-wave scattering length. We also discuss how to incorporate the usual boson-boson
T matrix in mean field approximation to obtain the total ground-state properties of the system. The next-order term beyond mean field stems from the boson-fermion interaction and is proportional to
aBFkF. The total ground-state energy density to this order is the sum of the kinetic energy of the free fermions, the boson-boson mean-field interaction, the usual mean-field contribution to the boson-fermion interaction energy, and the first boson-fermion correction beyond mean field. We also compute the bosonic and the fermionic chemical potentials, the compressibilities, and the modification to the induced fermion-fermion interaction. We discuss the behavior of the total ground-state energy and the importance of the correction beyond mean field for various parameter regimes, in particular considering mixtures of
6Li and
7Li and of
3He and
4He. Moreover, we determine the modification of the induced fermion-fermion interaction due to the effects beyond mean field. We show that there is no effect on the depletion of the Bose condensate to first order in the boson-fermion scattering length
aBF.In Chapter 9 we applied the second quantization to the Hartree–Fock method. In this chapter we provide the basic aspects of the thermal Green function method, which is convenient for systematic perturbative expansion beyond the mean-field approximation. Several problems in quantum statistical mechanics can be elegantly and compactly described in terms of the so-called Matsubara Green functions (Matsubara (1955); Abrikosov et al. (1963)). Hence, if on one hand, the density matrix description, often used in the previous chapters, has a more direct physical interpretation, on the other hand the Green function method is far more powerful and takes advantage of a handy diagrammatic perturbation expansion. In the following chapters, we rely on the Green function for a more complete treatment of some many-body problems. To be self contained we introduce here this technical chapter with a short presentation of the by now standard Green function method. For a comprehensive treatment we refer the reader to the many books available and in particular to Abrikosov et al. (1963). The Matsubara Green function The main aim of quantum statistical mechanics is the evaluation of the partition function (see Eqs. (6.31) and (6.32)), e.g. in the grand canonical ensemble In the case of quantum gases one can evaluate Ƶ exactly as we did in Chapter 7. In general, the evaluation of the partition function is a difficult task and it is useful to use perturbation theory to obtain approximate solutions. To set up a perturbation theory, one begins by splitting the Hamiltonian as H = H 0 + H i, where H 0 represents the “easy” part, which can be solved exactly. In the second step, one could naturally think of expanding the exponential of the interacting part H i in order to write the partition function as a perturbative series. However, such a procedure requires extreme care due to the non-commutative nature of operators in quantum mechanics. The way of solving this problem is via the “ time ” ordering of operators in each term of the perturbative expansion. It can be shown, as we shall see in the present chapter, that the quantum statistical average of any time-ordered sequence of operators with respect to the unperturbed Hamiltonian H 0 can be expressed in terms of elementary quantum statistical averages of two field operators. Such elementary averages are called Green functions and in this section we introduce the basic definitions concerning them.