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International Journal of Modern Physics B

THE UNIVERSAL EQUATIONS OF MOTION FOR NONLINEAR FIELDS IN DIFFERENT BASES DESCRIBING STRONGLY INTERACTING MANY-BODY SYSTEMS WITH TWO-BODY INTERACTIONS

J.M. Dixon, Jack A. Tuszyński

Published Jun 30, 1995• 4 citations
Publisher

Abstract

A brief account of the Method of Coherent Structures (MCS) is presented using a plane-wave basis to define a quantum field. It is also demonstrated that the form of the quantum field equations, obtained by MCS, although highly nonlinear for many-body systems with two-body interactions, is independent of the basis of states used for the definition of the field.

DOI
10.1142/s0217979295000690

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Journal of Physics A Mathematical and General•Nov 21, 1989•36 citations
A brief account of the Method of Coherent Structures (MCS) is presented using a plane-wave basis to define a quantum field. It is also demonstrated that the form of the quantum field equations, obtained by MCS, although highly nonlinear for many-body systems with two-body interactions, is independent of the basis of states used for the definition of the field.
Spin components have been explicitly included in an accurate analytical approach for the study of coherent structures in many-body systems. Equations of motion for annihilators and creators are transformed into non-linear equations of motion for a spin-dependent quantum field. It is demonstrated, using commutation relations for the quantum field operators, that spin-up, psi+, and spin-down, psi-, fields may be decoupled. The equation of motion for psi+ is shown to have a form identical to the equation of motion for psi+ which, in turn, is equivalent to that for a field with no specific spin component. The latter equations have the same form for both bosons and fermions.
Numerous examples in condensed matter physics possess an effective Hamiltonian structure involving two-body anharmonic interactions between elementary excitations. The effective second-quantised Hamiltonian used to model these systems is essentially of the same form except for the particulars of the interaction parameters and the statistics of the particles which are interacting. The author investigates such systems in the limit of strongly anharmonic interactions which may possibly lead to symmetry breaking. He introduces quantum field operators and calculates their equations of motion. The interaction coefficients are expanded in a Taylor series about the critical mode to second order to illustrate the method, but he gives the expansion to infinite order for completeness. It is demonstrated that both in first and second order the equations of motion are reduced to a system of highly nonlinear coupled partial differential equations (PDE) for the field operator. He shows that the form of the second-order equation is exact due to renormalisation theory, in the case where the Hamiltonian coefficients represent operators in real space which are isotropic, and incorporates symmetries and topologies which would appear in an expansion to infinite order. This special situation arises in a very wide range of physical examples. These equations can be solved exactly using a particular ansatz in conjunction with recent mathematical developments in the field of non-linear analysis. In particular, the method of symmetry reduction for PDE provides a complete set of symmetry variables for the type of equations used in the study.
A k-dependent state has been constructed from two pairs of mutually commuting Fermi-Dirac operators by replacing the Grassmann numbers in the conventional definition of a fermion coherent state. The Heisenberg time-dependent form of this state was then combined with a particular spin state to give a resultant state \G], the spin component containing a series of unknown k-dependent weights. A classical c-number field, phi(c),(r, t), was then defined by an expectation value of the quantum field, psi, using \G] and an energy and time-dependent phase. It is demonstrated that the form of the equation of motion for phi(c)(r, t) is identical to that for psi. (C) 1998 Elsevier Science B.V.
For pt.I see ibid. vol.22, p.4877 (1989). Previously (I) the author considered a second-quantised Hamiltonian which can be used to model a large number of important strongly interacting many-body systems. He has shown that the dynamics of such systems can be described exactly using quantum fields. The equations of motion derived in this way are highly nonlinear partial differential equations (PDE). He now uses a standard field theoretical approach where initially the fields are treated as classical functions. He studies the solutions of these equations which remarkably, may be obtained making full use of very recent mathematical discoveries (the symmetry reduction method). He finds exact solutions of the equations of motion for the classical field for each of the four main cases separately found in the preceding paper. These cases may correspond to different physical situations and the last, the second-order case, corresponds to the most general physical situation. Subsequently he outlines possible quantisation procedures for these fields. He then briefly discusses the type of boundary conditions which may be applied to both the classical field and the quantum fluctuations which arise.