The coherent-state formalism is used to develop a theory of fluctuations and irreversible processes in a superfluid Bose system. Equations of motion for the density matrix are derived both for the case of an isolated system and for the system in interaction with a heat bath. The formalism allows one to go easily to the classical limit and thus to obtain equations for the probability distribution of the superfluid order parameter. The results are in a form suitable for the calculation of the decay of superflow near the $\ensuremath{\lambda}$ point and other similar phenomena.
The coherent-state formalism is used to develop a theory of fluctuations and irreversible processes in a superfluid Bose system. Equations of motion for the density matrix are derived both for the case of an isolated system and for the system in interaction with a heat bath. The formalism allows one to go easily to the classical limit and thus to obtain equations for the probability distribution of the superfluid order parameter. The results are in a form suitable for the calculation of the decay of superflow near the \ensuremathλ point and other similar phenomena.
It is shown that the coherent-state representation of a many-boson wave function may be identified with the order-parameter function conventionally used to describe a superfluid. The statistical mechanics of the many-boson system is reformulated in terms of the coherent states, and a theory of the Ginzburg-Landau form is recovered in an obvious approximation. The formalism is particularly useful for describing metastable states of finite superflow and the fluctuations which may cause spontaneous decay of such states.
In any system, no matter how dreadfully off equilibrium it may be, this is the chemical potential. It is quite important that f, the mean value of the wave field, is a complex order parameter. The foundations of the subject of quantum coherence were laid long before there were any acceptable microscopic theories of superconducting electron gas in metals phenomena. The proposal O. Penrose and L. Onsager made was that one could essentially define superfluidity of a Bose system such as liquid He as a state in which the density matrix of the system factorized in a certain special way. If the energy were independent of the relative phase of the quantum field operator on the two sides the author would not have such a mean value because on the average the two phases would fluctuate completely independently. It is a different coherence, not a superposition of the original ones but a different coherent state.
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.
A formal apparatus is developed for dealing with quantum fluctuations in a driven (nonequilibrium) state. Phenomenological equations for the decay of the system from an arbitrary state to the driven state are presumed known. By assuming that at some initial time the system-reservoir density matrix can be factored, we establish that the spectrum of fluctuations is given by the Fourier transform of the solution of the phenomenological equations, i.e., the regression of fluctuations obey the "macroscopic" equations of motion even in the nonequilibrium case.
The renormalization group transformation near four space-dimensions is extended to irreversible processes of a quantum Bose system in terms of a coherent-state formalism. Recursion relations for a kinetic equation are obtained explicitly up to O(ε2), where ε≡4-d and d denotes the space dimensionality. It is shown that terms representing the quantum effects become “irrelevant” by renormalization group scale transformation which can be carried out in the case of quantum fields due to coherent-state formalism. The complex parameter S introduced by Abrahams and Tsuneto phenomenologically is obtained from the microscopic point of view by eliminating microscopic (non-critical) fluctuations. It is also shown that S is driven to the TDGL type at the fixed point, so that the dynamical critical exponent z agrees with that obtained by Halperin, Hohenberg and Ma in the TDGL model.