Relativistic field formulation of simultaneous quantum measurement

The simultaneous measurement of Dirac field operators is formulated in analogy to the work of von Neumann and Arthurs-Kelly. Meter fields are coupled to the system field with a relativistically invariant bilinear interaction. Measurement of vacuum meter field expectation values provides for the simultaneous measurement of noncommuting system components. It is shown that two meter coupling allows for a simultaneous minimum in the variance of the subsequent meter measurements. A pseudoscalar self-interaction of the Dirac field is shown to allow simultaneous measurement of positive energy field operators with negative energy meters. The simultaneous measurement ofn noncommuting field operators is obtained by coupling the system ton fermionic fields. Also, in this paper the related concept of mutual simultaneous measurement is developed. This requires that any operators in the enlarged Hilbert space are measurable by the remaining fields as meters. System embedding into a larger Hilbert space results in added noise due to the zero point motion of the meter fields. By the negentropy principle of Brillouin, the added noise is equivalent to entropy. A criterion determining the interaction among fields is that the averaged added noise in the components of each quantum field is minimized. This criterion defines an optimum fermionic mass matrix through the determination of the entangling interaction.1. This work was sponsored by the Department of the Air Force under contract F19628-90-C-0002.     

Integrable Euler equations on SO(4) and their physical applications

For the Lie algebra SO(4) (and other six dimensional Lie algebras) we find some Euler's equations which have an additional fourth order integral and are algebraically integrable (in terms of elliptic functions) in a one parameter set of orbits. Integrable Euler's equations having an additional second order integral and generalizing Steklov's case are presented. Equations for rotation of a rigid body havingn ellipsoid cavities filled with the ideal incompressible fluid being in a state of homogeneous vortex motion are derived. It is shown that the obtained equations are Euler's equations for the Lie algebra of the groupG _n+1=SO(3) × ... × SO(3). New physical applications of Euler's equations on SO(4) are discussed.     

Regular Basis and R-Matrices for the ^(su)(n)k Knizhnik–Zamolodchikov Equation

Dynamical R-matrix relations are derived for the group-valued chiral vertex operators in the SU(n) WZNW model from the KZ equation for a general four-point function including two step operators. They fit the exchange relations of the U _q (sl _n ) covariant quantum matrix algebra derived previously by solving the dynamical Yang–Baxter equation. As a byproduct, we extend the regular basis introduced earlier for SU(2) chiral fields to SU(n) step operators and display the corresponding triangular matrix representation of the braid group.     

Serre-type relations for special linear Lie superalgebras

It is pointed out that, for m, n 2, the naive Serre presentation corresponding to the simplest Cartan matrix of sl(m, n) does not define the Lie superalgebra sl(m, n) but a larger algebra s(m, n) of which sl(m, n) is a nontrivial quotient. The supplementary relations for the generators are found and the definition of the q-deformed universal enveloping algebra of sl(m, n) is modified accordingly.     

The Principal SO(1,2) Subalgebra of a Hyperbolic Kac–Moody Algebra

The analog of the principal SO(3) subalgebra of a finite-dimensional simple Lie algebra can be defined for any hyperbolic Kac–Moody algebra g(A) associated with a symmetrizable Cartan matrix A, and coincides with the non-compact algebra SO(1,2). We exhibit the decomposition of g(A) into representations of SO(1,2). With the exception of the adjoint SO(1,2) algebra itself, all of these representations are unitary. We compute the Casimir eigenvalues; the associated exponents are complex and noninteger.     

Abelianizing Vertex Algebras

To every vertex algebra V we associate a canonical decreasing sequence of subspaces and prove that the associated graded vector space gr(V) is naturally a vertex Poisson algebra, in particular a commutative vertex algebra. We establish a relation between this decreasing sequence and the sequence C_n introduced by Zhu. By using the (classical) algebra gr(V), we prove that for any vertex algebra V, C₂-cofiniteness implies C_n-cofiniteness for all n≥2. We further use gr(V) to study generating subspaces of certain types for lower truncated ℤ-graded vertex algebras.Partially supported by an NSA grant     

A superalgebra morphism of U q [osp(1/2N)] onto the deformed oscillator superalgebra W q (N)

We prove that the deformed oscillator superalgebra W _q (n) (which in the Fock representation is generated essentially byn pairs ofq-bosons) is a factor algebra of the quantized universal enveloping algebra U _q [osp(1/2n)]. We write down aq-analog of the Cartan-Weyl basis for the deformed osp(1/2n) and also give an oscillator realization of all Cartan-Weyl generators.     

On integrable systems close to the toda lattice

We study the generalized discrete self-trapping (DST) system formulated in terms of the u(n) Lie-Poisson algebra as well as its noncompact analog given on the gl(n) algebra. The Hamiltonian is a quadratic-linear function of the algebra generators where the quadratic part consists of the squared generators of the Cartan subalgebra only: $$H = \sum\limits_{i = 1}^n {\frac{{\gamma _i }}{2}A_{ii}^2 + } \sum\limits_{i,j = 1}^n {m_{ij} } A_{ij} $$ Two integrable cases are discovered: one for the u(n) case and the other for the gl(n) case. The correspondingL-operators (2 × 2 andn ×n) are found which give the Lax representation for these systems. The integrable model on the gl(n) algebra looks like the Toda lattice because in this case,m _ij=c _iδ_ij-1. The corresponding 2 × 2L-operator satisfies the Sklyanin algebra.     

Nonstandard q-deformation of the Euclidean Lie algebra and its representations

A nonstandard q-deformed Euclidean algebra U _q(iso _n ), based on the definition of the twisted q-deformed algebra U _qso_n) (different from the Drinfeld–Jimbo algebra U _q(so _n )), is defined. Infinite dimensional representations R of U _q(iso _n ) are described. Explicit formulas for operators of these representations in the orthonormal basis are given. The spectra of the operators R(T _n) corresponding to a q-analogue of the infinitesimal operator of shifts along the n-th axis are described. Contrary to the case of the classical Euclidean Lie algebra iso _n , these spectra are discrete and spectral points have one point of accumulation.     

Construction of the elliptic Gaudin system based on Lie algebra

Gaudin model is a very important integrable model in both quantum field theory and condensed matter physics. The integrability of Gaudin models is related to classical r-matrices of simple Lie algebras and semi-simple Lie algebra. Since most of the constructions of Gaudin models works concerned mainly on rational and trigonometric Gaudin algebras or just in a particular Lie algebra as an alternative to the matrix entry calculations often presented, in this paper we give our calculations in terms of a basis of the typical Lie algebra, A _n , B _n , C _n , D _n , and we calculate a classical r-matrix for the elliptic Gaudin system with spin.      

Matrix elements for the representations of SO(n) and SO0(n, 1)

Matrix elements of the unitary irreducible representations of the group SO(n) of class higher then 1 (with respect to SO(n−1)) in Gel'fand-Zetlin basis are obtained in explicit form. They are represented as polynomials in cosθ and sinθ of the order equal to the first coordinate of the highest weight. Making use of them the representation matrix elements for the group SO₀(n, 1) in SO(n) basis are calculated.     

Breaking of the SO(n) symmetry with vector and adjoint higgses

The spontaneous symmetry breaking of SO(n) is investigated by studying the most general quartic SO(n)-invariant Higgs potential with two multiplets of scalars belonging to a vector and to an adjoint representation. In the most general cases largest residual symmetry is found to be SU(l), SU(l-1), SO(l), SO(l-1) or SO(n-2) wherel is the rank of SO(n). In particular, the breaking of SO(n) into SU(n ₁)×SU(n ₂) is found to occur only in special cases.     

SO(n,1) Wigner rotation as an SL(2,R) problem

We show that the composition of not only two SO(3,1) boosts, but also that of two SO(n,1) boosts for anyn 2, is basically an SO(2,1) problem and hence can be analysed completely using SL(2,R) matrices. By computing the expression for the Thomas/Wigner angle directly using SL(2,R) matrices we show that this approach results in considerable economy of algebra.     

Bäcklund transformations and soliton-type solutions for σ models with values in real Grassmannian spaces

A Bäcklund transformation for the two-dimensional σ model with values in oriented real Grassmannian spaces is constructed using the known Bäcklund transformation for the SO(n) principal models. The construction provides a natural way to linearize the differential system of the Bäcklund transformation. In the case of the S ⁿ≈SO(n+1/SOn) model, the Backlund transformation reduces to that of Pohlmeyer. The one-soliton solution for S ²～SO(3)/SO(2) is obtained analytically and plots of one-soliton and two-soliton solutions are displayed.     

Multi-parameter deformed fermionic oscillators

In this paper, we study a quantum group covariant deformed fermion algebra. This system can be formulated in n dimensions and posesses two deformation parameters. The undeformed fermion algebra is obtained when both deformation parameters are unity. When both parameters are zero the deformed fermionic oscillator algebra reduces to the orthofermion algebra. If the quantum group symmetry is not preserved, then the number of parameters in n dimensions can be increased to 2n-2. Received: 6 December 2001 / Revised version: 18 June 2002 / Published online: 20 September 2002     

Unification Theory of Different Causal Algebras and Its Applications to Theoretical Physics

This paper gives a generalization of group theory, i.e. a unification theory of different causal algebras, and its applications to theoretical physics. We propose left and right causal algebras, left and right causal decomposition algebras, causal algebra and causal decomposition algebras in terms of quantitative causal principle. The causal algebraic system of containing left (or right) identity I _jL (or I _jR ) is called as the left (or right) causal algebra, and associative law is deduced. Furthermore the applications of the new algebraic systems are given in theoretical physics, specially in the reactions of containing supersymmetric particles, we generally obtain the invariance of supersymmetric parity of multiplying property. In the reactions of particles of high energy, there may be no identity, but there are special inverse elements, which make that the relative algebra be not group, however, the causal algebra given in this paper is just a tool of severely and directly describing the real reactions of particle physics. And it is deduced that the causal decomposition algebra is equivalent to group.     

Non-Abelian magnetic monopoles

It is shown that a general, irreducible, SU(n), Sp(n), SO(2n) monopole with maximal symmetry breaking is determined by its spectral data. For SU(n) with minimal symmetry breaking the spectral data is defined and also shown to determine the monopole.Research supported in part by NSF Grant 8120790     

Quantum Gates and Quantum Algorithms with Clifford Algebra Technique

We use the Clifford algebra technique (J. Math. Phys. 43:5782, 2002; J. Math. Phys. 44:4817, 2003), that is nilpotents and projectors which are binomials of the Clifford algebra objects γ ^a with the property {γ ^a ,γ ^b }₊=2η ^ab , for representing quantum gates and quantum algorithms needed in quantum computers in a simple and an elegant way. We identify n-qubits with the spinor representations of the group SO(1,3) for a system of n spinors. Representations are expressed in terms of products of projectors and nilpotents; we pay attention also on the nonrelativistic limit. An algorithm for extracting a particular information out of a general superposition of 2 ⁿ qubit states is presented. It reproduces for a particular choice of the initial state the Grover’s algorithm (Proc. 28th Annual ACM Symp. Theory Comput. 212, 1996).     

Nonstandard GLh(n) quantum groups and contraction of covariant q-bosonic algebras

GL_h(n) × GL_h(m)-covariant h-bosonic algebras are built by contracting the GL_q(n) × GL_q(m)-covariant q-bosonic algebras considered by the present author some years ago. Their defining relations are written in terms of the corresponding R _h-matrices. Whenever n = 2, and m = 1 or 2, it is proved by using U_h(sl(2)) Clebsch-Gordan coefficients that they can also be expressed in terms of coupled commutators in a way entirely similar to the classical case. Some U_h(sl(2)) rank-(1/2) irreducible tensor operators, recently constructed by Aizawa in terms of standard bosonic operators, are shown to provide a realization of the h-bosonic algebra corresponding to n = 2 and m = 1.     

On the lie algebraic structure of a new field quantization

It is shown that a new field quatization algebra with 2n generators is isomorphic to the O(2n, 1) algebra. The SL(2, C) algebra is realized by the new quantization algebra with two generators only (n = 1).