Clifford algebra,symmetries, and vectors

The realization of a Clifford algebra in laboratory space is considered and it is demonstrated that the elements of the algebra cannot, as often assumed, be directly identified with vectors in this space, but, rather, that they form the parametric space of the symmetry operations of the Euclidean group as performed in the laboratory space. Details of this parametrization are established and expressions are given that determine the action of the Euclidean-group operations (screws included) on laboratory-space vectors in terms of the elements of the Clifford algebra. A discussion of Clifford vectors, bivectors, and pseudoscalars and their relation to the Gibbs vectors is provided. The correct definition of axial and polar vectors within the Clifford algebra is carefully discussed. It is shown how simple it is to generate finite point groups in 4-dimensional space by means of the Clifford algebra. © 1996 John Wiley & Sons, Inc.     

On the structure of the Clifford algebra unitary group adapted states in the many‐electron correlation problem

An attempt has been made to understand the structure of the Clifford algebra unitary group adapted many‐particle states from the conventional symmetric group point of view. Emphasizing the symmetric group result that the consideration of the spin‐independent Hamiltonian matrix over the many‐particle configuration functions (CFs) entails a particular subspace of their spatial parts only, attention is confined entirely in this subspace. Question of adapting the functions therein to the unitary group subduction chain is then shown to bring out an interesting lead to the Clifford algebra unitary group approach (CAUGA) states, thus underlining the motive and the essential gains of the CAUGA formulation. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 607–614, 2000     

Generalized P,Q and G conditions

The N-Representability Problem entails characterizing the set of second order reduced states that are contractions of N-electron states of the Fermion Fock algebra. This problem is formulated in the form of finding the conditions that a positive linear functional defined on a subspace of this algebra must satisfy in order to be extended to the whole algebra. As this algebra is a w*-algebra one can utilize a theorem by Kadison that shows it is sufficient to consider the values of linear functionals on projectors contained in the subspace in order to determine whether they have positive extensions. Thus we find the form of projectors belonging to the subspace of one and two particle operators and subsequently show that the extension conditions needed in the N-Representability Problem correspond to generalized P, Q and G conditions plus the additional constraints that the functionals be dispersion free on the number operator and their values on one particle operators determined by their values on two particle operators.     

On the generation of point groups in spaces of various dimensions

In this paper the use of Clifford algebra in the parametrization of point groups in spaces of various dimensions is shown. Higher-dimensional spaces are of great interest especially when modulated crystals or quasicrystals are studied. While the quaternion units, which are useful to parametrize rotations in 3 dimensions, can be identified with rotations, the basic Clifford units may be regarded as mirrors from which all proper and improper symmetry operations can be generated. The practical implementation of this method of parametrization is demonstrated for the group of the hypercube in the 4-dimensional space, and generalisations to spaces of dimensions higher than 4 are suggested.     

Kratzer potential algebraic representation and matrix elements recurrence formulae

Generalized recurrence relations for the calculation of multipole matrix elements for Kratzer potential wave functions are obtained operationally. These formulas have been determined by using a non-analytical procedure based on the algebraic representation of the Kratzer eigenfunctions along with the usual ladder properties and commutation relations. For that, the creation and annihilation operators are adequately derived by means of an alternative approach to the factorization method and the exact expressions for matrix elements are achieved with the aid of a relationship between the ladder operators associated with the bra and theket. The proposed algebraic approach as well as the formulas for the calculation of matrix elements thus derived are quite simple and direct when compared with other alternative expressions already obtained analytically or pseudo-algebraically by means of the hypervirial theorem commutator algebra.     

Usage de l'Algèbre de Lie su (n) dans l'Etude des Systèmes Quantiques à n Etats. II. Transformation de l'Espace des Observables,Problèmes d'Evolution

In the previous paper we examined, for a quantum system, the relation between its n-dimensional state space and the su (n) Lie algebra. The present paper is devoted to relations between unitary transformations in the state space and orthogonal transformations in Lie's algebra. Two cases can happen. First, the transformations are independently chosen in the two spaces; this amounts to changing the former relation. On the other hand, the relation is maintained and the unitary operators are then related to some of the orthogonal operators. This second case is used to study the evolution operators.     

Application of geometric algebra for the description of polymer conformations

In this paper a Clifford algebra-based method is applied to calculate polymer chain conformations. The approach enables the calculation of the position of an atom in space with the knowledge of the bond length (l), valence angle (theta), and rotation angle (phi) of each of the preceding bonds in the chain. Hence, the set of geometrical parameters {l(i),theta(i),phi(i)} yields all the position coordinates p(i) of the main chain atoms. Moreover, the method allows the calculation of side chain conformations and the computation of rotations of chain segments. With these features it is, in principle, possible to generate conformations of any type of chemical structure. This method is proposed as an alternative for the classical approach by matrix algebra. It is more straightforward and its final symbolic representation considerably simpler than that of matrix algebra. Approaches for realistic modeling by means of incorporation of energetic considerations can be combined with it. This article, however, is entirely focused at showing the suitable mathematical framework on which further developments and applications can be built.     

Approach of the associated Laguerre functions to the su(1,1) coherent states for some quantum solvable models

Using second‐order differential operators as a realization of the su(1,1) Lie algebra by the associated Laguerre functions, it is shown that the quantum states of the Calogero‐Sutherland, half‐oscillator and radial part of a 3D harmonic oscillator constitute the unitary representations for the same algebra. This su(1,1) Lie algebra symmetry leads to derivation of the Barut‐Girardello and Klauder‐Perelomov coherent states for those models. The explicit compact forms of these coherent states are calculated. Also, to realize the resolution of the identity, their corresponding positive definite measures on the complex plane are obtained in terms of the known functions. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Diagram approach to group algebraic methods

Class sum theory, the duality with IRREP methods and tensor operators in the group algebra are discussed by generalizing the diagrammatic approach of conventional IRREP theory to include group label manipulation. Concepts such as invariant nodes and Jucys–Levinson–Vanagas reduction theorems generalize straightforwardly. The results are capable of unique simplification for certain nodes, when the group rearrangement theorem is useable or when a class sum is performed. A duality transformation (between IRREP –partner and class–element labels) emerges as an important concept.     

Second quantization for systems with a constant number of particles in the Dirac notation

The relation between the completeness condition for an appropriate one-particle basis set and the occupation number representation (second quantization) is shown for the time-independent case. The explicit expressions for the basic symmetric operators are derived in the Dirac bra–ket notation. The physical meaning of these operators, the algebra as well as the connections with the one-electron density matrix and with the projector on the Fermi sea in the one-electron approximation, follow directly from these expressions. The generalization for a nonorthogonal basis and the algebra for corresponding basic operators are formulated. The connection with the notion of the molecular diagrams of different kinds for the nonorthogonal atomic orbitals is shown. The Mulliken populations and the Chirgwin–Coulson bond orders are equal to the diagonal and offdiagonal elements of the molecular diagram 1, respectively. The matrix elements of the projector on the Fermi sea in the one-electron approximation in the representation of nonorthogonal atomic orbitals are elements of the molecular diagram 2.     

Scale‐adaptive tensor algebra for local many‐body methods of electronic structure theory

While the formalism of multiresolution analysis, based on wavelets and adaptive integral representations of operators, is actively progressing in electronic structure theory (mostly on the independent‐particle level and, recently, second‐order perturbation theory), the concepts of multiresolution and adaptivity can also be utilized within the traditional formulation of correlated (many‐particle) theory based on second quantization and the corresponding (generally nonorthogonal) tensor algebra. In this article, we present a formalism called scale‐adaptive tensor algebra, which introduces an adaptive representation of tensors of many‐body operators via the local adjustment of the basis set quality. Given a series of locally supported fragment bases of a progressively lower quality, we formulate the explicit rules for tensor algebra operations dealing with adaptively resolved tensor operands. The formalism suggested is expected to enhance the applicability of certain local correlated many‐body methods of electronic structure theory, for example, those directly based on atomic orbitals (or any other localized basis functions in general). © 2014 Wiley Periodicals, Inc.     

Explicit representation of Gel'fand–Tsetlin states in clifford algebra unitary group approach

A explicit expression for the unitary group Clebsch–Gordan coefficients, which couple two fully antisymmetric single-column states into the two-column Gel'fand–Tsetlin states, is given in terms of isoscalar factors for the canonical subgroup chain U(n) ? U(n – 1) ? …? ? U(1). The isoscalar factors are expressed through the step numbers labeling canonical basis states and enable a straightforward construction of Gel'fand–Tsetlin states in the Clifford algebra unitary group approach, without the use of the tables for the symmetric group outer-product reduction coefficients.     

An algebraic approach to bound states of simple one-electron systems

A very general realization of the so (2, 1) algebra, which easily follows from the basic commutation relations that are satisfied by any pair of mutually conjugate generalized coordinates and momenta, is constructed. Using special cases of this general realization, and closely following the well known derivation of the eigenvalue spectra of the angular-momentum operators, based on the so (3) algebra, we derive the energy spectrum for the N-dimensional isotropic harmonic oscillator, and for both the nonrelativistic and the relativistic N-dimensional hydrogen atom. Special attention is given to a simple derivation of the form of these Hamiltonians in terms of the so(2, 1) algebra generators. In particular, the usually exploited tilting transformation is avoided, and the whole derivation is presented in an extremely simple and straightforward way. The present approach stresses the similarity and mutual relationship between the systems studied and, in addition to introducing some novel techniques and providing considerable insight into the overall structure of these problems, also has a definite pedagogical value.     

Symmetric group graphical approach to the direct configuration interaction method

An approach to the configuration interaction method based on symmetric groups (SGA ) is developed. The formalism is an alternative of the unitary group approach (UGA ). In many aspects the present formulation seems to be superior to UGA . In particular, in SGA the orbital and the spin parts of the configuration state functions may be processed separately. In consequence its graphical formulation is much simpler and the coupling constant expressions are more compact than the UGA analogs. A special emphasis is put on direct CI implementations. In addition to formulas for coupling constants, explicit expressions allowing for separation of external and internal space contributions are also presented.     

A Laplace transform approach to find the exact solution of the $$$$-dimensional Schrödinger equation with Mie-type potentials and construction of Ladder operators

The second order \(N\)-dimensional Schrödinger equation with Mie-type potentials is reduced to a first order differential equation by using the Laplace transformation. Exact bound state solutions are obtained using convolution theorem. The Ladder operators are also constructed for the Mie-type potentials in \(N\)-dimensions. Lie algebra associated with these operators are studied and it is found that they satisfy the commutation relations for the SU(1,1) group.     

四原子分子振转相互作用的Lie代数方法

利用Lie代数方法研究了四原子分子振转相互作用，在代数框架内首次给出四 原子分子振转相互作用的张量算子非对角矩阵元的表达式，利用这些表达式对线型 四原子分子HCCF振转相互作用的l-doubling进行了计算。     

Multiconfigurational Green's function approach with quasidegenerate perturbation theory

The new approach to approximation of polarization propagator (PP) for electronic states of atoms and molecules with reference state wave function (RSWF) constructed on the base of a multidimensional model space is presented. This approach exploiting the apparatus of the quasidegenerate perturbation theory (QDPT) is realized in the zeroth QDPT order and through the first one. The original complete system of excitation operators introduced in the approach is consistent with the RSWF by the perturbation order. This factor in conjunction with the flexibility of the RSWF creates the capabilities of balanced accounting of correlation and quasidegeneracy effects at different locations of nuclei in a molecule and for all the states concerned. In this way, the transition characteristics in electronic shells of molecules in a wide area of nuclear geometry parameters may be appropriately evaluated. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Some comments on the properties of unitary transformations on linear spaces having an indefinite metric and the connection with the theory of spin

After a brief review of the history of the discovery of the spin, some fundamental properties of linear spaces having an indefinite metric are being discussed. The study starts with an elementary survey of the theory of matrices and their stability problem. It is emphasized that—by a similarity transformation—all matrices may be brought to classical canonical form characterized by the diagonal elements called eigenvalues, their multiplicities, their Jordan blocks, and their Segré characteristics. In connection with the reduced Cayley-Hamilton equation, the existence of the product projection operators and their main properties is briefly discussed. Particular attention is paid to the concept of a basis for the linear space and the associated metric matrix, which is self-adjoint and may be brought to diagonal form with the eigenvalues ± 1 by a unitary transformation, which reveals the indices of inertia, p and q. The Minkowski space having p = 3 and q = 1 is used as an example. After this introduction, some properties of linear operators defined on an indefinite space are discussed, and it is pointed out that self-adjoint operators and unitary operators may now have a rather peculiar and unexpected behavior, and the special Lorentz transformations are used as an example. It is then shown that these features are of essential importance in studying rotations as special cases of unitary transformations defined on an indefinite space. The rotations are here defined by means of their reduced Cayley-Hamilton equation, and their properties are studied by means of the associated product projection operators, which are idempotent, mutually exclusive, and form a resolution of the identity. In a previous article, it was shown that, in a positive definite space, there is a close connection between the requirement that all rotations around an external axis form a group and the existence of an anticommutator algebra leading to the concept of spinors. The rotations are expressed in the exponential form U=exp(iO), where O is a self-adjoint operator which is independent of any choice basis, coordinate system, etc., and which is, hence, a true invariant. It is shown that this approach may now be extended also to indefinite spaces and may lead to operators O which are both rotationally and relativistically invariant. In this connection, the full Lorentz transformations are given a particularly simple form. The article may be considered as a simple exercise in linear algebra, in which the mathematical connection between the concept of rotations and the existence of spinors is strongly emphasized. © John Wiley & Sons, Inc.