Complex germ method in fock space. II. Asymptotic solutions corresponding to finite-dimensional isotropic manifolds

The Hodge operator ("star" operator) plays an important role in the theory of differential forms, where it serves as a tool for the switching between the exterior derivative and co-derivative. In the theory of many-electron systems involving a finite-dimensional fermionic Fock space, one can define the Hodge operator as a unique (i.e., invariant with respect to linear transformations of the spin-orbital basis set) antilinear operator. The similarity transformation based on the Hodge operator results in the switching between the fermion creation and annihilation operators. The present paper gives a self-contained account on the algebraic structures which are necessary for the construction of the Hodge operator: the fermionic Fock space, the corresponding Grassmann algebra, and the generalized creation and annihilation operators. The Hodge operator is then defined, and its properties are reviewed. It is shown how the notion of the Hodge operator can be employed in a construction of the electronic time-reversal operator.

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Holomorphic Fock spaces for positive linear transformations

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-14995 ◽

2006 ◽

Vol 98 (2) ◽

pp. 262 ◽

Cited By ~ 3

Author(s):

R. Fabec ◽

G. Ólafsson ◽

A.N. Sengupta

Keyword(s):

Gaussian Measure ◽

Reproducing Kernel ◽

Fock Space ◽

Inner Product ◽

Fock Spaces ◽

Linear Transformations ◽

Positive Real ◽

Bargmann Transform ◽

Finite Dimensional ◽

Real Linear

Suppose $A$ is a positive real linear transformation on a finite dimensional complex inner product space $V$. The reproducing kernel for the Fock space of square integrable holomorphic functions on $V$ relative to the Gaussian measure $d\mu_A(z)=\frac {\sqrt{\det A}} {\pi^n}e^{-\Re\langle Az,z\rangle}\,dz$ is described in terms of the linear and antilinear decomposition of the linear operator $A$. Moreover, if $A$ commutes with a conjugation on $V$, then a restriction mapping to the real vectors in $V$ is polarized to obtain a Segal-Bargmann transform, which we also study in the Gaussian-measure setting.

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An application of the Maslov complex germ method to the one-dimensional nonlocal Fisher–KPP equation

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887818501025 ◽

2018 ◽

Vol 15 (06) ◽

pp. 1850102 ◽

Cited By ~ 4

Author(s):

A. V. Shapovalov ◽

A. Yu. Trifonov

Keyword(s):

Nonlinear Equation ◽

Linear Equation ◽

Algebraic Equations ◽

Semiclassical Asymptotics ◽

Complex Germ ◽

Kpp Equation ◽

One Dimensional ◽

Maslov Complex Germ ◽

The One ◽

Complex Germ Method

A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the one-dimensional nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher–KPP equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.

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Unbounded Perturbations of Boson Equilibrium States in Fock Space

Zeitschrift für Naturforschung A ◽

10.1515/zna-1991-0401 ◽

1991 ◽

Vol 46 (4) ◽

pp. 293-303

Author(s):

Reinhard Honegger

Keyword(s):

Fock Space ◽

Relativistic Quantum ◽

Trace Class ◽

Coupling Term ◽

Relativistic Quantum Theory ◽

Interaction Operator ◽

Gas Radiation ◽

Finite Dimensional ◽

Interacting Systems ◽

Photon Interactions

Abstract Based on the linear coupling term in the non-relativistic quantum theory of matter-photon interactions, the coupling of a finite dimensional quantum system (finitely many finite-level atoms) with the boson gas (radiation field) in thermal equilibrium by means of perturbation theoretical methods is calculated. For the perturbation term of the Dyson expansion the unbounded interaction operator is taken. By a detailed analysis of the perturbation integrals and series, it is possible to derive the trace-class property of e-ßH , where H is the corresponding hamiltonian of the interacting systems and ß the inverse temperature

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Modified Pauli matrix and spin algebra from the Z2-decomposition of the parafermionic Fock space

Modern Physics Letters A ◽

10.1142/s0217732315500224 ◽

2015 ◽

Vol 30 (05) ◽

pp. 1550022

Author(s):

Won Sang Chung

Keyword(s):

Fock Space ◽

Projection Operators ◽

Finite Dimensional Algebra ◽

Dimensional Algebra ◽

Finite Dimensional ◽

Pauli Matrices ◽

Pauli Matrix

In this paper the Z2-decomposition of the bosonic Fock space and the projection operators for each subspace are considered. Using these, we obtain two kinds of step operators obeying trilinear relation which resembles the parafermion algebra. The Z2-decomposition of the parafermionic Fock space is also investigated and a new kind of finite-dimensional algebra obeying the trilinear relation which resembles the parafermion algebra of order p. Finally, the modified Pauli matrices and the modified spin algebra are introduced from the Z2-decomposition of the parafermionic Fock space.

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Maslov complex germ method for systems with second-class constraints

Theoretical and Mathematical Physics ◽

10.1134/s0040577916030053 ◽

2016 ◽

Vol 186 (3) ◽

pp. 365-373

Author(s):

O. Yu. Shvedov

Keyword(s):

Complex Germ ◽

Maslov Complex Germ ◽

Complex Germ Method

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