Basic Commutation Research Articles

Basic Commutators in Weights Six and Seven as Relators

Charles Sims has asked whether or not the lower central subgroup γ n (F) of a free group F coincides with the normal closure in F of the basic commutators of weight n. This question has a positive answer in weights at most 5, but remains an open question in general. In earlier work with Gaglione and Spellman, it was shown that γ n (F) is the normal closure in F of the basic commutators of weights n through 2n − 4. Here, we specialize to the case where F has rank 2 and show that γ6(F) is the normal closure in F of the basic commutators of weights 6 and 7.

An Alternative Approach For Teaching The Basic Dc-dc Converters By A Generalization Of The Basic Pwm Commutation Cell

This paper proposes a new approach for teaching the basic DC-DC converters, aiming a better understanding of these converters by the student uninitiated in the subject. The proposed strategy is to begin by presenting the basic PWM commutation cell and make a generalization of its characteristics, showing that the buck, boost and buck-boost converters have many similarities. This procedure reduces the number of equations and waveforms to be studied. An interesting effect, only present in the boost converter, is the double region of continuous conduction mode (DRCCM). Simulation results show that the mathematical procedure is consistent and easy for designs.

On left multipliers and the commutativity of prime rings

AbstractLet

On the Order of Nilpotent Multipliers of Finite p-Groups#

ABSTRACT Let G be a finite p-group of the order p n . Berkovich (1991) proved that G is an elementary abelian p-group if and only if the order of its Schur multiplier, M(G), is at the maximum case. In this article, we first find the upper bound p χ c+1(n) for the order of the c-nilpotent multiplier of G, M (c) (G), where χc+1(i) is the number of basic commutators of weight c + 1 on i letters. Second, we obtain the structure of G, in an abelian case, where , for all 0 ≤ t ≤ n − 1. Finally, by putting a condition on the kernel of the left natural map of the generalized Stallings-Stammbach five–term exact sequence, we show that an arbitrary finite p-group with the c-nilpotent multiplier of maximum order is an elementary abelian p-group.

Basic commutators as relators

Basic commutators as relators

Remarks about UV regularization of basic commutators in string theories

The zeta regularization of basic commutators in string theories, recently proposed by Hwang, Marnelius and Saltsidis, is generalized to the string models with non-trivial vacuums. It is shown that implementation of this regularization implies the cancellation of dangerous terms in the commutators between Virasoro generators, which break the Jacobi identity.

Quantum mechanics as a geometric phase: phase-space interferometers

It is shown that basic quantum commutation relations are equivalent to a geometric phase. We propose two interferometric arrangements that are able to measure this quantum geometric phase via interference in phase space.

Multiloop string amplitudes with -field and noncommutative QFT

The multiloop amplitudes for the bosonic string in presence of a constant B-field are built by using the basic commutation relations for the open string zero modes and oscillators. The open string Green function on the annulus is obtained from the one loop scattering amplitude among N tachyons. For higher loops, it is necessary to use the so called three Reggeon vertex, which describes the emission from the open string of another string and not simply of a tachyon. We find that the modifications to the three (and multi) Reggeon vertex due to the B-field only affect the zero modes and can be written in a simple and elegant way. Therefore we can easily sew these vertices together and write the general expression for the multiloop N-Reggeon vertex, which contains any loop string amplitude, in presence of the B-field. The field theory limit is also considered in some examples at two loops and reproduces exactly the results of a noncommutative scalar field theory.

Commutators in real interpolation with quasi‐power parameters

The basic higher order commutator theorem is formulated for the real interpolation methods associated with the quasi‐power parameters, that is, the function spaces on which Hardy inequalities are valid. This theorem unifies and extends various results given by Cwikel, Jawerth, Milman, Rochberg, and others, and incorporates some results of Kalton to the context of commutator estimates for the real interpolation methods.

A general BRST approach to string theories with zeta function regularizations

We propose a new general BRST approach to string and string-like theories that have a wider range of applicability than, e.g., the conventional conformal field theory method. The method involves a simple general regularization of all basic commutators, which makes all divergent sums expressible in terms of zeta functions from which finite values then may be extracted in a rigorous manner. The method is particularly useful in order to investigate possible state space representations to a given model. The method is applied to three string models: The ordinary bosonic string, the tensionless string, and the conformal tensionless string. We also investigate different state spaces for these models. The tensionless string models are treated in detail. Although we mostly rederive known results, they appear in a new fashion that deepens our understanding of these models. Furthermore, we believe that our treatment is more rigorous than most of the previous ones. In the case of the conformal tensionless string we find a new solution for d=4.

Remarks about UV regularization of basic commutators in string theories

Recently proposed by Hwang, Marnelius and Saltsidis zeta regularization of basic commutators in string theories is generalized to the string models with non-trivial vacuums. It is shown that implementation of this regularization implies the cancellation of dangerous terms in the commutators between Virasoro generators, which break Jacobi identity.

Algebra of observables for identical particles in one dimension

The algebra of observables for identical particles on a line is formulated starting from postulated basic commutation relations. A realization of this algebra in the Calogero model was previously known. New realizations are presented here in terms of differentiation operators and in terms of SU(N)-invariant observables of the Hermitian matrix models. Some particular structure properties of the algebra are briefly discussed.

Non-Hermitian quantum canonical variables and the generalized ladder operators.

Quantum canonical transformations of the second kind and the non-Hermitian realizations of the basic canonical commutation relations are investigated with a special interest in the generalization of the conventional ladder operators. The operator ordering problem is shown to be resolved when the non-Hermitian realizations for the canonical variables which can not be measured simultaneously with the energy are chosen for the canonical quantizations. Another merit of the non-Hermitian representations is that it naturally allows us to introduce the generalized ladder operators with which one can solve eigenvalue problems quite neatly.

Quantum-Mechanically Allowed Form of Force in General Relativistic Particle Motion

We reformulate Tanimura's general relativistic version of Feynman's consideration. Under the basic commutation relations, the requirement that velocity and momentum of a point particle are transformed as ‘quantum’ vectors restricts strongly the equation of motion, shown to be Heisenberg-type one including scalar, vector and gravitational fields covariantly.

The systematic derivation of realizations of generators of Lie algebras

Realizations of the generators of the SO(3) and SO(2,1) Lie algebras are derived systematically using only the basic commutation relations.

Internal geometry of hadron resonances

Motivated by previous work on high-energy quantum mechanics, a simple model is devised to study the internal geometry of hadron resonances. In this model we assume new basic canonical commutation relations between the (internal) coordinate and momentum operators of the hadronic quantum system. By systematically imposing Lie algebra commutation relations between these and other observables, we discuss the free and bound particle problems, identifying in each case the corresponding internal symmetries. For the bound particle problem, which models quark confinement, this symmetry turns out to be characterized by Dirac's two-oscillator representation of theO(3, 2) de Sitter group.

An Alternative Lorentz-Type Symmetry of the Theory of Chiral Bosons with Interactions

We canonically quantize the extended theory of chiral bosons described by the Floreanini-Jackiw Lagrangian to include an interaction potential, and obtain that the commutation relations of interacting chiral bosons coincide with those of free ones. We then construct an alternative Lorentz-type symmetry for this theory in terms of the basic commutation relation.

Multi-mode, multi-photon coherent states

A two-mode photon operator satisfying the canonical boson commutation relations is used to define two-mode coherent states for the Weyl, SU(2) and SU(1,l) groups. In all these cases squeezing is determined by an appropriate correlation, which we evaluate. The generalisation to multi-mode, many-photon states is indicated. In this letter we introduce a formalism for describing coherent states, tailored to the multi-mode case. We shall present the two-mode case for definiteness, although the generalisation to several modes will be obvious. Similarly, we show at the end of this note how to accommodate multi-photon states with little extra effort. States associ- ated with the Weyl group (two-mode coherent states), and the groups SU.(2) and SU(1,l) will be given; and we show that (for a specific representation) the last case reduces to the two-mode sqaeezed states of Caves and Schumaker (1986). In the conventional quantum optics approach, a mode of the electromagnetic field is represented by a creation (annihilation) operator a(at), with commutation relation (a, a') = 1. Our starting point in considering the two-mode case is to introduce a two-mode operator A analogous to the single-mode operator a. We do this by defining (Katriel and Hummer 1981) A = (max(Al, A2) + 1)'2ala2 , (1) Here al, a2 are the (annihilation) operators referring to each mode; they obey the basic commutation relation (ai, ait) = 6, but commute with each other. The number operators AI, A2 are defined in the usual way, Ai = aitai and functions of Ai will be evaluated in simultaneous eigenstates of rii, where of course they equal the functions of the corresponding eigenvalues. Now, we may show easily that (A, A') = 1. We note that AA = AtA = min(A1, h2). (The generalisation of (1) to three (or more) modes, which we shall not pursue, is immediate on replacing one of the two single-mode operators by a two-mode one.) Note that all the states defined here give variance-free single-mode operators; since At always creates equal numbers of 1 and 2 modes

Identifications in a free group

Let γ n ( G) denote the nth term of the lower central series of a group G and R a normal subgroup of the free group F. Then it is shown that γ s ( R) ∩ γ n ( F) is the isolator in R of the product of all [ R ∩ γ i 1 ( F), R ∩ γ i 2 ( F), … R ∩ γ i s ( F)] such that i 1 + i 2 + ∣ + i s = n; and that (1 + f nɽ) ∩ F is the isolator in R of the product of all [ R ∩ γ i 1 ( F), R ∩ γ i 2 ( F), …, R ∩ γ i t ( F)], t ≥ 2, such that for each j, 1 ≤ j ≤ t, i 1 + i 2 + ∣ + i ̂ j + ∣ + i t = n . (Here f = Ker ( Z F → Z , and ɽ = Ker ( Z F → Z (F/R))). The latter result is a solution for Fox's problem. A theory of basic commutators relative to the normal subgroup R of F is developed.

Quantum-mechanical sum rules and gauge invariance: A study of the HF molecule.

The perturbed Hamiltonian for magnetic dipole transitions is rewritten in terms of the torque operator, instead of the angular momentum operator, which, owing to its nondifferential form, permits tactical advantages in actual calculations of magnetic susceptibility. The translational gauge invariance of the magnetic properties is used to obtain a large series of sum rules involving linear and angular momenta and torque, force, and position operators. These are found to be very general quantum-mechanical relations, restating in a synthetic and unitary form the Thomas-Reiche-Kuhn sum rule, the basic operator commutation properties, the hypervirial theorem, and the conservation of the current-density vector, which are reduced to the same theoretical framework. Accurate calculations of the magnetic properties of the HF molecule, based on the equation-of-motion approach, reveal that the gauge-invariant sum rules can be used for rigorous tests of the quality of approximate molecular wave functions.