A theory is formulated for the ground state of a degenerate Bose fluid in thermal equilibrium. The fluctuations in the condensate are treated in terms of the fluctuation operators C = a 0 -α, and thir conjugate, α 0 being the annihilation operator for the zero momentum state and the order parameter α the quasiaverage of α 0 . The superfluid ground state is viewed as energy lowering by coherent excitations. The coherent state is defined in terms of the normal modes in the superfluid phase. The superfluid ground state is not the ground state of the type studied by Glassgold and Sauermann.
A theory is formulated for the ground state of a degenerate Bose fluid in thermal equilibrium. The fluctuations in the condensate are treated in terms of the fluctuation operators C = a 0 -α, and thir conjugate, α 0 being the annihilation operator for the zero momentum state and the order parameter α the quasiaverage of α 0 . The superfluid ground state is viewed as energy lowering by coherent excitations. The coherent state is defined in terms of the normal modes in the superfluid phase. The superfluid ground state is not the ground state of the type studied by Glassgold and Sauermann.
A general theory is formulated for the fluctuations in the condensate of a degenerate boson fluid in homogeneous thermodynamic equilibrium. The point of view adopted is based on Bogoliubov's description of the condensate in terms of quasiaverages and on his "Principle of Diminishing Correlation." An essential aspect of the theory is the consistent treatment of the fluctuation operators c and ${c}^{\ifmmode\dagger\else\textdagger\fi{}}$, where $c={a}_{0}\ensuremath{-}\ensuremath{\alpha}$. a0 is the annihilation operator for the zero momentum state, and the order parameter $\ensuremath{\alpha}$ is the quasiaverage of a0. After the Hamiltonian for the fluctuation operators is obtained, a stability condition is derived, which we then demonstrate is satisfied under very general conditions. The energy spectrum of the condensate fluctuations is shown to be discrete, and explicit expressions are derived for the quantum states of the condensate. The ground state of the condensate is not a coherent state of the type studied by Glauber. A detailed analysis is given of the fluctuations in the order parameter and in the number of particles in the condensate. The physical properties of the fluctuations are determined by two parameters, which are the averages of the second derivatives of the Hamiltonian with respect to the order parameter. These quantities are shown to be related to the one-particle Green's functions introduced by Beliaev. The implications of this general theory are also worked out for Bogoliubov's low-density, weak-coupling model. The limit of the ideal Bose gas is then shown to possess pathological features, namely an unnormalizable ground state and infinite fluctuations, in contrast to the interacting fluid where all properties are well-behaved.
The coherent-state representation of the density operator for an ideal Bose gas in a thermal equilibrium is introduced. It is shown that a pure coherent state can be an eigenstate of the Hamiltonian of the non-interacting boson system when the symmetry breaking term is added to it. However, at the limit of the vanishing “classical field” only zero-energy states can be a coherent state. It is also proved that the pure coherent state is the lowest energy state among all the zero-momentum states. The temperature dependence of the coherent state is then discussed. The diminution of the coherent state can clearly be seen as the temperature rises from zero to the critical temperature.
A theory of superfluidity (S.F.) is developed from first principles using two novel concepts, (1) that of a 'superfluid Ensemble' (S.E.) i.e. a 'Restricted Ensemble' constructed from a 'Separable Phase Space' (S.P.S.) admitting independent configurations, at least one set of which are statistically equivalent. (2) The notion of 'Dynamical Equivalence' (D.E.), satisfied if and only if (i) all dynamical symmetries are rearranged (not broken) for two Lagrangian formulations of the same problem and if (ii) the expectation values of all the constants of motion are the same, even if their functional expressions are not. The dynamical variables (d.v.) of the S.P.S. are defined from the ('q' and 'c' number) fields of the most general 'Linear Coherent State Representation', more general than those of Glauber and Bogoljubov-Valatin combined. Three independent pairs of d.v. are obtained.
D.E. is proven for the Ideal Bose Gas and for a non-linear, interacting zero order Bose problem (I.Z.O.P.). An exact relation is obtained from the action principle, ensuring the cancellation of 'low and high order dangerous diagrams'. From this it follows that D.E. for the exact interacting problem must be demonstrable at infinite order of perturbation, in the finite volume limit. The I.Z.O.P. is posed in the Random Phase Approximation (R.P.A.), free from 'anomalous averages' and solved for the three branches of the excitation spectrum in a pure state description; the lowest branch is gapless, whilst the upper two coincide and show a gap. The standard strategy of linearization is found to be faulty.
The partition functions for both superfluid and non-superfluid ensembles are obtained for the I.Z.O.P. in the R.P.A. The coincident upper two branches (in a pure state description) split into a band in thermal equilibrium for the superfluid ensemble, in agreement with an upper band recently observed experimentally. O.D.L.R.O. is found in the second reduced density matrix, but ruled out in the first. Integral equations are obtained in thermal equilibrium - for the I.Z.O.P. in the R.P.A.; which differ, however, from those of existing approaches for the same problem. Most existing theories of S.F. are in fact shown not to predict superfluid behaviour. The present theory is applicable to arbitrary Bose or Fermi systems, whether superfluid or not. O.D.L.R.O. is found to be sufficient for SF. No a priori assumption is made as to the occurrence of Bose-Einstein condensation, its existence being here contingent on the solution of the integral equation; in any case, it is not to be associated with O.D.L.R.O. or S.F.
A theory of superfluidity (S.F.) is developed from first principles using two novel concepts, (1) that of a 'superfluid Ensemble' (S.E.) i.e. a 'Restricted Ensemble' constructed from a 'Separable Phase Space' (S.P.S.) admitting independent configurations, at least one set of which are statistically equivalent. (2) The notion of 'Dynamical Equivalence' (D.E.), satisfied if and only if (i) all dynamical symmetries are rearranged (not broken) for two Lagrangian formulations of the same problem and if (ii) the expectation values of all the constants of motion are the same, even if their functional expressions are not. The dynamical variables (d.v.) of the S.P.S. are defined from the ('q' and 'c' number) fields of the most general 'Linear Coherent State Representation', more general than those of Glauber and Bogoljubov-Valatin combined. Three independent pairs of d.v. are obtained. D.E. is proven for the Ideal Bose Gas and for a non-linear, interacting zero order Bose problem (I.Z.O.P.). An exact relation is obtained from the action principle, ensuring the cancellation of 'low and high order dangerous diagrams'. From this it follows that D.E. for the exact interacting problem must be demonstrable at infinite order of perturbation, in the finite volume limit. The I.Z.O.P. is posed in the Random Phase Approximation (R.P.A.), free from 'anomalous averages' and solved for the three branches of the excitation spectrum in a pure state description; the lowest branch is gapless, whilst the upper two coincide and show a gap. The standard strategy of linearization is found to be faulty. The partition functions for both superfluid and non-superfluid ensembles are obtained for the I.Z.O.P. in the R.P.A. The coincident upper two branches (in a pure state description) split into a band in thermal equilibrium for the superfluid ensemble, in agreement with an upper band recently observed experimentally. O.D.L.R.O. is found in the second reduced density matrix, but ruled out in the first. Integral equations are obtained in thermal equilibrium - for the I.Z.O.P. in the R.P.A.; which differ, however, from those of existing approaches for the same problem. Most existing theories of S.F. are in fact shown not to predict superfluid behaviour. The present theory is applicable to arbitrary Bose or Fermi systems, whether superfluid or not. O.D.L.R.O. is found to be sufficient for SF. No a priori assumption is made as to the occurrence of Bose-Einstein condensation, its existence being here contingent on the solution of the integral equation; in any case, it is not to be associated with O.D.L.R.O. or S.F.
The phenomena of superfluidity and superconductivity are intimately connected with the existence of a condensate, a macroscopically occupied quantum state. Such condensates occur in a variety of different physical systems, as described in Chapter 1. The foundation for the description of superfluidity is a picture of the system as being comprised of a condensate and elementary excitations. In Chapter 8 we have seen how physical properties such as the energy and the density of a Bose–Einstein condensed system may be expressed in terms of a contribution from the condensate, plus one from the elementary excitations, and in this chapter we shall consider further developments of this basic idea to other situations. As a first application, we determine the critical velocity for creation of an excitation in a homogeneous system (Sec. 10.1). Following that, we show how to express the momentum density in terms of the velocity of the condensate and the distribution function for excitations. This provides the basis for a two-component description, the two components being the condensate and the thermal excitations (Sec. 10.2). In the past, this framework has proved to be very effective in describing the properties of superfluids and superconductors, and in Sec. 10.3 we apply it to dynamical processes. To describe the state of a superfluid, one must specify the condensate velocity, in addition to the variables needed to characterize the state of an ordinary fluid. As a consequence, the collective behaviour of a superfluid is richer than that of an ordinary one. Collective modes are most simply examined when excitations collide frequently enough that they are in local thermodynamic equilibrium.