General Commutation Research Articles

Bose-like oscillator in fractional-dimensional space

The system of a fractional-dimensional Bose-like oscillator of one degree of freedom whose canonical variables satisfy the general Wigner commutation relations is investigated. Momentum-position uncertainty relations are obtained. For states without definite parity the well known one-dimensional Heisenberg uncertainty principle is retained, but for odd and even states momentum-position uncertainty inequalities depending on the dimension of the space are obtained. The motions of both the free particle and the harmonic oscillator in a fractional-dimensional space are studied through the probability density function. The existence of compression (spread) of the probability density for dimensions D 1) is shown. Fractional-dimensional Bose-like operators are also deduced. They together with the reflection operator form an R-deformed Heisenberg algebra with a deformation parameter depending on the dimension of the space.

Fractional Dimensional Momentum Operator for a System of One Degree of Freedom

By introducing a Cartesian-like pseudocoordinate, and starting with the general Wigner commutation relations for the canonical variables of a Bose-like oscillator of a single degree of freedom, an expression for the momentum operator is obtained. The wave function that describes the motion of a free particle in a fractional dimensional space is also found.

Wick Product for Commutation Relations Connected with Yang-Baxter Operators and New Constructions of Factors

We analyze a certain class of von Neumann algebras generated by selfadjoint elements \(\), for \(\) satisfying the general commutation relations: $$$$ Such algebras can be continuously embedded into some closure of the set of finite linear combinations of vectors \(\), where \(\) is an orthonormal basis of a Hilbert space \(\). The operator which represents the vector \(\) is denoted by \(\) and called the “Wick product” of the operators \(\). We describe explicitly the form of this product. Also, we estimate the operator norm of \(\) for \(\). Finally we apply these two results and prove that under the assumption \(\) all the von Neumann algebras considered are II 1 factors.

Congruence Modularity Implies the Arguesian Law for Single Algebras with a Difference Term

Recently, a generalization of commutator theory has been developed for algebraic systems belonging to a congruence modular variety. This general commutator theory is used here both to provide a very simple proof of a classical result by R. Freese and B. Jónsson and to solve a problem raised by S. Tschantz. Some other known results are improved or are given a simple proof. Moreover, we pursue the goal of extending commutator theory to a somewhat larger setting; thus we give a more general version of the result by Freese and Jónsson (partially solving a problem we raised in P. Lipparini, 1994, Trans. Amer. Math. Soc.346, 177–202).

The Commutation Theorem for Tensor Products over von Neumann Algebras

A general commutation theorem is proved for tensor products of von Neumann algebras over common von Neumann subalgebras. Roughly speaking, if the non-common parts of two von Neumann algebras M1 and M2 on the same Hilbert space are appropriately separated by commuting type I von Neumann algebras N1 and N2, then the commutant of the von Neumann algebra generated by M1 and M2 is generated by the relative commutants M′1∩N1 and M′2∩N2, as well as by the intersection of the commutants of all concerned von Neumann algebras. This theorem extends both Tomita's classical commutation theorem and a splitting result in tensor products, proved recently in the factor case by L. Ge and R. V. Kadison. Applications are given to a decomposition criterion in ordinary tensor products and to a partial solution of a conjecture of S. Popa concerning the maximal injectivity of tensor products of maximal injective von Neumann subalgebras.

Representations of general commutation relations

General commutation relations involving creation, annihilation, and particle number operators are considered. Such commutation relations arise in the context of nonstandard Poisson brackets. All possible types of irreducible representations in which the particle number operator or the product of the creation and annihilation operators has a basis of orthonormal eigenvectors are constructed. The irreducible representations that involve the particle number operator reduce to one of four types and those that do not involve the particle number operator reduce to one of five types.

Positive Representations of General Commutation Relations Allowing Wick Ordering

We consider the problem of representing in Hilbert space commutation relations of the form [GRAPHICS] where the T kl ij are essentially arbitrary scalar coefficients. Examples comprise the q-canonical commutation relations introduced by Greenberg, Bozejko, and Speicher, and the twisted canonical (anti-)commutation relations studied by Pusz and Woronowicz, as well as the quantum group S νU (2). Using these relations, any polynomial in the generators a i and their adjoints can uniquely be written in "Wick ordered form" in which all starred generators are to the left of all unstarred ones. In this general framework we define the Fock representation, as well as coherent representations. We develop criteria for the natural scalar product in the associated representation spaces to be positive definite and for the relations to have representations by bounded operators in a Hilbert space. We characterize the relations between the generators a i (not involving a * i ) which are compatible with the basic relations. The relations may also be interpreted as defining a non-commutative differential calculus. For generic coefficients T kl ij , however, all differential forms of degree 2 and higher vanish. We exhibit conditions for this not to be the case and relate them to the ideal structure of the Wick algebra and conditions of positivity. We show that the differential calculus is compatible with the involution iff the coefficients T define a representation of the braid group. This condition is also shown to imply improved bounds for the positivity of the Fock representation. Finally, we study the KMS states of the group of gauge transformations defined by a j → exp( it) a j .

CURRENT ALGEBRA AND CONFORMAL FIELD THEORY ON A FIGURE EIGHT

We examine the dynamics of a free massless scalar field on a figure eight network. Upon requiring the scalar field to have a well-defined value at the junction of the network, it is seen that the conserved currents of the theory satisfy Kirchhoff’s law, that is that the current flowing into the junction equals the current flowing out. We obtain the corresponding current algebra and show that, unlike on a circle, the left- and right-moving currents on the figure eight do not in general commute in quantum theory. Since a free scalar field theory on a one-dimensional spatial manifold exhibits conformal symmetry, it is natural to ask whether an analogous symmetry can be defined for the figure eight. We find that, unlike in the case of a manifold, the action plus boundary conditions for the network are not invariant under separate conformal transformations associated with left- and right-movers. Instead, the system is, at best, invariant under only a single set of transformations. Its conserved current is also found to satisfy Kirchhoff’s law at the junction. We obtain the associated conserved charges, and show that they generate a Virasoro algebra. Its conformal anomaly (central charge) is computed for special values of the parameters characterizing the network.

Skew power series rings with general commutation formula

The noncommutative product in a skew power series ring Λ in an indeterminate X with coefficients in a skewfield K is entirely determined from a commutation law between X and any element of K. Different types of commutation rules are described, with some resulting properties for Λ and its skewfield of fractions.

A General Approach to Deriving the Statistical Distribution Function

In this paper a general approach to obtaining the statistical distribution function is given. In the specific case of free field, the statistical distribution functions are derived under a basis set of general commutation relations by using the double–time Green's functions in the first, second and third order decoupling approximations respectively.

Some general commutation rules and identities for three-dimensional quantum-mechanical operators

A general proof of several formulas used in three-dimensional quantum problems is presented. By considering the two vector operators A and B having Cartesian components that satisfy the canonical commutation relations—as the position and momentum operators do—we obtain commutation rules and relations for the projection onto each another. All of these rules and relations have symmetry under A↔B exchange. This symmetry suggests, in the particular case of position and momentum operators, a new operator, the projection of r onto p, which seems to be useful in dissipative quantum problems.

M n as a 0, 1-Sublattice of Con A Does not Force the Term Condition

For every n > 3 there exists a finite nonabelian algebra whose congruence lattice has Mn as a 0, 1-sublattice. This answers a question of R. McKenzie and D. Hobby. DEFINITION 0.1. Suppose L and L1 are bounded lattices. A copy of L (in L1) is any sublattice L' of L1 which is isomorphic to L. L' is a 0, 1-copy of L if it includes the least and greatest elements of L1; in this case L' is also called a 0, 1-sublattice of L1. DEFINITION 0. 2. For n > 1, Mn is the finite lattice of height 2 having exactly n atoms. For example, M6 is Suppose G is a group and N(G) is its lattice of normal subgroups. There is a trivial proof, using the commutator operation on normal subgroups, that if N(G) has a 0,1-copy of M3 then G is abelian. This same proof extends, via the general commutator theory of universal algebra, to any algebra A belonging to a variety whose congruence lattices satisfy the modular law. Here N(G) is replaced by Con A (the lattice of congruence relations of A), while 'abelian' means the following 'term condition': DEFINITION 0 . 3. An algebra A is abelian if for every n > 1, every (n + 1)-ary term t(x, Y1 ... Yn) in the language of A, and all a, b, c a .... acn dla ... adn E Al tA (a, c) = tA (a, d) iff tA(bC) = tA(b, d). In their forthcoming book on tame congruence theory [1], R. McKenzie and D. Hobby ask whether the above phenomenon Con A having M3 as a 0, 1-sublattice Received by the editors September 28, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 08A30; Secondary 06B99, 08A40.

PRODUCT OPERATOR FORMALISM FOR SPIN SYSTEMS WITH MAGNETIC EQUIVALENCE

A complete set of symmetry-adapted product operator was introduced to analyse multipulse FT-NMR experiments for spin-1/2 I2S and I3S systems. General cyclic commutation relations exist among the symmetrized product operators. These commutation relations govern the evolution rules of different order coherences under the action of the Hamiltonian in various stage of pulse sequence. The use of symmetry-adapted product operator algebra simplified the analysis of experiments and revealed more evident physical meaning. The procedure was demonstrated by the treatment of DEPT experiment performed on spin-1/2 I2S and l3S systems.

Erratum

Proc. R. Soc. Lond. A 375, 397–408 (1981) Truncation theorems for spinodals and critical points of mean-field models for polydisperse polymer solutions By P. Irvine and M. Gordon It is regretted that the statement of theorem 1 in (8 a ) of this paper is incorrect. The correct form is (32), whose proof is given and which was used exclusively for numerical checks. Theorem 1 is not, however, equivalent (as wrongly stated) to (32). The ‘theorem’ is incorrect because the operator O [ ] used does not in general commute with matrix multiplication. Thanks are due to Professor M. T. Raetzsch (Technical University of Leuna-Merseburg) for pointing out that (8 a ) gave her incorrect results. The usefulness of our work is unaffected as long as (32) is used in place of (8 a ).

Commutation of Prison Sentences: Practice, Promise, and Limitation

Executive clemency procedures offer an ancient but still modern mechanism for righting injustice and coping with problems arising when the judicial system fails to adjust to social needs. As the indeterminate sentence gives way to determinate sanctions and prison crowding grows acute, clemency, particularly in the form of commutation of sentences of prisoners, offers one means of adaptation to correctional population pressures. This article examines the legal and administrative structures, policies, and actual uses of sentence commutation in the United States. A survey of state commutation practices finds that there is wide variation in structure and policy among the states, that in general commutation is granted quite sparingly, that in response to sentencing changes states have both increased and decreased its use, and that political and practical considerations limit the use of commutation as a prison populaton safety valve.

Algebras with general commutation relations and their applications. II. Unitary-nonlinear operator equations

Various aspects of the calculus of functions of ordered self-adjoint operators are considered. Passage to the commutative limit in the case of general nonlinear commutation relations is studied. An asymptotic solution of the Cauchy problem and asymptotically self-similar solutions are constructed for unitary-nonlinear operator equations. Asymptotic solutions are found for the Hartree equation with Coulomb interaction.

Algebras with general commutation relations and their applications. I. Pseudodifferential equations with increasing coefficients

Operators of the regular representation for commutation relations of general form are constructed. On the basis of the technique developed, conditions for the existence of a mixed (with respect to smoothness and rate of growth at infinity) asymptotic expression for a solution of a pseudodifferential equation with increasing coefficients are obtained. The work contains a survey of the general theory of quasiinversion of functions of ordered operators.

Remarks on the uncertainty relation for a Bose-like oscillator

The uncertainty relation is discussed for the canonical variablesq andp of a Bose-like oscillator that satisfy the general commutation relations. It is shown that the approach usually adopted for the case of the canonical commutation relation does not enable us to find the final expression of the required relation. It is possible, however, to prove a lemma concerning the possible minimum value of Δq·Δp.

Application of the Infinitesimal Operators of Translation and Rotation : No.2. Extension of the Technique to Continuum Mechanics

The infinitesimal operators translation and rotation are useful to derive the basic dynamic quantities and corresponding conservation equations fro the energy from and its conservation equation. this technique is applied effectively to continuum mechanics. The general commutation rules between the infinitesimal operators of rheonomic coordinate transformations and the differential with respect to time and space are formulated.

A general commutator equation in the algebra of spin operators

An algebraic method of spin operators was developed by Jha and Valatin to solve (a) the Hamiltonian of the isotropic and anisotropic xy model in a one-dimensional lattice of N spin 1/2 particles and (b) the partition function of the Ising model in the absence of magnetic field in two dimensions. The pair of fermion operators used to explain BCS theory in superconductivity were shown to be related to a set of spin operators of Jha and Valatin in a very simple way. Onsager’s Lie algebra for diagonalizing the partition function of the Ising model was found to be included within the said commutator algebra of the spin operators. The structure constants of the algebra are so simple as to allow the entire algebra to be casted in one general commutator equation. In the present paper, the author presents the proof of a general equation from which all sets of commutator relationships existing among the elements of the algebra directly follow.