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THE ROLE OF SPIN IN THE DEVELOPMENT OF COHERENT STRUCTURES IN MANY-BODY SYSTEMS

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Published Sep 10, 2026• 0 citations
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Abstract

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.

DOI
10.82444/warw.33555559

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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.
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.
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A system of harmonic oscillators in the presence of interaction, and with an arbitrary number of degrees of freedom, is considered. The most general form of the Hamiltonian is derived under the restriction that the states which are initially coherent remain coherent at all times. The equation of motion for the annihilation operator, obtained by using this Hamiltonian, is solved, and the frequency spectrum of the annihilation operator is discussed. By giving specific examples it is shown that, in general, the annihilation operators (or their eigenvalues) contain positive as well as negative frequency components and hence are not analytic signals. Some special cases are also considered where the annihilation operators are analytic signals.
In the analytic-Bargmann representation associated with the harmonic oscillator and spin coherent states, the wavefunction as entire complex functions can be factorized in terms of their zeros in a unique way. The Schrödinger equation of motion for the wavefunction is turned to a system of equations for its zeros. The motion of these zeros as a non-linear flow of points is studied and interpreted for linear and non-linear bosonic and spin Hamiltonians. Attention is given to the study of the zeros of the Jaynes-Cummings model and to its finite analoque. Numerical solutions are derived and discussed.