The Hodge Operator in Fermionic Fock Space

2005 ◽  
Vol 70 (7) ◽  
pp. 979-1016 ◽  
Author(s):  
Leszek Z. Stolarczyk
Keyword(s):  
Fock Space ◽  
Differential Forms ◽  
Grassmann Algebra ◽  
Basis Set ◽  
Electron Systems ◽  
Linear Transformations ◽  
Spin Orbital ◽  
Creation And Annihilation Operators ◽  
Finite Dimensional ◽  
Hodge Operator

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.

scholarly journals Holomorphic Fock spaces for positive linear transformations

2006 ◽  
Vol 98 (2) ◽  
pp. 262 ◽  
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.

The incomplete basis‐set problem. V. Application of CIBS to many‐electron systems

1982 ◽  
Vol 77 (6) ◽  
pp. 3068-3080 ◽  
Author(s):  
Keith McDowell ◽  
Lynn Lewis
Keyword(s):  
Basis Set ◽  
Electron Systems

Purity And Copurity in Systems of Linear Transformations

1976 ◽  
Vol 28 (4) ◽  
pp. 889-896
Author(s):  
Frank Zorzitto
Keyword(s):  
Vector Spaces ◽  
Finite Codimension ◽  
Linear Transformations ◽  
Complex Vector ◽  
Finite Dimensional ◽  
Quotient System

Consider a system of N linear transformations A1, … , AN: V → W, where F and IF are complex vector spaces. Denote it for short by (F, W). A pair of subspaces X ⊂ V, Y ⊂ W such that determines a subsystem (X, Y) and a quotient system (V/X, W/Y) (with the induced transformations). The subsystem (X, Y) is of finite codimension in (V, W) if and only if V/X and W / Y are finite-dimensional. It is a direct summand of (V, W) in case there exist supplementary subspaces P of X in F and Q of F in IF such that (P, Q) is a subsystem.

THE REGULAR PART OF A SEMIGROUP OF LINEAR TRANSFORMATIONS WITH RESTRICTED RANGE

2017 ◽  
Vol 103 (3) ◽  
pp. 402-419 ◽  
Author(s):  
WORACHEAD SOMMANEE ◽  
KRITSADA SANGKHANAN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Restricted Range ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional ◽  
Idempotent Rank ◽  
Image Position

Let$V$be a vector space and let$T(V)$denote the semigroup (under composition) of all linear transformations from$V$into$V$. For a fixed subspace$W$of$V$, let$T(V,W)$be the semigroup consisting of all linear transformations from$V$into$W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’,Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$is the largest regular subsemigroup of$T(V,W)$and characterized Green’s relations on$T(V,W)$. In this paper, we determine all the maximal regular subsemigroups of$Q$when$W$is a finite-dimensional subspace of$V$over a finite field. Moreover, we compute the rank and idempotent rank of$Q$when$W$is an$n$-dimensional subspace of an$m$-dimensional vector space$V$over a finite field$F$.

scholarly journals κ-DEFORMED STATISTICS AND CLASSICAL FOUR-MOMENTUM ADDITION LAW

2008 ◽  
Vol 23 (09) ◽  
pp. 653-665 ◽  
Author(s):  
MARCIN DASZKIEWICZ ◽  
JERZY LUKIERSKI ◽  
MARIUSZ WORONOWICZ
Keyword(s):  
Minkowski Space ◽  
Fock Space ◽  
Scalar Fields ◽  
Cross Product ◽  
Creation And Annihilation Operators ◽  
Multiplication Rule ◽  
Symmetry Properties ◽  
Deformed Algebra ◽  
Free Scalar

We consider κ-deformed relativistic symmetries described algebraically by modified Majid–Ruegg bi-cross-product basis and investigate the quantization of field oscillators for the κ-deformed free scalar fields on κ-Minkowski space. By modification of standard multiplication rule, we postulate the κ-deformed algebra of bosonic creation and annihilation operators. Our algebra permits one to define the n-particle states with classical addition law for the four-momentum in a way which is not in contradiction with the nonsymmetric quantum four-momentum co-product. We introduce κ-deformed Fock space generated by our κ-deformed oscillators which satisfy the standard algebraic relations with modified κ-multiplication rule. We show that such a κ-deformed bosonic Fock space is endowed with the conventional bosonic symmetry properties. Finally we discuss the role of κ-deformed algebra of oscillators in field-theoretic noncommutative framework.

scholarly journals Isomorphisms of semigroups of transformations

1983 ◽  
Vol 6 (3) ◽  
pp. 487-501
Author(s):  
A. Sita Rama Murti
Keyword(s):  
Ring Theory ◽  
Vector Spaces ◽  
Linear Transformations ◽  
Finite Dimensional

IfMis a centered operand over a semigroupS, the suboperands ofMcontaining zero are characterized in terms ofS-homomorphisms ofM. Some properties of centered operands over a semigroup with zero are studied.AΔ-centralizerCof a setMand the semigroupS(C,Δ)of transformations ofMoverCare introduced, whereΔis a subset ofM. WhenΔ=M,Mis a faithful and irreducible centered operand overS(C,Δ). Theorems concerning the isomorphisms of semigroups of transformations of setsMioverΔi-centralizersCi,i=1,2are obtained, and the following theorem in ring theory is deduced: LetLi,i=1,2be the rings of linear transformations of vector spaces(Mi,Di)not necessarily finite dimensional. Thenfis an isomorphism ofL1→L2if and only if there exists a1−1semilinear transformationhofM1ontoM2such thatfT=hTh−1for allT∈L1.

scholarly journals Linear functionals on hypervector spaces

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 3031-3043
Author(s):  
O.R. Dehghan
Keyword(s):  
Vector Space ◽  
Linear Algebra ◽  
Linear Functional ◽  
Dual Basis ◽  
Linear Functionals ◽  
Linear Transformations ◽  
Important Special Case ◽  
Finite Dimensional ◽  
Key Topics ◽  
Special Case

The study of linear functionals, as an important special case of linear transformations, is one of the key topics in linear algebra and plays a significant role in analysis. In this paper we generalize the crucial results from the classical theory and study main properties of linear functionals on hypervector spaces. In this way, we obtain the dual basis of a given basis for a finite-dimensional hypervector space. Moreover, we investigate the relation between linear functionals and subhyperspaces and conclude the dimension of the vector space of all linear functionals over a hypervector space, the dimension of sum of two subhyperspaces and the dimension of the annihilator of a subhyperspace, under special conditions. Also, we show that every superhyperspace is the kernel of a linear functional. Finally, we check out whether every basis for the vector space of all linear functionals over a hypervector space V is the dual of some basis for V.

Sign in / Sign up

Export Citation Format

Share Document