It is shown explicitly how a Bogoliubov type transformation acting on coherent states gives rise to states where the mean values of the field strengths keep the original values while the mean values of the squared strengths or the correlation functions can, within some bounds be prescribed independently.
The exciton system is described by means of specially constructed coherent states (CS), according to an idea of L.V. Keldysh long ago. The system is analyzed from the geometric point of view and from the relevant symmetry group contained in the displacement operator as generador of the CS system in the Klauder–Perelomov sense. Explicitly important quantities involving the quantum statistics of the system (partition function, mean density n and the Yuen limit) are calculated and briefly discussed in relation to the known case of pure bosons (e.g. photons). The Bogoliubov type transformations for the field operators of the Wannier–Mott type exciton system are presented in an exact way (in sharp contrast with the case of [11] where only a rough approximation was given) showing how these transformations induce nonlinearity (cubic at first order) in the Schrodinger equation of the system. The conditions for a hydrodynamic description of the original Keldysh model, as the possibility of a BEC (Bose–Einstein Condensation), are discussed.
The authors extend their previous work on coherent paired states associated with the Lie group SU(1,1). Whereas the earlier states were defined with respect to a single type of canonical boson or (linear) quantum harmonic oscillator, the new states are defined in terms of two distinct types of bosons or oscillators. The new coherent states may again, on the one hand, be viewed as ordinary (Glauber) coherent states in the two-boson Hilbert space spanned by arbitrary numbers of two distinct Bogoliubov quasiparticles associated with the original bosons via a generalised Bogoliubov transformation. Alternatively, expressed wholly in terms of the original bosons these new coherent states are reached from the ordinary coherent states via a unitary (pairing) transformation which is shown to be associated with the entire so-called discrete series of representations of the group SU(1,1). As an important illustration of the use of these states and transformations, they study in detail a rather general class of quantum Lagrangians which includes the damped linear harmonic oscillator. They thereby illustrate their possible usefulness in applications to quantum many-body or field-theoretic processes involving fluctuation-dissipation phenomena in general.
We discuss the quantum mechanics of a particle restricted to the half-line >x 0 with potential energy a = V x 2 for a -< < 1 4 0. It is known that two scale-invariant theories may be defined.By regularizing the near-origin behavior of the potential by a finite square well with variable width b and depth g, it is shown how these two scale-invariant theories occupy fixed points in the resulting (b, g)space of Hamiltonians.A renormalization group (RG) flow exists in this space and scaling variables are shown to exist in a neighborhood of the fixed points.Consequently, the propagator of the regulated theory enjoys homogeneous scaling laws close to the fixed points.Using RG arguments it is possible to discern the functional form of the propagator for long distances and long imaginary times, thus demonstrating the extent to which fixed points control the behavior of the cut-off theory.By keeping the width fixed and varying only the well depth, we show how the mean position of a bound state diverges as g approaches a critical value.It is proven that the exponent characterizing the divergence is universal in the sense that its value is independent of the choice of regulator.Two classical interpretations of the results are discussed: standard Brownian motion on the real line, and the free energy of a certain one-dimensional chain of particles with prescribed boundary conditions.In the former example, V appears as part of an expectation value in the Feynman-Kac formula.In the latter example, V appears as the background potential for the chain, and the loss of extensivity is dictated by a universal power law.