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.
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.
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.
Introduction In this chapter we apply temporal discretization to mechanical systems described by anticommuting variables rather than the commuting variables normally used in classical mechanics (CM). The anticommuting numbers representing such variables are called Grassmann(ian) numbers by mathematicians, after Hermann G. Grassmann (1809–1877), who did some pioneering work in linear algebra. On that account we shall refer to anticommuting numbers as g-numbers , in contrast to c-numbers ( c for commuting 1 ) when we refer to ordinary real or complex variables. The significant difference between g-numbers and c-numbers is that, where as any two c numbers x and y satisfy the commutative multiplication rule xy = + yx , any two g-numbers θ and ϕ satisfy the anticommutation rule θ ϕ = –ϕ θ . G-numbers and c-numbers can be multiplied together in any order, i.e., g-numbers and c-numbers commute. The product zθ of a c-number z and a g-number θ is a g-number. The values of classical variables in ordinary CM are c-numbers, so we may refer to such variables as c-variables. It is possible to construct forms of CM where the classical variables are g-numbers, in which case we shall refer to such variables as g-variables. G-variables should not be thought of as quantized versions of c-variables. They are just as classical as c-variables but much less familiar in terms of their applications to the real world. G-variables should not be confused with Dirac's q-numbers either.
A simple prescription for quantizing fields is developed and applied to the sytsem of interacting spins in an electromagnetic field. The classical equations of motion are written in the standard form $\ensuremath{-}i(\frac{d}{\mathrm{dt}})|V〉+\mathfrak{M}|V〉=0$, where the state vector ∣V〉 has the state variables, such as the generalized coordinates and momenta, as its components, and the time development operator M contains derivative, integral, and matrix operators in general. By diagonalizing M and replacing the resulting classical variables with operators obeying boson commutation relations, the system is quantized, and the Hamiltonian and Heisenberg equations of motion are automatically cast into the harmonic-oscillator form. Thus the quantization is reduced to the simple problem of diagonalizing a matrix. This avoids the laborious process of finding a Lagrangian density (by trial and error in general), determining the canonical variables, forming the Hamiltonian density, and inventing transformations to creation and annihilation operators. Problems with zero-frequency modes, which can be annoying in the standard treatment, are automatically eliminated by the prescription. The bookkeeping of forward and backward waves, different branches of dispersion curves, etc., is simplified. In addition to affording a straightforward and simple method of quantizing fields, the method also gives the necessary and sufficient conditions for a Hamiltonian to be transformable into harmonic-oscillator form, gives a direct relation between the classical equations of motion and the Heisenberg quantum-mechanical equations, and gives a simple method of determining the constants of the motion.
The manifold of one-particle operators has a rich group structure. It is the Lie algebra of the complex Lie group GL(2s,), (2s = dimension of one-particle spin orbital basis) realized as second-quantized operators acting in fermion fock space. This representation is reduced by the subspaces referring to N particles (N = 0, 1, 2,…). One can associate families of coherent states within each of these subspaces with real forms, Gr, of GL(2s,). These families have isotropy groups, Hr, which are associated with subgroups of these real forms (isotropy groups are the groups of transformations that leave a state invariant). The cosets GL(2s,)/Hr can be given an exponential parameterization that labels the coherent states. This label space is a complex vector space which is in correspondence with the nonlinear manifold of coherent states in the hilbert space ℋ︁N. These complex variables can be made time dependent, and by utilizing the time-dependent variation principle one can obtain a nonlinear approximation to the time-dependent Schrödinger equation which has the form of classical equations of motion in a generalized phase space 𝒫 = Grc/Hrc. This classical dynamics can be given a Hilbert space description, where the Hilbert space is a subspace of square integrable holomorphic functions f: ↦ 𝒫 → and is isomorphic with ℋ︁N. Different pairs {Gr,Hr} are associated with different families of coherent states. Some examples of these constructions are given, and the structure of the approximate time evolution is discussed and related to time-dependent approximation schemes and generalized random phase approximations (RPA).
Influenced by Klauder’s investigations on the same subject, we study the question of correspondence principle for Dirac fields, looking for its formulation without use of Grassman algebras. We prove that with each Fermi operator (the series with respect to asymptotic free fields): Ω (ψ,ψ̄): one can associate the functional ΩC(ψC, ψ̄C) with respect to classical spinor fields. Here the projector 1F and the Hilbert (Fock) space ℱF=1FℱB are given such that the identity 1F: ΩC(ψB, ψ̄B): 1FℱF = :Ω (ψ, ψ̄):ℱF defines the mediating boson level, where coherent state expectation values of operator expressions are in order: 〈:ΩC(ψB, ψ̄B):〉=ΩC(ψC, ψ̄C). For proofs we employ functional differentiation (resp. integration) methods, especially in connection with the use of functional representations of the CCR and CAR algebras.
We implement in systems of fermions the formalism of pseudoclassical paths that we recently developed for systems of bosons and show that quantum states of fermionic fields can be described, in the Heisenberg picture, as linear combinations of randomly distributed paths that do not interfere between themselves and obey classical Dirac equations. Every physical observable is assigned a time-dependent value on each path in a way that respects the anticommutative algebra between quantum operators, and we observe that these values on paths do not necessarily satisfy the usual algebraic relations between classical observables. We use these pseudoclassical paths to define the dynamics of quantum fluctuations in systems of fermions and show that, as we found for systems of bosons, the dynamics of fluctuations of a wide class of observables that we call ``collective'' observables can be approximately described in terms of classical stochastic concepts. Finally, we apply this formalism to describe the dynamics of local fluctuations of globally conserved fermion numbers.